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- Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved - - Arbitrary precision integer arithmetic library - */ - -#include "mpi.h" -#include -#include -#include - -#if MP_DEBUG -#include - -#define DIAG(T,V) {fprintf(stderr,T);mp_print(V,stderr);fputc('\n',stderr);} -#else -#define DIAG(T,V) -#endif - -/* - If MP_LOGTAB is not defined, use the math library to compute the - logarithms on the fly. Otherwise, use the static table below. - Pick which works best for your system. - */ -#if MP_LOGTAB - -/* {{{ s_logv_2[] - log table for 2 in various bases */ - -/* - A table of the logs of 2 for various bases (the 0 and 1 entries of - this table are meaningless and should not be referenced). - - This table is used to compute output lengths for the mp_toradix() - function. Since a number n in radix r takes up about log_r(n) - digits, we estimate the output size by taking the least integer - greater than log_r(n), where: - - log_r(n) = log_2(n) * log_r(2) - - This table, therefore, is a table of log_r(2) for 2 <= r <= 36, - which are the output bases supported. - */ - -#include "logtab.h" - -/* }}} */ -#define LOG_V_2(R) s_logv_2[(R)] - -#else - -#include -#define LOG_V_2(R) (log(2.0)/log(R)) - -#endif - -/* Default precision for newly created mp_int's */ -static unsigned int s_mp_defprec = MP_DEFPREC; - -/* {{{ Digit arithmetic macros */ - -/* - When adding and multiplying digits, the results can be larger than - can be contained in an mp_digit. Thus, an mp_word is used. These - macros mask off the upper and lower digits of the mp_word (the - mp_word may be more than 2 mp_digits wide, but we only concern - ourselves with the low-order 2 mp_digits) - - If your mp_word DOES have more than 2 mp_digits, you need to - uncomment the first line, and comment out the second. - */ - -/* #define CARRYOUT(W) (((W)>>DIGIT_BIT)&MP_DIGIT_MAX) */ -#define CARRYOUT(W) ((W)>>DIGIT_BIT) -#define ACCUM(W) ((W)&MP_DIGIT_MAX) - -/* }}} */ - -/* {{{ Comparison constants */ - -#define MP_LT -1 -#define MP_EQ 0 -#define MP_GT 1 - -/* }}} */ - -/* {{{ Constant strings */ - -/* Constant strings returned by mp_strerror() */ -static const char *const mp_err_string[] = { - "unknown result code", /* say what? */ - "boolean true", /* MP_OKAY, MP_YES */ - "boolean false", /* MP_NO */ - "out of memory", /* MP_MEM */ - "argument out of range", /* MP_RANGE */ - "invalid input parameter", /* MP_BADARG */ - "result is undefined" /* MP_UNDEF */ -}; - -/* Value to digit maps for radix conversion */ - -/* s_dmap_1 - standard digits and letters */ -static const char *s_dmap_1 = - "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/"; - -#if 0 -/* s_dmap_2 - base64 ordering for digits */ -static const char *s_dmap_2 = - "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/"; -#endif - -/* }}} */ - -/* {{{ Static function declarations */ - -/* - If MP_MACRO is false, these will be defined as actual functions; - otherwise, suitable macro definitions will be used. This works - around the fact that ANSI C89 doesn't support an 'inline' keyword - (although I hear C9x will ... about bloody time). At present, the - macro definitions are identical to the function bodies, but they'll - expand in place, instead of generating a function call. - - I chose these particular functions to be made into macros because - some profiling showed they are called a lot on a typical workload, - and yet they are primarily housekeeping. - */ -#if MP_MACRO == 0 - void s_mp_setz(mp_digit *dp, mp_size count); /* zero digits */ - void s_mp_copy(mp_digit *sp, mp_digit *dp, mp_size count); /* copy */ - void *s_mp_alloc(size_t nb, size_t ni); /* general allocator */ - void s_mp_free(void *ptr); /* general free function */ -#else - - /* Even if these are defined as macros, we need to respect the settings - of the MP_MEMSET and MP_MEMCPY configuration options... - */ - #if MP_MEMSET == 0 - #define s_mp_setz(dp, count) \ - {int ix;for(ix=0;ix<(count);ix++)(dp)[ix]=0;} - #else - #define s_mp_setz(dp, count) memset(dp, 0, (count) * sizeof(mp_digit)) - #endif /* MP_MEMSET */ - - #if MP_MEMCPY == 0 - #define s_mp_copy(sp, dp, count) \ - {int ix;for(ix=0;ix<(count);ix++)(dp)[ix]=(sp)[ix];} - #else - #define s_mp_copy(sp, dp, count) memcpy(dp, sp, (count) * sizeof(mp_digit)) - #endif /* MP_MEMCPY */ - - #define s_mp_alloc(nb, ni) calloc(nb, ni) - #define s_mp_free(ptr) {if(ptr) free(ptr);} -#endif /* MP_MACRO */ - -mp_err s_mp_grow(mp_int *mp, mp_size min); /* increase allocated size */ -mp_err s_mp_pad(mp_int *mp, mp_size min); /* left pad with zeroes */ - -void s_mp_clamp(mp_int *mp); /* clip leading zeroes */ - -void s_mp_exch(mp_int *a, mp_int *b); /* swap a and b in place */ - -mp_err s_mp_lshd(mp_int *mp, mp_size p); /* left-shift by p digits */ -void s_mp_rshd(mp_int *mp, mp_size p); /* right-shift by p digits */ -void s_mp_div_2d(mp_int *mp, mp_digit d); /* divide by 2^d in place */ -void s_mp_mod_2d(mp_int *mp, mp_digit d); /* modulo 2^d in place */ -mp_err s_mp_mul_2d(mp_int *mp, mp_digit d); /* multiply by 2^d in place*/ -void s_mp_div_2(mp_int *mp); /* divide by 2 in place */ -mp_err s_mp_mul_2(mp_int *mp); /* multiply by 2 in place */ -mp_digit s_mp_norm(mp_int *a, mp_int *b); /* normalize for division */ -mp_err s_mp_add_d(mp_int *mp, mp_digit d); /* unsigned digit addition */ -mp_err s_mp_sub_d(mp_int *mp, mp_digit d); /* unsigned digit subtract */ -mp_err s_mp_mul_d(mp_int *mp, mp_digit d); /* unsigned digit multiply */ -mp_err s_mp_div_d(mp_int *mp, mp_digit d, mp_digit *r); - /* unsigned digit divide */ -mp_err s_mp_reduce(mp_int *x, mp_int *m, mp_int *mu); - /* Barrett reduction */ -mp_err s_mp_add(mp_int *a, mp_int *b); /* magnitude addition */ -mp_err s_mp_sub(mp_int *a, mp_int *b); /* magnitude subtract */ -mp_err s_mp_mul(mp_int *a, mp_int *b); /* magnitude multiply */ -#if 0 -void s_mp_kmul(mp_digit *a, mp_digit *b, mp_digit *out, mp_size len); - /* multiply buffers in place */ -#endif -#if MP_SQUARE -mp_err s_mp_sqr(mp_int *a); /* magnitude square */ -#else -#define s_mp_sqr(a) s_mp_mul(a, a) -#endif -mp_err s_mp_div(mp_int *a, mp_int *b); /* magnitude divide */ -mp_err s_mp_2expt(mp_int *a, mp_digit k); /* a = 2^k */ -int s_mp_cmp(mp_int *a, mp_int *b); /* magnitude comparison */ -int s_mp_cmp_d(mp_int *a, mp_digit d); /* magnitude digit compare */ -int s_mp_ispow2(mp_int *v); /* is v a power of 2? */ -int s_mp_ispow2d(mp_digit d); /* is d a power of 2? */ - -int s_mp_tovalue(char ch, int r); /* convert ch to value */ -char s_mp_todigit(int val, int r, int low); /* convert val to digit */ -int s_mp_outlen(int bits, int r); /* output length in bytes */ - -/* }}} */ - -/* {{{ Default precision manipulation */ - -unsigned int mp_get_prec(void) -{ - return s_mp_defprec; - -} /* end mp_get_prec() */ - -void mp_set_prec(unsigned int prec) -{ - if(prec == 0) - s_mp_defprec = MP_DEFPREC; - else - s_mp_defprec = prec; - -} /* end mp_set_prec() */ - -/* }}} */ - -/*------------------------------------------------------------------------*/ -/* {{{ mp_init(mp) */ - -/* - mp_init(mp) - - Initialize a new zero-valued mp_int. Returns MP_OKAY if successful, - MP_MEM if memory could not be allocated for the structure. - */ - -mp_err mp_init(mp_int *mp) -{ - return mp_init_size(mp, s_mp_defprec); - -} /* end mp_init() */ - -/* }}} */ - -/* {{{ mp_init_array(mp[], count) */ - -mp_err mp_init_array(mp_int mp[], int count) -{ - mp_err res; - int pos; - - ARGCHK(mp !=NULL && count > 0, MP_BADARG); - - for(pos = 0; pos < count; ++pos) { - if((res = mp_init(&mp[pos])) != MP_OKAY) - goto CLEANUP; - } - - return MP_OKAY; - - CLEANUP: - while(--pos >= 0) - mp_clear(&mp[pos]); - - return res; - -} /* end mp_init_array() */ - -/* }}} */ - -/* {{{ mp_init_size(mp, prec) */ - -/* - mp_init_size(mp, prec) - - Initialize a new zero-valued mp_int with at least the given - precision; returns MP_OKAY if successful, or MP_MEM if memory could - not be allocated for the structure. - */ - -mp_err mp_init_size(mp_int *mp, mp_size prec) -{ - ARGCHK(mp != NULL && prec > 0, MP_BADARG); - - if((DIGITS(mp) = s_mp_alloc(prec, sizeof(mp_digit))) == NULL) - return MP_MEM; - - SIGN(mp) = MP_ZPOS; - USED(mp) = 1; - ALLOC(mp) = prec; - - return MP_OKAY; - -} /* end mp_init_size() */ - -/* }}} */ - -/* {{{ mp_init_copy(mp, from) */ - -/* - mp_init_copy(mp, from) - - Initialize mp as an exact copy of from. Returns MP_OKAY if - successful, MP_MEM if memory could not be allocated for the new - structure. - */ - -mp_err mp_init_copy(mp_int *mp, mp_int *from) -{ - ARGCHK(mp != NULL && from != NULL, MP_BADARG); - - if(mp == from) - return MP_OKAY; - - if((DIGITS(mp) = s_mp_alloc(USED(from), sizeof(mp_digit))) == NULL) - return MP_MEM; - - s_mp_copy(DIGITS(from), DIGITS(mp), USED(from)); - USED(mp) = USED(from); - ALLOC(mp) = USED(from); - SIGN(mp) = SIGN(from); - - return MP_OKAY; - -} /* end mp_init_copy() */ - -/* }}} */ - -/* {{{ mp_copy(from, to) */ - -/* - mp_copy(from, to) - - Copies the mp_int 'from' to the mp_int 'to'. It is presumed that - 'to' has already been initialized (if not, use mp_init_copy() - instead). If 'from' and 'to' are identical, nothing happens. - */ - -mp_err mp_copy(mp_int *from, mp_int *to) -{ - ARGCHK(from != NULL && to != NULL, MP_BADARG); - - if(from == to) - return MP_OKAY; - - { /* copy */ - mp_digit *tmp; - - /* - If the allocated buffer in 'to' already has enough space to hold - all the used digits of 'from', we'll re-use it to avoid hitting - the memory allocater more than necessary; otherwise, we'd have - to grow anyway, so we just allocate a hunk and make the copy as - usual - */ - if(ALLOC(to) >= USED(from)) { - s_mp_setz(DIGITS(to) + USED(from), ALLOC(to) - USED(from)); - s_mp_copy(DIGITS(from), DIGITS(to), USED(from)); - - } else { - if((tmp = s_mp_alloc(USED(from), sizeof(mp_digit))) == NULL) - return MP_MEM; - - s_mp_copy(DIGITS(from), tmp, USED(from)); - - if(DIGITS(to) != NULL) { -#if MP_CRYPTO - s_mp_setz(DIGITS(to), ALLOC(to)); -#endif - s_mp_free(DIGITS(to)); - } - - DIGITS(to) = tmp; - ALLOC(to) = USED(from); - } - - /* Copy the precision and sign from the original */ - USED(to) = USED(from); - SIGN(to) = SIGN(from); - } /* end copy */ - - return MP_OKAY; - -} /* end mp_copy() */ - -/* }}} */ - -/* {{{ mp_exch(mp1, mp2) */ - -/* - mp_exch(mp1, mp2) - - Exchange mp1 and mp2 without allocating any intermediate memory - (well, unless you count the stack space needed for this call and the - locals it creates...). This cannot fail. - */ - -void mp_exch(mp_int *mp1, mp_int *mp2) -{ -#if MP_ARGCHK == 2 - assert(mp1 != NULL && mp2 != NULL); -#else - if(mp1 == NULL || mp2 == NULL) - return; -#endif - - s_mp_exch(mp1, mp2); - -} /* end mp_exch() */ - -/* }}} */ - -/* {{{ mp_clear(mp) */ - -/* - mp_clear(mp) - - Release the storage used by an mp_int, and void its fields so that - if someone calls mp_clear() again for the same int later, we won't - get tollchocked. - */ - -void mp_clear(mp_int *mp) -{ - if(mp == NULL) - return; - - if(DIGITS(mp) != NULL) { -#if MP_CRYPTO - s_mp_setz(DIGITS(mp), ALLOC(mp)); -#endif - s_mp_free(DIGITS(mp)); - DIGITS(mp) = NULL; - } - - USED(mp) = 0; - ALLOC(mp) = 0; - -} /* end mp_clear() */ - -/* }}} */ - -/* {{{ mp_clear_array(mp[], count) */ - -void mp_clear_array(mp_int mp[], int count) -{ - ARGCHK(mp != NULL && count > 0, MP_BADARG); - - while(--count >= 0) - mp_clear(&mp[count]); - -} /* end mp_clear_array() */ - -/* }}} */ - -/* {{{ mp_zero(mp) */ - -/* - mp_zero(mp) - - Set mp to zero. Does not change the allocated size of the structure, - and therefore cannot fail (except on a bad argument, which we ignore) - */ -void mp_zero(mp_int *mp) -{ - if(mp == NULL) - return; - - s_mp_setz(DIGITS(mp), ALLOC(mp)); - USED(mp) = 1; - SIGN(mp) = MP_ZPOS; - -} /* end mp_zero() */ - -/* }}} */ - -/* {{{ mp_set(mp, d) */ - -void mp_set(mp_int *mp, mp_digit d) -{ - if(mp == NULL) - return; - - mp_zero(mp); - DIGIT(mp, 0) = d; - -} /* end mp_set() */ - -/* }}} */ - -/* {{{ mp_set_int(mp, z) */ - -mp_err mp_set_int(mp_int *mp, long z) -{ - int ix; - unsigned long v = abs(z); - mp_err res; - - ARGCHK(mp != NULL, MP_BADARG); - - mp_zero(mp); - if(z == 0) - return MP_OKAY; /* shortcut for zero */ - - for(ix = sizeof(long) - 1; ix >= 0; ix--) { - - if((res = s_mp_mul_2d(mp, CHAR_BIT)) != MP_OKAY) - return res; - - res = s_mp_add_d(mp, - (mp_digit)((v >> (ix * CHAR_BIT)) & UCHAR_MAX)); - if(res != MP_OKAY) - return res; - - } - - if(z < 0) - SIGN(mp) = MP_NEG; - - return MP_OKAY; - -} /* end mp_set_int() */ - -/* }}} */ - -/*------------------------------------------------------------------------*/ -/* {{{ Digit arithmetic */ - -/* {{{ mp_add_d(a, d, b) */ - -/* - mp_add_d(a, d, b) - - Compute the sum b = a + d, for a single digit d. Respects the sign of - its primary addend (single digits are unsigned anyway). - */ - -mp_err mp_add_d(mp_int *a, mp_digit d, mp_int *b) -{ - mp_err res = MP_OKAY; - - ARGCHK(a != NULL && b != NULL, MP_BADARG); - - if((res = mp_copy(a, b)) != MP_OKAY) - return res; - - if(SIGN(b) == MP_ZPOS) { - res = s_mp_add_d(b, d); - } else if(s_mp_cmp_d(b, d) >= 0) { - res = s_mp_sub_d(b, d); - } else { - SIGN(b) = MP_ZPOS; - - DIGIT(b, 0) = d - DIGIT(b, 0); - } - - return res; - -} /* end mp_add_d() */ - -/* }}} */ - -/* {{{ mp_sub_d(a, d, b) */ - -/* - mp_sub_d(a, d, b) - - Compute the difference b = a - d, for a single digit d. Respects the - sign of its subtrahend (single digits are unsigned anyway). - */ - -mp_err mp_sub_d(mp_int *a, mp_digit d, mp_int *b) -{ - mp_err res; - - ARGCHK(a != NULL && b != NULL, MP_BADARG); - - if((res = mp_copy(a, b)) != MP_OKAY) - return res; - - if(SIGN(b) == MP_NEG) { - if((res = s_mp_add_d(b, d)) != MP_OKAY) - return res; - - } else if(s_mp_cmp_d(b, d) >= 0) { - if((res = s_mp_sub_d(b, d)) != MP_OKAY) - return res; - - } else { - mp_neg(b, b); - - DIGIT(b, 0) = d - DIGIT(b, 0); - SIGN(b) = MP_NEG; - } - - if(s_mp_cmp_d(b, 0) == 0) - SIGN(b) = MP_ZPOS; - - return MP_OKAY; - -} /* end mp_sub_d() */ - -/* }}} */ - -/* {{{ mp_mul_d(a, d, b) */ - -/* - mp_mul_d(a, d, b) - - Compute the product b = a * d, for a single digit d. Respects the sign - of its multiplicand (single digits are unsigned anyway) - */ - -mp_err mp_mul_d(mp_int *a, mp_digit d, mp_int *b) -{ - mp_err res; - - ARGCHK(a != NULL && b != NULL, MP_BADARG); - - if(d == 0) { - mp_zero(b); - return MP_OKAY; - } - - if((res = mp_copy(a, b)) != MP_OKAY) - return res; - - res = s_mp_mul_d(b, d); - - return res; - -} /* end mp_mul_d() */ - -/* }}} */ - -/* {{{ mp_mul_2(a, c) */ - -mp_err mp_mul_2(mp_int *a, mp_int *c) -{ - mp_err res; - - ARGCHK(a != NULL && c != NULL, MP_BADARG); - - if((res = mp_copy(a, c)) != MP_OKAY) - return res; - - return s_mp_mul_2(c); - -} /* end mp_mul_2() */ - -/* }}} */ - -/* {{{ mp_div_d(a, d, q, r) */ - -/* - mp_div_d(a, d, q, r) - - Compute the quotient q = a / d and remainder r = a mod d, for a - single digit d. Respects the sign of its divisor (single digits are - unsigned anyway). - */ - -mp_err mp_div_d(mp_int *a, mp_digit d, mp_int *q, mp_digit *r) -{ - mp_err res; - mp_digit rem; - int pow; - - ARGCHK(a != NULL, MP_BADARG); - - if(d == 0) - return MP_RANGE; - - /* Shortcut for powers of two ... */ - if((pow = s_mp_ispow2d(d)) >= 0) { - mp_digit mask; - - mask = (1 << pow) - 1; - rem = DIGIT(a, 0) & mask; - - if(q) { - mp_copy(a, q); - s_mp_div_2d(q, pow); - } - - if(r) - *r = rem; - - return MP_OKAY; - } - - /* - If the quotient is actually going to be returned, we'll try to - avoid hitting the memory allocator by copying the dividend into it - and doing the division there. This can't be any _worse_ than - always copying, and will sometimes be better (since it won't make - another copy) - - If it's not going to be returned, we need to allocate a temporary - to hold the quotient, which will just be discarded. - */ - if(q) { - if((res = mp_copy(a, q)) != MP_OKAY) - return res; - - res = s_mp_div_d(q, d, &rem); - if(s_mp_cmp_d(q, 0) == MP_EQ) - SIGN(q) = MP_ZPOS; - - } else { - mp_int qp; - - if((res = mp_init_copy(&qp, a)) != MP_OKAY) - return res; - - res = s_mp_div_d(&qp, d, &rem); - if(s_mp_cmp_d(&qp, 0) == 0) - SIGN(&qp) = MP_ZPOS; - - mp_clear(&qp); - } - - if(r) - *r = rem; - - return res; - -} /* end mp_div_d() */ - -/* }}} */ - -/* {{{ mp_div_2(a, c) */ - -/* - mp_div_2(a, c) - - Compute c = a / 2, disregarding the remainder. - */ - -mp_err mp_div_2(mp_int *a, mp_int *c) -{ - mp_err res; - - ARGCHK(a != NULL && c != NULL, MP_BADARG); - - if((res = mp_copy(a, c)) != MP_OKAY) - return res; - - s_mp_div_2(c); - - return MP_OKAY; - -} /* end mp_div_2() */ - -/* }}} */ - -/* {{{ mp_expt_d(a, d, b) */ - -mp_err mp_expt_d(mp_int *a, mp_digit d, mp_int *c) -{ - mp_int s, x; - mp_err res; - - ARGCHK(a != NULL && c != NULL, MP_BADARG); - - if((res = mp_init(&s)) != MP_OKAY) - return res; - if((res = mp_init_copy(&x, a)) != MP_OKAY) - goto X; - - DIGIT(&s, 0) = 1; - - while(d != 0) { - if(d & 1) { - if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; - } - - d >>= 1; - - if((res = s_mp_sqr(&x)) != MP_OKAY) - goto CLEANUP; - } - - s_mp_exch(&s, c); - -CLEANUP: - mp_clear(&x); -X: - mp_clear(&s); - - return res; - -} /* end mp_expt_d() */ - -/* }}} */ - -/* }}} */ - -/*------------------------------------------------------------------------*/ -/* {{{ Full arithmetic */ - -/* {{{ mp_abs(a, b) */ - -/* - mp_abs(a, b) - - Compute b = |a|. 'a' and 'b' may be identical. - */ - -mp_err mp_abs(mp_int *a, mp_int *b) -{ - mp_err res; - - ARGCHK(a != NULL && b != NULL, MP_BADARG); - - if((res = mp_copy(a, b)) != MP_OKAY) - return res; - - SIGN(b) = MP_ZPOS; - - return MP_OKAY; - -} /* end mp_abs() */ - -/* }}} */ - -/* {{{ mp_neg(a, b) */ - -/* - mp_neg(a, b) - - Compute b = -a. 'a' and 'b' may be identical. - */ - -mp_err mp_neg(mp_int *a, mp_int *b) -{ - mp_err res; - - ARGCHK(a != NULL && b != NULL, MP_BADARG); - - if((res = mp_copy(a, b)) != MP_OKAY) - return res; - - if(s_mp_cmp_d(b, 0) == MP_EQ) - SIGN(b) = MP_ZPOS; - else - SIGN(b) = (SIGN(b) == MP_NEG) ? MP_ZPOS : MP_NEG; - - return MP_OKAY; - -} /* end mp_neg() */ - -/* }}} */ - -/* {{{ mp_add(a, b, c) */ - -/* - mp_add(a, b, c) - - Compute c = a + b. All parameters may be identical. - */ - -mp_err mp_add(mp_int *a, mp_int *b, mp_int *c) -{ - mp_err res; - int cmp; - - ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); - - if(SIGN(a) == SIGN(b)) { /* same sign: add values, keep sign */ - - /* Commutativity of addition lets us do this in either order, - so we avoid having to use a temporary even if the result - is supposed to replace the output - */ - if(c == b) { - if((res = s_mp_add(c, a)) != MP_OKAY) - return res; - } else { - if(c != a && (res = mp_copy(a, c)) != MP_OKAY) - return res; - - if((res = s_mp_add(c, b)) != MP_OKAY) - return res; - } - - } else if((cmp = s_mp_cmp(a, b)) > 0) { /* different sign: a > b */ - - /* If the output is going to be clobbered, we will use a temporary - variable; otherwise, we'll do it without touching the memory - allocator at all, if possible - */ - if(c == b) { - mp_int tmp; - - if((res = mp_init_copy(&tmp, a)) != MP_OKAY) - return res; - if((res = s_mp_sub(&tmp, b)) != MP_OKAY) { - mp_clear(&tmp); - return res; - } - - s_mp_exch(&tmp, c); - mp_clear(&tmp); - - } else { - - if(c != a && (res = mp_copy(a, c)) != MP_OKAY) - return res; - if((res = s_mp_sub(c, b)) != MP_OKAY) - return res; - - } - - } else if(cmp == 0) { /* different sign, a == b */ - - mp_zero(c); - return MP_OKAY; - - } else { /* different sign: a < b */ - - /* See above... */ - if(c == a) { - mp_int tmp; - - if((res = mp_init_copy(&tmp, b)) != MP_OKAY) - return res; - if((res = s_mp_sub(&tmp, a)) != MP_OKAY) { - mp_clear(&tmp); - return res; - } - - s_mp_exch(&tmp, c); - mp_clear(&tmp); - - } else { - - if(c != b && (res = mp_copy(b, c)) != MP_OKAY) - return res; - if((res = s_mp_sub(c, a)) != MP_OKAY) - return res; - - } - } - - if(USED(c) == 1 && DIGIT(c, 0) == 0) - SIGN(c) = MP_ZPOS; - - return MP_OKAY; - -} /* end mp_add() */ - -/* }}} */ - -/* {{{ mp_sub(a, b, c) */ - -/* - mp_sub(a, b, c) - - Compute c = a - b. All parameters may be identical. - */ - -mp_err mp_sub(mp_int *a, mp_int *b, mp_int *c) -{ - mp_err res; - int cmp; - - ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); - - if(SIGN(a) != SIGN(b)) { - if(c == a) { - if((res = s_mp_add(c, b)) != MP_OKAY) - return res; - } else { - if(c != b && ((res = mp_copy(b, c)) != MP_OKAY)) - return res; - if((res = s_mp_add(c, a)) != MP_OKAY) - return res; - SIGN(c) = SIGN(a); - } - - } else if((cmp = s_mp_cmp(a, b)) > 0) { /* Same sign, a > b */ - if(c == b) { - mp_int tmp; - - if((res = mp_init_copy(&tmp, a)) != MP_OKAY) - return res; - if((res = s_mp_sub(&tmp, b)) != MP_OKAY) { - mp_clear(&tmp); - return res; - } - s_mp_exch(&tmp, c); - mp_clear(&tmp); - - } else { - if(c != a && ((res = mp_copy(a, c)) != MP_OKAY)) - return res; - - if((res = s_mp_sub(c, b)) != MP_OKAY) - return res; - } - - } else if(cmp == 0) { /* Same sign, equal magnitude */ - mp_zero(c); - return MP_OKAY; - - } else { /* Same sign, b > a */ - if(c == a) { - mp_int tmp; - - if((res = mp_init_copy(&tmp, b)) != MP_OKAY) - return res; - - if((res = s_mp_sub(&tmp, a)) != MP_OKAY) { - mp_clear(&tmp); - return res; - } - s_mp_exch(&tmp, c); - mp_clear(&tmp); - - } else { - if(c != b && ((res = mp_copy(b, c)) != MP_OKAY)) - return res; - - if((res = s_mp_sub(c, a)) != MP_OKAY) - return res; - } - - SIGN(c) = !SIGN(b); - } - - if(USED(c) == 1 && DIGIT(c, 0) == 0) - SIGN(c) = MP_ZPOS; - - return MP_OKAY; - -} /* end mp_sub() */ - -/* }}} */ - -/* {{{ mp_mul(a, b, c) */ - -/* - mp_mul(a, b, c) - - Compute c = a * b. All parameters may be identical. - */ - -mp_err mp_mul(mp_int *a, mp_int *b, mp_int *c) -{ - mp_err res; - mp_sign sgn; - - ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); - - sgn = (SIGN(a) == SIGN(b)) ? MP_ZPOS : MP_NEG; - - if(c == b) { - if((res = s_mp_mul(c, a)) != MP_OKAY) - return res; - - } else { - if((res = mp_copy(a, c)) != MP_OKAY) - return res; - - if((res = s_mp_mul(c, b)) != MP_OKAY) - return res; - } - - if(sgn == MP_ZPOS || s_mp_cmp_d(c, 0) == MP_EQ) - SIGN(c) = MP_ZPOS; - else - SIGN(c) = sgn; - - return MP_OKAY; - -} /* end mp_mul() */ - -/* }}} */ - -/* {{{ mp_mul_2d(a, d, c) */ - -/* - mp_mul_2d(a, d, c) - - Compute c = a * 2^d. a may be the same as c. - */ - -mp_err mp_mul_2d(mp_int *a, mp_digit d, mp_int *c) -{ - mp_err res; - - ARGCHK(a != NULL && c != NULL, MP_BADARG); - - if((res = mp_copy(a, c)) != MP_OKAY) - return res; - - if(d == 0) - return MP_OKAY; - - return s_mp_mul_2d(c, d); - -} /* end mp_mul() */ - -/* }}} */ - -/* {{{ mp_sqr(a, b) */ - -#if MP_SQUARE -mp_err mp_sqr(mp_int *a, mp_int *b) -{ - mp_err res; - - ARGCHK(a != NULL && b != NULL, MP_BADARG); - - if((res = mp_copy(a, b)) != MP_OKAY) - return res; - - if((res = s_mp_sqr(b)) != MP_OKAY) - return res; - - SIGN(b) = MP_ZPOS; - - return MP_OKAY; - -} /* end mp_sqr() */ -#endif - -/* }}} */ - -/* {{{ mp_div(a, b, q, r) */ - -/* - mp_div(a, b, q, r) - - Compute q = a / b and r = a mod b. Input parameters may be re-used - as output parameters. If q or r is NULL, that portion of the - computation will be discarded (although it will still be computed) - - Pay no attention to the hacker behind the curtain. - */ - -mp_err mp_div(mp_int *a, mp_int *b, mp_int *q, mp_int *r) -{ - mp_err res; - mp_int qtmp, rtmp; - int cmp; - - ARGCHK(a != NULL && b != NULL, MP_BADARG); - - if(mp_cmp_z(b) == MP_EQ) - return MP_RANGE; - - /* If a <= b, we can compute the solution without division, and - avoid any memory allocation - */ - if((cmp = s_mp_cmp(a, b)) < 0) { - if(r) { - if((res = mp_copy(a, r)) != MP_OKAY) - return res; - } - - if(q) - mp_zero(q); - - return MP_OKAY; - - } else if(cmp == 0) { - - /* Set quotient to 1, with appropriate sign */ - if(q) { - int qneg = (SIGN(a) != SIGN(b)); - - mp_set(q, 1); - if(qneg) - SIGN(q) = MP_NEG; - } - - if(r) - mp_zero(r); - - return MP_OKAY; - } - - /* If we get here, it means we actually have to do some division */ - - /* Set up some temporaries... */ - if((res = mp_init_copy(&qtmp, a)) != MP_OKAY) - return res; - if((res = mp_init_copy(&rtmp, b)) != MP_OKAY) - goto CLEANUP; - - if((res = s_mp_div(&qtmp, &rtmp)) != MP_OKAY) - goto CLEANUP; - - /* Compute the signs for the output */ - SIGN(&rtmp) = SIGN(a); /* Sr = Sa */ - if(SIGN(a) == SIGN(b)) - SIGN(&qtmp) = MP_ZPOS; /* Sq = MP_ZPOS if Sa = Sb */ - else - SIGN(&qtmp) = MP_NEG; /* Sq = MP_NEG if Sa != Sb */ - - if(s_mp_cmp_d(&qtmp, 0) == MP_EQ) - SIGN(&qtmp) = MP_ZPOS; - if(s_mp_cmp_d(&rtmp, 0) == MP_EQ) - SIGN(&rtmp) = MP_ZPOS; - - /* Copy output, if it is needed */ - if(q) - s_mp_exch(&qtmp, q); - - if(r) - s_mp_exch(&rtmp, r); - -CLEANUP: - mp_clear(&rtmp); - mp_clear(&qtmp); - - return res; - -} /* end mp_div() */ - -/* }}} */ - -/* {{{ mp_div_2d(a, d, q, r) */ - -mp_err mp_div_2d(mp_int *a, mp_digit d, mp_int *q, mp_int *r) -{ - mp_err res; - - ARGCHK(a != NULL, MP_BADARG); - - if(q) { - if((res = mp_copy(a, q)) != MP_OKAY) - return res; - - s_mp_div_2d(q, d); - } - - if(r) { - if((res = mp_copy(a, r)) != MP_OKAY) - return res; - - s_mp_mod_2d(r, d); - } - - return MP_OKAY; - -} /* end mp_div_2d() */ - -/* }}} */ - -/* {{{ mp_expt(a, b, c) */ - -/* - mp_expt(a, b, c) - - Compute c = a ** b, that is, raise a to the b power. Uses a - standard iterative square-and-multiply technique. - */ - -mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c) -{ - mp_int s, x; - mp_err res; - mp_digit d; - int dig, bit; - - ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); - - if(mp_cmp_z(b) < 0) - return MP_RANGE; - - if((res = mp_init(&s)) != MP_OKAY) - return res; - - mp_set(&s, 1); - - if((res = mp_init_copy(&x, a)) != MP_OKAY) - goto X; - - /* Loop over low-order digits in ascending order */ - for(dig = 0; dig < (USED(b) - 1); dig++) { - d = DIGIT(b, dig); - - /* Loop over bits of each non-maximal digit */ - for(bit = 0; bit < DIGIT_BIT; bit++) { - if(d & 1) { - if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; - } - - d >>= 1; - - if((res = s_mp_sqr(&x)) != MP_OKAY) - goto CLEANUP; - } - } - - /* Consider now the last digit... */ - d = DIGIT(b, dig); - - while(d) { - if(d & 1) { - if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; - } - - d >>= 1; - - if((res = s_mp_sqr(&x)) != MP_OKAY) - goto CLEANUP; - } - - if(mp_iseven(b)) - SIGN(&s) = SIGN(a); - - res = mp_copy(&s, c); - -CLEANUP: - mp_clear(&x); -X: - mp_clear(&s); - - return res; - -} /* end mp_expt() */ - -/* }}} */ - -/* {{{ mp_2expt(a, k) */ - -/* Compute a = 2^k */ - -mp_err mp_2expt(mp_int *a, mp_digit k) -{ - ARGCHK(a != NULL, MP_BADARG); - - return s_mp_2expt(a, k); - -} /* end mp_2expt() */ - -/* }}} */ - -/* {{{ mp_mod(a, m, c) */ - -/* - mp_mod(a, m, c) - - Compute c = a (mod m). Result will always be 0 <= c < m. - */ - -mp_err mp_mod(mp_int *a, mp_int *m, mp_int *c) -{ - mp_err res; - int mag; - - ARGCHK(a != NULL && m != NULL && c != NULL, MP_BADARG); - - if(SIGN(m) == MP_NEG) - return MP_RANGE; - - /* - If |a| > m, we need to divide to get the remainder and take the - absolute value. - - If |a| < m, we don't need to do any division, just copy and adjust - the sign (if a is negative). - - If |a| == m, we can simply set the result to zero. - - This order is intended to minimize the average path length of the - comparison chain on common workloads -- the most frequent cases are - that |a| != m, so we do those first. - */ - if((mag = s_mp_cmp(a, m)) > 0) { - if((res = mp_div(a, m, NULL, c)) != MP_OKAY) - return res; - - if(SIGN(c) == MP_NEG) { - if((res = mp_add(c, m, c)) != MP_OKAY) - return res; - } - - } else if(mag < 0) { - if((res = mp_copy(a, c)) != MP_OKAY) - return res; - - if(mp_cmp_z(a) < 0) { - if((res = mp_add(c, m, c)) != MP_OKAY) - return res; - - } - - } else { - mp_zero(c); - - } - - return MP_OKAY; - -} /* end mp_mod() */ - -/* }}} */ - -/* {{{ mp_mod_d(a, d, c) */ - -/* - mp_mod_d(a, d, c) - - Compute c = a (mod d). Result will always be 0 <= c < d - */ -mp_err mp_mod_d(mp_int *a, mp_digit d, mp_digit *c) -{ - mp_err res; - mp_digit rem; - - ARGCHK(a != NULL && c != NULL, MP_BADARG); - - if(s_mp_cmp_d(a, d) > 0) { - if((res = mp_div_d(a, d, NULL, &rem)) != MP_OKAY) - return res; - - } else { - if(SIGN(a) == MP_NEG) - rem = d - DIGIT(a, 0); - else - rem = DIGIT(a, 0); - } - - if(c) - *c = rem; - - return MP_OKAY; - -} /* end mp_mod_d() */ - -/* }}} */ - -/* {{{ mp_sqrt(a, b) */ - -/* - mp_sqrt(a, b) - - Compute the integer square root of a, and store the result in b. - Uses an integer-arithmetic version of Newton's iterative linear - approximation technique to determine this value; the result has the - following two properties: - - b^2 <= a - (b+1)^2 >= a - - It is a range error to pass a negative value. - */ -mp_err mp_sqrt(mp_int *a, mp_int *b) -{ - mp_int x, t; - mp_err res; - - ARGCHK(a != NULL && b != NULL, MP_BADARG); - - /* Cannot take square root of a negative value */ - if(SIGN(a) == MP_NEG) - return MP_RANGE; - - /* Special cases for zero and one, trivial */ - if(mp_cmp_d(a, 0) == MP_EQ || mp_cmp_d(a, 1) == MP_EQ) - return mp_copy(a, b); - - /* Initialize the temporaries we'll use below */ - if((res = mp_init_size(&t, USED(a))) != MP_OKAY) - return res; - - /* Compute an initial guess for the iteration as a itself */ - if((res = mp_init_copy(&x, a)) != MP_OKAY) - goto X; - -s_mp_rshd(&x, (USED(&x)/2)+1); -mp_add_d(&x, 1, &x); - - for(;;) { - /* t = (x * x) - a */ - mp_copy(&x, &t); /* can't fail, t is big enough for original x */ - if((res = mp_sqr(&t, &t)) != MP_OKAY || - (res = mp_sub(&t, a, &t)) != MP_OKAY) - goto CLEANUP; - - /* t = t / 2x */ - s_mp_mul_2(&x); - if((res = mp_div(&t, &x, &t, NULL)) != MP_OKAY) - goto CLEANUP; - s_mp_div_2(&x); - - /* Terminate the loop, if the quotient is zero */ - if(mp_cmp_z(&t) == MP_EQ) - break; - - /* x = x - t */ - if((res = mp_sub(&x, &t, &x)) != MP_OKAY) - goto CLEANUP; - - } - - /* Copy result to output parameter */ - mp_sub_d(&x, 1, &x); - s_mp_exch(&x, b); - - CLEANUP: - mp_clear(&x); - X: - mp_clear(&t); - - return res; - -} /* end mp_sqrt() */ - -/* }}} */ - -/* }}} */ - -/*------------------------------------------------------------------------*/ -/* {{{ Modular arithmetic */ - -#if MP_MODARITH -/* {{{ mp_addmod(a, b, m, c) */ - -/* - mp_addmod(a, b, m, c) - - Compute c = (a + b) mod m - */ - -mp_err mp_addmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) -{ - mp_err res; - - ARGCHK(a != NULL && b != NULL && m != NULL && c != NULL, MP_BADARG); - - if((res = mp_add(a, b, c)) != MP_OKAY) - return res; - if((res = mp_mod(c, m, c)) != MP_OKAY) - return res; - - return MP_OKAY; - -} - -/* }}} */ - -/* {{{ mp_submod(a, b, m, c) */ - -/* - mp_submod(a, b, m, c) - - Compute c = (a - b) mod m - */ - -mp_err mp_submod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) -{ - mp_err res; - - ARGCHK(a != NULL && b != NULL && m != NULL && c != NULL, MP_BADARG); - - if((res = mp_sub(a, b, c)) != MP_OKAY) - return res; - if((res = mp_mod(c, m, c)) != MP_OKAY) - return res; - - return MP_OKAY; - -} - -/* }}} */ - -/* {{{ mp_mulmod(a, b, m, c) */ - -/* - mp_mulmod(a, b, m, c) - - Compute c = (a * b) mod m - */ - -mp_err mp_mulmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) -{ - mp_err res; - - ARGCHK(a != NULL && b != NULL && m != NULL && c != NULL, MP_BADARG); - - if((res = mp_mul(a, b, c)) != MP_OKAY) - return res; - if((res = mp_mod(c, m, c)) != MP_OKAY) - return res; - - return MP_OKAY; - -} - -/* }}} */ - -/* {{{ mp_sqrmod(a, m, c) */ - -#if MP_SQUARE -mp_err mp_sqrmod(mp_int *a, mp_int *m, mp_int *c) -{ - mp_err res; - - ARGCHK(a != NULL && m != NULL && c != NULL, MP_BADARG); - - if((res = mp_sqr(a, c)) != MP_OKAY) - return res; - if((res = mp_mod(c, m, c)) != MP_OKAY) - return res; - - return MP_OKAY; - -} /* end mp_sqrmod() */ -#endif - -/* }}} */ - -/* {{{ mp_exptmod(a, b, m, c) */ - -/* - mp_exptmod(a, b, m, c) - - Compute c = (a ** b) mod m. Uses a standard square-and-multiply - method with modular reductions at each step. (This is basically the - same code as mp_expt(), except for the addition of the reductions) - - The modular reductions are done using Barrett's algorithm (see - s_mp_reduce() below for details) - */ - -mp_err mp_exptmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) -{ - mp_int s, x, mu; - mp_err res; - mp_digit d, *db = DIGITS(b); - mp_size ub = USED(b); - int dig, bit; - - ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); - - if(mp_cmp_z(b) < 0 || mp_cmp_z(m) <= 0) - return MP_RANGE; - - if((res = mp_init(&s)) != MP_OKAY) - return res; - if((res = mp_init_copy(&x, a)) != MP_OKAY) - goto X; - if((res = mp_mod(&x, m, &x)) != MP_OKAY || - (res = mp_init(&mu)) != MP_OKAY) - goto MU; - - mp_set(&s, 1); - - /* mu = b^2k / m */ - s_mp_add_d(&mu, 1); - s_mp_lshd(&mu, 2 * USED(m)); - if((res = mp_div(&mu, m, &mu, NULL)) != MP_OKAY) - goto CLEANUP; - - /* Loop over digits of b in ascending order, except highest order */ - for(dig = 0; dig < (ub - 1); dig++) { - d = *db++; - - /* Loop over the bits of the lower-order digits */ - for(bit = 0; bit < DIGIT_BIT; bit++) { - if(d & 1) { - if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; - if((res = s_mp_reduce(&s, m, &mu)) != MP_OKAY) - goto CLEANUP; - } - - d >>= 1; - - if((res = s_mp_sqr(&x)) != MP_OKAY) - goto CLEANUP; - if((res = s_mp_reduce(&x, m, &mu)) != MP_OKAY) - goto CLEANUP; - } - } - - /* Now do the last digit... */ - d = *db; - - while(d) { - if(d & 1) { - if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; - if((res = s_mp_reduce(&s, m, &mu)) != MP_OKAY) - goto CLEANUP; - } - - d >>= 1; - - if((res = s_mp_sqr(&x)) != MP_OKAY) - goto CLEANUP; - if((res = s_mp_reduce(&x, m, &mu)) != MP_OKAY) - goto CLEANUP; - } - - s_mp_exch(&s, c); - - CLEANUP: - mp_clear(&mu); - MU: - mp_clear(&x); - X: - mp_clear(&s); - - return res; - -} /* end mp_exptmod() */ - -/* }}} */ - -/* {{{ mp_exptmod_d(a, d, m, c) */ - -mp_err mp_exptmod_d(mp_int *a, mp_digit d, mp_int *m, mp_int *c) -{ - mp_int s, x; - mp_err res; - - ARGCHK(a != NULL && c != NULL, MP_BADARG); - - if((res = mp_init(&s)) != MP_OKAY) - return res; - if((res = mp_init_copy(&x, a)) != MP_OKAY) - goto X; - - mp_set(&s, 1); - - while(d != 0) { - if(d & 1) { - if((res = s_mp_mul(&s, &x)) != MP_OKAY || - (res = mp_mod(&s, m, &s)) != MP_OKAY) - goto CLEANUP; - } - - d /= 2; - - if((res = s_mp_sqr(&x)) != MP_OKAY || - (res = mp_mod(&x, m, &x)) != MP_OKAY) - goto CLEANUP; - } - - s_mp_exch(&s, c); - -CLEANUP: - mp_clear(&x); -X: - mp_clear(&s); - - return res; - -} /* end mp_exptmod_d() */ - -/* }}} */ -#endif /* if MP_MODARITH */ - -/* }}} */ - -/*------------------------------------------------------------------------*/ -/* {{{ Comparison functions */ - -/* {{{ mp_cmp_z(a) */ - -/* - mp_cmp_z(a) - - Compare a <=> 0. Returns <0 if a<0, 0 if a=0, >0 if a>0. - */ - -int mp_cmp_z(mp_int *a) -{ - if(SIGN(a) == MP_NEG) - return MP_LT; - else if(USED(a) == 1 && DIGIT(a, 0) == 0) - return MP_EQ; - else - return MP_GT; - -} /* end mp_cmp_z() */ - -/* }}} */ - -/* {{{ mp_cmp_d(a, d) */ - -/* - mp_cmp_d(a, d) - - Compare a <=> d. Returns <0 if a0 if a>d - */ - -int mp_cmp_d(mp_int *a, mp_digit d) -{ - ARGCHK(a != NULL, MP_EQ); - - if(SIGN(a) == MP_NEG) - return MP_LT; - - return s_mp_cmp_d(a, d); - -} /* end mp_cmp_d() */ - -/* }}} */ - -/* {{{ mp_cmp(a, b) */ - -int mp_cmp(mp_int *a, mp_int *b) -{ - ARGCHK(a != NULL && b != NULL, MP_EQ); - - if(SIGN(a) == SIGN(b)) { - int mag; - - if((mag = s_mp_cmp(a, b)) == MP_EQ) - return MP_EQ; - - if(SIGN(a) == MP_ZPOS) - return mag; - else - return -mag; - - } else if(SIGN(a) == MP_ZPOS) { - return MP_GT; - } else { - return MP_LT; - } - -} /* end mp_cmp() */ - -/* }}} */ - -/* {{{ mp_cmp_mag(a, b) */ - -/* - mp_cmp_mag(a, b) - - Compares |a| <=> |b|, and returns an appropriate comparison result - */ - -int mp_cmp_mag(mp_int *a, mp_int *b) -{ - ARGCHK(a != NULL && b != NULL, MP_EQ); - - return s_mp_cmp(a, b); - -} /* end mp_cmp_mag() */ - -/* }}} */ - -/* {{{ mp_cmp_int(a, z) */ - -/* - This just converts z to an mp_int, and uses the existing comparison - routines. This is sort of inefficient, but it's not clear to me how - frequently this wil get used anyway. For small positive constants, - you can always use mp_cmp_d(), and for zero, there is mp_cmp_z(). - */ -int mp_cmp_int(mp_int *a, long z) -{ - mp_int tmp; - int out; - - ARGCHK(a != NULL, MP_EQ); - - mp_init(&tmp); mp_set_int(&tmp, z); - out = mp_cmp(a, &tmp); - mp_clear(&tmp); - - return out; - -} /* end mp_cmp_int() */ - -/* }}} */ - -/* {{{ mp_isodd(a) */ - -/* - mp_isodd(a) - - Returns a true (non-zero) value if a is odd, false (zero) otherwise. - */ -int mp_isodd(mp_int *a) -{ - ARGCHK(a != NULL, 0); - - return (DIGIT(a, 0) & 1); - -} /* end mp_isodd() */ - -/* }}} */ - -/* {{{ mp_iseven(a) */ - -int mp_iseven(mp_int *a) -{ - return !mp_isodd(a); - -} /* end mp_iseven() */ - -/* }}} */ - -/* }}} */ - -/*------------------------------------------------------------------------*/ -/* {{{ Number theoretic functions */ - -#if MP_NUMTH -/* {{{ mp_gcd(a, b, c) */ - -/* - Like the old mp_gcd() function, except computes the GCD using the - binary algorithm due to Josef Stein in 1961 (via Knuth). - */ -mp_err mp_gcd(mp_int *a, mp_int *b, mp_int *c) -{ - mp_err res; - mp_int u, v, t; - mp_size k = 0; - - ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); - - if(mp_cmp_z(a) == MP_EQ && mp_cmp_z(b) == MP_EQ) - return MP_RANGE; - if(mp_cmp_z(a) == MP_EQ) { - return mp_copy(b, c); - } else if(mp_cmp_z(b) == MP_EQ) { - return mp_copy(a, c); - } - - if((res = mp_init(&t)) != MP_OKAY) - return res; - if((res = mp_init_copy(&u, a)) != MP_OKAY) - goto U; - if((res = mp_init_copy(&v, b)) != MP_OKAY) - goto V; - - SIGN(&u) = MP_ZPOS; - SIGN(&v) = MP_ZPOS; - - /* Divide out common factors of 2 until at least 1 of a, b is even */ - while(mp_iseven(&u) && mp_iseven(&v)) { - s_mp_div_2(&u); - s_mp_div_2(&v); - ++k; - } - - /* Initialize t */ - if(mp_isodd(&u)) { - if((res = mp_copy(&v, &t)) != MP_OKAY) - goto CLEANUP; - - /* t = -v */ - if(SIGN(&v) == MP_ZPOS) - SIGN(&t) = MP_NEG; - else - SIGN(&t) = MP_ZPOS; - - } else { - if((res = mp_copy(&u, &t)) != MP_OKAY) - goto CLEANUP; - - } - - for(;;) { - while(mp_iseven(&t)) { - s_mp_div_2(&t); - } - - if(mp_cmp_z(&t) == MP_GT) { - if((res = mp_copy(&t, &u)) != MP_OKAY) - goto CLEANUP; - - } else { - if((res = mp_copy(&t, &v)) != MP_OKAY) - goto CLEANUP; - - /* v = -t */ - if(SIGN(&t) == MP_ZPOS) - SIGN(&v) = MP_NEG; - else - SIGN(&v) = MP_ZPOS; - } - - if((res = mp_sub(&u, &v, &t)) != MP_OKAY) - goto CLEANUP; - - if(s_mp_cmp_d(&t, 0) == MP_EQ) - break; - } - - s_mp_2expt(&v, k); /* v = 2^k */ - res = mp_mul(&u, &v, c); /* c = u * v */ - - CLEANUP: - mp_clear(&v); - V: - mp_clear(&u); - U: - mp_clear(&t); - - return res; - -} /* end mp_bgcd() */ - -/* }}} */ - -/* {{{ mp_lcm(a, b, c) */ - -/* We compute the least common multiple using the rule: - - ab = [a, b](a, b) - - ... by computing the product, and dividing out the gcd. - */ - -mp_err mp_lcm(mp_int *a, mp_int *b, mp_int *c) -{ - mp_int gcd, prod; - mp_err res; - - ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); - - /* Set up temporaries */ - if((res = mp_init(&gcd)) != MP_OKAY) - return res; - if((res = mp_init(&prod)) != MP_OKAY) - goto GCD; - - if((res = mp_mul(a, b, &prod)) != MP_OKAY) - goto CLEANUP; - if((res = mp_gcd(a, b, &gcd)) != MP_OKAY) - goto CLEANUP; - - res = mp_div(&prod, &gcd, c, NULL); - - CLEANUP: - mp_clear(&prod); - GCD: - mp_clear(&gcd); - - return res; - -} /* end mp_lcm() */ - -/* }}} */ - -/* {{{ mp_xgcd(a, b, g, x, y) */ - -/* - mp_xgcd(a, b, g, x, y) - - Compute g = (a, b) and values x and y satisfying Bezout's identity - (that is, ax + by = g). This uses the extended binary GCD algorithm - based on the Stein algorithm used for mp_gcd() - */ - -mp_err mp_xgcd(mp_int *a, mp_int *b, mp_int *g, mp_int *x, mp_int *y) -{ - mp_int gx, xc, yc, u, v, A, B, C, D; - mp_int *clean[9]; - mp_err res; - int last = -1; - - if(mp_cmp_z(b) == 0) - return MP_RANGE; - - /* Initialize all these variables we need */ - if((res = mp_init(&u)) != MP_OKAY) goto CLEANUP; - clean[++last] = &u; - if((res = mp_init(&v)) != MP_OKAY) goto CLEANUP; - clean[++last] = &v; - if((res = mp_init(&gx)) != MP_OKAY) goto CLEANUP; - clean[++last] = &gx; - if((res = mp_init(&A)) != MP_OKAY) goto CLEANUP; - clean[++last] = &A; - if((res = mp_init(&B)) != MP_OKAY) goto CLEANUP; - clean[++last] = &B; - if((res = mp_init(&C)) != MP_OKAY) goto CLEANUP; - clean[++last] = &C; - if((res = mp_init(&D)) != MP_OKAY) goto CLEANUP; - clean[++last] = &D; - if((res = mp_init_copy(&xc, a)) != MP_OKAY) goto CLEANUP; - clean[++last] = &xc; - mp_abs(&xc, &xc); - if((res = mp_init_copy(&yc, b)) != MP_OKAY) goto CLEANUP; - clean[++last] = &yc; - mp_abs(&yc, &yc); - - mp_set(&gx, 1); - - /* Divide by two until at least one of them is even */ - while(mp_iseven(&xc) && mp_iseven(&yc)) { - s_mp_div_2(&xc); - s_mp_div_2(&yc); - if((res = s_mp_mul_2(&gx)) != MP_OKAY) - goto CLEANUP; - } - - mp_copy(&xc, &u); - mp_copy(&yc, &v); - mp_set(&A, 1); mp_set(&D, 1); - - /* Loop through binary GCD algorithm */ - for(;;) { - while(mp_iseven(&u)) { - s_mp_div_2(&u); - - if(mp_iseven(&A) && mp_iseven(&B)) { - s_mp_div_2(&A); s_mp_div_2(&B); - } else { - if((res = mp_add(&A, &yc, &A)) != MP_OKAY) goto CLEANUP; - s_mp_div_2(&A); - if((res = mp_sub(&B, &xc, &B)) != MP_OKAY) goto CLEANUP; - s_mp_div_2(&B); - } - } - - while(mp_iseven(&v)) { - s_mp_div_2(&v); - - if(mp_iseven(&C) && mp_iseven(&D)) { - s_mp_div_2(&C); s_mp_div_2(&D); - } else { - if((res = mp_add(&C, &yc, &C)) != MP_OKAY) goto CLEANUP; - s_mp_div_2(&C); - if((res = mp_sub(&D, &xc, &D)) != MP_OKAY) goto CLEANUP; - s_mp_div_2(&D); - } - } - - if(mp_cmp(&u, &v) >= 0) { - if((res = mp_sub(&u, &v, &u)) != MP_OKAY) goto CLEANUP; - if((res = mp_sub(&A, &C, &A)) != MP_OKAY) goto CLEANUP; - if((res = mp_sub(&B, &D, &B)) != MP_OKAY) goto CLEANUP; - - } else { - if((res = mp_sub(&v, &u, &v)) != MP_OKAY) goto CLEANUP; - if((res = mp_sub(&C, &A, &C)) != MP_OKAY) goto CLEANUP; - if((res = mp_sub(&D, &B, &D)) != MP_OKAY) goto CLEANUP; - - } - - /* If we're done, copy results to output */ - if(mp_cmp_z(&u) == 0) { - if(x) - if((res = mp_copy(&C, x)) != MP_OKAY) goto CLEANUP; - - if(y) - if((res = mp_copy(&D, y)) != MP_OKAY) goto CLEANUP; - - if(g) - if((res = mp_mul(&gx, &v, g)) != MP_OKAY) goto CLEANUP; - - break; - } - } - - CLEANUP: - while(last >= 0) - mp_clear(clean[last--]); - - return res; - -} /* end mp_xgcd() */ - -/* }}} */ - -/* {{{ mp_invmod(a, m, c) */ - -/* - mp_invmod(a, m, c) - - Compute c = a^-1 (mod m), if there is an inverse for a (mod m). - This is equivalent to the question of whether (a, m) = 1. If not, - MP_UNDEF is returned, and there is no inverse. - */ - -mp_err mp_invmod(mp_int *a, mp_int *m, mp_int *c) -{ - mp_int g, x; - mp_err res; - - ARGCHK(a && m && c, MP_BADARG); - - if(mp_cmp_z(a) == 0 || mp_cmp_z(m) == 0) - return MP_RANGE; - - if((res = mp_init(&g)) != MP_OKAY) - return res; - if((res = mp_init(&x)) != MP_OKAY) - goto X; - - if((res = mp_xgcd(a, m, &g, &x, NULL)) != MP_OKAY) - goto CLEANUP; - - if(mp_cmp_d(&g, 1) != MP_EQ) { - res = MP_UNDEF; - goto CLEANUP; - } - - res = mp_mod(&x, m, c); - SIGN(c) = SIGN(a); - -CLEANUP: - mp_clear(&x); -X: - mp_clear(&g); - - return res; - -} /* end mp_invmod() */ - -/* }}} */ -#endif /* if MP_NUMTH */ - -/* }}} */ - -/*------------------------------------------------------------------------*/ -/* {{{ mp_print(mp, ofp) */ - -#if MP_IOFUNC -/* - mp_print(mp, ofp) - - Print a textual representation of the given mp_int on the output - stream 'ofp'. Output is generated using the internal radix. - */ - -void mp_print(mp_int *mp, FILE *ofp) -{ - int ix; - - if(mp == NULL || ofp == NULL) - return; - - fputc((SIGN(mp) == MP_NEG) ? '-' : '+', ofp); - - for(ix = USED(mp) - 1; ix >= 0; ix--) { - fprintf(ofp, DIGIT_FMT, DIGIT(mp, ix)); - } - -} /* end mp_print() */ - -#endif /* if MP_IOFUNC */ - -/* }}} */ - -/*------------------------------------------------------------------------*/ -/* {{{ More I/O Functions */ - -/* {{{ mp_read_signed_bin(mp, str, len) */ - -/* - mp_read_signed_bin(mp, str, len) - - Read in a raw value (base 256) into the given mp_int - */ - -mp_err mp_read_signed_bin(mp_int *mp, unsigned char *str, int len) -{ - mp_err res; - - ARGCHK(mp != NULL && str != NULL && len > 0, MP_BADARG); - - if((res = mp_read_unsigned_bin(mp, str + 1, len - 1)) == MP_OKAY) { - /* Get sign from first byte */ - if(str[0]) - SIGN(mp) = MP_NEG; - else - SIGN(mp) = MP_ZPOS; - } - - return res; - -} /* end mp_read_signed_bin() */ - -/* }}} */ - -/* {{{ mp_signed_bin_size(mp) */ - -int mp_signed_bin_size(mp_int *mp) -{ - ARGCHK(mp != NULL, 0); - - return mp_unsigned_bin_size(mp) + 1; - -} /* end mp_signed_bin_size() */ - -/* }}} */ - -/* {{{ mp_to_signed_bin(mp, str) */ - -mp_err mp_to_signed_bin(mp_int *mp, unsigned char *str) -{ - ARGCHK(mp != NULL && str != NULL, MP_BADARG); - - /* Caller responsible for allocating enough memory (use mp_raw_size(mp)) */ - str[0] = (char)SIGN(mp); - - return mp_to_unsigned_bin(mp, str + 1); - -} /* end mp_to_signed_bin() */ - -/* }}} */ - -/* {{{ mp_read_unsigned_bin(mp, str, len) */ - -/* - mp_read_unsigned_bin(mp, str, len) - - Read in an unsigned value (base 256) into the given mp_int - */ - -mp_err mp_read_unsigned_bin(mp_int *mp, unsigned char *str, int len) -{ - int ix; - mp_err res; - - ARGCHK(mp != NULL && str != NULL && len > 0, MP_BADARG); - - mp_zero(mp); - - for(ix = 0; ix < len; ix++) { - if((res = s_mp_mul_2d(mp, CHAR_BIT)) != MP_OKAY) - return res; - - if((res = mp_add_d(mp, str[ix], mp)) != MP_OKAY) - return res; - } - - return MP_OKAY; - -} /* end mp_read_unsigned_bin() */ - -/* }}} */ - -/* {{{ mp_unsigned_bin_size(mp) */ - -int mp_unsigned_bin_size(mp_int *mp) -{ - mp_digit topdig; - int count; - - ARGCHK(mp != NULL, 0); - - /* Special case for the value zero */ - if(USED(mp) == 1 && DIGIT(mp, 0) == 0) - return 1; - - count = (USED(mp) - 1) * sizeof(mp_digit); - topdig = DIGIT(mp, USED(mp) - 1); - - while(topdig != 0) { - ++count; - topdig >>= CHAR_BIT; - } - - return count; - -} /* end mp_unsigned_bin_size() */ - -/* }}} */ - -/* {{{ mp_to_unsigned_bin(mp, str) */ - -mp_err mp_to_unsigned_bin(mp_int *mp, unsigned char *str) -{ - mp_digit *dp, *end, d; - unsigned char *spos; - - ARGCHK(mp != NULL && str != NULL, MP_BADARG); - - dp = DIGITS(mp); - end = dp + USED(mp) - 1; - spos = str; - - /* Special case for zero, quick test */ - if(dp == end && *dp == 0) { - *str = '\0'; - return MP_OKAY; - } - - /* Generate digits in reverse order */ - while(dp < end) { - int ix; - - d = *dp; - for(ix = 0; ix < sizeof(mp_digit); ++ix) { - *spos = d & UCHAR_MAX; - d >>= CHAR_BIT; - ++spos; - } - - ++dp; - } - - /* Now handle last digit specially, high order zeroes are not written */ - d = *end; - while(d != 0) { - *spos = d & UCHAR_MAX; - d >>= CHAR_BIT; - ++spos; - } - - /* Reverse everything to get digits in the correct order */ - while(--spos > str) { - unsigned char t = *str; - *str = *spos; - *spos = t; - - ++str; - } - - return MP_OKAY; - -} /* end mp_to_unsigned_bin() */ - -/* }}} */ - -/* {{{ mp_count_bits(mp) */ - -int mp_count_bits(mp_int *mp) -{ - int len; - mp_digit d; - - ARGCHK(mp != NULL, MP_BADARG); - - len = DIGIT_BIT * (USED(mp) - 1); - d = DIGIT(mp, USED(mp) - 1); - - while(d != 0) { - ++len; - d >>= 1; - } - - return len; - -} /* end mp_count_bits() */ - -/* }}} */ - -/* {{{ mp_read_radix(mp, str, radix) */ - -/* - mp_read_radix(mp, str, radix) - - Read an integer from the given string, and set mp to the resulting - value. The input is presumed to be in base 10. Leading non-digit - characters are ignored, and the function reads until a non-digit - character or the end of the string. - */ - -mp_err mp_read_radix(mp_int *mp, unsigned char *str, int radix) -{ - int ix = 0, val = 0; - mp_err res; - mp_sign sig = MP_ZPOS; - - ARGCHK(mp != NULL && str != NULL && radix >= 2 && radix <= MAX_RADIX, - MP_BADARG); - - mp_zero(mp); - - /* Skip leading non-digit characters until a digit or '-' or '+' */ - while(str[ix] && - (s_mp_tovalue(str[ix], radix) < 0) && - str[ix] != '-' && - str[ix] != '+') { - ++ix; - } - - if(str[ix] == '-') { - sig = MP_NEG; - ++ix; - } else if(str[ix] == '+') { - sig = MP_ZPOS; /* this is the default anyway... */ - ++ix; - } - - while((val = s_mp_tovalue(str[ix], radix)) >= 0) { - if((res = s_mp_mul_d(mp, radix)) != MP_OKAY) - return res; - if((res = s_mp_add_d(mp, val)) != MP_OKAY) - return res; - ++ix; - } - - if(s_mp_cmp_d(mp, 0) == MP_EQ) - SIGN(mp) = MP_ZPOS; - else - SIGN(mp) = sig; - - return MP_OKAY; - -} /* end mp_read_radix() */ - -/* }}} */ - -/* {{{ mp_radix_size(mp, radix) */ - -int mp_radix_size(mp_int *mp, int radix) -{ - int len; - ARGCHK(mp != NULL, 0); - - len = s_mp_outlen(mp_count_bits(mp), radix) + 1; /* for NUL terminator */ - - if(mp_cmp_z(mp) < 0) - ++len; /* for sign */ - - return len; - -} /* end mp_radix_size() */ - -/* }}} */ - -/* {{{ mp_value_radix_size(num, qty, radix) */ - -/* num = number of digits - qty = number of bits per digit - radix = target base - - Return the number of digits in the specified radix that would be - needed to express 'num' digits of 'qty' bits each. - */ -int mp_value_radix_size(int num, int qty, int radix) -{ - ARGCHK(num >= 0 && qty > 0 && radix >= 2 && radix <= MAX_RADIX, 0); - - return s_mp_outlen(num * qty, radix); - -} /* end mp_value_radix_size() */ - -/* }}} */ - -/* {{{ mp_toradix(mp, str, radix) */ - -mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix) -{ - int ix, pos = 0; - - ARGCHK(mp != NULL && str != NULL, MP_BADARG); - ARGCHK(radix > 1 && radix <= MAX_RADIX, MP_RANGE); - - if(mp_cmp_z(mp) == MP_EQ) { - str[0] = '0'; - str[1] = '\0'; - } else { - mp_err res; - mp_int tmp; - mp_sign sgn; - mp_digit rem, rdx = (mp_digit)radix; - char ch; - - if((res = mp_init_copy(&tmp, mp)) != MP_OKAY) - return res; - - /* Save sign for later, and take absolute value */ - sgn = SIGN(&tmp); SIGN(&tmp) = MP_ZPOS; - - /* Generate output digits in reverse order */ - while(mp_cmp_z(&tmp) != 0) { - if((res = s_mp_div_d(&tmp, rdx, &rem)) != MP_OKAY) { - mp_clear(&tmp); - return res; - } - - /* Generate digits, use capital letters */ - ch = s_mp_todigit(rem, radix, 0); - - str[pos++] = ch; - } - - /* Add - sign if original value was negative */ - if(sgn == MP_NEG) - str[pos++] = '-'; - - /* Add trailing NUL to end the string */ - str[pos--] = '\0'; - - /* Reverse the digits and sign indicator */ - ix = 0; - while(ix < pos) { - char tmp = str[ix]; - - str[ix] = str[pos]; - str[pos] = tmp; - ++ix; - --pos; - } - - mp_clear(&tmp); - } - - return MP_OKAY; - -} /* end mp_toradix() */ - -/* }}} */ - -/* {{{ mp_char2value(ch, r) */ - -int mp_char2value(char ch, int r) -{ - return s_mp_tovalue(ch, r); - -} /* end mp_tovalue() */ - -/* }}} */ - -/* }}} */ - -/* {{{ mp_strerror(ec) */ - -/* - mp_strerror(ec) - - Return a string describing the meaning of error code 'ec'. The - string returned is allocated in static memory, so the caller should - not attempt to modify or free the memory associated with this - string. - */ -const char *mp_strerror(mp_err ec) -{ - int aec = (ec < 0) ? -ec : ec; - - /* Code values are negative, so the senses of these comparisons - are accurate */ - if(ec < MP_LAST_CODE || ec > MP_OKAY) { - return mp_err_string[0]; /* unknown error code */ - } else { - return mp_err_string[aec + 1]; - } - -} /* end mp_strerror() */ - -/* }}} */ - -/*========================================================================*/ -/*------------------------------------------------------------------------*/ -/* Static function definitions (internal use only) */ - -/* {{{ Memory management */ - -/* {{{ s_mp_grow(mp, min) */ - -/* Make sure there are at least 'min' digits allocated to mp */ -mp_err s_mp_grow(mp_int *mp, mp_size min) -{ - if(min > ALLOC(mp)) { - mp_digit *tmp; - - /* Set min to next nearest default precision block size */ - min = ((min + (s_mp_defprec - 1)) / s_mp_defprec) * s_mp_defprec; - - if((tmp = s_mp_alloc(min, sizeof(mp_digit))) == NULL) - return MP_MEM; - - s_mp_copy(DIGITS(mp), tmp, USED(mp)); - -#if MP_CRYPTO - s_mp_setz(DIGITS(mp), ALLOC(mp)); -#endif - s_mp_free(DIGITS(mp)); - DIGITS(mp) = tmp; - ALLOC(mp) = min; - } - - return MP_OKAY; - -} /* end s_mp_grow() */ - -/* }}} */ - -/* {{{ s_mp_pad(mp, min) */ - -/* Make sure the used size of mp is at least 'min', growing if needed */ -mp_err s_mp_pad(mp_int *mp, mp_size min) -{ - if(min > USED(mp)) { - mp_err res; - - /* Make sure there is room to increase precision */ - if(min > ALLOC(mp) && (res = s_mp_grow(mp, min)) != MP_OKAY) - return res; - - /* Increase precision; should already be 0-filled */ - USED(mp) = min; - } - - return MP_OKAY; - -} /* end s_mp_pad() */ - -/* }}} */ - -/* {{{ s_mp_setz(dp, count) */ - -#if MP_MACRO == 0 -/* Set 'count' digits pointed to by dp to be zeroes */ -void s_mp_setz(mp_digit *dp, mp_size count) -{ -#if MP_MEMSET == 0 - int ix; - - for(ix = 0; ix < count; ix++) - dp[ix] = 0; -#else - memset(dp, 0, count * sizeof(mp_digit)); -#endif - -} /* end s_mp_setz() */ -#endif - -/* }}} */ - -/* {{{ s_mp_copy(sp, dp, count) */ - -#if MP_MACRO == 0 -/* Copy 'count' digits from sp to dp */ -void s_mp_copy(mp_digit *sp, mp_digit *dp, mp_size count) -{ -#if MP_MEMCPY == 0 - int ix; - - for(ix = 0; ix < count; ix++) - dp[ix] = sp[ix]; -#else - memcpy(dp, sp, count * sizeof(mp_digit)); -#endif - -} /* end s_mp_copy() */ -#endif - -/* }}} */ - -/* {{{ s_mp_alloc(nb, ni) */ - -#if MP_MACRO == 0 -/* Allocate ni records of nb bytes each, and return a pointer to that */ -void *s_mp_alloc(size_t nb, size_t ni) -{ - return calloc(nb, ni); - -} /* end s_mp_alloc() */ -#endif - -/* }}} */ - -/* {{{ s_mp_free(ptr) */ - -#if MP_MACRO == 0 -/* Free the memory pointed to by ptr */ -void s_mp_free(void *ptr) -{ - if(ptr) - free(ptr); - -} /* end s_mp_free() */ -#endif - -/* }}} */ - -/* {{{ s_mp_clamp(mp) */ - -/* Remove leading zeroes from the given value */ -void s_mp_clamp(mp_int *mp) -{ - mp_size du = USED(mp); - mp_digit *zp = DIGITS(mp) + du - 1; - - while(du > 1 && !*zp--) - --du; - - USED(mp) = du; - -} /* end s_mp_clamp() */ - - -/* }}} */ - -/* {{{ s_mp_exch(a, b) */ - -/* Exchange the data for a and b; (b, a) = (a, b) */ -void s_mp_exch(mp_int *a, mp_int *b) -{ - mp_int tmp; - - tmp = *a; - *a = *b; - *b = tmp; - -} /* end s_mp_exch() */ - -/* }}} */ - -/* }}} */ - -/* {{{ Arithmetic helpers */ - -/* {{{ s_mp_lshd(mp, p) */ - -/* - Shift mp leftward by p digits, growing if needed, and zero-filling - the in-shifted digits at the right end. This is a convenient - alternative to multiplication by powers of the radix - */ - -mp_err s_mp_lshd(mp_int *mp, mp_size p) -{ - mp_err res; - mp_size pos; - mp_digit *dp; - int ix; - - if(p == 0) - return MP_OKAY; - - if((res = s_mp_pad(mp, USED(mp) + p)) != MP_OKAY) - return res; - - pos = USED(mp) - 1; - dp = DIGITS(mp); - - /* Shift all the significant figures over as needed */ - for(ix = pos - p; ix >= 0; ix--) - dp[ix + p] = dp[ix]; - - /* Fill the bottom digits with zeroes */ - for(ix = 0; ix < p; ix++) - dp[ix] = 0; - - return MP_OKAY; - -} /* end s_mp_lshd() */ - -/* }}} */ - -/* {{{ s_mp_rshd(mp, p) */ - -/* - Shift mp rightward by p digits. Maintains the invariant that - digits above the precision are all zero. Digits shifted off the - end are lost. Cannot fail. - */ - -void s_mp_rshd(mp_int *mp, mp_size p) -{ - mp_size ix; - mp_digit *dp; - - if(p == 0) - return; - - /* Shortcut when all digits are to be shifted off */ - if(p >= USED(mp)) { - s_mp_setz(DIGITS(mp), ALLOC(mp)); - USED(mp) = 1; - SIGN(mp) = MP_ZPOS; - return; - } - - /* Shift all the significant figures over as needed */ - dp = DIGITS(mp); - for(ix = p; ix < USED(mp); ix++) - dp[ix - p] = dp[ix]; - - /* Fill the top digits with zeroes */ - ix -= p; - while(ix < USED(mp)) - dp[ix++] = 0; - - /* Strip off any leading zeroes */ - s_mp_clamp(mp); - -} /* end s_mp_rshd() */ - -/* }}} */ - -/* {{{ s_mp_div_2(mp) */ - -/* Divide by two -- take advantage of radix properties to do it fast */ -void s_mp_div_2(mp_int *mp) -{ - s_mp_div_2d(mp, 1); - -} /* end s_mp_div_2() */ - -/* }}} */ - -/* {{{ s_mp_mul_2(mp) */ - -mp_err s_mp_mul_2(mp_int *mp) -{ - int ix; - mp_digit kin = 0, kout, *dp = DIGITS(mp); - mp_err res; - - /* Shift digits leftward by 1 bit */ - for(ix = 0; ix < USED(mp); ix++) { - kout = (dp[ix] >> (DIGIT_BIT - 1)) & 1; - dp[ix] = (dp[ix] << 1) | kin; - - kin = kout; - } - - /* Deal with rollover from last digit */ - if(kin) { - if(ix >= ALLOC(mp)) { - if((res = s_mp_grow(mp, ALLOC(mp) + 1)) != MP_OKAY) - return res; - dp = DIGITS(mp); - } - - dp[ix] = kin; - USED(mp) += 1; - } - - return MP_OKAY; - -} /* end s_mp_mul_2() */ - -/* }}} */ - -/* {{{ s_mp_mod_2d(mp, d) */ - -/* - Remainder the integer by 2^d, where d is a number of bits. This - amounts to a bitwise AND of the value, and does not require the full - division code - */ -void s_mp_mod_2d(mp_int *mp, mp_digit d) -{ - unsigned int ndig = (d / DIGIT_BIT), nbit = (d % DIGIT_BIT); - unsigned int ix; - mp_digit dmask, *dp = DIGITS(mp); - - if(ndig >= USED(mp)) - return; - - /* Flush all the bits above 2^d in its digit */ - dmask = (1 << nbit) - 1; - dp[ndig] &= dmask; - - /* Flush all digits above the one with 2^d in it */ - for(ix = ndig + 1; ix < USED(mp); ix++) - dp[ix] = 0; - - s_mp_clamp(mp); - -} /* end s_mp_mod_2d() */ - -/* }}} */ - -/* {{{ s_mp_mul_2d(mp, d) */ - -/* - Multiply by the integer 2^d, where d is a number of bits. This - amounts to a bitwise shift of the value, and does not require the - full multiplication code. - */ -mp_err s_mp_mul_2d(mp_int *mp, mp_digit d) -{ - mp_err res; - mp_digit save, next, mask, *dp; - mp_size used; - int ix; - - if((res = s_mp_lshd(mp, d / DIGIT_BIT)) != MP_OKAY) - return res; - - dp = DIGITS(mp); used = USED(mp); - d %= DIGIT_BIT; - - mask = (1 << d) - 1; - - /* If the shift requires another digit, make sure we've got one to - work with */ - if((dp[used - 1] >> (DIGIT_BIT - d)) & mask) { - if((res = s_mp_grow(mp, used + 1)) != MP_OKAY) - return res; - dp = DIGITS(mp); - } - - /* Do the shifting... */ - save = 0; - for(ix = 0; ix < used; ix++) { - next = (dp[ix] >> (DIGIT_BIT - d)) & mask; - dp[ix] = (dp[ix] << d) | save; - save = next; - } - - /* If, at this point, we have a nonzero carryout into the next - digit, we'll increase the size by one digit, and store it... - */ - if(save) { - dp[used] = save; - USED(mp) += 1; - } - - s_mp_clamp(mp); - return MP_OKAY; - -} /* end s_mp_mul_2d() */ - -/* }}} */ - -/* {{{ s_mp_div_2d(mp, d) */ - -/* - Divide the integer by 2^d, where d is a number of bits. This - amounts to a bitwise shift of the value, and does not require the - full division code (used in Barrett reduction, see below) - */ -void s_mp_div_2d(mp_int *mp, mp_digit d) -{ - int ix; - mp_digit save, next, mask, *dp = DIGITS(mp); - - s_mp_rshd(mp, d / DIGIT_BIT); - d %= DIGIT_BIT; - - mask = (1 << d) - 1; - - save = 0; - for(ix = USED(mp) - 1; ix >= 0; ix--) { - next = dp[ix] & mask; - dp[ix] = (dp[ix] >> d) | (save << (DIGIT_BIT - d)); - save = next; - } - - s_mp_clamp(mp); - -} /* end s_mp_div_2d() */ - -/* }}} */ - -/* {{{ s_mp_norm(a, b) */ - -/* - s_mp_norm(a, b) - - Normalize a and b for division, where b is the divisor. In order - that we might make good guesses for quotient digits, we want the - leading digit of b to be at least half the radix, which we - accomplish by multiplying a and b by a constant. This constant is - returned (so that it can be divided back out of the remainder at the - end of the division process). - - We multiply by the smallest power of 2 that gives us a leading digit - at least half the radix. By choosing a power of 2, we simplify the - multiplication and division steps to simple shifts. - */ -mp_digit s_mp_norm(mp_int *a, mp_int *b) -{ - mp_digit t, d = 0; - - t = DIGIT(b, USED(b) - 1); - while(t < (RADIX / 2)) { - t <<= 1; - ++d; - } - - if(d != 0) { - s_mp_mul_2d(a, d); - s_mp_mul_2d(b, d); - } - - return d; - -} /* end s_mp_norm() */ - -/* }}} */ - -/* }}} */ - -/* {{{ Primitive digit arithmetic */ - -/* {{{ s_mp_add_d(mp, d) */ - -/* Add d to |mp| in place */ -mp_err s_mp_add_d(mp_int *mp, mp_digit d) /* unsigned digit addition */ -{ - mp_word w, k = 0; - mp_size ix = 1, used = USED(mp); - mp_digit *dp = DIGITS(mp); - - w = dp[0] + d; - dp[0] = ACCUM(w); - k = CARRYOUT(w); - - while(ix < used && k) { - w = dp[ix] + k; - dp[ix] = ACCUM(w); - k = CARRYOUT(w); - ++ix; - } - - if(k != 0) { - mp_err res; - - if((res = s_mp_pad(mp, USED(mp) + 1)) != MP_OKAY) - return res; - - DIGIT(mp, ix) = k; - } - - return MP_OKAY; - -} /* end s_mp_add_d() */ - -/* }}} */ - -/* {{{ s_mp_sub_d(mp, d) */ - -/* Subtract d from |mp| in place, assumes |mp| > d */ -mp_err s_mp_sub_d(mp_int *mp, mp_digit d) /* unsigned digit subtract */ -{ - mp_word w, b = 0; - mp_size ix = 1, used = USED(mp); - mp_digit *dp = DIGITS(mp); - - /* Compute initial subtraction */ - w = (RADIX + dp[0]) - d; - b = CARRYOUT(w) ? 0 : 1; - dp[0] = ACCUM(w); - - /* Propagate borrows leftward */ - while(b && ix < used) { - w = (RADIX + dp[ix]) - b; - b = CARRYOUT(w) ? 0 : 1; - dp[ix] = ACCUM(w); - ++ix; - } - - /* Remove leading zeroes */ - s_mp_clamp(mp); - - /* If we have a borrow out, it's a violation of the input invariant */ - if(b) - return MP_RANGE; - else - return MP_OKAY; - -} /* end s_mp_sub_d() */ - -/* }}} */ - -/* {{{ s_mp_mul_d(a, d) */ - -/* Compute a = a * d, single digit multiplication */ -mp_err s_mp_mul_d(mp_int *a, mp_digit d) -{ - mp_word w, k = 0; - mp_size ix, max; - mp_err res; - mp_digit *dp = DIGITS(a); - - /* - Single-digit multiplication will increase the precision of the - output by at most one digit. However, we can detect when this - will happen -- if the high-order digit of a, times d, gives a - two-digit result, then the precision of the result will increase; - otherwise it won't. We use this fact to avoid calling s_mp_pad() - unless absolutely necessary. - */ - max = USED(a); - w = dp[max - 1] * d; - if(CARRYOUT(w) != 0) { - if((res = s_mp_pad(a, max + 1)) != MP_OKAY) - return res; - dp = DIGITS(a); - } - - for(ix = 0; ix < max; ix++) { - w = (dp[ix] * d) + k; - dp[ix] = ACCUM(w); - k = CARRYOUT(w); - } - - /* If there is a precision increase, take care of it here; the above - test guarantees we have enough storage to do this safely. - */ - if(k) { - dp[max] = k; - USED(a) = max + 1; - } - - s_mp_clamp(a); - - return MP_OKAY; - -} /* end s_mp_mul_d() */ - -/* }}} */ - -/* {{{ s_mp_div_d(mp, d, r) */ - -/* - s_mp_div_d(mp, d, r) - - Compute the quotient mp = mp / d and remainder r = mp mod d, for a - single digit d. If r is null, the remainder will be discarded. - */ - -mp_err s_mp_div_d(mp_int *mp, mp_digit d, mp_digit *r) -{ - mp_word w = 0, t; - mp_int quot; - mp_err res; - mp_digit *dp = DIGITS(mp), *qp; - int ix; - - if(d == 0) - return MP_RANGE; - - /* Make room for the quotient */ - if((res = mp_init_size(", USED(mp))) != MP_OKAY) - return res; - - USED(") = USED(mp); /* so clamping will work below */ - qp = DIGITS("); - - /* Divide without subtraction */ - for(ix = USED(mp) - 1; ix >= 0; ix--) { - w = (w << DIGIT_BIT) | dp[ix]; - - if(w >= d) { - t = w / d; - w = w % d; - } else { - t = 0; - } - - qp[ix] = t; - } - - /* Deliver the remainder, if desired */ - if(r) - *r = w; - - s_mp_clamp("); - mp_exch(", mp); - mp_clear("); - - return MP_OKAY; - -} /* end s_mp_div_d() */ - -/* }}} */ - -/* }}} */ - -/* {{{ Primitive full arithmetic */ - -/* {{{ s_mp_add(a, b) */ - -/* Compute a = |a| + |b| */ -mp_err s_mp_add(mp_int *a, mp_int *b) /* magnitude addition */ -{ - mp_word w = 0; - mp_digit *pa, *pb; - mp_size ix, used = USED(b); - mp_err res; - - /* Make sure a has enough precision for the output value */ - if((used > USED(a)) && (res = s_mp_pad(a, used)) != MP_OKAY) - return res; - - /* - Add up all digits up to the precision of b. If b had initially - the same precision as a, or greater, we took care of it by the - padding step above, so there is no problem. If b had initially - less precision, we'll have to make sure the carry out is duly - propagated upward among the higher-order digits of the sum. - */ - pa = DIGITS(a); - pb = DIGITS(b); - for(ix = 0; ix < used; ++ix) { - w += *pa + *pb++; - *pa++ = ACCUM(w); - w = CARRYOUT(w); - } - - /* If we run out of 'b' digits before we're actually done, make - sure the carries get propagated upward... - */ - used = USED(a); - while(w && ix < used) { - w += *pa; - *pa++ = ACCUM(w); - w = CARRYOUT(w); - ++ix; - } - - /* If there's an overall carry out, increase precision and include - it. We could have done this initially, but why touch the memory - allocator unless we're sure we have to? - */ - if(w) { - if((res = s_mp_pad(a, used + 1)) != MP_OKAY) - return res; - - DIGIT(a, ix) = w; /* pa may not be valid after s_mp_pad() call */ - } - - return MP_OKAY; - -} /* end s_mp_add() */ - -/* }}} */ - -/* {{{ s_mp_sub(a, b) */ - -/* Compute a = |a| - |b|, assumes |a| >= |b| */ -mp_err s_mp_sub(mp_int *a, mp_int *b) /* magnitude subtract */ -{ - mp_word w = 0; - mp_digit *pa, *pb; - mp_size ix, used = USED(b); - - /* - Subtract and propagate borrow. Up to the precision of b, this - accounts for the digits of b; after that, we just make sure the - carries get to the right place. This saves having to pad b out to - the precision of a just to make the loops work right... - */ - pa = DIGITS(a); - pb = DIGITS(b); - - for(ix = 0; ix < used; ++ix) { - w = (RADIX + *pa) - w - *pb++; - *pa++ = ACCUM(w); - w = CARRYOUT(w) ? 0 : 1; - } - - used = USED(a); - while(ix < used) { - w = RADIX + *pa - w; - *pa++ = ACCUM(w); - w = CARRYOUT(w) ? 0 : 1; - ++ix; - } - - /* Clobber any leading zeroes we created */ - s_mp_clamp(a); - - /* - If there was a borrow out, then |b| > |a| in violation - of our input invariant. We've already done the work, - but we'll at least complain about it... - */ - if(w) - return MP_RANGE; - else - return MP_OKAY; - -} /* end s_mp_sub() */ - -/* }}} */ - -mp_err s_mp_reduce(mp_int *x, mp_int *m, mp_int *mu) -{ - mp_int q; - mp_err res; - mp_size um = USED(m); - - if((res = mp_init_copy(&q, x)) != MP_OKAY) - return res; - - s_mp_rshd(&q, um - 1); /* q1 = x / b^(k-1) */ - s_mp_mul(&q, mu); /* q2 = q1 * mu */ - s_mp_rshd(&q, um + 1); /* q3 = q2 / b^(k+1) */ - - /* x = x mod b^(k+1), quick (no division) */ - s_mp_mod_2d(x, (mp_digit)(DIGIT_BIT * (um + 1))); - - /* q = q * m mod b^(k+1), quick (no division), uses the short multiplier */ -#ifndef SHRT_MUL - s_mp_mul(&q, m); - s_mp_mod_2d(&q, (mp_digit)(DIGIT_BIT * (um + 1))); -#else - s_mp_mul_dig(&q, m, um + 1); -#endif - - /* x = x - q */ - if((res = mp_sub(x, &q, x)) != MP_OKAY) - goto CLEANUP; - - /* If x < 0, add b^(k+1) to it */ - if(mp_cmp_z(x) < 0) { - mp_set(&q, 1); - if((res = s_mp_lshd(&q, um + 1)) != MP_OKAY) - goto CLEANUP; - if((res = mp_add(x, &q, x)) != MP_OKAY) - goto CLEANUP; - } - - /* Back off if it's too big */ - while(mp_cmp(x, m) >= 0) { - if((res = s_mp_sub(x, m)) != MP_OKAY) - break; - } - - CLEANUP: - mp_clear(&q); - - return res; - -} /* end s_mp_reduce() */ - - - -/* {{{ s_mp_mul(a, b) */ - -/* Compute a = |a| * |b| */ -mp_err s_mp_mul(mp_int *a, mp_int *b) -{ - mp_word w, k = 0; - mp_int tmp; - mp_err res; - mp_size ix, jx, ua = USED(a), ub = USED(b); - mp_digit *pa, *pb, *pt, *pbt; - - if((res = mp_init_size(&tmp, ua + ub)) != MP_OKAY) - return res; - - /* This has the effect of left-padding with zeroes... */ - USED(&tmp) = ua + ub; - - /* We're going to need the base value each iteration */ - pbt = DIGITS(&tmp); - - /* Outer loop: Digits of b */ - - pb = DIGITS(b); - for(ix = 0; ix < ub; ++ix, ++pb) { - if(*pb == 0) - continue; - - /* Inner product: Digits of a */ - pa = DIGITS(a); - for(jx = 0; jx < ua; ++jx, ++pa) { - pt = pbt + ix + jx; - w = *pb * *pa + k + *pt; - *pt = ACCUM(w); - k = CARRYOUT(w); - } - - pbt[ix + jx] = k; - k = 0; - } - - s_mp_clamp(&tmp); - s_mp_exch(&tmp, a); - - mp_clear(&tmp); - - return MP_OKAY; - -} /* end s_mp_mul() */ - -/* }}} */ - -/* {{{ s_mp_kmul(a, b, out, len) */ - -#if 0 -void s_mp_kmul(mp_digit *a, mp_digit *b, mp_digit *out, mp_size len) -{ - mp_word w, k = 0; - mp_size ix, jx; - mp_digit *pa, *pt; - - for(ix = 0; ix < len; ++ix, ++b) { - if(*b == 0) - continue; - - pa = a; - for(jx = 0; jx < len; ++jx, ++pa) { - pt = out + ix + jx; - w = *b * *pa + k + *pt; - *pt = ACCUM(w); - k = CARRYOUT(w); - } - - out[ix + jx] = k; - k = 0; - } - -} /* end s_mp_kmul() */ -#endif - -/* }}} */ - -/* {{{ s_mp_sqr(a) */ - -/* - Computes the square of a, in place. This can be done more - efficiently than a general multiplication, because many of the - computation steps are redundant when squaring. The inner product - step is a bit more complicated, but we save a fair number of - iterations of the multiplication loop. - */ -#if MP_SQUARE -mp_err s_mp_sqr(mp_int *a) -{ - mp_word w, k = 0; - mp_int tmp; - mp_err res; - mp_size ix, jx, kx, used = USED(a); - mp_digit *pa1, *pa2, *pt, *pbt; - - if((res = mp_init_size(&tmp, 2 * used)) != MP_OKAY) - return res; - - /* Left-pad with zeroes */ - USED(&tmp) = 2 * used; - - /* We need the base value each time through the loop */ - pbt = DIGITS(&tmp); - - pa1 = DIGITS(a); - for(ix = 0; ix < used; ++ix, ++pa1) { - if(*pa1 == 0) - continue; - - w = DIGIT(&tmp, ix + ix) + (*pa1 * *pa1); - - pbt[ix + ix] = ACCUM(w); - k = CARRYOUT(w); - - /* - The inner product is computed as: - - (C, S) = t[i,j] + 2 a[i] a[j] + C - - This can overflow what can be represented in an mp_word, and - since C arithmetic does not provide any way to check for - overflow, we have to check explicitly for overflow conditions - before they happen. - */ - for(jx = ix + 1, pa2 = DIGITS(a) + jx; jx < used; ++jx, ++pa2) { - mp_word u = 0, v; - - /* Store this in a temporary to avoid indirections later */ - pt = pbt + ix + jx; - - /* Compute the multiplicative step */ - w = *pa1 * *pa2; - - /* If w is more than half MP_WORD_MAX, the doubling will - overflow, and we need to record a carry out into the next - word */ - u = (w >> (MP_WORD_BIT - 1)) & 1; - - /* Double what we've got, overflow will be ignored as defined - for C arithmetic (we've already noted if it is to occur) - */ - w *= 2; - - /* Compute the additive step */ - v = *pt + k; - - /* If we do not already have an overflow carry, check to see - if the addition will cause one, and set the carry out if so - */ - u |= ((MP_WORD_MAX - v) < w); - - /* Add in the rest, again ignoring overflow */ - w += v; - - /* Set the i,j digit of the output */ - *pt = ACCUM(w); - - /* Save carry information for the next iteration of the loop. - This is why k must be an mp_word, instead of an mp_digit */ - k = CARRYOUT(w) | (u << DIGIT_BIT); - - } /* for(jx ...) */ - - /* Set the last digit in the cycle and reset the carry */ - k = DIGIT(&tmp, ix + jx) + k; - pbt[ix + jx] = ACCUM(k); - k = CARRYOUT(k); - - /* If we are carrying out, propagate the carry to the next digit - in the output. This may cascade, so we have to be somewhat - circumspect -- but we will have enough precision in the output - that we won't overflow - */ - kx = 1; - while(k) { - k = pbt[ix + jx + kx] + 1; - pbt[ix + jx + kx] = ACCUM(k); - k = CARRYOUT(k); - ++kx; - } - } /* for(ix ...) */ - - s_mp_clamp(&tmp); - s_mp_exch(&tmp, a); - - mp_clear(&tmp); - - return MP_OKAY; - -} /* end s_mp_sqr() */ -#endif - -/* }}} */ - -/* {{{ s_mp_div(a, b) */ - -/* - s_mp_div(a, b) - - Compute a = a / b and b = a mod b. Assumes b > a. - */ - -mp_err s_mp_div(mp_int *a, mp_int *b) -{ - mp_int quot, rem, t; - mp_word q; - mp_err res; - mp_digit d; - int ix; - - if(mp_cmp_z(b) == 0) - return MP_RANGE; - - /* Shortcut if b is power of two */ - if((ix = s_mp_ispow2(b)) >= 0) { - mp_copy(a, b); /* need this for remainder */ - s_mp_div_2d(a, (mp_digit)ix); - s_mp_mod_2d(b, (mp_digit)ix); - - return MP_OKAY; - } - - /* Allocate space to store the quotient */ - if((res = mp_init_size(", USED(a))) != MP_OKAY) - return res; - - /* A working temporary for division */ - if((res = mp_init_size(&t, USED(a))) != MP_OKAY) - goto T; - - /* Allocate space for the remainder */ - if((res = mp_init_size(&rem, USED(a))) != MP_OKAY) - goto REM; - - /* Normalize to optimize guessing */ - d = s_mp_norm(a, b); - - /* Perform the division itself...woo! */ - ix = USED(a) - 1; - - while(ix >= 0) { - /* Find a partial substring of a which is at least b */ - while(s_mp_cmp(&rem, b) < 0 && ix >= 0) { - if((res = s_mp_lshd(&rem, 1)) != MP_OKAY) - goto CLEANUP; - - if((res = s_mp_lshd(", 1)) != MP_OKAY) - goto CLEANUP; - - DIGIT(&rem, 0) = DIGIT(a, ix); - s_mp_clamp(&rem); - --ix; - } - - /* If we didn't find one, we're finished dividing */ - if(s_mp_cmp(&rem, b) < 0) - break; - - /* Compute a guess for the next quotient digit */ - q = DIGIT(&rem, USED(&rem) - 1); - if(q <= DIGIT(b, USED(b) - 1) && USED(&rem) > 1) - q = (q << DIGIT_BIT) | DIGIT(&rem, USED(&rem) - 2); - - q /= DIGIT(b, USED(b) - 1); - - /* The guess can be as much as RADIX + 1 */ - if(q >= RADIX) - q = RADIX - 1; - - /* See what that multiplies out to */ - mp_copy(b, &t); - if((res = s_mp_mul_d(&t, q)) != MP_OKAY) - goto CLEANUP; - - /* - If it's too big, back it off. We should not have to do this - more than once, or, in rare cases, twice. Knuth describes a - method by which this could be reduced to a maximum of once, but - I didn't implement that here. - */ - while(s_mp_cmp(&t, &rem) > 0) { - --q; - s_mp_sub(&t, b); - } - - /* At this point, q should be the right next digit */ - if((res = s_mp_sub(&rem, &t)) != MP_OKAY) - goto CLEANUP; - - /* - Include the digit in the quotient. We allocated enough memory - for any quotient we could ever possibly get, so we should not - have to check for failures here - */ - DIGIT(", 0) = q; - } - - /* Denormalize remainder */ - if(d != 0) - s_mp_div_2d(&rem, d); - - s_mp_clamp("); - s_mp_clamp(&rem); - - /* Copy quotient back to output */ - s_mp_exch(", a); - - /* Copy remainder back to output */ - s_mp_exch(&rem, b); - -CLEANUP: - mp_clear(&rem); -REM: - mp_clear(&t); -T: - mp_clear("); - - return res; - -} /* end s_mp_div() */ - -/* }}} */ - -/* {{{ s_mp_2expt(a, k) */ - -mp_err s_mp_2expt(mp_int *a, mp_digit k) -{ - mp_err res; - mp_size dig, bit; - - dig = k / DIGIT_BIT; - bit = k % DIGIT_BIT; - - mp_zero(a); - if((res = s_mp_pad(a, dig + 1)) != MP_OKAY) - return res; - - DIGIT(a, dig) |= (1 << bit); - - return MP_OKAY; - -} /* end s_mp_2expt() */ - -/* }}} */ - - -/* }}} */ - -/* }}} */ - -/* {{{ Primitive comparisons */ - -/* {{{ s_mp_cmp(a, b) */ - -/* Compare |a| <=> |b|, return 0 if equal, <0 if a0 if a>b */ -int s_mp_cmp(mp_int *a, mp_int *b) -{ - mp_size ua = USED(a), ub = USED(b); - - if(ua > ub) - return MP_GT; - else if(ua < ub) - return MP_LT; - else { - int ix = ua - 1; - mp_digit *ap = DIGITS(a) + ix, *bp = DIGITS(b) + ix; - - while(ix >= 0) { - if(*ap > *bp) - return MP_GT; - else if(*ap < *bp) - return MP_LT; - - --ap; --bp; --ix; - } - - return MP_EQ; - } - -} /* end s_mp_cmp() */ - -/* }}} */ - -/* {{{ s_mp_cmp_d(a, d) */ - -/* Compare |a| <=> d, return 0 if equal, <0 if a0 if a>d */ -int s_mp_cmp_d(mp_int *a, mp_digit d) -{ - mp_size ua = USED(a); - mp_digit *ap = DIGITS(a); - - if(ua > 1) - return MP_GT; - - if(*ap < d) - return MP_LT; - else if(*ap > d) - return MP_GT; - else - return MP_EQ; - -} /* end s_mp_cmp_d() */ - -/* }}} */ - -/* {{{ s_mp_ispow2(v) */ - -/* - Returns -1 if the value is not a power of two; otherwise, it returns - k such that v = 2^k, i.e. lg(v). - */ -int s_mp_ispow2(mp_int *v) -{ - mp_digit d, *dp; - mp_size uv = USED(v); - int extra = 0, ix; - - d = DIGIT(v, uv - 1); /* most significant digit of v */ - - while(d && ((d & 1) == 0)) { - d >>= 1; - ++extra; - } - - if(d == 1) { - ix = uv - 2; - dp = DIGITS(v) + ix; - - while(ix >= 0) { - if(*dp) - return -1; /* not a power of two */ - - --dp; --ix; - } - - return ((uv - 1) * DIGIT_BIT) + extra; - } - - return -1; - -} /* end s_mp_ispow2() */ - -/* }}} */ - -/* {{{ s_mp_ispow2d(d) */ - -int s_mp_ispow2d(mp_digit d) -{ - int pow = 0; - - while((d & 1) == 0) { - ++pow; d >>= 1; - } - - if(d == 1) - return pow; - - return -1; - -} /* end s_mp_ispow2d() */ - -/* }}} */ - -/* }}} */ - -/* {{{ Primitive I/O helpers */ - -/* {{{ s_mp_tovalue(ch, r) */ - -/* - Convert the given character to its digit value, in the given radix. - If the given character is not understood in the given radix, -1 is - returned. Otherwise the digit's numeric value is returned. - - The results will be odd if you use a radix < 2 or > 62, you are - expected to know what you're up to. - */ -int s_mp_tovalue(char ch, int r) -{ - int val, xch; - - if(r > 36) - xch = ch; - else - xch = toupper(ch); - - if(isdigit(xch)) - val = xch - '0'; - else if(isupper(xch)) - val = xch - 'A' + 10; - else if(islower(xch)) - val = xch - 'a' + 36; - else if(xch == '+') - val = 62; - else if(xch == '/') - val = 63; - else - return -1; - - if(val < 0 || val >= r) - return -1; - - return val; - -} /* end s_mp_tovalue() */ - -/* }}} */ - -/* {{{ s_mp_todigit(val, r, low) */ - -/* - Convert val to a radix-r digit, if possible. If val is out of range - for r, returns zero. Otherwise, returns an ASCII character denoting - the value in the given radix. - - The results may be odd if you use a radix < 2 or > 64, you are - expected to know what you're doing. - */ - -char s_mp_todigit(int val, int r, int low) -{ - char ch; - - if(val < 0 || val >= r) - return 0; - - ch = s_dmap_1[val]; - - if(r <= 36 && low) - ch = tolower(ch); - - return ch; - -} /* end s_mp_todigit() */ - -/* }}} */ - -/* {{{ s_mp_outlen(bits, radix) */ - -/* - Return an estimate for how long a string is needed to hold a radix - r representation of a number with 'bits' significant bits. - - Does not include space for a sign or a NUL terminator. - */ -int s_mp_outlen(int bits, int r) -{ - return (int)((double)bits * LOG_V_2(r)); - -} /* end s_mp_outlen() */ - -/* }}} */ - -/* }}} */ - -/*------------------------------------------------------------------------*/ -/* HERE THERE BE DRAGONS */ -/* crc==4242132123, version==2, Sat Feb 02 06:43:52 2002 */ diff --git a/libtommath/mtest/mpi.h b/libtommath/mtest/mpi.h deleted file mode 100644 index 211421f..0000000 --- a/libtommath/mtest/mpi.h +++ /dev/null @@ -1,225 +0,0 @@ -/* - mpi.h - - by Michael J. Fromberger - Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved - - Arbitrary precision integer arithmetic library - */ - -#ifndef _H_MPI_ -#define _H_MPI_ - -#include "mpi-config.h" - -#define MP_LT -1 -#define MP_EQ 0 -#define MP_GT 1 - -#if MP_DEBUG -#undef MP_IOFUNC -#define MP_IOFUNC 1 -#endif - -#if MP_IOFUNC -#include -#include -#endif - -#include - -#define MP_NEG 1 -#define MP_ZPOS 0 - -/* Included for compatibility... */ -#define NEG MP_NEG -#define ZPOS MP_ZPOS - -#define MP_OKAY 0 /* no error, all is well */ -#define MP_YES 0 /* yes (boolean result) */ -#define MP_NO -1 /* no (boolean result) */ -#define MP_MEM -2 /* out of memory */ -#define MP_RANGE -3 /* argument out of range */ -#define MP_BADARG -4 /* invalid parameter */ -#define MP_UNDEF -5 /* answer is undefined */ -#define MP_LAST_CODE MP_UNDEF - -#include "mpi-types.h" - -/* Included for compatibility... */ -#define DIGIT_BIT MP_DIGIT_BIT -#define DIGIT_MAX MP_DIGIT_MAX - -/* Macros for accessing the mp_int internals */ -#define SIGN(MP) ((MP)->sign) -#define USED(MP) ((MP)->used) -#define ALLOC(MP) ((MP)->alloc) -#define DIGITS(MP) ((MP)->dp) -#define DIGIT(MP,N) (MP)->dp[(N)] - -#if MP_ARGCHK == 1 -#define ARGCHK(X,Y) {if(!(X)){return (Y);}} -#elif MP_ARGCHK == 2 -#include -#define ARGCHK(X,Y) assert(X) -#else -#define ARGCHK(X,Y) /* */ -#endif - -/* This defines the maximum I/O base (minimum is 2) */ -#define MAX_RADIX 64 - -typedef struct { - mp_sign sign; /* sign of this quantity */ - mp_size alloc; /* how many digits allocated */ - mp_size used; /* how many digits used */ - mp_digit *dp; /* the digits themselves */ -} mp_int; - -/*------------------------------------------------------------------------*/ -/* Default precision */ - -unsigned int mp_get_prec(void); -void mp_set_prec(unsigned int prec); - -/*------------------------------------------------------------------------*/ -/* Memory management */ - -mp_err mp_init(mp_int *mp); -mp_err mp_init_array(mp_int mp[], int count); -mp_err mp_init_size(mp_int *mp, mp_size prec); -mp_err mp_init_copy(mp_int *mp, mp_int *from); -mp_err mp_copy(mp_int *from, mp_int *to); -void mp_exch(mp_int *mp1, mp_int *mp2); -void mp_clear(mp_int *mp); -void mp_clear_array(mp_int mp[], int count); -void mp_zero(mp_int *mp); -void mp_set(mp_int *mp, mp_digit d); -mp_err mp_set_int(mp_int *mp, long z); -mp_err mp_shrink(mp_int *a); - - -/*------------------------------------------------------------------------*/ -/* Single digit arithmetic */ - -mp_err mp_add_d(mp_int *a, mp_digit d, mp_int *b); -mp_err mp_sub_d(mp_int *a, mp_digit d, mp_int *b); -mp_err mp_mul_d(mp_int *a, mp_digit d, mp_int *b); -mp_err mp_mul_2(mp_int *a, mp_int *c); -mp_err mp_div_d(mp_int *a, mp_digit d, mp_int *q, mp_digit *r); -mp_err mp_div_2(mp_int *a, mp_int *c); -mp_err mp_expt_d(mp_int *a, mp_digit d, mp_int *c); - -/*------------------------------------------------------------------------*/ -/* Sign manipulations */ - -mp_err mp_abs(mp_int *a, mp_int *b); -mp_err mp_neg(mp_int *a, mp_int *b); - -/*------------------------------------------------------------------------*/ -/* Full arithmetic */ - -mp_err mp_add(mp_int *a, mp_int *b, mp_int *c); -mp_err mp_sub(mp_int *a, mp_int *b, mp_int *c); -mp_err mp_mul(mp_int *a, mp_int *b, mp_int *c); -mp_err mp_mul_2d(mp_int *a, mp_digit d, mp_int *c); -#if MP_SQUARE -mp_err mp_sqr(mp_int *a, mp_int *b); -#else -#define mp_sqr(a, b) mp_mul(a, a, b) -#endif -mp_err mp_div(mp_int *a, mp_int *b, mp_int *q, mp_int *r); -mp_err mp_div_2d(mp_int *a, mp_digit d, mp_int *q, mp_int *r); -mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c); -mp_err mp_2expt(mp_int *a, mp_digit k); -mp_err mp_sqrt(mp_int *a, mp_int *b); - -/*------------------------------------------------------------------------*/ -/* Modular arithmetic */ - -#if MP_MODARITH -mp_err mp_mod(mp_int *a, mp_int *m, mp_int *c); -mp_err mp_mod_d(mp_int *a, mp_digit d, mp_digit *c); -mp_err mp_addmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c); -mp_err mp_submod(mp_int *a, mp_int *b, mp_int *m, mp_int *c); -mp_err mp_mulmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c); -#if MP_SQUARE -mp_err mp_sqrmod(mp_int *a, mp_int *m, mp_int *c); -#else -#define mp_sqrmod(a, m, c) mp_mulmod(a, a, m, c) -#endif -mp_err mp_exptmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c); -mp_err mp_exptmod_d(mp_int *a, mp_digit d, mp_int *m, mp_int *c); -#endif /* MP_MODARITH */ - -/*------------------------------------------------------------------------*/ -/* Comparisons */ - -int mp_cmp_z(mp_int *a); -int mp_cmp_d(mp_int *a, mp_digit d); -int mp_cmp(mp_int *a, mp_int *b); -int mp_cmp_mag(mp_int *a, mp_int *b); -int mp_cmp_int(mp_int *a, long z); -int mp_isodd(mp_int *a); -int mp_iseven(mp_int *a); - -/*------------------------------------------------------------------------*/ -/* Number theoretic */ - -#if MP_NUMTH -mp_err mp_gcd(mp_int *a, mp_int *b, mp_int *c); -mp_err mp_lcm(mp_int *a, mp_int *b, mp_int *c); -mp_err mp_xgcd(mp_int *a, mp_int *b, mp_int *g, mp_int *x, mp_int *y); -mp_err mp_invmod(mp_int *a, mp_int *m, mp_int *c); -#endif /* end MP_NUMTH */ - -/*------------------------------------------------------------------------*/ -/* Input and output */ - -#if MP_IOFUNC -void mp_print(mp_int *mp, FILE *ofp); -#endif /* end MP_IOFUNC */ - -/*------------------------------------------------------------------------*/ -/* Base conversion */ - -#define BITS 1 -#define BYTES CHAR_BIT - -mp_err mp_read_signed_bin(mp_int *mp, unsigned char *str, int len); -int mp_signed_bin_size(mp_int *mp); -mp_err mp_to_signed_bin(mp_int *mp, unsigned char *str); - -mp_err mp_read_unsigned_bin(mp_int *mp, unsigned char *str, int len); -int mp_unsigned_bin_size(mp_int *mp); -mp_err mp_to_unsigned_bin(mp_int *mp, unsigned char *str); - -int mp_count_bits(mp_int *mp); - -#if MP_COMPAT_MACROS -#define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len)) -#define mp_raw_size(mp) mp_signed_bin_size(mp) -#define mp_toraw(mp, str) mp_to_signed_bin((mp), (str)) -#define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len)) -#define mp_mag_size(mp) mp_unsigned_bin_size(mp) -#define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str)) -#endif - -mp_err mp_read_radix(mp_int *mp, unsigned char *str, int radix); -int mp_radix_size(mp_int *mp, int radix); -int mp_value_radix_size(int num, int qty, int radix); -mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix); - -int mp_char2value(char ch, int r); - -#define mp_tobinary(M, S) mp_toradix((M), (S), 2) -#define mp_tooctal(M, S) mp_toradix((M), (S), 8) -#define mp_todecimal(M, S) mp_toradix((M), (S), 10) -#define mp_tohex(M, S) mp_toradix((M), (S), 16) - -/*------------------------------------------------------------------------*/ -/* Error strings */ - -const char *mp_strerror(mp_err ec); - -#endif /* end _H_MPI_ */ diff --git a/libtommath/mtest/mtest.c b/libtommath/mtest/mtest.c deleted file mode 100644 index d46f456..0000000 --- a/libtommath/mtest/mtest.c +++ /dev/null @@ -1,304 +0,0 @@ -/* makes a bignum test harness with NUM tests per operation - * - * the output is made in the following format [one parameter per line] - -operation -operand1 -operand2 -[... operandN] -result1 -result2 -[... resultN] - -So for example "a * b mod n" would be - -mulmod -a -b -n -a*b mod n - -e.g. if a=3, b=4 n=11 then - -mulmod -3 -4 -11 -1 - - */ - -#ifdef MP_8BIT -#define THE_MASK 127 -#else -#define THE_MASK 32767 -#endif - -#include -#include -#include -#include "mpi.c" - -FILE *rng; - -void rand_num(mp_int *a) -{ - int n, size; - unsigned char buf[2048]; - - size = 1 + ((fgetc(rng)<<8) + fgetc(rng)) % 101; - buf[0] = (fgetc(rng)&1)?1:0; - fread(buf+1, 1, size, rng); - while (buf[1] == 0) buf[1] = fgetc(rng); - mp_read_raw(a, buf, 1+size); -} - -void rand_num2(mp_int *a) -{ - int n, size; - unsigned char buf[2048]; - - size = 10 + ((fgetc(rng)<<8) + fgetc(rng)) % 101; - buf[0] = (fgetc(rng)&1)?1:0; - fread(buf+1, 1, size, rng); - while (buf[1] == 0) buf[1] = fgetc(rng); - mp_read_raw(a, buf, 1+size); -} - -#define mp_to64(a, b) mp_toradix(a, b, 64) - -int main(void) -{ - int n, tmp; - mp_int a, b, c, d, e; - clock_t t1; - char buf[4096]; - - mp_init(&a); - mp_init(&b); - mp_init(&c); - mp_init(&d); - mp_init(&e); - - - /* initial (2^n - 1)^2 testing, makes sure the comba multiplier works [it has the new carry code] */ -/* - mp_set(&a, 1); - for (n = 1; n < 8192; n++) { - mp_mul(&a, &a, &c); - printf("mul\n"); - mp_to64(&a, buf); - printf("%s\n%s\n", buf, buf); - mp_to64(&c, buf); - printf("%s\n", buf); - - mp_add_d(&a, 1, &a); - mp_mul_2(&a, &a); - mp_sub_d(&a, 1, &a); - } -*/ - - rng = fopen("/dev/urandom", "rb"); - if (rng == NULL) { - rng = fopen("/dev/random", "rb"); - if (rng == NULL) { - fprintf(stderr, "\nWarning: stdin used as random source\n\n"); - rng = stdin; - } - } - - t1 = clock(); - for (;;) { -#if 0 - if (clock() - t1 > CLOCKS_PER_SEC) { - sleep(2); - t1 = clock(); - } -#endif - n = fgetc(rng) % 15; - - if (n == 0) { - /* add tests */ - rand_num(&a); - rand_num(&b); - mp_add(&a, &b, &c); - printf("add\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 1) { - /* sub tests */ - rand_num(&a); - rand_num(&b); - mp_sub(&a, &b, &c); - printf("sub\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 2) { - /* mul tests */ - rand_num(&a); - rand_num(&b); - mp_mul(&a, &b, &c); - printf("mul\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 3) { - /* div tests */ - rand_num(&a); - rand_num(&b); - mp_div(&a, &b, &c, &d); - printf("div\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - mp_to64(&d, buf); - printf("%s\n", buf); - } else if (n == 4) { - /* sqr tests */ - rand_num(&a); - mp_sqr(&a, &b); - printf("sqr\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 5) { - /* mul_2d test */ - rand_num(&a); - mp_copy(&a, &b); - n = fgetc(rng) & 63; - mp_mul_2d(&b, n, &b); - mp_to64(&a, buf); - printf("mul2d\n"); - printf("%s\n", buf); - printf("%d\n", n); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 6) { - /* div_2d test */ - rand_num(&a); - mp_copy(&a, &b); - n = fgetc(rng) & 63; - mp_div_2d(&b, n, &b, NULL); - mp_to64(&a, buf); - printf("div2d\n"); - printf("%s\n", buf); - printf("%d\n", n); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 7) { - /* gcd test */ - rand_num(&a); - rand_num(&b); - a.sign = MP_ZPOS; - b.sign = MP_ZPOS; - mp_gcd(&a, &b, &c); - printf("gcd\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 8) { - /* lcm test */ - rand_num(&a); - rand_num(&b); - a.sign = MP_ZPOS; - b.sign = MP_ZPOS; - mp_lcm(&a, &b, &c); - printf("lcm\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 9) { - /* exptmod test */ - rand_num2(&a); - rand_num2(&b); - rand_num2(&c); -// if (c.dp[0]&1) mp_add_d(&c, 1, &c); - a.sign = b.sign = c.sign = 0; - mp_exptmod(&a, &b, &c, &d); - printf("expt\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - mp_to64(&d, buf); - printf("%s\n", buf); - } else if (n == 10) { - /* invmod test */ - rand_num2(&a); - rand_num2(&b); - b.sign = MP_ZPOS; - a.sign = MP_ZPOS; - mp_gcd(&a, &b, &c); - if (mp_cmp_d(&c, 1) != 0) continue; - if (mp_cmp_d(&b, 1) == 0) continue; - mp_invmod(&a, &b, &c); - printf("invmod\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 11) { - rand_num(&a); - mp_mul_2(&a, &a); - mp_div_2(&a, &b); - printf("div2\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 12) { - rand_num2(&a); - mp_mul_2(&a, &b); - printf("mul2\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 13) { - rand_num2(&a); - tmp = abs(rand()) & THE_MASK; - mp_add_d(&a, tmp, &b); - printf("add_d\n"); - mp_to64(&a, buf); - printf("%s\n%d\n", buf, tmp); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 14) { - rand_num2(&a); - tmp = abs(rand()) & THE_MASK; - mp_sub_d(&a, tmp, &b); - printf("sub_d\n"); - mp_to64(&a, buf); - printf("%s\n%d\n", buf, tmp); - mp_to64(&b, buf); - printf("%s\n", buf); - } - } - fclose(rng); - return 0; -} diff --git a/libtommath/pics/design_process.sxd b/libtommath/pics/design_process.sxd deleted file mode 100644 index 7414dbb..0000000 Binary files a/libtommath/pics/design_process.sxd and /dev/null differ diff --git a/libtommath/pics/design_process.tif b/libtommath/pics/design_process.tif deleted file mode 100644 index 4a0c012..0000000 Binary files a/libtommath/pics/design_process.tif and /dev/null differ diff --git a/libtommath/pics/expt_state.sxd b/libtommath/pics/expt_state.sxd deleted file mode 100644 index 6518404..0000000 Binary files a/libtommath/pics/expt_state.sxd and /dev/null differ diff --git a/libtommath/pics/expt_state.tif b/libtommath/pics/expt_state.tif deleted file mode 100644 index 0aaee39..0000000 Binary files a/libtommath/pics/expt_state.tif and /dev/null differ diff --git a/libtommath/pics/makefile b/libtommath/pics/makefile deleted file mode 100644 index 3ecb02f..0000000 --- a/libtommath/pics/makefile +++ /dev/null @@ -1,35 +0,0 @@ -# makes the images... yeah - -default: pses - -design_process.ps: design_process.tif - tiff2ps -s -e design_process.tif > design_process.ps - -sliding_window.ps: sliding_window.tif - tiff2ps -s -e sliding_window.tif > sliding_window.ps - -expt_state.ps: expt_state.tif - tiff2ps -s -e expt_state.tif > expt_state.ps - -primality.ps: primality.tif - tiff2ps -s -e primality.tif > primality.ps - -design_process.pdf: design_process.ps - epstopdf design_process.ps - -sliding_window.pdf: sliding_window.ps - epstopdf sliding_window.ps - -expt_state.pdf: expt_state.ps - epstopdf expt_state.ps - -primality.pdf: primality.ps - epstopdf primality.ps - - -pses: sliding_window.ps expt_state.ps primality.ps design_process.ps -pdfes: sliding_window.pdf expt_state.pdf primality.pdf design_process.pdf - -clean: - rm -rf *.ps *.pdf .xvpics - \ No newline at end of file diff --git a/libtommath/pics/primality.tif b/libtommath/pics/primality.tif deleted file mode 100644 index 83aafe0..0000000 Binary files a/libtommath/pics/primality.tif and /dev/null differ diff --git a/libtommath/pics/radix.sxd b/libtommath/pics/radix.sxd deleted file mode 100644 index b9eb9a0..0000000 Binary files a/libtommath/pics/radix.sxd and /dev/null differ diff --git a/libtommath/pics/sliding_window.sxd b/libtommath/pics/sliding_window.sxd deleted file mode 100644 index 91e7c0d..0000000 Binary files a/libtommath/pics/sliding_window.sxd and /dev/null differ diff --git a/libtommath/pics/sliding_window.tif b/libtommath/pics/sliding_window.tif deleted file mode 100644 index bb4cb96..0000000 Binary files a/libtommath/pics/sliding_window.tif and /dev/null differ diff --git a/libtommath/poster.out b/libtommath/poster.out deleted file mode 100644 index e69de29..0000000 diff --git a/libtommath/poster.pdf b/libtommath/poster.pdf deleted file mode 100644 index 1f705cf..0000000 Binary files a/libtommath/poster.pdf and /dev/null differ diff --git a/libtommath/pre_gen/mpi.c b/libtommath/pre_gen/mpi.c deleted file mode 100644 index d2224c0..0000000 --- a/libtommath/pre_gen/mpi.c +++ /dev/null @@ -1,9048 +0,0 @@ -/* Start: bn_error.c */ -#include -#ifdef BN_ERROR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -static const struct { - int code; - char *msg; -} msgs[] = { - { MP_OKAY, "Successful" }, - { MP_MEM, "Out of heap" }, - { MP_VAL, "Value out of range" } -}; - -/* return a char * string for a given code */ -char *mp_error_to_string(int code) -{ - int x; - - /* scan the lookup table for the given message */ - for (x = 0; x < (int)(sizeof(msgs) / sizeof(msgs[0])); x++) { - if (msgs[x].code == code) { - return msgs[x].msg; - } - } - - /* generic reply for invalid code */ - return "Invalid error code"; -} - -#endif - -/* End: bn_error.c */ - -/* Start: bn_fast_mp_invmod.c */ -#include -#ifdef BN_FAST_MP_INVMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* computes the modular inverse via binary extended euclidean algorithm, - * that is c = 1/a mod b - * - * Based on slow invmod except this is optimized for the case where b is - * odd as per HAC Note 14.64 on pp. 610 - */ -int fast_mp_invmod (mp_int * a, mp_int * b, mp_int * c) -{ - mp_int x, y, u, v, B, D; - int res, neg; - - /* 2. [modified] b must be odd */ - if (mp_iseven (b) == 1) { - return MP_VAL; - } - - /* init all our temps */ - if ((res = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) { - return res; - } - - /* x == modulus, y == value to invert */ - if ((res = mp_copy (b, &x)) != MP_OKAY) { - goto LBL_ERR; - } - - /* we need y = |a| */ - if ((res = mp_mod (a, b, &y)) != MP_OKAY) { - goto LBL_ERR; - } - - /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ - if ((res = mp_copy (&x, &u)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_copy (&y, &v)) != MP_OKAY) { - goto LBL_ERR; - } - mp_set (&D, 1); - -top: - /* 4. while u is even do */ - while (mp_iseven (&u) == 1) { - /* 4.1 u = u/2 */ - if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { - goto LBL_ERR; - } - /* 4.2 if B is odd then */ - if (mp_isodd (&B) == 1) { - if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } - /* B = B/2 */ - if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* 5. while v is even do */ - while (mp_iseven (&v) == 1) { - /* 5.1 v = v/2 */ - if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { - goto LBL_ERR; - } - /* 5.2 if D is odd then */ - if (mp_isodd (&D) == 1) { - /* D = (D-x)/2 */ - if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - /* D = D/2 */ - if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* 6. if u >= v then */ - if (mp_cmp (&u, &v) != MP_LT) { - /* u = u - v, B = B - D */ - if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } else { - /* v - v - u, D = D - B */ - if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* if not zero goto step 4 */ - if (mp_iszero (&u) == 0) { - goto top; - } - - /* now a = C, b = D, gcd == g*v */ - - /* if v != 1 then there is no inverse */ - if (mp_cmp_d (&v, 1) != MP_EQ) { - res = MP_VAL; - goto LBL_ERR; - } - - /* b is now the inverse */ - neg = a->sign; - while (D.sign == MP_NEG) { - if ((res = mp_add (&D, b, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - mp_exch (&D, c); - c->sign = neg; - res = MP_OKAY; - -LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL); - return res; -} -#endif - -/* End: bn_fast_mp_invmod.c */ - -/* Start: bn_fast_mp_montgomery_reduce.c */ -#include -#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* computes xR**-1 == x (mod N) via Montgomery Reduction - * - * This is an optimized implementation of montgomery_reduce - * which uses the comba method to quickly calculate the columns of the - * reduction. - * - * Based on Algorithm 14.32 on pp.601 of HAC. -*/ -int fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) -{ - int ix, res, olduse; - mp_word W[MP_WARRAY]; - - /* get old used count */ - olduse = x->used; - - /* grow a as required */ - if (x->alloc < n->used + 1) { - if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) { - return res; - } - } - - /* first we have to get the digits of the input into - * an array of double precision words W[...] - */ - { - register mp_word *_W; - register mp_digit *tmpx; - - /* alias for the W[] array */ - _W = W; - - /* alias for the digits of x*/ - tmpx = x->dp; - - /* copy the digits of a into W[0..a->used-1] */ - for (ix = 0; ix < x->used; ix++) { - *_W++ = *tmpx++; - } - - /* zero the high words of W[a->used..m->used*2] */ - for (; ix < n->used * 2 + 1; ix++) { - *_W++ = 0; - } - } - - /* now we proceed to zero successive digits - * from the least significant upwards - */ - for (ix = 0; ix < n->used; ix++) { - /* mu = ai * m' mod b - * - * We avoid a double precision multiplication (which isn't required) - * by casting the value down to a mp_digit. Note this requires - * that W[ix-1] have the carry cleared (see after the inner loop) - */ - register mp_digit mu; - mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK); - - /* a = a + mu * m * b**i - * - * This is computed in place and on the fly. The multiplication - * by b**i is handled by offseting which columns the results - * are added to. - * - * Note the comba method normally doesn't handle carries in the - * inner loop In this case we fix the carry from the previous - * column since the Montgomery reduction requires digits of the - * result (so far) [see above] to work. This is - * handled by fixing up one carry after the inner loop. The - * carry fixups are done in order so after these loops the - * first m->used words of W[] have the carries fixed - */ - { - register int iy; - register mp_digit *tmpn; - register mp_word *_W; - - /* alias for the digits of the modulus */ - tmpn = n->dp; - - /* Alias for the columns set by an offset of ix */ - _W = W + ix; - - /* inner loop */ - for (iy = 0; iy < n->used; iy++) { - *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++); - } - } - - /* now fix carry for next digit, W[ix+1] */ - W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT); - } - - /* now we have to propagate the carries and - * shift the words downward [all those least - * significant digits we zeroed]. - */ - { - register mp_digit *tmpx; - register mp_word *_W, *_W1; - - /* nox fix rest of carries */ - - /* alias for current word */ - _W1 = W + ix; - - /* alias for next word, where the carry goes */ - _W = W + ++ix; - - for (; ix <= n->used * 2 + 1; ix++) { - *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT); - } - - /* copy out, A = A/b**n - * - * The result is A/b**n but instead of converting from an - * array of mp_word to mp_digit than calling mp_rshd - * we just copy them in the right order - */ - - /* alias for destination word */ - tmpx = x->dp; - - /* alias for shifted double precision result */ - _W = W + n->used; - - for (ix = 0; ix < n->used + 1; ix++) { - *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK)); - } - - /* zero oldused digits, if the input a was larger than - * m->used+1 we'll have to clear the digits - */ - for (; ix < olduse; ix++) { - *tmpx++ = 0; - } - } - - /* set the max used and clamp */ - x->used = n->used + 1; - mp_clamp (x); - - /* if A >= m then A = A - m */ - if (mp_cmp_mag (x, n) != MP_LT) { - return s_mp_sub (x, n, x); - } - return MP_OKAY; -} -#endif - -/* End: bn_fast_mp_montgomery_reduce.c */ - -/* Start: bn_fast_s_mp_mul_digs.c */ -#include -#ifdef BN_FAST_S_MP_MUL_DIGS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* Fast (comba) multiplier - * - * This is the fast column-array [comba] multiplier. It is - * designed to compute the columns of the product first - * then handle the carries afterwards. This has the effect - * of making the nested loops that compute the columns very - * simple and schedulable on super-scalar processors. - * - * This has been modified to produce a variable number of - * digits of output so if say only a half-product is required - * you don't have to compute the upper half (a feature - * required for fast Barrett reduction). - * - * Based on Algorithm 14.12 on pp.595 of HAC. - * - */ -int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -{ - int olduse, res, pa, ix, iz; - mp_digit W[MP_WARRAY]; - register mp_word _W; - - /* grow the destination as required */ - if (c->alloc < digs) { - if ((res = mp_grow (c, digs)) != MP_OKAY) { - return res; - } - } - - /* number of output digits to produce */ - pa = MIN(digs, a->used + b->used); - - /* clear the carry */ - _W = 0; - for (ix = 0; ix < pa; ix++) { - int tx, ty; - int iy; - mp_digit *tmpx, *tmpy; - - /* get offsets into the two bignums */ - ty = MIN(b->used-1, ix); - tx = ix - ty; - - /* setup temp aliases */ - tmpx = a->dp + tx; - tmpy = b->dp + ty; - - /* this is the number of times the loop will iterrate, essentially - while (tx++ < a->used && ty-- >= 0) { ... } - */ - iy = MIN(a->used-tx, ty+1); - - /* execute loop */ - for (iz = 0; iz < iy; ++iz) { - _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); - - } - - /* store term */ - W[ix] = ((mp_digit)_W) & MP_MASK; - - /* make next carry */ - _W = _W >> ((mp_word)DIGIT_BIT); - } - - /* setup dest */ - olduse = c->used; - c->used = pa; - - { - register mp_digit *tmpc; - tmpc = c->dp; - for (ix = 0; ix < pa+1; ix++) { - /* now extract the previous digit [below the carry] */ - *tmpc++ = W[ix]; - } - - /* clear unused digits [that existed in the old copy of c] */ - for (; ix < olduse; ix++) { - *tmpc++ = 0; - } - } - mp_clamp (c); - return MP_OKAY; -} -#endif - -/* End: bn_fast_s_mp_mul_digs.c */ - -/* Start: bn_fast_s_mp_mul_high_digs.c */ -#include -#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* this is a modified version of fast_s_mul_digs that only produces - * output digits *above* digs. See the comments for fast_s_mul_digs - * to see how it works. - * - * This is used in the Barrett reduction since for one of the multiplications - * only the higher digits were needed. This essentially halves the work. - * - * Based on Algorithm 14.12 on pp.595 of HAC. - */ -int fast_s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -{ - int olduse, res, pa, ix, iz; - mp_digit W[MP_WARRAY]; - mp_word _W; - - /* grow the destination as required */ - pa = a->used + b->used; - if (c->alloc < pa) { - if ((res = mp_grow (c, pa)) != MP_OKAY) { - return res; - } - } - - /* number of output digits to produce */ - pa = a->used + b->used; - _W = 0; - for (ix = digs; ix < pa; ix++) { - int tx, ty, iy; - mp_digit *tmpx, *tmpy; - - /* get offsets into the two bignums */ - ty = MIN(b->used-1, ix); - tx = ix - ty; - - /* setup temp aliases */ - tmpx = a->dp + tx; - tmpy = b->dp + ty; - - /* this is the number of times the loop will iterrate, essentially its - while (tx++ < a->used && ty-- >= 0) { ... } - */ - iy = MIN(a->used-tx, ty+1); - - /* execute loop */ - for (iz = 0; iz < iy; iz++) { - _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); - } - - /* store term */ - W[ix] = ((mp_digit)_W) & MP_MASK; - - /* make next carry */ - _W = _W >> ((mp_word)DIGIT_BIT); - } - - /* setup dest */ - olduse = c->used; - c->used = pa; - - { - register mp_digit *tmpc; - - tmpc = c->dp + digs; - for (ix = digs; ix < pa; ix++) { - /* now extract the previous digit [below the carry] */ - *tmpc++ = W[ix]; - } - - /* clear unused digits [that existed in the old copy of c] */ - for (; ix < olduse; ix++) { - *tmpc++ = 0; - } - } - mp_clamp (c); - return MP_OKAY; -} -#endif - -/* End: bn_fast_s_mp_mul_high_digs.c */ - -/* Start: bn_fast_s_mp_sqr.c */ -#include -#ifdef BN_FAST_S_MP_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* the jist of squaring... - * you do like mult except the offset of the tmpx [one that - * starts closer to zero] can't equal the offset of tmpy. - * So basically you set up iy like before then you min it with - * (ty-tx) so that it never happens. You double all those - * you add in the inner loop - -After that loop you do the squares and add them in. -*/ - -int fast_s_mp_sqr (mp_int * a, mp_int * b) -{ - int olduse, res, pa, ix, iz; - mp_digit W[MP_WARRAY], *tmpx; - mp_word W1; - - /* grow the destination as required */ - pa = a->used + a->used; - if (b->alloc < pa) { - if ((res = mp_grow (b, pa)) != MP_OKAY) { - return res; - } - } - - /* number of output digits to produce */ - W1 = 0; - for (ix = 0; ix < pa; ix++) { - int tx, ty, iy; - mp_word _W; - mp_digit *tmpy; - - /* clear counter */ - _W = 0; - - /* get offsets into the two bignums */ - ty = MIN(a->used-1, ix); - tx = ix - ty; - - /* setup temp aliases */ - tmpx = a->dp + tx; - tmpy = a->dp + ty; - - /* this is the number of times the loop will iterrate, essentially - while (tx++ < a->used && ty-- >= 0) { ... } - */ - iy = MIN(a->used-tx, ty+1); - - /* now for squaring tx can never equal ty - * we halve the distance since they approach at a rate of 2x - * and we have to round because odd cases need to be executed - */ - iy = MIN(iy, (ty-tx+1)>>1); - - /* execute loop */ - for (iz = 0; iz < iy; iz++) { - _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); - } - - /* double the inner product and add carry */ - _W = _W + _W + W1; - - /* even columns have the square term in them */ - if ((ix&1) == 0) { - _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]); - } - - /* store it */ - W[ix] = (mp_digit)(_W & MP_MASK); - - /* make next carry */ - W1 = _W >> ((mp_word)DIGIT_BIT); - } - - /* setup dest */ - olduse = b->used; - b->used = a->used+a->used; - - { - mp_digit *tmpb; - tmpb = b->dp; - for (ix = 0; ix < pa; ix++) { - *tmpb++ = W[ix] & MP_MASK; - } - - /* clear unused digits [that existed in the old copy of c] */ - for (; ix < olduse; ix++) { - *tmpb++ = 0; - } - } - mp_clamp (b); - return MP_OKAY; -} -#endif - -/* End: bn_fast_s_mp_sqr.c */ - -/* Start: bn_mp_2expt.c */ -#include -#ifdef BN_MP_2EXPT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* computes a = 2**b - * - * Simple algorithm which zeroes the int, grows it then just sets one bit - * as required. - */ -int -mp_2expt (mp_int * a, int b) -{ - int res; - - /* zero a as per default */ - mp_zero (a); - - /* grow a to accomodate the single bit */ - if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) { - return res; - } - - /* set the used count of where the bit will go */ - a->used = b / DIGIT_BIT + 1; - - /* put the single bit in its place */ - a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT); - - return MP_OKAY; -} -#endif - -/* End: bn_mp_2expt.c */ - -/* Start: bn_mp_abs.c */ -#include -#ifdef BN_MP_ABS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* b = |a| - * - * Simple function copies the input and fixes the sign to positive - */ -int -mp_abs (mp_int * a, mp_int * b) -{ - int res; - - /* copy a to b */ - if (a != b) { - if ((res = mp_copy (a, b)) != MP_OKAY) { - return res; - } - } - - /* force the sign of b to positive */ - b->sign = MP_ZPOS; - - return MP_OKAY; -} -#endif - -/* End: bn_mp_abs.c */ - -/* Start: bn_mp_add.c */ -#include -#ifdef BN_MP_ADD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* high level addition (handles signs) */ -int mp_add (mp_int * a, mp_int * b, mp_int * c) -{ - int sa, sb, res; - - /* get sign of both inputs */ - sa = a->sign; - sb = b->sign; - - /* handle two cases, not four */ - if (sa == sb) { - /* both positive or both negative */ - /* add their magnitudes, copy the sign */ - c->sign = sa; - res = s_mp_add (a, b, c); - } else { - /* one positive, the other negative */ - /* subtract the one with the greater magnitude from */ - /* the one of the lesser magnitude. The result gets */ - /* the sign of the one with the greater magnitude. */ - if (mp_cmp_mag (a, b) == MP_LT) { - c->sign = sb; - res = s_mp_sub (b, a, c); - } else { - c->sign = sa; - res = s_mp_sub (a, b, c); - } - } - return res; -} - -#endif - -/* End: bn_mp_add.c */ - -/* Start: bn_mp_add_d.c */ -#include -#ifdef BN_MP_ADD_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* single digit addition */ -int -mp_add_d (mp_int * a, mp_digit b, mp_int * c) -{ - int res, ix, oldused; - mp_digit *tmpa, *tmpc, mu; - - /* grow c as required */ - if (c->alloc < a->used + 1) { - if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) { - return res; - } - } - - /* if a is negative and |a| >= b, call c = |a| - b */ - if (a->sign == MP_NEG && (a->used > 1 || a->dp[0] >= b)) { - /* temporarily fix sign of a */ - a->sign = MP_ZPOS; - - /* c = |a| - b */ - res = mp_sub_d(a, b, c); - - /* fix sign */ - a->sign = c->sign = MP_NEG; - - /* clamp */ - mp_clamp(c); - - return res; - } - - /* old number of used digits in c */ - oldused = c->used; - - /* sign always positive */ - c->sign = MP_ZPOS; - - /* source alias */ - tmpa = a->dp; - - /* destination alias */ - tmpc = c->dp; - - /* if a is positive */ - if (a->sign == MP_ZPOS) { - /* add digit, after this we're propagating - * the carry. - */ - *tmpc = *tmpa++ + b; - mu = *tmpc >> DIGIT_BIT; - *tmpc++ &= MP_MASK; - - /* now handle rest of the digits */ - for (ix = 1; ix < a->used; ix++) { - *tmpc = *tmpa++ + mu; - mu = *tmpc >> DIGIT_BIT; - *tmpc++ &= MP_MASK; - } - /* set final carry */ - ix++; - *tmpc++ = mu; - - /* setup size */ - c->used = a->used + 1; - } else { - /* a was negative and |a| < b */ - c->used = 1; - - /* the result is a single digit */ - if (a->used == 1) { - *tmpc++ = b - a->dp[0]; - } else { - *tmpc++ = b; - } - - /* setup count so the clearing of oldused - * can fall through correctly - */ - ix = 1; - } - - /* now zero to oldused */ - while (ix++ < oldused) { - *tmpc++ = 0; - } - mp_clamp(c); - - return MP_OKAY; -} - -#endif - -/* End: bn_mp_add_d.c */ - -/* Start: bn_mp_addmod.c */ -#include -#ifdef BN_MP_ADDMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* d = a + b (mod c) */ -int -mp_addmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) -{ - int res; - mp_int t; - - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } - - if ((res = mp_add (a, b, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - res = mp_mod (&t, c, d); - mp_clear (&t); - return res; -} -#endif - -/* End: bn_mp_addmod.c */ - -/* Start: bn_mp_and.c */ -#include -#ifdef BN_MP_AND_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* AND two ints together */ -int -mp_and (mp_int * a, mp_int * b, mp_int * c) -{ - int res, ix, px; - mp_int t, *x; - - if (a->used > b->used) { - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - px = b->used; - x = b; - } else { - if ((res = mp_init_copy (&t, b)) != MP_OKAY) { - return res; - } - px = a->used; - x = a; - } - - for (ix = 0; ix < px; ix++) { - t.dp[ix] &= x->dp[ix]; - } - - /* zero digits above the last from the smallest mp_int */ - for (; ix < t.used; ix++) { - t.dp[ix] = 0; - } - - mp_clamp (&t); - mp_exch (c, &t); - mp_clear (&t); - return MP_OKAY; -} -#endif - -/* End: bn_mp_and.c */ - -/* Start: bn_mp_clamp.c */ -#include -#ifdef BN_MP_CLAMP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* trim unused digits - * - * This is used to ensure that leading zero digits are - * trimed and the leading "used" digit will be non-zero - * Typically very fast. Also fixes the sign if there - * are no more leading digits - */ -void -mp_clamp (mp_int * a) -{ - /* decrease used while the most significant digit is - * zero. - */ - while (a->used > 0 && a->dp[a->used - 1] == 0) { - --(a->used); - } - - /* reset the sign flag if used == 0 */ - if (a->used == 0) { - a->sign = MP_ZPOS; - } -} -#endif - -/* End: bn_mp_clamp.c */ - -/* Start: bn_mp_clear.c */ -#include -#ifdef BN_MP_CLEAR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* clear one (frees) */ -void -mp_clear (mp_int * a) -{ - int i; - - /* only do anything if a hasn't been freed previously */ - if (a->dp != NULL) { - /* first zero the digits */ - for (i = 0; i < a->used; i++) { - a->dp[i] = 0; - } - - /* free ram */ - XFREE(a->dp); - - /* reset members to make debugging easier */ - a->dp = NULL; - a->alloc = a->used = 0; - a->sign = MP_ZPOS; - } -} -#endif - -/* End: bn_mp_clear.c */ - -/* Start: bn_mp_clear_multi.c */ -#include -#ifdef BN_MP_CLEAR_MULTI_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ -#include - -void mp_clear_multi(mp_int *mp, ...) -{ - mp_int* next_mp = mp; - va_list args; - va_start(args, mp); - while (next_mp != NULL) { - mp_clear(next_mp); - next_mp = va_arg(args, mp_int*); - } - va_end(args); -} -#endif - -/* End: bn_mp_clear_multi.c */ - -/* Start: bn_mp_cmp.c */ -#include -#ifdef BN_MP_CMP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* compare two ints (signed)*/ -int -mp_cmp (mp_int * a, mp_int * b) -{ - /* compare based on sign */ - if (a->sign != b->sign) { - if (a->sign == MP_NEG) { - return MP_LT; - } else { - return MP_GT; - } - } - - /* compare digits */ - if (a->sign == MP_NEG) { - /* if negative compare opposite direction */ - return mp_cmp_mag(b, a); - } else { - return mp_cmp_mag(a, b); - } -} -#endif - -/* End: bn_mp_cmp.c */ - -/* Start: bn_mp_cmp_d.c */ -#include -#ifdef BN_MP_CMP_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* compare a digit */ -int mp_cmp_d(mp_int * a, mp_digit b) -{ - /* compare based on sign */ - if (a->sign == MP_NEG) { - return MP_LT; - } - - /* compare based on magnitude */ - if (a->used > 1) { - return MP_GT; - } - - /* compare the only digit of a to b */ - if (a->dp[0] > b) { - return MP_GT; - } else if (a->dp[0] < b) { - return MP_LT; - } else { - return MP_EQ; - } -} -#endif - -/* End: bn_mp_cmp_d.c */ - -/* Start: bn_mp_cmp_mag.c */ -#include -#ifdef BN_MP_CMP_MAG_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* compare maginitude of two ints (unsigned) */ -int mp_cmp_mag (mp_int * a, mp_int * b) -{ - int n; - mp_digit *tmpa, *tmpb; - - /* compare based on # of non-zero digits */ - if (a->used > b->used) { - return MP_GT; - } - - if (a->used < b->used) { - return MP_LT; - } - - /* alias for a */ - tmpa = a->dp + (a->used - 1); - - /* alias for b */ - tmpb = b->dp + (a->used - 1); - - /* compare based on digits */ - for (n = 0; n < a->used; ++n, --tmpa, --tmpb) { - if (*tmpa > *tmpb) { - return MP_GT; - } - - if (*tmpa < *tmpb) { - return MP_LT; - } - } - return MP_EQ; -} -#endif - -/* End: bn_mp_cmp_mag.c */ - -/* Start: bn_mp_cnt_lsb.c */ -#include -#ifdef BN_MP_CNT_LSB_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -static const int lnz[16] = { - 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0 -}; - -/* Counts the number of lsbs which are zero before the first zero bit */ -int mp_cnt_lsb(mp_int *a) -{ - int x; - mp_digit q, qq; - - /* easy out */ - if (mp_iszero(a) == 1) { - return 0; - } - - /* scan lower digits until non-zero */ - for (x = 0; x < a->used && a->dp[x] == 0; x++); - q = a->dp[x]; - x *= DIGIT_BIT; - - /* now scan this digit until a 1 is found */ - if ((q & 1) == 0) { - do { - qq = q & 15; - x += lnz[qq]; - q >>= 4; - } while (qq == 0); - } - return x; -} - -#endif - -/* End: bn_mp_cnt_lsb.c */ - -/* Start: bn_mp_copy.c */ -#include -#ifdef BN_MP_COPY_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* copy, b = a */ -int -mp_copy (mp_int * a, mp_int * b) -{ - int res, n; - - /* if dst == src do nothing */ - if (a == b) { - return MP_OKAY; - } - - /* grow dest */ - if (b->alloc < a->used) { - if ((res = mp_grow (b, a->used)) != MP_OKAY) { - return res; - } - } - - /* zero b and copy the parameters over */ - { - register mp_digit *tmpa, *tmpb; - - /* pointer aliases */ - - /* source */ - tmpa = a->dp; - - /* destination */ - tmpb = b->dp; - - /* copy all the digits */ - for (n = 0; n < a->used; n++) { - *tmpb++ = *tmpa++; - } - - /* clear high digits */ - for (; n < b->used; n++) { - *tmpb++ = 0; - } - } - - /* copy used count and sign */ - b->used = a->used; - b->sign = a->sign; - return MP_OKAY; -} -#endif - -/* End: bn_mp_copy.c */ - -/* Start: bn_mp_count_bits.c */ -#include -#ifdef BN_MP_COUNT_BITS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* returns the number of bits in an int */ -int -mp_count_bits (mp_int * a) -{ - int r; - mp_digit q; - - /* shortcut */ - if (a->used == 0) { - return 0; - } - - /* get number of digits and add that */ - r = (a->used - 1) * DIGIT_BIT; - - /* take the last digit and count the bits in it */ - q = a->dp[a->used - 1]; - while (q > ((mp_digit) 0)) { - ++r; - q >>= ((mp_digit) 1); - } - return r; -} -#endif - -/* End: bn_mp_count_bits.c */ - -/* Start: bn_mp_div.c */ -#include -#ifdef BN_MP_DIV_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -#ifdef BN_MP_DIV_SMALL - -/* slower bit-bang division... also smaller */ -int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d) -{ - mp_int ta, tb, tq, q; - int res, n, n2; - - /* is divisor zero ? */ - if (mp_iszero (b) == 1) { - return MP_VAL; - } - - /* if a < b then q=0, r = a */ - if (mp_cmp_mag (a, b) == MP_LT) { - if (d != NULL) { - res = mp_copy (a, d); - } else { - res = MP_OKAY; - } - if (c != NULL) { - mp_zero (c); - } - return res; - } - - /* init our temps */ - if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL) != MP_OKAY)) { - return res; - } - - - mp_set(&tq, 1); - n = mp_count_bits(a) - mp_count_bits(b); - if (((res = mp_abs(a, &ta)) != MP_OKAY) || - ((res = mp_abs(b, &tb)) != MP_OKAY) || - ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) || - ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) { - goto LBL_ERR; - } - - while (n-- >= 0) { - if (mp_cmp(&tb, &ta) != MP_GT) { - if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) || - ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) { - goto LBL_ERR; - } - } - if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) || - ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) { - goto LBL_ERR; - } - } - - /* now q == quotient and ta == remainder */ - n = a->sign; - n2 = (a->sign == b->sign ? MP_ZPOS : MP_NEG); - if (c != NULL) { - mp_exch(c, &q); - c->sign = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2; - } - if (d != NULL) { - mp_exch(d, &ta); - d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n; - } -LBL_ERR: - mp_clear_multi(&ta, &tb, &tq, &q, NULL); - return res; -} - -#else - -/* integer signed division. - * c*b + d == a [e.g. a/b, c=quotient, d=remainder] - * HAC pp.598 Algorithm 14.20 - * - * Note that the description in HAC is horribly - * incomplete. For example, it doesn't consider - * the case where digits are removed from 'x' in - * the inner loop. It also doesn't consider the - * case that y has fewer than three digits, etc.. - * - * The overall algorithm is as described as - * 14.20 from HAC but fixed to treat these cases. -*/ -int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d) -{ - mp_int q, x, y, t1, t2; - int res, n, t, i, norm, neg; - - /* is divisor zero ? */ - if (mp_iszero (b) == 1) { - return MP_VAL; - } - - /* if a < b then q=0, r = a */ - if (mp_cmp_mag (a, b) == MP_LT) { - if (d != NULL) { - res = mp_copy (a, d); - } else { - res = MP_OKAY; - } - if (c != NULL) { - mp_zero (c); - } - return res; - } - - if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) { - return res; - } - q.used = a->used + 2; - - if ((res = mp_init (&t1)) != MP_OKAY) { - goto LBL_Q; - } - - if ((res = mp_init (&t2)) != MP_OKAY) { - goto LBL_T1; - } - - if ((res = mp_init_copy (&x, a)) != MP_OKAY) { - goto LBL_T2; - } - - if ((res = mp_init_copy (&y, b)) != MP_OKAY) { - goto LBL_X; - } - - /* fix the sign */ - neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; - x.sign = y.sign = MP_ZPOS; - - /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */ - norm = mp_count_bits(&y) % DIGIT_BIT; - if (norm < (int)(DIGIT_BIT-1)) { - norm = (DIGIT_BIT-1) - norm; - if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) { - goto LBL_Y; - } - if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) { - goto LBL_Y; - } - } else { - norm = 0; - } - - /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ - n = x.used - 1; - t = y.used - 1; - - /* while (x >= y*b**n-t) do { q[n-t] += 1; x -= y*b**{n-t} } */ - if ((res = mp_lshd (&y, n - t)) != MP_OKAY) { /* y = y*b**{n-t} */ - goto LBL_Y; - } - - while (mp_cmp (&x, &y) != MP_LT) { - ++(q.dp[n - t]); - if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) { - goto LBL_Y; - } - } - - /* reset y by shifting it back down */ - mp_rshd (&y, n - t); - - /* step 3. for i from n down to (t + 1) */ - for (i = n; i >= (t + 1); i--) { - if (i > x.used) { - continue; - } - - /* step 3.1 if xi == yt then set q{i-t-1} to b-1, - * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */ - if (x.dp[i] == y.dp[t]) { - q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1); - } else { - mp_word tmp; - tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT); - tmp |= ((mp_word) x.dp[i - 1]); - tmp /= ((mp_word) y.dp[t]); - if (tmp > (mp_word) MP_MASK) - tmp = MP_MASK; - q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK)); - } - - /* while (q{i-t-1} * (yt * b + y{t-1})) > - xi * b**2 + xi-1 * b + xi-2 - - do q{i-t-1} -= 1; - */ - q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK; - do { - q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK; - - /* find left hand */ - mp_zero (&t1); - t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1]; - t1.dp[1] = y.dp[t]; - t1.used = 2; - if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) { - goto LBL_Y; - } - - /* find right hand */ - t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2]; - t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1]; - t2.dp[2] = x.dp[i]; - t2.used = 3; - } while (mp_cmp_mag(&t1, &t2) == MP_GT); - - /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */ - if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) { - goto LBL_Y; - } - - if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { - goto LBL_Y; - } - - if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) { - goto LBL_Y; - } - - /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */ - if (x.sign == MP_NEG) { - if ((res = mp_copy (&y, &t1)) != MP_OKAY) { - goto LBL_Y; - } - if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { - goto LBL_Y; - } - if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) { - goto LBL_Y; - } - - q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK; - } - } - - /* now q is the quotient and x is the remainder - * [which we have to normalize] - */ - - /* get sign before writing to c */ - x.sign = x.used == 0 ? MP_ZPOS : a->sign; - - if (c != NULL) { - mp_clamp (&q); - mp_exch (&q, c); - c->sign = neg; - } - - if (d != NULL) { - mp_div_2d (&x, norm, &x, NULL); - mp_exch (&x, d); - } - - res = MP_OKAY; - -LBL_Y:mp_clear (&y); -LBL_X:mp_clear (&x); -LBL_T2:mp_clear (&t2); -LBL_T1:mp_clear (&t1); -LBL_Q:mp_clear (&q); - return res; -} - -#endif - -#endif - -/* End: bn_mp_div.c */ - -/* Start: bn_mp_div_2.c */ -#include -#ifdef BN_MP_DIV_2_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* b = a/2 */ -int mp_div_2(mp_int * a, mp_int * b) -{ - int x, res, oldused; - - /* copy */ - if (b->alloc < a->used) { - if ((res = mp_grow (b, a->used)) != MP_OKAY) { - return res; - } - } - - oldused = b->used; - b->used = a->used; - { - register mp_digit r, rr, *tmpa, *tmpb; - - /* source alias */ - tmpa = a->dp + b->used - 1; - - /* dest alias */ - tmpb = b->dp + b->used - 1; - - /* carry */ - r = 0; - for (x = b->used - 1; x >= 0; x--) { - /* get the carry for the next iteration */ - rr = *tmpa & 1; - - /* shift the current digit, add in carry and store */ - *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1)); - - /* forward carry to next iteration */ - r = rr; - } - - /* zero excess digits */ - tmpb = b->dp + b->used; - for (x = b->used; x < oldused; x++) { - *tmpb++ = 0; - } - } - b->sign = a->sign; - mp_clamp (b); - return MP_OKAY; -} -#endif - -/* End: bn_mp_div_2.c */ - -/* Start: bn_mp_div_2d.c */ -#include -#ifdef BN_MP_DIV_2D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* shift right by a certain bit count (store quotient in c, optional remainder in d) */ -int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d) -{ - mp_digit D, r, rr; - int x, res; - mp_int t; - - - /* if the shift count is <= 0 then we do no work */ - if (b <= 0) { - res = mp_copy (a, c); - if (d != NULL) { - mp_zero (d); - } - return res; - } - - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } - - /* get the remainder */ - if (d != NULL) { - if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - } - - /* copy */ - if ((res = mp_copy (a, c)) != MP_OKAY) { - mp_clear (&t); - return res; - } - - /* shift by as many digits in the bit count */ - if (b >= (int)DIGIT_BIT) { - mp_rshd (c, b / DIGIT_BIT); - } - - /* shift any bit count < DIGIT_BIT */ - D = (mp_digit) (b % DIGIT_BIT); - if (D != 0) { - register mp_digit *tmpc, mask, shift; - - /* mask */ - mask = (((mp_digit)1) << D) - 1; - - /* shift for lsb */ - shift = DIGIT_BIT - D; - - /* alias */ - tmpc = c->dp + (c->used - 1); - - /* carry */ - r = 0; - for (x = c->used - 1; x >= 0; x--) { - /* get the lower bits of this word in a temp */ - rr = *tmpc & mask; - - /* shift the current word and mix in the carry bits from the previous word */ - *tmpc = (*tmpc >> D) | (r << shift); - --tmpc; - - /* set the carry to the carry bits of the current word found above */ - r = rr; - } - } - mp_clamp (c); - if (d != NULL) { - mp_exch (&t, d); - } - mp_clear (&t); - return MP_OKAY; -} -#endif - -/* End: bn_mp_div_2d.c */ - -/* Start: bn_mp_div_3.c */ -#include -#ifdef BN_MP_DIV_3_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* divide by three (based on routine from MPI and the GMP manual) */ -int -mp_div_3 (mp_int * a, mp_int *c, mp_digit * d) -{ - mp_int q; - mp_word w, t; - mp_digit b; - int res, ix; - - /* b = 2**DIGIT_BIT / 3 */ - b = (((mp_word)1) << ((mp_word)DIGIT_BIT)) / ((mp_word)3); - - if ((res = mp_init_size(&q, a->used)) != MP_OKAY) { - return res; - } - - q.used = a->used; - q.sign = a->sign; - w = 0; - for (ix = a->used - 1; ix >= 0; ix--) { - w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); - - if (w >= 3) { - /* multiply w by [1/3] */ - t = (w * ((mp_word)b)) >> ((mp_word)DIGIT_BIT); - - /* now subtract 3 * [w/3] from w, to get the remainder */ - w -= t+t+t; - - /* fixup the remainder as required since - * the optimization is not exact. - */ - while (w >= 3) { - t += 1; - w -= 3; - } - } else { - t = 0; - } - q.dp[ix] = (mp_digit)t; - } - - /* [optional] store the remainder */ - if (d != NULL) { - *d = (mp_digit)w; - } - - /* [optional] store the quotient */ - if (c != NULL) { - mp_clamp(&q); - mp_exch(&q, c); - } - mp_clear(&q); - - return res; -} - -#endif - -/* End: bn_mp_div_3.c */ - -/* Start: bn_mp_div_d.c */ -#include -#ifdef BN_MP_DIV_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -static int s_is_power_of_two(mp_digit b, int *p) -{ - int x; - - /* fast return if no power of two */ - if ((b==0) || (b & (b-1))) { - return 0; - } - - for (x = 0; x < DIGIT_BIT; x++) { - if (b == (((mp_digit)1)<dp[0] & ((((mp_digit)1)<used)) != MP_OKAY) { - return res; - } - - q.used = a->used; - q.sign = a->sign; - w = 0; - for (ix = a->used - 1; ix >= 0; ix--) { - w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); - - if (w >= b) { - t = (mp_digit)(w / b); - w -= ((mp_word)t) * ((mp_word)b); - } else { - t = 0; - } - q.dp[ix] = (mp_digit)t; - } - - if (d != NULL) { - *d = (mp_digit)w; - } - - if (c != NULL) { - mp_clamp(&q); - mp_exch(&q, c); - } - mp_clear(&q); - - return res; -} - -#endif - -/* End: bn_mp_div_d.c */ - -/* Start: bn_mp_dr_is_modulus.c */ -#include -#ifdef BN_MP_DR_IS_MODULUS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* determines if a number is a valid DR modulus */ -int mp_dr_is_modulus(mp_int *a) -{ - int ix; - - /* must be at least two digits */ - if (a->used < 2) { - return 0; - } - - /* must be of the form b**k - a [a <= b] so all - * but the first digit must be equal to -1 (mod b). - */ - for (ix = 1; ix < a->used; ix++) { - if (a->dp[ix] != MP_MASK) { - return 0; - } - } - return 1; -} - -#endif - -/* End: bn_mp_dr_is_modulus.c */ - -/* Start: bn_mp_dr_reduce.c */ -#include -#ifdef BN_MP_DR_REDUCE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* reduce "x" in place modulo "n" using the Diminished Radix algorithm. - * - * Based on algorithm from the paper - * - * "Generating Efficient Primes for Discrete Log Cryptosystems" - * Chae Hoon Lim, Pil Joong Lee, - * POSTECH Information Research Laboratories - * - * The modulus must be of a special format [see manual] - * - * Has been modified to use algorithm 7.10 from the LTM book instead - * - * Input x must be in the range 0 <= x <= (n-1)**2 - */ -int -mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k) -{ - int err, i, m; - mp_word r; - mp_digit mu, *tmpx1, *tmpx2; - - /* m = digits in modulus */ - m = n->used; - - /* ensure that "x" has at least 2m digits */ - if (x->alloc < m + m) { - if ((err = mp_grow (x, m + m)) != MP_OKAY) { - return err; - } - } - -/* top of loop, this is where the code resumes if - * another reduction pass is required. - */ -top: - /* aliases for digits */ - /* alias for lower half of x */ - tmpx1 = x->dp; - - /* alias for upper half of x, or x/B**m */ - tmpx2 = x->dp + m; - - /* set carry to zero */ - mu = 0; - - /* compute (x mod B**m) + k * [x/B**m] inline and inplace */ - for (i = 0; i < m; i++) { - r = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu; - *tmpx1++ = (mp_digit)(r & MP_MASK); - mu = (mp_digit)(r >> ((mp_word)DIGIT_BIT)); - } - - /* set final carry */ - *tmpx1++ = mu; - - /* zero words above m */ - for (i = m + 1; i < x->used; i++) { - *tmpx1++ = 0; - } - - /* clamp, sub and return */ - mp_clamp (x); - - /* if x >= n then subtract and reduce again - * Each successive "recursion" makes the input smaller and smaller. - */ - if (mp_cmp_mag (x, n) != MP_LT) { - s_mp_sub(x, n, x); - goto top; - } - return MP_OKAY; -} -#endif - -/* End: bn_mp_dr_reduce.c */ - -/* Start: bn_mp_dr_setup.c */ -#include -#ifdef BN_MP_DR_SETUP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* determines the setup value */ -void mp_dr_setup(mp_int *a, mp_digit *d) -{ - /* the casts are required if DIGIT_BIT is one less than - * the number of bits in a mp_digit [e.g. DIGIT_BIT==31] - */ - *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - - ((mp_word)a->dp[0])); -} - -#endif - -/* End: bn_mp_dr_setup.c */ - -/* Start: bn_mp_exch.c */ -#include -#ifdef BN_MP_EXCH_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* swap the elements of two integers, for cases where you can't simply swap the - * mp_int pointers around - */ -void -mp_exch (mp_int * a, mp_int * b) -{ - mp_int t; - - t = *a; - *a = *b; - *b = t; -} -#endif - -/* End: bn_mp_exch.c */ - -/* Start: bn_mp_expt_d.c */ -#include -#ifdef BN_MP_EXPT_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* calculate c = a**b using a square-multiply algorithm */ -int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) -{ - int res, x; - mp_int g; - - if ((res = mp_init_copy (&g, a)) != MP_OKAY) { - return res; - } - - /* set initial result */ - mp_set (c, 1); - - for (x = 0; x < (int) DIGIT_BIT; x++) { - /* square */ - if ((res = mp_sqr (c, c)) != MP_OKAY) { - mp_clear (&g); - return res; - } - - /* if the bit is set multiply */ - if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) { - if ((res = mp_mul (c, &g, c)) != MP_OKAY) { - mp_clear (&g); - return res; - } - } - - /* shift to next bit */ - b <<= 1; - } - - mp_clear (&g); - return MP_OKAY; -} -#endif - -/* End: bn_mp_expt_d.c */ - -/* Start: bn_mp_exptmod.c */ -#include -#ifdef BN_MP_EXPTMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - - -/* this is a shell function that calls either the normal or Montgomery - * exptmod functions. Originally the call to the montgomery code was - * embedded in the normal function but that wasted alot of stack space - * for nothing (since 99% of the time the Montgomery code would be called) - */ -int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) -{ - int dr; - - /* modulus P must be positive */ - if (P->sign == MP_NEG) { - return MP_VAL; - } - - /* if exponent X is negative we have to recurse */ - if (X->sign == MP_NEG) { -#ifdef BN_MP_INVMOD_C - mp_int tmpG, tmpX; - int err; - - /* first compute 1/G mod P */ - if ((err = mp_init(&tmpG)) != MP_OKAY) { - return err; - } - if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) { - mp_clear(&tmpG); - return err; - } - - /* now get |X| */ - if ((err = mp_init(&tmpX)) != MP_OKAY) { - mp_clear(&tmpG); - return err; - } - if ((err = mp_abs(X, &tmpX)) != MP_OKAY) { - mp_clear_multi(&tmpG, &tmpX, NULL); - return err; - } - - /* and now compute (1/G)**|X| instead of G**X [X < 0] */ - err = mp_exptmod(&tmpG, &tmpX, P, Y); - mp_clear_multi(&tmpG, &tmpX, NULL); - return err; -#else - /* no invmod */ - return MP_VAL; -#endif - } - -/* modified diminished radix reduction */ -#if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_C) && defined(BN_S_MP_EXPTMOD_C) - if (mp_reduce_is_2k_l(P) == MP_YES) { - return s_mp_exptmod(G, X, P, Y, 1); - } -#endif - -#ifdef BN_MP_DR_IS_MODULUS_C - /* is it a DR modulus? */ - dr = mp_dr_is_modulus(P); -#else - /* default to no */ - dr = 0; -#endif - -#ifdef BN_MP_REDUCE_IS_2K_C - /* if not, is it a unrestricted DR modulus? */ - if (dr == 0) { - dr = mp_reduce_is_2k(P) << 1; - } -#endif - - /* if the modulus is odd or dr != 0 use the montgomery method */ -#ifdef BN_MP_EXPTMOD_FAST_C - if (mp_isodd (P) == 1 || dr != 0) { - return mp_exptmod_fast (G, X, P, Y, dr); - } else { -#endif -#ifdef BN_S_MP_EXPTMOD_C - /* otherwise use the generic Barrett reduction technique */ - return s_mp_exptmod (G, X, P, Y, 0); -#else - /* no exptmod for evens */ - return MP_VAL; -#endif -#ifdef BN_MP_EXPTMOD_FAST_C - } -#endif -} - -#endif - -/* End: bn_mp_exptmod.c */ - -/* Start: bn_mp_exptmod_fast.c */ -#include -#ifdef BN_MP_EXPTMOD_FAST_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85 - * - * Uses a left-to-right k-ary sliding window to compute the modular exponentiation. - * The value of k changes based on the size of the exponent. - * - * Uses Montgomery or Diminished Radix reduction [whichever appropriate] - */ - -#ifdef MP_LOW_MEM - #define TAB_SIZE 32 -#else - #define TAB_SIZE 256 -#endif - -int mp_exptmod_fast (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode) -{ - mp_int M[TAB_SIZE], res; - mp_digit buf, mp; - int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; - - /* use a pointer to the reduction algorithm. This allows us to use - * one of many reduction algorithms without modding the guts of - * the code with if statements everywhere. - */ - int (*redux)(mp_int*,mp_int*,mp_digit); - - /* find window size */ - x = mp_count_bits (X); - if (x <= 7) { - winsize = 2; - } else if (x <= 36) { - winsize = 3; - } else if (x <= 140) { - winsize = 4; - } else if (x <= 450) { - winsize = 5; - } else if (x <= 1303) { - winsize = 6; - } else if (x <= 3529) { - winsize = 7; - } else { - winsize = 8; - } - -#ifdef MP_LOW_MEM - if (winsize > 5) { - winsize = 5; - } -#endif - - /* init M array */ - /* init first cell */ - if ((err = mp_init(&M[1])) != MP_OKAY) { - return err; - } - - /* now init the second half of the array */ - for (x = 1<<(winsize-1); x < (1 << winsize); x++) { - if ((err = mp_init(&M[x])) != MP_OKAY) { - for (y = 1<<(winsize-1); y < x; y++) { - mp_clear (&M[y]); - } - mp_clear(&M[1]); - return err; - } - } - - /* determine and setup reduction code */ - if (redmode == 0) { -#ifdef BN_MP_MONTGOMERY_SETUP_C - /* now setup montgomery */ - if ((err = mp_montgomery_setup (P, &mp)) != MP_OKAY) { - goto LBL_M; - } -#else - err = MP_VAL; - goto LBL_M; -#endif - - /* automatically pick the comba one if available (saves quite a few calls/ifs) */ -#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C - if (((P->used * 2 + 1) < MP_WARRAY) && - P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - redux = fast_mp_montgomery_reduce; - } else -#endif - { -#ifdef BN_MP_MONTGOMERY_REDUCE_C - /* use slower baseline Montgomery method */ - redux = mp_montgomery_reduce; -#else - err = MP_VAL; - goto LBL_M; -#endif - } - } else if (redmode == 1) { -#if defined(BN_MP_DR_SETUP_C) && defined(BN_MP_DR_REDUCE_C) - /* setup DR reduction for moduli of the form B**k - b */ - mp_dr_setup(P, &mp); - redux = mp_dr_reduce; -#else - err = MP_VAL; - goto LBL_M; -#endif - } else { -#if defined(BN_MP_REDUCE_2K_SETUP_C) && defined(BN_MP_REDUCE_2K_C) - /* setup DR reduction for moduli of the form 2**k - b */ - if ((err = mp_reduce_2k_setup(P, &mp)) != MP_OKAY) { - goto LBL_M; - } - redux = mp_reduce_2k; -#else - err = MP_VAL; - goto LBL_M; -#endif - } - - /* setup result */ - if ((err = mp_init (&res)) != MP_OKAY) { - goto LBL_M; - } - - /* create M table - * - - * - * The first half of the table is not computed though accept for M[0] and M[1] - */ - - if (redmode == 0) { -#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C - /* now we need R mod m */ - if ((err = mp_montgomery_calc_normalization (&res, P)) != MP_OKAY) { - goto LBL_RES; - } -#else - err = MP_VAL; - goto LBL_RES; -#endif - - /* now set M[1] to G * R mod m */ - if ((err = mp_mulmod (G, &res, P, &M[1])) != MP_OKAY) { - goto LBL_RES; - } - } else { - mp_set(&res, 1); - if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) { - goto LBL_RES; - } - } - - /* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */ - if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { - goto LBL_RES; - } - - for (x = 0; x < (winsize - 1); x++) { - if ((err = mp_sqr (&M[1 << (winsize - 1)], &M[1 << (winsize - 1)])) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&M[1 << (winsize - 1)], P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - - /* create upper table */ - for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { - if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&M[x], P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - - /* set initial mode and bit cnt */ - mode = 0; - bitcnt = 1; - buf = 0; - digidx = X->used - 1; - bitcpy = 0; - bitbuf = 0; - - for (;;) { - /* grab next digit as required */ - if (--bitcnt == 0) { - /* if digidx == -1 we are out of digits so break */ - if (digidx == -1) { - break; - } - /* read next digit and reset bitcnt */ - buf = X->dp[digidx--]; - bitcnt = (int)DIGIT_BIT; - } - - /* grab the next msb from the exponent */ - y = (mp_digit)(buf >> (DIGIT_BIT - 1)) & 1; - buf <<= (mp_digit)1; - - /* if the bit is zero and mode == 0 then we ignore it - * These represent the leading zero bits before the first 1 bit - * in the exponent. Technically this opt is not required but it - * does lower the # of trivial squaring/reductions used - */ - if (mode == 0 && y == 0) { - continue; - } - - /* if the bit is zero and mode == 1 then we square */ - if (mode == 1 && y == 0) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - continue; - } - - /* else we add it to the window */ - bitbuf |= (y << (winsize - ++bitcpy)); - mode = 2; - - if (bitcpy == winsize) { - /* ok window is filled so square as required and multiply */ - /* square first */ - for (x = 0; x < winsize; x++) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - - /* then multiply */ - if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - - /* empty window and reset */ - bitcpy = 0; - bitbuf = 0; - mode = 1; - } - } - - /* if bits remain then square/multiply */ - if (mode == 2 && bitcpy > 0) { - /* square then multiply if the bit is set */ - for (x = 0; x < bitcpy; x++) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - - /* get next bit of the window */ - bitbuf <<= 1; - if ((bitbuf & (1 << winsize)) != 0) { - /* then multiply */ - if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - } - } - - if (redmode == 0) { - /* fixup result if Montgomery reduction is used - * recall that any value in a Montgomery system is - * actually multiplied by R mod n. So we have - * to reduce one more time to cancel out the factor - * of R. - */ - if ((err = redux(&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - - /* swap res with Y */ - mp_exch (&res, Y); - err = MP_OKAY; -LBL_RES:mp_clear (&res); -LBL_M: - mp_clear(&M[1]); - for (x = 1<<(winsize-1); x < (1 << winsize); x++) { - mp_clear (&M[x]); - } - return err; -} -#endif - -/* End: bn_mp_exptmod_fast.c */ - -/* Start: bn_mp_exteuclid.c */ -#include -#ifdef BN_MP_EXTEUCLID_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* Extended euclidean algorithm of (a, b) produces - a*u1 + b*u2 = u3 - */ -int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) -{ - mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp; - int err; - - if ((err = mp_init_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL)) != MP_OKAY) { - return err; - } - - /* initialize, (u1,u2,u3) = (1,0,a) */ - mp_set(&u1, 1); - if ((err = mp_copy(a, &u3)) != MP_OKAY) { goto _ERR; } - - /* initialize, (v1,v2,v3) = (0,1,b) */ - mp_set(&v2, 1); - if ((err = mp_copy(b, &v3)) != MP_OKAY) { goto _ERR; } - - /* loop while v3 != 0 */ - while (mp_iszero(&v3) == MP_NO) { - /* q = u3/v3 */ - if ((err = mp_div(&u3, &v3, &q, NULL)) != MP_OKAY) { goto _ERR; } - - /* (t1,t2,t3) = (u1,u2,u3) - (v1,v2,v3)q */ - if ((err = mp_mul(&v1, &q, &tmp)) != MP_OKAY) { goto _ERR; } - if ((err = mp_sub(&u1, &tmp, &t1)) != MP_OKAY) { goto _ERR; } - if ((err = mp_mul(&v2, &q, &tmp)) != MP_OKAY) { goto _ERR; } - if ((err = mp_sub(&u2, &tmp, &t2)) != MP_OKAY) { goto _ERR; } - if ((err = mp_mul(&v3, &q, &tmp)) != MP_OKAY) { goto _ERR; } - if ((err = mp_sub(&u3, &tmp, &t3)) != MP_OKAY) { goto _ERR; } - - /* (u1,u2,u3) = (v1,v2,v3) */ - if ((err = mp_copy(&v1, &u1)) != MP_OKAY) { goto _ERR; } - if ((err = mp_copy(&v2, &u2)) != MP_OKAY) { goto _ERR; } - if ((err = mp_copy(&v3, &u3)) != MP_OKAY) { goto _ERR; } - - /* (v1,v2,v3) = (t1,t2,t3) */ - if ((err = mp_copy(&t1, &v1)) != MP_OKAY) { goto _ERR; } - if ((err = mp_copy(&t2, &v2)) != MP_OKAY) { goto _ERR; } - if ((err = mp_copy(&t3, &v3)) != MP_OKAY) { goto _ERR; } - } - - /* make sure U3 >= 0 */ - if (u3.sign == MP_NEG) { - mp_neg(&u1, &u1); - mp_neg(&u2, &u2); - mp_neg(&u3, &u3); - } - - /* copy result out */ - if (U1 != NULL) { mp_exch(U1, &u1); } - if (U2 != NULL) { mp_exch(U2, &u2); } - if (U3 != NULL) { mp_exch(U3, &u3); } - - err = MP_OKAY; -_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL); - return err; -} -#endif - -/* End: bn_mp_exteuclid.c */ - -/* Start: bn_mp_fread.c */ -#include -#ifdef BN_MP_FREAD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* read a bigint from a file stream in ASCII */ -int mp_fread(mp_int *a, int radix, FILE *stream) -{ - int err, ch, neg, y; - - /* clear a */ - mp_zero(a); - - /* if first digit is - then set negative */ - ch = fgetc(stream); - if (ch == '-') { - neg = MP_NEG; - ch = fgetc(stream); - } else { - neg = MP_ZPOS; - } - - for (;;) { - /* find y in the radix map */ - for (y = 0; y < radix; y++) { - if (mp_s_rmap[y] == ch) { - break; - } - } - if (y == radix) { - break; - } - - /* shift up and add */ - if ((err = mp_mul_d(a, radix, a)) != MP_OKAY) { - return err; - } - if ((err = mp_add_d(a, y, a)) != MP_OKAY) { - return err; - } - - ch = fgetc(stream); - } - if (mp_cmp_d(a, 0) != MP_EQ) { - a->sign = neg; - } - - return MP_OKAY; -} - -#endif - -/* End: bn_mp_fread.c */ - -/* Start: bn_mp_fwrite.c */ -#include -#ifdef BN_MP_FWRITE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -int mp_fwrite(mp_int *a, int radix, FILE *stream) -{ - char *buf; - int err, len, x; - - if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) { - return err; - } - - buf = OPT_CAST(char) XMALLOC (len); - if (buf == NULL) { - return MP_MEM; - } - - if ((err = mp_toradix(a, buf, radix)) != MP_OKAY) { - XFREE (buf); - return err; - } - - for (x = 0; x < len; x++) { - if (fputc(buf[x], stream) == EOF) { - XFREE (buf); - return MP_VAL; - } - } - - XFREE (buf); - return MP_OKAY; -} - -#endif - -/* End: bn_mp_fwrite.c */ - -/* Start: bn_mp_gcd.c */ -#include -#ifdef BN_MP_GCD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* Greatest Common Divisor using the binary method */ -int mp_gcd (mp_int * a, mp_int * b, mp_int * c) -{ - mp_int u, v; - int k, u_lsb, v_lsb, res; - - /* either zero than gcd is the largest */ - if (mp_iszero (a) == MP_YES) { - return mp_abs (b, c); - } - if (mp_iszero (b) == MP_YES) { - return mp_abs (a, c); - } - - /* get copies of a and b we can modify */ - if ((res = mp_init_copy (&u, a)) != MP_OKAY) { - return res; - } - - if ((res = mp_init_copy (&v, b)) != MP_OKAY) { - goto LBL_U; - } - - /* must be positive for the remainder of the algorithm */ - u.sign = v.sign = MP_ZPOS; - - /* B1. Find the common power of two for u and v */ - u_lsb = mp_cnt_lsb(&u); - v_lsb = mp_cnt_lsb(&v); - k = MIN(u_lsb, v_lsb); - - if (k > 0) { - /* divide the power of two out */ - if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) { - goto LBL_V; - } - - if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) { - goto LBL_V; - } - } - - /* divide any remaining factors of two out */ - if (u_lsb != k) { - if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) { - goto LBL_V; - } - } - - if (v_lsb != k) { - if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) { - goto LBL_V; - } - } - - while (mp_iszero(&v) == 0) { - /* make sure v is the largest */ - if (mp_cmp_mag(&u, &v) == MP_GT) { - /* swap u and v to make sure v is >= u */ - mp_exch(&u, &v); - } - - /* subtract smallest from largest */ - if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) { - goto LBL_V; - } - - /* Divide out all factors of two */ - if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) { - goto LBL_V; - } - } - - /* multiply by 2**k which we divided out at the beginning */ - if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) { - goto LBL_V; - } - c->sign = MP_ZPOS; - res = MP_OKAY; -LBL_V:mp_clear (&u); -LBL_U:mp_clear (&v); - return res; -} -#endif - -/* End: bn_mp_gcd.c */ - -/* Start: bn_mp_get_int.c */ -#include -#ifdef BN_MP_GET_INT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* get the lower 32-bits of an mp_int */ -unsigned long mp_get_int(mp_int * a) -{ - int i; - unsigned long res; - - if (a->used == 0) { - return 0; - } - - /* get number of digits of the lsb we have to read */ - i = MIN(a->used,(int)((sizeof(unsigned long)*CHAR_BIT+DIGIT_BIT-1)/DIGIT_BIT))-1; - - /* get most significant digit of result */ - res = DIGIT(a,i); - - while (--i >= 0) { - res = (res << DIGIT_BIT) | DIGIT(a,i); - } - - /* force result to 32-bits always so it is consistent on non 32-bit platforms */ - return res & 0xFFFFFFFFUL; -} -#endif - -/* End: bn_mp_get_int.c */ - -/* Start: bn_mp_grow.c */ -#include -#ifdef BN_MP_GROW_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* grow as required */ -int mp_grow (mp_int * a, int size) -{ - int i; - mp_digit *tmp; - - /* if the alloc size is smaller alloc more ram */ - if (a->alloc < size) { - /* ensure there are always at least MP_PREC digits extra on top */ - size += (MP_PREC * 2) - (size % MP_PREC); - - /* reallocate the array a->dp - * - * We store the return in a temporary variable - * in case the operation failed we don't want - * to overwrite the dp member of a. - */ - tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size); - if (tmp == NULL) { - /* reallocation failed but "a" is still valid [can be freed] */ - return MP_MEM; - } - - /* reallocation succeeded so set a->dp */ - a->dp = tmp; - - /* zero excess digits */ - i = a->alloc; - a->alloc = size; - for (; i < a->alloc; i++) { - a->dp[i] = 0; - } - } - return MP_OKAY; -} -#endif - -/* End: bn_mp_grow.c */ - -/* Start: bn_mp_init.c */ -#include -#ifdef BN_MP_INIT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* init a new mp_int */ -int mp_init (mp_int * a) -{ - int i; - - /* allocate memory required and clear it */ - a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC); - if (a->dp == NULL) { - return MP_MEM; - } - - /* set the digits to zero */ - for (i = 0; i < MP_PREC; i++) { - a->dp[i] = 0; - } - - /* set the used to zero, allocated digits to the default precision - * and sign to positive */ - a->used = 0; - a->alloc = MP_PREC; - a->sign = MP_ZPOS; - - return MP_OKAY; -} -#endif - -/* End: bn_mp_init.c */ - -/* Start: bn_mp_init_copy.c */ -#include -#ifdef BN_MP_INIT_COPY_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* creates "a" then copies b into it */ -int mp_init_copy (mp_int * a, mp_int * b) -{ - int res; - - if ((res = mp_init (a)) != MP_OKAY) { - return res; - } - return mp_copy (b, a); -} -#endif - -/* End: bn_mp_init_copy.c */ - -/* Start: bn_mp_init_multi.c */ -#include -#ifdef BN_MP_INIT_MULTI_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ -#include - -int mp_init_multi(mp_int *mp, ...) -{ - mp_err res = MP_OKAY; /* Assume ok until proven otherwise */ - int n = 0; /* Number of ok inits */ - mp_int* cur_arg = mp; - va_list args; - - va_start(args, mp); /* init args to next argument from caller */ - while (cur_arg != NULL) { - if (mp_init(cur_arg) != MP_OKAY) { - /* Oops - error! Back-track and mp_clear what we already - succeeded in init-ing, then return error. - */ - va_list clean_args; - - /* end the current list */ - va_end(args); - - /* now start cleaning up */ - cur_arg = mp; - va_start(clean_args, mp); - while (n--) { - mp_clear(cur_arg); - cur_arg = va_arg(clean_args, mp_int*); - } - va_end(clean_args); - res = MP_MEM; - break; - } - n++; - cur_arg = va_arg(args, mp_int*); - } - va_end(args); - return res; /* Assumed ok, if error flagged above. */ -} - -#endif - -/* End: bn_mp_init_multi.c */ - -/* Start: bn_mp_init_set.c */ -#include -#ifdef BN_MP_INIT_SET_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* initialize and set a digit */ -int mp_init_set (mp_int * a, mp_digit b) -{ - int err; - if ((err = mp_init(a)) != MP_OKAY) { - return err; - } - mp_set(a, b); - return err; -} -#endif - -/* End: bn_mp_init_set.c */ - -/* Start: bn_mp_init_set_int.c */ -#include -#ifdef BN_MP_INIT_SET_INT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* initialize and set a digit */ -int mp_init_set_int (mp_int * a, unsigned long b) -{ - int err; - if ((err = mp_init(a)) != MP_OKAY) { - return err; - } - return mp_set_int(a, b); -} -#endif - -/* End: bn_mp_init_set_int.c */ - -/* Start: bn_mp_init_size.c */ -#include -#ifdef BN_MP_INIT_SIZE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* init an mp_init for a given size */ -int mp_init_size (mp_int * a, int size) -{ - int x; - - /* pad size so there are always extra digits */ - size += (MP_PREC * 2) - (size % MP_PREC); - - /* alloc mem */ - a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size); - if (a->dp == NULL) { - return MP_MEM; - } - - /* set the members */ - a->used = 0; - a->alloc = size; - a->sign = MP_ZPOS; - - /* zero the digits */ - for (x = 0; x < size; x++) { - a->dp[x] = 0; - } - - return MP_OKAY; -} -#endif - -/* End: bn_mp_init_size.c */ - -/* Start: bn_mp_invmod.c */ -#include -#ifdef BN_MP_INVMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* hac 14.61, pp608 */ -int mp_invmod (mp_int * a, mp_int * b, mp_int * c) -{ - /* b cannot be negative */ - if (b->sign == MP_NEG || mp_iszero(b) == 1) { - return MP_VAL; - } - -#ifdef BN_FAST_MP_INVMOD_C - /* if the modulus is odd we can use a faster routine instead */ - if (mp_isodd (b) == 1) { - return fast_mp_invmod (a, b, c); - } -#endif - -#ifdef BN_MP_INVMOD_SLOW_C - return mp_invmod_slow(a, b, c); -#endif - - return MP_VAL; -} -#endif - -/* End: bn_mp_invmod.c */ - -/* Start: bn_mp_invmod_slow.c */ -#include -#ifdef BN_MP_INVMOD_SLOW_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* hac 14.61, pp608 */ -int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c) -{ - mp_int x, y, u, v, A, B, C, D; - int res; - - /* b cannot be negative */ - if (b->sign == MP_NEG || mp_iszero(b) == 1) { - return MP_VAL; - } - - /* init temps */ - if ((res = mp_init_multi(&x, &y, &u, &v, - &A, &B, &C, &D, NULL)) != MP_OKAY) { - return res; - } - - /* x = a, y = b */ - if ((res = mp_mod(a, b, &x)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_copy (b, &y)) != MP_OKAY) { - goto LBL_ERR; - } - - /* 2. [modified] if x,y are both even then return an error! */ - if (mp_iseven (&x) == 1 && mp_iseven (&y) == 1) { - res = MP_VAL; - goto LBL_ERR; - } - - /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ - if ((res = mp_copy (&x, &u)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_copy (&y, &v)) != MP_OKAY) { - goto LBL_ERR; - } - mp_set (&A, 1); - mp_set (&D, 1); - -top: - /* 4. while u is even do */ - while (mp_iseven (&u) == 1) { - /* 4.1 u = u/2 */ - if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { - goto LBL_ERR; - } - /* 4.2 if A or B is odd then */ - if (mp_isodd (&A) == 1 || mp_isodd (&B) == 1) { - /* A = (A+y)/2, B = (B-x)/2 */ - if ((res = mp_add (&A, &y, &A)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } - /* A = A/2, B = B/2 */ - if ((res = mp_div_2 (&A, &A)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* 5. while v is even do */ - while (mp_iseven (&v) == 1) { - /* 5.1 v = v/2 */ - if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { - goto LBL_ERR; - } - /* 5.2 if C or D is odd then */ - if (mp_isodd (&C) == 1 || mp_isodd (&D) == 1) { - /* C = (C+y)/2, D = (D-x)/2 */ - if ((res = mp_add (&C, &y, &C)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - /* C = C/2, D = D/2 */ - if ((res = mp_div_2 (&C, &C)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* 6. if u >= v then */ - if (mp_cmp (&u, &v) != MP_LT) { - /* u = u - v, A = A - C, B = B - D */ - if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&A, &C, &A)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } else { - /* v - v - u, C = C - A, D = D - B */ - if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&C, &A, &C)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* if not zero goto step 4 */ - if (mp_iszero (&u) == 0) - goto top; - - /* now a = C, b = D, gcd == g*v */ - - /* if v != 1 then there is no inverse */ - if (mp_cmp_d (&v, 1) != MP_EQ) { - res = MP_VAL; - goto LBL_ERR; - } - - /* if its too low */ - while (mp_cmp_d(&C, 0) == MP_LT) { - if ((res = mp_add(&C, b, &C)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* too big */ - while (mp_cmp_mag(&C, b) != MP_LT) { - if ((res = mp_sub(&C, b, &C)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* C is now the inverse */ - mp_exch (&C, c); - res = MP_OKAY; -LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL); - return res; -} -#endif - -/* End: bn_mp_invmod_slow.c */ - -/* Start: bn_mp_is_square.c */ -#include -#ifdef BN_MP_IS_SQUARE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* Check if remainders are possible squares - fast exclude non-squares */ -static const char rem_128[128] = { - 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 -}; - -static const char rem_105[105] = { - 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, - 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, - 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, - 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, - 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, - 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1 -}; - -/* Store non-zero to ret if arg is square, and zero if not */ -int mp_is_square(mp_int *arg,int *ret) -{ - int res; - mp_digit c; - mp_int t; - unsigned long r; - - /* Default to Non-square :) */ - *ret = MP_NO; - - if (arg->sign == MP_NEG) { - return MP_VAL; - } - - /* digits used? (TSD) */ - if (arg->used == 0) { - return MP_OKAY; - } - - /* First check mod 128 (suppose that DIGIT_BIT is at least 7) */ - if (rem_128[127 & DIGIT(arg,0)] == 1) { - return MP_OKAY; - } - - /* Next check mod 105 (3*5*7) */ - if ((res = mp_mod_d(arg,105,&c)) != MP_OKAY) { - return res; - } - if (rem_105[c] == 1) { - return MP_OKAY; - } - - - if ((res = mp_init_set_int(&t,11L*13L*17L*19L*23L*29L*31L)) != MP_OKAY) { - return res; - } - if ((res = mp_mod(arg,&t,&t)) != MP_OKAY) { - goto ERR; - } - r = mp_get_int(&t); - /* Check for other prime modules, note it's not an ERROR but we must - * free "t" so the easiest way is to goto ERR. We know that res - * is already equal to MP_OKAY from the mp_mod call - */ - if ( (1L<<(r%11)) & 0x5C4L ) goto ERR; - if ( (1L<<(r%13)) & 0x9E4L ) goto ERR; - if ( (1L<<(r%17)) & 0x5CE8L ) goto ERR; - if ( (1L<<(r%19)) & 0x4F50CL ) goto ERR; - if ( (1L<<(r%23)) & 0x7ACCA0L ) goto ERR; - if ( (1L<<(r%29)) & 0xC2EDD0CL ) goto ERR; - if ( (1L<<(r%31)) & 0x6DE2B848L ) goto ERR; - - /* Final check - is sqr(sqrt(arg)) == arg ? */ - if ((res = mp_sqrt(arg,&t)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sqr(&t,&t)) != MP_OKAY) { - goto ERR; - } - - *ret = (mp_cmp_mag(&t,arg) == MP_EQ) ? MP_YES : MP_NO; -ERR:mp_clear(&t); - return res; -} -#endif - -/* End: bn_mp_is_square.c */ - -/* Start: bn_mp_jacobi.c */ -#include -#ifdef BN_MP_JACOBI_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* computes the jacobi c = (a | n) (or Legendre if n is prime) - * HAC pp. 73 Algorithm 2.149 - */ -int mp_jacobi (mp_int * a, mp_int * p, int *c) -{ - mp_int a1, p1; - int k, s, r, res; - mp_digit residue; - - /* if p <= 0 return MP_VAL */ - if (mp_cmp_d(p, 0) != MP_GT) { - return MP_VAL; - } - - /* step 1. if a == 0, return 0 */ - if (mp_iszero (a) == 1) { - *c = 0; - return MP_OKAY; - } - - /* step 2. if a == 1, return 1 */ - if (mp_cmp_d (a, 1) == MP_EQ) { - *c = 1; - return MP_OKAY; - } - - /* default */ - s = 0; - - /* step 3. write a = a1 * 2**k */ - if ((res = mp_init_copy (&a1, a)) != MP_OKAY) { - return res; - } - - if ((res = mp_init (&p1)) != MP_OKAY) { - goto LBL_A1; - } - - /* divide out larger power of two */ - k = mp_cnt_lsb(&a1); - if ((res = mp_div_2d(&a1, k, &a1, NULL)) != MP_OKAY) { - goto LBL_P1; - } - - /* step 4. if e is even set s=1 */ - if ((k & 1) == 0) { - s = 1; - } else { - /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */ - residue = p->dp[0] & 7; - - if (residue == 1 || residue == 7) { - s = 1; - } else if (residue == 3 || residue == 5) { - s = -1; - } - } - - /* step 5. if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */ - if ( ((p->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) { - s = -s; - } - - /* if a1 == 1 we're done */ - if (mp_cmp_d (&a1, 1) == MP_EQ) { - *c = s; - } else { - /* n1 = n mod a1 */ - if ((res = mp_mod (p, &a1, &p1)) != MP_OKAY) { - goto LBL_P1; - } - if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) { - goto LBL_P1; - } - *c = s * r; - } - - /* done */ - res = MP_OKAY; -LBL_P1:mp_clear (&p1); -LBL_A1:mp_clear (&a1); - return res; -} -#endif - -/* End: bn_mp_jacobi.c */ - -/* Start: bn_mp_karatsuba_mul.c */ -#include -#ifdef BN_MP_KARATSUBA_MUL_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* c = |a| * |b| using Karatsuba Multiplication using - * three half size multiplications - * - * Let B represent the radix [e.g. 2**DIGIT_BIT] and - * let n represent half of the number of digits in - * the min(a,b) - * - * a = a1 * B**n + a0 - * b = b1 * B**n + b0 - * - * Then, a * b => - a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0 - * - * Note that a1b1 and a0b0 are used twice and only need to be - * computed once. So in total three half size (half # of - * digit) multiplications are performed, a0b0, a1b1 and - * (a1+b1)(a0+b0) - * - * Note that a multiplication of half the digits requires - * 1/4th the number of single precision multiplications so in - * total after one call 25% of the single precision multiplications - * are saved. Note also that the call to mp_mul can end up back - * in this function if the a0, a1, b0, or b1 are above the threshold. - * This is known as divide-and-conquer and leads to the famous - * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than - * the standard O(N**2) that the baseline/comba methods use. - * Generally though the overhead of this method doesn't pay off - * until a certain size (N ~ 80) is reached. - */ -int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c) -{ - mp_int x0, x1, y0, y1, t1, x0y0, x1y1; - int B, err; - - /* default the return code to an error */ - err = MP_MEM; - - /* min # of digits */ - B = MIN (a->used, b->used); - - /* now divide in two */ - B = B >> 1; - - /* init copy all the temps */ - if (mp_init_size (&x0, B) != MP_OKAY) - goto ERR; - if (mp_init_size (&x1, a->used - B) != MP_OKAY) - goto X0; - if (mp_init_size (&y0, B) != MP_OKAY) - goto X1; - if (mp_init_size (&y1, b->used - B) != MP_OKAY) - goto Y0; - - /* init temps */ - if (mp_init_size (&t1, B * 2) != MP_OKAY) - goto Y1; - if (mp_init_size (&x0y0, B * 2) != MP_OKAY) - goto T1; - if (mp_init_size (&x1y1, B * 2) != MP_OKAY) - goto X0Y0; - - /* now shift the digits */ - x0.used = y0.used = B; - x1.used = a->used - B; - y1.used = b->used - B; - - { - register int x; - register mp_digit *tmpa, *tmpb, *tmpx, *tmpy; - - /* we copy the digits directly instead of using higher level functions - * since we also need to shift the digits - */ - tmpa = a->dp; - tmpb = b->dp; - - tmpx = x0.dp; - tmpy = y0.dp; - for (x = 0; x < B; x++) { - *tmpx++ = *tmpa++; - *tmpy++ = *tmpb++; - } - - tmpx = x1.dp; - for (x = B; x < a->used; x++) { - *tmpx++ = *tmpa++; - } - - tmpy = y1.dp; - for (x = B; x < b->used; x++) { - *tmpy++ = *tmpb++; - } - } - - /* only need to clamp the lower words since by definition the - * upper words x1/y1 must have a known number of digits - */ - mp_clamp (&x0); - mp_clamp (&y0); - - /* now calc the products x0y0 and x1y1 */ - /* after this x0 is no longer required, free temp [x0==t2]! */ - if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY) - goto X1Y1; /* x0y0 = x0*y0 */ - if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY) - goto X1Y1; /* x1y1 = x1*y1 */ - - /* now calc x1+x0 and y1+y0 */ - if (s_mp_add (&x1, &x0, &t1) != MP_OKAY) - goto X1Y1; /* t1 = x1 - x0 */ - if (s_mp_add (&y1, &y0, &x0) != MP_OKAY) - goto X1Y1; /* t2 = y1 - y0 */ - if (mp_mul (&t1, &x0, &t1) != MP_OKAY) - goto X1Y1; /* t1 = (x1 + x0) * (y1 + y0) */ - - /* add x0y0 */ - if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY) - goto X1Y1; /* t2 = x0y0 + x1y1 */ - if (s_mp_sub (&t1, &x0, &t1) != MP_OKAY) - goto X1Y1; /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */ - - /* shift by B */ - if (mp_lshd (&t1, B) != MP_OKAY) - goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))< -#ifdef BN_MP_KARATSUBA_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* Karatsuba squaring, computes b = a*a using three - * half size squarings - * - * See comments of karatsuba_mul for details. It - * is essentially the same algorithm but merely - * tuned to perform recursive squarings. - */ -int mp_karatsuba_sqr (mp_int * a, mp_int * b) -{ - mp_int x0, x1, t1, t2, x0x0, x1x1; - int B, err; - - err = MP_MEM; - - /* min # of digits */ - B = a->used; - - /* now divide in two */ - B = B >> 1; - - /* init copy all the temps */ - if (mp_init_size (&x0, B) != MP_OKAY) - goto ERR; - if (mp_init_size (&x1, a->used - B) != MP_OKAY) - goto X0; - - /* init temps */ - if (mp_init_size (&t1, a->used * 2) != MP_OKAY) - goto X1; - if (mp_init_size (&t2, a->used * 2) != MP_OKAY) - goto T1; - if (mp_init_size (&x0x0, B * 2) != MP_OKAY) - goto T2; - if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY) - goto X0X0; - - { - register int x; - register mp_digit *dst, *src; - - src = a->dp; - - /* now shift the digits */ - dst = x0.dp; - for (x = 0; x < B; x++) { - *dst++ = *src++; - } - - dst = x1.dp; - for (x = B; x < a->used; x++) { - *dst++ = *src++; - } - } - - x0.used = B; - x1.used = a->used - B; - - mp_clamp (&x0); - - /* now calc the products x0*x0 and x1*x1 */ - if (mp_sqr (&x0, &x0x0) != MP_OKAY) - goto X1X1; /* x0x0 = x0*x0 */ - if (mp_sqr (&x1, &x1x1) != MP_OKAY) - goto X1X1; /* x1x1 = x1*x1 */ - - /* now calc (x1+x0)**2 */ - if (s_mp_add (&x1, &x0, &t1) != MP_OKAY) - goto X1X1; /* t1 = x1 - x0 */ - if (mp_sqr (&t1, &t1) != MP_OKAY) - goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */ - - /* add x0y0 */ - if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY) - goto X1X1; /* t2 = x0x0 + x1x1 */ - if (s_mp_sub (&t1, &t2, &t1) != MP_OKAY) - goto X1X1; /* t1 = (x1+x0)**2 - (x0x0 + x1x1) */ - - /* shift by B */ - if (mp_lshd (&t1, B) != MP_OKAY) - goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))< -#ifdef BN_MP_LCM_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* computes least common multiple as |a*b|/(a, b) */ -int mp_lcm (mp_int * a, mp_int * b, mp_int * c) -{ - int res; - mp_int t1, t2; - - - if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) { - return res; - } - - /* t1 = get the GCD of the two inputs */ - if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) { - goto LBL_T; - } - - /* divide the smallest by the GCD */ - if (mp_cmp_mag(a, b) == MP_LT) { - /* store quotient in t2 such that t2 * b is the LCM */ - if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) { - goto LBL_T; - } - res = mp_mul(b, &t2, c); - } else { - /* store quotient in t2 such that t2 * a is the LCM */ - if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) { - goto LBL_T; - } - res = mp_mul(a, &t2, c); - } - - /* fix the sign to positive */ - c->sign = MP_ZPOS; - -LBL_T: - mp_clear_multi (&t1, &t2, NULL); - return res; -} -#endif - -/* End: bn_mp_lcm.c */ - -/* Start: bn_mp_lshd.c */ -#include -#ifdef BN_MP_LSHD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* shift left a certain amount of digits */ -int mp_lshd (mp_int * a, int b) -{ - int x, res; - - /* if its less than zero return */ - if (b <= 0) { - return MP_OKAY; - } - - /* grow to fit the new digits */ - if (a->alloc < a->used + b) { - if ((res = mp_grow (a, a->used + b)) != MP_OKAY) { - return res; - } - } - - { - register mp_digit *top, *bottom; - - /* increment the used by the shift amount then copy upwards */ - a->used += b; - - /* top */ - top = a->dp + a->used - 1; - - /* base */ - bottom = a->dp + a->used - 1 - b; - - /* much like mp_rshd this is implemented using a sliding window - * except the window goes the otherway around. Copying from - * the bottom to the top. see bn_mp_rshd.c for more info. - */ - for (x = a->used - 1; x >= b; x--) { - *top-- = *bottom--; - } - - /* zero the lower digits */ - top = a->dp; - for (x = 0; x < b; x++) { - *top++ = 0; - } - } - return MP_OKAY; -} -#endif - -/* End: bn_mp_lshd.c */ - -/* Start: bn_mp_mod.c */ -#include -#ifdef BN_MP_MOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* c = a mod b, 0 <= c < b */ -int -mp_mod (mp_int * a, mp_int * b, mp_int * c) -{ - mp_int t; - int res; - - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } - - if ((res = mp_div (a, b, NULL, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - - if (t.sign != b->sign) { - res = mp_add (b, &t, c); - } else { - res = MP_OKAY; - mp_exch (&t, c); - } - - mp_clear (&t); - return res; -} -#endif - -/* End: bn_mp_mod.c */ - -/* Start: bn_mp_mod_2d.c */ -#include -#ifdef BN_MP_MOD_2D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* calc a value mod 2**b */ -int -mp_mod_2d (mp_int * a, int b, mp_int * c) -{ - int x, res; - - /* if b is <= 0 then zero the int */ - if (b <= 0) { - mp_zero (c); - return MP_OKAY; - } - - /* if the modulus is larger than the value than return */ - if (b >= (int) (a->used * DIGIT_BIT)) { - res = mp_copy (a, c); - return res; - } - - /* copy */ - if ((res = mp_copy (a, c)) != MP_OKAY) { - return res; - } - - /* zero digits above the last digit of the modulus */ - for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x++) { - c->dp[x] = 0; - } - /* clear the digit that is not completely outside/inside the modulus */ - c->dp[b / DIGIT_BIT] &= - (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digit) 1)); - mp_clamp (c); - return MP_OKAY; -} -#endif - -/* End: bn_mp_mod_2d.c */ - -/* Start: bn_mp_mod_d.c */ -#include -#ifdef BN_MP_MOD_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -int -mp_mod_d (mp_int * a, mp_digit b, mp_digit * c) -{ - return mp_div_d(a, b, NULL, c); -} -#endif - -/* End: bn_mp_mod_d.c */ - -/* Start: bn_mp_montgomery_calc_normalization.c */ -#include -#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* - * shifts with subtractions when the result is greater than b. - * - * The method is slightly modified to shift B unconditionally upto just under - * the leading bit of b. This saves alot of multiple precision shifting. - */ -int mp_montgomery_calc_normalization (mp_int * a, mp_int * b) -{ - int x, bits, res; - - /* how many bits of last digit does b use */ - bits = mp_count_bits (b) % DIGIT_BIT; - - if (b->used > 1) { - if ((res = mp_2expt (a, (b->used - 1) * DIGIT_BIT + bits - 1)) != MP_OKAY) { - return res; - } - } else { - mp_set(a, 1); - bits = 1; - } - - - /* now compute C = A * B mod b */ - for (x = bits - 1; x < (int)DIGIT_BIT; x++) { - if ((res = mp_mul_2 (a, a)) != MP_OKAY) { - return res; - } - if (mp_cmp_mag (a, b) != MP_LT) { - if ((res = s_mp_sub (a, b, a)) != MP_OKAY) { - return res; - } - } - } - - return MP_OKAY; -} -#endif - -/* End: bn_mp_montgomery_calc_normalization.c */ - -/* Start: bn_mp_montgomery_reduce.c */ -#include -#ifdef BN_MP_MONTGOMERY_REDUCE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* computes xR**-1 == x (mod N) via Montgomery Reduction */ -int -mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) -{ - int ix, res, digs; - mp_digit mu; - - /* can the fast reduction [comba] method be used? - * - * Note that unlike in mul you're safely allowed *less* - * than the available columns [255 per default] since carries - * are fixed up in the inner loop. - */ - digs = n->used * 2 + 1; - if ((digs < MP_WARRAY) && - n->used < - (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - return fast_mp_montgomery_reduce (x, n, rho); - } - - /* grow the input as required */ - if (x->alloc < digs) { - if ((res = mp_grow (x, digs)) != MP_OKAY) { - return res; - } - } - x->used = digs; - - for (ix = 0; ix < n->used; ix++) { - /* mu = ai * rho mod b - * - * The value of rho must be precalculated via - * montgomery_setup() such that - * it equals -1/n0 mod b this allows the - * following inner loop to reduce the - * input one digit at a time - */ - mu = (mp_digit) (((mp_word)x->dp[ix]) * ((mp_word)rho) & MP_MASK); - - /* a = a + mu * m * b**i */ - { - register int iy; - register mp_digit *tmpn, *tmpx, u; - register mp_word r; - - /* alias for digits of the modulus */ - tmpn = n->dp; - - /* alias for the digits of x [the input] */ - tmpx = x->dp + ix; - - /* set the carry to zero */ - u = 0; - - /* Multiply and add in place */ - for (iy = 0; iy < n->used; iy++) { - /* compute product and sum */ - r = ((mp_word)mu) * ((mp_word)*tmpn++) + - ((mp_word) u) + ((mp_word) * tmpx); - - /* get carry */ - u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); - - /* fix digit */ - *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK)); - } - /* At this point the ix'th digit of x should be zero */ - - - /* propagate carries upwards as required*/ - while (u) { - *tmpx += u; - u = *tmpx >> DIGIT_BIT; - *tmpx++ &= MP_MASK; - } - } - } - - /* at this point the n.used'th least - * significant digits of x are all zero - * which means we can shift x to the - * right by n.used digits and the - * residue is unchanged. - */ - - /* x = x/b**n.used */ - mp_clamp(x); - mp_rshd (x, n->used); - - /* if x >= n then x = x - n */ - if (mp_cmp_mag (x, n) != MP_LT) { - return s_mp_sub (x, n, x); - } - - return MP_OKAY; -} -#endif - -/* End: bn_mp_montgomery_reduce.c */ - -/* Start: bn_mp_montgomery_setup.c */ -#include -#ifdef BN_MP_MONTGOMERY_SETUP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* setups the montgomery reduction stuff */ -int -mp_montgomery_setup (mp_int * n, mp_digit * rho) -{ - mp_digit x, b; - -/* fast inversion mod 2**k - * - * Based on the fact that - * - * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) - * => 2*X*A - X*X*A*A = 1 - * => 2*(1) - (1) = 1 - */ - b = n->dp[0]; - - if ((b & 1) == 0) { - return MP_VAL; - } - - x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */ - x *= 2 - b * x; /* here x*a==1 mod 2**8 */ -#if !defined(MP_8BIT) - x *= 2 - b * x; /* here x*a==1 mod 2**16 */ -#endif -#if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT)) - x *= 2 - b * x; /* here x*a==1 mod 2**32 */ -#endif -#ifdef MP_64BIT - x *= 2 - b * x; /* here x*a==1 mod 2**64 */ -#endif - - /* rho = -1/m mod b */ - *rho = (unsigned long)(((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK; - - return MP_OKAY; -} -#endif - -/* End: bn_mp_montgomery_setup.c */ - -/* Start: bn_mp_mul.c */ -#include -#ifdef BN_MP_MUL_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* high level multiplication (handles sign) */ -int mp_mul (mp_int * a, mp_int * b, mp_int * c) -{ - int res, neg; - neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; - - /* use Toom-Cook? */ -#ifdef BN_MP_TOOM_MUL_C - if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) { - res = mp_toom_mul(a, b, c); - } else -#endif -#ifdef BN_MP_KARATSUBA_MUL_C - /* use Karatsuba? */ - if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) { - res = mp_karatsuba_mul (a, b, c); - } else -#endif - { - /* can we use the fast multiplier? - * - * The fast multiplier can be used if the output will - * have less than MP_WARRAY digits and the number of - * digits won't affect carry propagation - */ - int digs = a->used + b->used + 1; - -#ifdef BN_FAST_S_MP_MUL_DIGS_C - if ((digs < MP_WARRAY) && - MIN(a->used, b->used) <= - (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - res = fast_s_mp_mul_digs (a, b, c, digs); - } else -#endif -#ifdef BN_S_MP_MUL_DIGS_C - res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */ -#else - res = MP_VAL; -#endif - - } - c->sign = (c->used > 0) ? neg : MP_ZPOS; - return res; -} -#endif - -/* End: bn_mp_mul.c */ - -/* Start: bn_mp_mul_2.c */ -#include -#ifdef BN_MP_MUL_2_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* b = a*2 */ -int mp_mul_2(mp_int * a, mp_int * b) -{ - int x, res, oldused; - - /* grow to accomodate result */ - if (b->alloc < a->used + 1) { - if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) { - return res; - } - } - - oldused = b->used; - b->used = a->used; - - { - register mp_digit r, rr, *tmpa, *tmpb; - - /* alias for source */ - tmpa = a->dp; - - /* alias for dest */ - tmpb = b->dp; - - /* carry */ - r = 0; - for (x = 0; x < a->used; x++) { - - /* get what will be the *next* carry bit from the - * MSB of the current digit - */ - rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1)); - - /* now shift up this digit, add in the carry [from the previous] */ - *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK; - - /* copy the carry that would be from the source - * digit into the next iteration - */ - r = rr; - } - - /* new leading digit? */ - if (r != 0) { - /* add a MSB which is always 1 at this point */ - *tmpb = 1; - ++(b->used); - } - - /* now zero any excess digits on the destination - * that we didn't write to - */ - tmpb = b->dp + b->used; - for (x = b->used; x < oldused; x++) { - *tmpb++ = 0; - } - } - b->sign = a->sign; - return MP_OKAY; -} -#endif - -/* End: bn_mp_mul_2.c */ - -/* Start: bn_mp_mul_2d.c */ -#include -#ifdef BN_MP_MUL_2D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* shift left by a certain bit count */ -int mp_mul_2d (mp_int * a, int b, mp_int * c) -{ - mp_digit d; - int res; - - /* copy */ - if (a != c) { - if ((res = mp_copy (a, c)) != MP_OKAY) { - return res; - } - } - - if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) { - if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) { - return res; - } - } - - /* shift by as many digits in the bit count */ - if (b >= (int)DIGIT_BIT) { - if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) { - return res; - } - } - - /* shift any bit count < DIGIT_BIT */ - d = (mp_digit) (b % DIGIT_BIT); - if (d != 0) { - register mp_digit *tmpc, shift, mask, r, rr; - register int x; - - /* bitmask for carries */ - mask = (((mp_digit)1) << d) - 1; - - /* shift for msbs */ - shift = DIGIT_BIT - d; - - /* alias */ - tmpc = c->dp; - - /* carry */ - r = 0; - for (x = 0; x < c->used; x++) { - /* get the higher bits of the current word */ - rr = (*tmpc >> shift) & mask; - - /* shift the current word and OR in the carry */ - *tmpc = ((*tmpc << d) | r) & MP_MASK; - ++tmpc; - - /* set the carry to the carry bits of the current word */ - r = rr; - } - - /* set final carry */ - if (r != 0) { - c->dp[(c->used)++] = r; - } - } - mp_clamp (c); - return MP_OKAY; -} -#endif - -/* End: bn_mp_mul_2d.c */ - -/* Start: bn_mp_mul_d.c */ -#include -#ifdef BN_MP_MUL_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* multiply by a digit */ -int -mp_mul_d (mp_int * a, mp_digit b, mp_int * c) -{ - mp_digit u, *tmpa, *tmpc; - mp_word r; - int ix, res, olduse; - - /* make sure c is big enough to hold a*b */ - if (c->alloc < a->used + 1) { - if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) { - return res; - } - } - - /* get the original destinations used count */ - olduse = c->used; - - /* set the sign */ - c->sign = a->sign; - - /* alias for a->dp [source] */ - tmpa = a->dp; - - /* alias for c->dp [dest] */ - tmpc = c->dp; - - /* zero carry */ - u = 0; - - /* compute columns */ - for (ix = 0; ix < a->used; ix++) { - /* compute product and carry sum for this term */ - r = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b); - - /* mask off higher bits to get a single digit */ - *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK)); - - /* send carry into next iteration */ - u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); - } - - /* store final carry [if any] and increment ix offset */ - *tmpc++ = u; - ++ix; - - /* now zero digits above the top */ - while (ix++ < olduse) { - *tmpc++ = 0; - } - - /* set used count */ - c->used = a->used + 1; - mp_clamp(c); - - return MP_OKAY; -} -#endif - -/* End: bn_mp_mul_d.c */ - -/* Start: bn_mp_mulmod.c */ -#include -#ifdef BN_MP_MULMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* d = a * b (mod c) */ -int mp_mulmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) -{ - int res; - mp_int t; - - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } - - if ((res = mp_mul (a, b, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - res = mp_mod (&t, c, d); - mp_clear (&t); - return res; -} -#endif - -/* End: bn_mp_mulmod.c */ - -/* Start: bn_mp_n_root.c */ -#include -#ifdef BN_MP_N_ROOT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* find the n'th root of an integer - * - * Result found such that (c)**b <= a and (c+1)**b > a - * - * This algorithm uses Newton's approximation - * x[i+1] = x[i] - f(x[i])/f'(x[i]) - * which will find the root in log(N) time where - * each step involves a fair bit. This is not meant to - * find huge roots [square and cube, etc]. - */ -int mp_n_root (mp_int * a, mp_digit b, mp_int * c) -{ - mp_int t1, t2, t3; - int res, neg; - - /* input must be positive if b is even */ - if ((b & 1) == 0 && a->sign == MP_NEG) { - return MP_VAL; - } - - if ((res = mp_init (&t1)) != MP_OKAY) { - return res; - } - - if ((res = mp_init (&t2)) != MP_OKAY) { - goto LBL_T1; - } - - if ((res = mp_init (&t3)) != MP_OKAY) { - goto LBL_T2; - } - - /* if a is negative fudge the sign but keep track */ - neg = a->sign; - a->sign = MP_ZPOS; - - /* t2 = 2 */ - mp_set (&t2, 2); - - do { - /* t1 = t2 */ - if ((res = mp_copy (&t2, &t1)) != MP_OKAY) { - goto LBL_T3; - } - - /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */ - - /* t3 = t1**(b-1) */ - if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) { - goto LBL_T3; - } - - /* numerator */ - /* t2 = t1**b */ - if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) { - goto LBL_T3; - } - - /* t2 = t1**b - a */ - if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) { - goto LBL_T3; - } - - /* denominator */ - /* t3 = t1**(b-1) * b */ - if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) { - goto LBL_T3; - } - - /* t3 = (t1**b - a)/(b * t1**(b-1)) */ - if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) { - goto LBL_T3; - } - - if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) { - goto LBL_T3; - } - } while (mp_cmp (&t1, &t2) != MP_EQ); - - /* result can be off by a few so check */ - for (;;) { - if ((res = mp_expt_d (&t1, b, &t2)) != MP_OKAY) { - goto LBL_T3; - } - - if (mp_cmp (&t2, a) == MP_GT) { - if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) { - goto LBL_T3; - } - } else { - break; - } - } - - /* reset the sign of a first */ - a->sign = neg; - - /* set the result */ - mp_exch (&t1, c); - - /* set the sign of the result */ - c->sign = neg; - - res = MP_OKAY; - -LBL_T3:mp_clear (&t3); -LBL_T2:mp_clear (&t2); -LBL_T1:mp_clear (&t1); - return res; -} -#endif - -/* End: bn_mp_n_root.c */ - -/* Start: bn_mp_neg.c */ -#include -#ifdef BN_MP_NEG_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* b = -a */ -int mp_neg (mp_int * a, mp_int * b) -{ - int res; - if (a != b) { - if ((res = mp_copy (a, b)) != MP_OKAY) { - return res; - } - } - - if (mp_iszero(b) != MP_YES) { - b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; - } else { - b->sign = MP_ZPOS; - } - - return MP_OKAY; -} -#endif - -/* End: bn_mp_neg.c */ - -/* Start: bn_mp_or.c */ -#include -#ifdef BN_MP_OR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* OR two ints together */ -int mp_or (mp_int * a, mp_int * b, mp_int * c) -{ - int res, ix, px; - mp_int t, *x; - - if (a->used > b->used) { - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - px = b->used; - x = b; - } else { - if ((res = mp_init_copy (&t, b)) != MP_OKAY) { - return res; - } - px = a->used; - x = a; - } - - for (ix = 0; ix < px; ix++) { - t.dp[ix] |= x->dp[ix]; - } - mp_clamp (&t); - mp_exch (c, &t); - mp_clear (&t); - return MP_OKAY; -} -#endif - -/* End: bn_mp_or.c */ - -/* Start: bn_mp_prime_fermat.c */ -#include -#ifdef BN_MP_PRIME_FERMAT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* performs one Fermat test. - * - * If "a" were prime then b**a == b (mod a) since the order of - * the multiplicative sub-group would be phi(a) = a-1. That means - * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a). - * - * Sets result to 1 if the congruence holds, or zero otherwise. - */ -int mp_prime_fermat (mp_int * a, mp_int * b, int *result) -{ - mp_int t; - int err; - - /* default to composite */ - *result = MP_NO; - - /* ensure b > 1 */ - if (mp_cmp_d(b, 1) != MP_GT) { - return MP_VAL; - } - - /* init t */ - if ((err = mp_init (&t)) != MP_OKAY) { - return err; - } - - /* compute t = b**a mod a */ - if ((err = mp_exptmod (b, a, a, &t)) != MP_OKAY) { - goto LBL_T; - } - - /* is it equal to b? */ - if (mp_cmp (&t, b) == MP_EQ) { - *result = MP_YES; - } - - err = MP_OKAY; -LBL_T:mp_clear (&t); - return err; -} -#endif - -/* End: bn_mp_prime_fermat.c */ - -/* Start: bn_mp_prime_is_divisible.c */ -#include -#ifdef BN_MP_PRIME_IS_DIVISIBLE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* determines if an integers is divisible by one - * of the first PRIME_SIZE primes or not - * - * sets result to 0 if not, 1 if yes - */ -int mp_prime_is_divisible (mp_int * a, int *result) -{ - int err, ix; - mp_digit res; - - /* default to not */ - *result = MP_NO; - - for (ix = 0; ix < PRIME_SIZE; ix++) { - /* what is a mod LBL_prime_tab[ix] */ - if ((err = mp_mod_d (a, ltm_prime_tab[ix], &res)) != MP_OKAY) { - return err; - } - - /* is the residue zero? */ - if (res == 0) { - *result = MP_YES; - return MP_OKAY; - } - } - - return MP_OKAY; -} -#endif - -/* End: bn_mp_prime_is_divisible.c */ - -/* Start: bn_mp_prime_is_prime.c */ -#include -#ifdef BN_MP_PRIME_IS_PRIME_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* performs a variable number of rounds of Miller-Rabin - * - * Probability of error after t rounds is no more than - - * - * Sets result to 1 if probably prime, 0 otherwise - */ -int mp_prime_is_prime (mp_int * a, int t, int *result) -{ - mp_int b; - int ix, err, res; - - /* default to no */ - *result = MP_NO; - - /* valid value of t? */ - if (t <= 0 || t > PRIME_SIZE) { - return MP_VAL; - } - - /* is the input equal to one of the primes in the table? */ - for (ix = 0; ix < PRIME_SIZE; ix++) { - if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) { - *result = 1; - return MP_OKAY; - } - } - - /* first perform trial division */ - if ((err = mp_prime_is_divisible (a, &res)) != MP_OKAY) { - return err; - } - - /* return if it was trivially divisible */ - if (res == MP_YES) { - return MP_OKAY; - } - - /* now perform the miller-rabin rounds */ - if ((err = mp_init (&b)) != MP_OKAY) { - return err; - } - - for (ix = 0; ix < t; ix++) { - /* set the prime */ - mp_set (&b, ltm_prime_tab[ix]); - - if ((err = mp_prime_miller_rabin (a, &b, &res)) != MP_OKAY) { - goto LBL_B; - } - - if (res == MP_NO) { - goto LBL_B; - } - } - - /* passed the test */ - *result = MP_YES; -LBL_B:mp_clear (&b); - return err; -} -#endif - -/* End: bn_mp_prime_is_prime.c */ - -/* Start: bn_mp_prime_miller_rabin.c */ -#include -#ifdef BN_MP_PRIME_MILLER_RABIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* Miller-Rabin test of "a" to the base of "b" as described in - * HAC pp. 139 Algorithm 4.24 - * - * Sets result to 0 if definitely composite or 1 if probably prime. - * Randomly the chance of error is no more than 1/4 and often - * very much lower. - */ -int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) -{ - mp_int n1, y, r; - int s, j, err; - - /* default */ - *result = MP_NO; - - /* ensure b > 1 */ - if (mp_cmp_d(b, 1) != MP_GT) { - return MP_VAL; - } - - /* get n1 = a - 1 */ - if ((err = mp_init_copy (&n1, a)) != MP_OKAY) { - return err; - } - if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) { - goto LBL_N1; - } - - /* set 2**s * r = n1 */ - if ((err = mp_init_copy (&r, &n1)) != MP_OKAY) { - goto LBL_N1; - } - - /* count the number of least significant bits - * which are zero - */ - s = mp_cnt_lsb(&r); - - /* now divide n - 1 by 2**s */ - if ((err = mp_div_2d (&r, s, &r, NULL)) != MP_OKAY) { - goto LBL_R; - } - - /* compute y = b**r mod a */ - if ((err = mp_init (&y)) != MP_OKAY) { - goto LBL_R; - } - if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) { - goto LBL_Y; - } - - /* if y != 1 and y != n1 do */ - if (mp_cmp_d (&y, 1) != MP_EQ && mp_cmp (&y, &n1) != MP_EQ) { - j = 1; - /* while j <= s-1 and y != n1 */ - while ((j <= (s - 1)) && mp_cmp (&y, &n1) != MP_EQ) { - if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) { - goto LBL_Y; - } - - /* if y == 1 then composite */ - if (mp_cmp_d (&y, 1) == MP_EQ) { - goto LBL_Y; - } - - ++j; - } - - /* if y != n1 then composite */ - if (mp_cmp (&y, &n1) != MP_EQ) { - goto LBL_Y; - } - } - - /* probably prime now */ - *result = MP_YES; -LBL_Y:mp_clear (&y); -LBL_R:mp_clear (&r); -LBL_N1:mp_clear (&n1); - return err; -} -#endif - -/* End: bn_mp_prime_miller_rabin.c */ - -/* Start: bn_mp_prime_next_prime.c */ -#include -#ifdef BN_MP_PRIME_NEXT_PRIME_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* finds the next prime after the number "a" using "t" trials - * of Miller-Rabin. - * - * bbs_style = 1 means the prime must be congruent to 3 mod 4 - */ -int mp_prime_next_prime(mp_int *a, int t, int bbs_style) -{ - int err, res, x, y; - mp_digit res_tab[PRIME_SIZE], step, kstep; - mp_int b; - - /* ensure t is valid */ - if (t <= 0 || t > PRIME_SIZE) { - return MP_VAL; - } - - /* force positive */ - a->sign = MP_ZPOS; - - /* simple algo if a is less than the largest prime in the table */ - if (mp_cmp_d(a, ltm_prime_tab[PRIME_SIZE-1]) == MP_LT) { - /* find which prime it is bigger than */ - for (x = PRIME_SIZE - 2; x >= 0; x--) { - if (mp_cmp_d(a, ltm_prime_tab[x]) != MP_LT) { - if (bbs_style == 1) { - /* ok we found a prime smaller or - * equal [so the next is larger] - * - * however, the prime must be - * congruent to 3 mod 4 - */ - if ((ltm_prime_tab[x + 1] & 3) != 3) { - /* scan upwards for a prime congruent to 3 mod 4 */ - for (y = x + 1; y < PRIME_SIZE; y++) { - if ((ltm_prime_tab[y] & 3) == 3) { - mp_set(a, ltm_prime_tab[y]); - return MP_OKAY; - } - } - } - } else { - mp_set(a, ltm_prime_tab[x + 1]); - return MP_OKAY; - } - } - } - /* at this point a maybe 1 */ - if (mp_cmp_d(a, 1) == MP_EQ) { - mp_set(a, 2); - return MP_OKAY; - } - /* fall through to the sieve */ - } - - /* generate a prime congruent to 3 mod 4 or 1/3 mod 4? */ - if (bbs_style == 1) { - kstep = 4; - } else { - kstep = 2; - } - - /* at this point we will use a combination of a sieve and Miller-Rabin */ - - if (bbs_style == 1) { - /* if a mod 4 != 3 subtract the correct value to make it so */ - if ((a->dp[0] & 3) != 3) { - if ((err = mp_sub_d(a, (a->dp[0] & 3) + 1, a)) != MP_OKAY) { return err; }; - } - } else { - if (mp_iseven(a) == 1) { - /* force odd */ - if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) { - return err; - } - } - } - - /* generate the restable */ - for (x = 1; x < PRIME_SIZE; x++) { - if ((err = mp_mod_d(a, ltm_prime_tab[x], res_tab + x)) != MP_OKAY) { - return err; - } - } - - /* init temp used for Miller-Rabin Testing */ - if ((err = mp_init(&b)) != MP_OKAY) { - return err; - } - - for (;;) { - /* skip to the next non-trivially divisible candidate */ - step = 0; - do { - /* y == 1 if any residue was zero [e.g. cannot be prime] */ - y = 0; - - /* increase step to next candidate */ - step += kstep; - - /* compute the new residue without using division */ - for (x = 1; x < PRIME_SIZE; x++) { - /* add the step to each residue */ - res_tab[x] += kstep; - - /* subtract the modulus [instead of using division] */ - if (res_tab[x] >= ltm_prime_tab[x]) { - res_tab[x] -= ltm_prime_tab[x]; - } - - /* set flag if zero */ - if (res_tab[x] == 0) { - y = 1; - } - } - } while (y == 1 && step < ((((mp_digit)1)<= ((((mp_digit)1)< -#ifdef BN_MP_PRIME_RABIN_MILLER_TRIALS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - - -static const struct { - int k, t; -} sizes[] = { -{ 128, 28 }, -{ 256, 16 }, -{ 384, 10 }, -{ 512, 7 }, -{ 640, 6 }, -{ 768, 5 }, -{ 896, 4 }, -{ 1024, 4 } -}; - -/* returns # of RM trials required for a given bit size */ -int mp_prime_rabin_miller_trials(int size) -{ - int x; - - for (x = 0; x < (int)(sizeof(sizes)/(sizeof(sizes[0]))); x++) { - if (sizes[x].k == size) { - return sizes[x].t; - } else if (sizes[x].k > size) { - return (x == 0) ? sizes[0].t : sizes[x - 1].t; - } - } - return sizes[x-1].t + 1; -} - - -#endif - -/* End: bn_mp_prime_rabin_miller_trials.c */ - -/* Start: bn_mp_prime_random_ex.c */ -#include -#ifdef BN_MP_PRIME_RANDOM_EX_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* makes a truly random prime of a given size (bits), - * - * Flags are as follows: - * - * LTM_PRIME_BBS - make prime congruent to 3 mod 4 - * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS) - * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero - * LTM_PRIME_2MSB_ON - make the 2nd highest bit one - * - * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can - * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself - * so it can be NULL - * - */ - -/* This is possibly the mother of all prime generation functions, muahahahahaha! */ -int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat) -{ - unsigned char *tmp, maskAND, maskOR_msb, maskOR_lsb; - int res, err, bsize, maskOR_msb_offset; - - /* sanity check the input */ - if (size <= 1 || t <= 0) { - return MP_VAL; - } - - /* LTM_PRIME_SAFE implies LTM_PRIME_BBS */ - if (flags & LTM_PRIME_SAFE) { - flags |= LTM_PRIME_BBS; - } - - /* calc the byte size */ - bsize = (size>>3) + ((size&7)?1:0); - - /* we need a buffer of bsize bytes */ - tmp = OPT_CAST(unsigned char) XMALLOC(bsize); - if (tmp == NULL) { - return MP_MEM; - } - - /* calc the maskAND value for the MSbyte*/ - maskAND = ((size&7) == 0) ? 0xFF : (0xFF >> (8 - (size & 7))); - - /* calc the maskOR_msb */ - maskOR_msb = 0; - maskOR_msb_offset = ((size & 7) == 1) ? 1 : 0; - if (flags & LTM_PRIME_2MSB_ON) { - maskOR_msb |= 0x80 >> ((9 - size) & 7); - } - - /* get the maskOR_lsb */ - maskOR_lsb = 1; - if (flags & LTM_PRIME_BBS) { - maskOR_lsb |= 3; - } - - do { - /* read the bytes */ - if (cb(tmp, bsize, dat) != bsize) { - err = MP_VAL; - goto error; - } - - /* work over the MSbyte */ - tmp[0] &= maskAND; - tmp[0] |= 1 << ((size - 1) & 7); - - /* mix in the maskORs */ - tmp[maskOR_msb_offset] |= maskOR_msb; - tmp[bsize-1] |= maskOR_lsb; - - /* read it in */ - if ((err = mp_read_unsigned_bin(a, tmp, bsize)) != MP_OKAY) { goto error; } - - /* is it prime? */ - if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { goto error; } - if (res == MP_NO) { - continue; - } - - if (flags & LTM_PRIME_SAFE) { - /* see if (a-1)/2 is prime */ - if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) { goto error; } - if ((err = mp_div_2(a, a)) != MP_OKAY) { goto error; } - - /* is it prime? */ - if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { goto error; } - } - } while (res == MP_NO); - - if (flags & LTM_PRIME_SAFE) { - /* restore a to the original value */ - if ((err = mp_mul_2(a, a)) != MP_OKAY) { goto error; } - if ((err = mp_add_d(a, 1, a)) != MP_OKAY) { goto error; } - } - - err = MP_OKAY; -error: - XFREE(tmp); - return err; -} - - -#endif - -/* End: bn_mp_prime_random_ex.c */ - -/* Start: bn_mp_radix_size.c */ -#include -#ifdef BN_MP_RADIX_SIZE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* returns size of ASCII reprensentation */ -int mp_radix_size (mp_int * a, int radix, int *size) -{ - int res, digs; - mp_int t; - mp_digit d; - - *size = 0; - - /* special case for binary */ - if (radix == 2) { - *size = mp_count_bits (a) + (a->sign == MP_NEG ? 1 : 0) + 1; - return MP_OKAY; - } - - /* make sure the radix is in range */ - if (radix < 2 || radix > 64) { - return MP_VAL; - } - - if (mp_iszero(a) == MP_YES) { - *size = 2; - return MP_OKAY; - } - - /* digs is the digit count */ - digs = 0; - - /* if it's negative add one for the sign */ - if (a->sign == MP_NEG) { - ++digs; - } - - /* init a copy of the input */ - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - - /* force temp to positive */ - t.sign = MP_ZPOS; - - /* fetch out all of the digits */ - while (mp_iszero (&t) == MP_NO) { - if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { - mp_clear (&t); - return res; - } - ++digs; - } - mp_clear (&t); - - /* return digs + 1, the 1 is for the NULL byte that would be required. */ - *size = digs + 1; - return MP_OKAY; -} - -#endif - -/* End: bn_mp_radix_size.c */ - -/* Start: bn_mp_radix_smap.c */ -#include -#ifdef BN_MP_RADIX_SMAP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* chars used in radix conversions */ -const char *mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/"; -#endif - -/* End: bn_mp_radix_smap.c */ - -/* Start: bn_mp_rand.c */ -#include -#ifdef BN_MP_RAND_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* makes a pseudo-random int of a given size */ -int -mp_rand (mp_int * a, int digits) -{ - int res; - mp_digit d; - - mp_zero (a); - if (digits <= 0) { - return MP_OKAY; - } - - /* first place a random non-zero digit */ - do { - d = ((mp_digit) abs (rand ())) & MP_MASK; - } while (d == 0); - - if ((res = mp_add_d (a, d, a)) != MP_OKAY) { - return res; - } - - while (--digits > 0) { - if ((res = mp_lshd (a, 1)) != MP_OKAY) { - return res; - } - - if ((res = mp_add_d (a, ((mp_digit) abs (rand ())), a)) != MP_OKAY) { - return res; - } - } - - return MP_OKAY; -} -#endif - -/* End: bn_mp_rand.c */ - -/* Start: bn_mp_read_radix.c */ -#include -#ifdef BN_MP_READ_RADIX_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* read a string [ASCII] in a given radix */ -int mp_read_radix (mp_int * a, const char *str, int radix) -{ - int y, res, neg; - char ch; - - /* zero the digit bignum */ - mp_zero(a); - - /* make sure the radix is ok */ - if (radix < 2 || radix > 64) { - return MP_VAL; - } - - /* if the leading digit is a - * minus set the sign to negative. - */ - if (*str == '-') { - ++str; - neg = MP_NEG; - } else { - neg = MP_ZPOS; - } - - /* set the integer to the default of zero */ - mp_zero (a); - - /* process each digit of the string */ - while (*str) { - /* if the radix < 36 the conversion is case insensitive - * this allows numbers like 1AB and 1ab to represent the same value - * [e.g. in hex] - */ - ch = (char) ((radix < 36) ? toupper (*str) : *str); - for (y = 0; y < 64; y++) { - if (ch == mp_s_rmap[y]) { - break; - } - } - - /* if the char was found in the map - * and is less than the given radix add it - * to the number, otherwise exit the loop. - */ - if (y < radix) { - if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) { - return res; - } - if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) { - return res; - } - } else { - break; - } - ++str; - } - - /* set the sign only if a != 0 */ - if (mp_iszero(a) != 1) { - a->sign = neg; - } - return MP_OKAY; -} -#endif - -/* End: bn_mp_read_radix.c */ - -/* Start: bn_mp_read_signed_bin.c */ -#include -#ifdef BN_MP_READ_SIGNED_BIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* read signed bin, big endian, first byte is 0==positive or 1==negative */ -int mp_read_signed_bin (mp_int * a, const unsigned char *b, int c) -{ - int res; - - /* read magnitude */ - if ((res = mp_read_unsigned_bin (a, b + 1, c - 1)) != MP_OKAY) { - return res; - } - - /* first byte is 0 for positive, non-zero for negative */ - if (b[0] == 0) { - a->sign = MP_ZPOS; - } else { - a->sign = MP_NEG; - } - - return MP_OKAY; -} -#endif - -/* End: bn_mp_read_signed_bin.c */ - -/* Start: bn_mp_read_unsigned_bin.c */ -#include -#ifdef BN_MP_READ_UNSIGNED_BIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* reads a unsigned char array, assumes the msb is stored first [big endian] */ -int mp_read_unsigned_bin (mp_int * a, const unsigned char *b, int c) -{ - int res; - - /* make sure there are at least two digits */ - if (a->alloc < 2) { - if ((res = mp_grow(a, 2)) != MP_OKAY) { - return res; - } - } - - /* zero the int */ - mp_zero (a); - - /* read the bytes in */ - while (c-- > 0) { - if ((res = mp_mul_2d (a, 8, a)) != MP_OKAY) { - return res; - } - -#ifndef MP_8BIT - a->dp[0] |= *b++; - a->used += 1; -#else - a->dp[0] = (*b & MP_MASK); - a->dp[1] |= ((*b++ >> 7U) & 1); - a->used += 2; -#endif - } - mp_clamp (a); - return MP_OKAY; -} -#endif - -/* End: bn_mp_read_unsigned_bin.c */ - -/* Start: bn_mp_reduce.c */ -#include -#ifdef BN_MP_REDUCE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* reduces x mod m, assumes 0 < x < m**2, mu is - * precomputed via mp_reduce_setup. - * From HAC pp.604 Algorithm 14.42 - */ -int mp_reduce (mp_int * x, mp_int * m, mp_int * mu) -{ - mp_int q; - int res, um = m->used; - - /* q = x */ - if ((res = mp_init_copy (&q, x)) != MP_OKAY) { - return res; - } - - /* q1 = x / b**(k-1) */ - mp_rshd (&q, um - 1); - - /* according to HAC this optimization is ok */ - if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) { - if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) { - goto CLEANUP; - } - } else { -#ifdef BN_S_MP_MUL_HIGH_DIGS_C - if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) { - goto CLEANUP; - } -#elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) - if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) { - goto CLEANUP; - } -#else - { - res = MP_VAL; - goto CLEANUP; - } -#endif - } - - /* q3 = q2 / b**(k+1) */ - mp_rshd (&q, um + 1); - - /* x = x mod b**(k+1), quick (no division) */ - if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) { - goto CLEANUP; - } - - /* q = q * m mod b**(k+1), quick (no division) */ - if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) { - goto CLEANUP; - } - - /* x = x - q */ - if ((res = mp_sub (x, &q, x)) != MP_OKAY) { - goto CLEANUP; - } - - /* If x < 0, add b**(k+1) to it */ - if (mp_cmp_d (x, 0) == MP_LT) { - mp_set (&q, 1); - if ((res = mp_lshd (&q, um + 1)) != MP_OKAY) - goto CLEANUP; - if ((res = mp_add (x, &q, x)) != MP_OKAY) - goto CLEANUP; - } - - /* Back off if it's too big */ - while (mp_cmp (x, m) != MP_LT) { - if ((res = s_mp_sub (x, m, x)) != MP_OKAY) { - goto CLEANUP; - } - } - -CLEANUP: - mp_clear (&q); - - return res; -} -#endif - -/* End: bn_mp_reduce.c */ - -/* Start: bn_mp_reduce_2k.c */ -#include -#ifdef BN_MP_REDUCE_2K_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* reduces a modulo n where n is of the form 2**p - d */ -int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d) -{ - mp_int q; - int p, res; - - if ((res = mp_init(&q)) != MP_OKAY) { - return res; - } - - p = mp_count_bits(n); -top: - /* q = a/2**p, a = a mod 2**p */ - if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) { - goto ERR; - } - - if (d != 1) { - /* q = q * d */ - if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) { - goto ERR; - } - } - - /* a = a + q */ - if ((res = s_mp_add(a, &q, a)) != MP_OKAY) { - goto ERR; - } - - if (mp_cmp_mag(a, n) != MP_LT) { - s_mp_sub(a, n, a); - goto top; - } - -ERR: - mp_clear(&q); - return res; -} - -#endif - -/* End: bn_mp_reduce_2k.c */ - -/* Start: bn_mp_reduce_2k_l.c */ -#include -#ifdef BN_MP_REDUCE_2K_L_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* reduces a modulo n where n is of the form 2**p - d - This differs from reduce_2k since "d" can be larger - than a single digit. -*/ -int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d) -{ - mp_int q; - int p, res; - - if ((res = mp_init(&q)) != MP_OKAY) { - return res; - } - - p = mp_count_bits(n); -top: - /* q = a/2**p, a = a mod 2**p */ - if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) { - goto ERR; - } - - /* q = q * d */ - if ((res = mp_mul(&q, d, &q)) != MP_OKAY) { - goto ERR; - } - - /* a = a + q */ - if ((res = s_mp_add(a, &q, a)) != MP_OKAY) { - goto ERR; - } - - if (mp_cmp_mag(a, n) != MP_LT) { - s_mp_sub(a, n, a); - goto top; - } - -ERR: - mp_clear(&q); - return res; -} - -#endif - -/* End: bn_mp_reduce_2k_l.c */ - -/* Start: bn_mp_reduce_2k_setup.c */ -#include -#ifdef BN_MP_REDUCE_2K_SETUP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* determines the setup value */ -int mp_reduce_2k_setup(mp_int *a, mp_digit *d) -{ - int res, p; - mp_int tmp; - - if ((res = mp_init(&tmp)) != MP_OKAY) { - return res; - } - - p = mp_count_bits(a); - if ((res = mp_2expt(&tmp, p)) != MP_OKAY) { - mp_clear(&tmp); - return res; - } - - if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) { - mp_clear(&tmp); - return res; - } - - *d = tmp.dp[0]; - mp_clear(&tmp); - return MP_OKAY; -} -#endif - -/* End: bn_mp_reduce_2k_setup.c */ - -/* Start: bn_mp_reduce_2k_setup_l.c */ -#include -#ifdef BN_MP_REDUCE_2K_SETUP_L_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* determines the setup value */ -int mp_reduce_2k_setup_l(mp_int *a, mp_int *d) -{ - int res; - mp_int tmp; - - if ((res = mp_init(&tmp)) != MP_OKAY) { - return res; - } - - if ((res = mp_2expt(&tmp, mp_count_bits(a))) != MP_OKAY) { - goto ERR; - } - - if ((res = s_mp_sub(&tmp, a, d)) != MP_OKAY) { - goto ERR; - } - -ERR: - mp_clear(&tmp); - return res; -} -#endif - -/* End: bn_mp_reduce_2k_setup_l.c */ - -/* Start: bn_mp_reduce_is_2k.c */ -#include -#ifdef BN_MP_REDUCE_IS_2K_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* determines if mp_reduce_2k can be used */ -int mp_reduce_is_2k(mp_int *a) -{ - int ix, iy, iw; - mp_digit iz; - - if (a->used == 0) { - return MP_NO; - } else if (a->used == 1) { - return MP_YES; - } else if (a->used > 1) { - iy = mp_count_bits(a); - iz = 1; - iw = 1; - - /* Test every bit from the second digit up, must be 1 */ - for (ix = DIGIT_BIT; ix < iy; ix++) { - if ((a->dp[iw] & iz) == 0) { - return MP_NO; - } - iz <<= 1; - if (iz > (mp_digit)MP_MASK) { - ++iw; - iz = 1; - } - } - } - return MP_YES; -} - -#endif - -/* End: bn_mp_reduce_is_2k.c */ - -/* Start: bn_mp_reduce_is_2k_l.c */ -#include -#ifdef BN_MP_REDUCE_IS_2K_L_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* determines if reduce_2k_l can be used */ -int mp_reduce_is_2k_l(mp_int *a) -{ - int ix, iy; - - if (a->used == 0) { - return MP_NO; - } else if (a->used == 1) { - return MP_YES; - } else if (a->used > 1) { - /* if more than half of the digits are -1 we're sold */ - for (iy = ix = 0; ix < a->used; ix++) { - if (a->dp[ix] == MP_MASK) { - ++iy; - } - } - return (iy >= (a->used/2)) ? MP_YES : MP_NO; - - } - return MP_NO; -} - -#endif - -/* End: bn_mp_reduce_is_2k_l.c */ - -/* Start: bn_mp_reduce_setup.c */ -#include -#ifdef BN_MP_REDUCE_SETUP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* pre-calculate the value required for Barrett reduction - * For a given modulus "b" it calulates the value required in "a" - */ -int mp_reduce_setup (mp_int * a, mp_int * b) -{ - int res; - - if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) { - return res; - } - return mp_div (a, b, a, NULL); -} -#endif - -/* End: bn_mp_reduce_setup.c */ - -/* Start: bn_mp_rshd.c */ -#include -#ifdef BN_MP_RSHD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* shift right a certain amount of digits */ -void mp_rshd (mp_int * a, int b) -{ - int x; - - /* if b <= 0 then ignore it */ - if (b <= 0) { - return; - } - - /* if b > used then simply zero it and return */ - if (a->used <= b) { - mp_zero (a); - return; - } - - { - register mp_digit *bottom, *top; - - /* shift the digits down */ - - /* bottom */ - bottom = a->dp; - - /* top [offset into digits] */ - top = a->dp + b; - - /* this is implemented as a sliding window where - * the window is b-digits long and digits from - * the top of the window are copied to the bottom - * - * e.g. - - b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> - /\ | ----> - \-------------------/ ----> - */ - for (x = 0; x < (a->used - b); x++) { - *bottom++ = *top++; - } - - /* zero the top digits */ - for (; x < a->used; x++) { - *bottom++ = 0; - } - } - - /* remove excess digits */ - a->used -= b; -} -#endif - -/* End: bn_mp_rshd.c */ - -/* Start: bn_mp_set.c */ -#include -#ifdef BN_MP_SET_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* set to a digit */ -void mp_set (mp_int * a, mp_digit b) -{ - mp_zero (a); - a->dp[0] = b & MP_MASK; - a->used = (a->dp[0] != 0) ? 1 : 0; -} -#endif - -/* End: bn_mp_set.c */ - -/* Start: bn_mp_set_int.c */ -#include -#ifdef BN_MP_SET_INT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* set a 32-bit const */ -int mp_set_int (mp_int * a, unsigned long b) -{ - int x, res; - - mp_zero (a); - - /* set four bits at a time */ - for (x = 0; x < 8; x++) { - /* shift the number up four bits */ - if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) { - return res; - } - - /* OR in the top four bits of the source */ - a->dp[0] |= (b >> 28) & 15; - - /* shift the source up to the next four bits */ - b <<= 4; - - /* ensure that digits are not clamped off */ - a->used += 1; - } - mp_clamp (a); - return MP_OKAY; -} -#endif - -/* End: bn_mp_set_int.c */ - -/* Start: bn_mp_shrink.c */ -#include -#ifdef BN_MP_SHRINK_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* shrink a bignum */ -int mp_shrink (mp_int * a) -{ - mp_digit *tmp; - int used = 1; - - if(a->used > 0) - used = a->used; - - if (a->alloc != used) { - if ((tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * used)) == NULL) { - return MP_MEM; - } - a->dp = tmp; - a->alloc = used; - } - return MP_OKAY; -} -#endif - -/* End: bn_mp_shrink.c */ - -/* Start: bn_mp_signed_bin_size.c */ -#include -#ifdef BN_MP_SIGNED_BIN_SIZE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* get the size for an signed equivalent */ -int mp_signed_bin_size (mp_int * a) -{ - return 1 + mp_unsigned_bin_size (a); -} -#endif - -/* End: bn_mp_signed_bin_size.c */ - -/* Start: bn_mp_sqr.c */ -#include -#ifdef BN_MP_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* computes b = a*a */ -int -mp_sqr (mp_int * a, mp_int * b) -{ - int res; - -#ifdef BN_MP_TOOM_SQR_C - /* use Toom-Cook? */ - if (a->used >= TOOM_SQR_CUTOFF) { - res = mp_toom_sqr(a, b); - /* Karatsuba? */ - } else -#endif -#ifdef BN_MP_KARATSUBA_SQR_C -if (a->used >= KARATSUBA_SQR_CUTOFF) { - res = mp_karatsuba_sqr (a, b); - } else -#endif - { -#ifdef BN_FAST_S_MP_SQR_C - /* can we use the fast comba multiplier? */ - if ((a->used * 2 + 1) < MP_WARRAY && - a->used < - (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) { - res = fast_s_mp_sqr (a, b); - } else -#endif -#ifdef BN_S_MP_SQR_C - res = s_mp_sqr (a, b); -#else - res = MP_VAL; -#endif - } - b->sign = MP_ZPOS; - return res; -} -#endif - -/* End: bn_mp_sqr.c */ - -/* Start: bn_mp_sqrmod.c */ -#include -#ifdef BN_MP_SQRMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* c = a * a (mod b) */ -int -mp_sqrmod (mp_int * a, mp_int * b, mp_int * c) -{ - int res; - mp_int t; - - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } - - if ((res = mp_sqr (a, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - res = mp_mod (&t, b, c); - mp_clear (&t); - return res; -} -#endif - -/* End: bn_mp_sqrmod.c */ - -/* Start: bn_mp_sqrt.c */ -#include - -#ifdef BN_MP_SQRT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* this function is less generic than mp_n_root, simpler and faster */ -int mp_sqrt(mp_int *arg, mp_int *ret) -{ - int res; - mp_int t1,t2; - - /* must be positive */ - if (arg->sign == MP_NEG) { - return MP_VAL; - } - - /* easy out */ - if (mp_iszero(arg) == MP_YES) { - mp_zero(ret); - return MP_OKAY; - } - - if ((res = mp_init_copy(&t1, arg)) != MP_OKAY) { - return res; - } - - if ((res = mp_init(&t2)) != MP_OKAY) { - goto E2; - } - - /* First approx. (not very bad for large arg) */ - mp_rshd (&t1,t1.used/2); - - /* t1 > 0 */ - if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) { - goto E1; - } - if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) { - goto E1; - } - if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) { - goto E1; - } - /* And now t1 > sqrt(arg) */ - do { - if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) { - goto E1; - } - if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) { - goto E1; - } - if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) { - goto E1; - } - /* t1 >= sqrt(arg) >= t2 at this point */ - } while (mp_cmp_mag(&t1,&t2) == MP_GT); - - mp_exch(&t1,ret); - -E1: mp_clear(&t2); -E2: mp_clear(&t1); - return res; -} - -#endif - -/* End: bn_mp_sqrt.c */ - -/* Start: bn_mp_sub.c */ -#include -#ifdef BN_MP_SUB_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* high level subtraction (handles signs) */ -int -mp_sub (mp_int * a, mp_int * b, mp_int * c) -{ - int sa, sb, res; - - sa = a->sign; - sb = b->sign; - - if (sa != sb) { - /* subtract a negative from a positive, OR */ - /* subtract a positive from a negative. */ - /* In either case, ADD their magnitudes, */ - /* and use the sign of the first number. */ - c->sign = sa; - res = s_mp_add (a, b, c); - } else { - /* subtract a positive from a positive, OR */ - /* subtract a negative from a negative. */ - /* First, take the difference between their */ - /* magnitudes, then... */ - if (mp_cmp_mag (a, b) != MP_LT) { - /* Copy the sign from the first */ - c->sign = sa; - /* The first has a larger or equal magnitude */ - res = s_mp_sub (a, b, c); - } else { - /* The result has the *opposite* sign from */ - /* the first number. */ - c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; - /* The second has a larger magnitude */ - res = s_mp_sub (b, a, c); - } - } - return res; -} - -#endif - -/* End: bn_mp_sub.c */ - -/* Start: bn_mp_sub_d.c */ -#include -#ifdef BN_MP_SUB_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* single digit subtraction */ -int -mp_sub_d (mp_int * a, mp_digit b, mp_int * c) -{ - mp_digit *tmpa, *tmpc, mu; - int res, ix, oldused; - - /* grow c as required */ - if (c->alloc < a->used + 1) { - if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) { - return res; - } - } - - /* if a is negative just do an unsigned - * addition [with fudged signs] - */ - if (a->sign == MP_NEG) { - a->sign = MP_ZPOS; - res = mp_add_d(a, b, c); - a->sign = c->sign = MP_NEG; - - /* clamp */ - mp_clamp(c); - - return res; - } - - /* setup regs */ - oldused = c->used; - tmpa = a->dp; - tmpc = c->dp; - - /* if a <= b simply fix the single digit */ - if ((a->used == 1 && a->dp[0] <= b) || a->used == 0) { - if (a->used == 1) { - *tmpc++ = b - *tmpa; - } else { - *tmpc++ = b; - } - ix = 1; - - /* negative/1digit */ - c->sign = MP_NEG; - c->used = 1; - } else { - /* positive/size */ - c->sign = MP_ZPOS; - c->used = a->used; - - /* subtract first digit */ - *tmpc = *tmpa++ - b; - mu = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1); - *tmpc++ &= MP_MASK; - - /* handle rest of the digits */ - for (ix = 1; ix < a->used; ix++) { - *tmpc = *tmpa++ - mu; - mu = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1); - *tmpc++ &= MP_MASK; - } - } - - /* zero excess digits */ - while (ix++ < oldused) { - *tmpc++ = 0; - } - mp_clamp(c); - return MP_OKAY; -} - -#endif - -/* End: bn_mp_sub_d.c */ - -/* Start: bn_mp_submod.c */ -#include -#ifdef BN_MP_SUBMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* d = a - b (mod c) */ -int -mp_submod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) -{ - int res; - mp_int t; - - - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } - - if ((res = mp_sub (a, b, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - res = mp_mod (&t, c, d); - mp_clear (&t); - return res; -} -#endif - -/* End: bn_mp_submod.c */ - -/* Start: bn_mp_to_signed_bin.c */ -#include -#ifdef BN_MP_TO_SIGNED_BIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* store in signed [big endian] format */ -int mp_to_signed_bin (mp_int * a, unsigned char *b) -{ - int res; - - if ((res = mp_to_unsigned_bin (a, b + 1)) != MP_OKAY) { - return res; - } - b[0] = (unsigned char) ((a->sign == MP_ZPOS) ? 0 : 1); - return MP_OKAY; -} -#endif - -/* End: bn_mp_to_signed_bin.c */ - -/* Start: bn_mp_to_signed_bin_n.c */ -#include -#ifdef BN_MP_TO_SIGNED_BIN_N_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* store in signed [big endian] format */ -int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen) -{ - if (*outlen < (unsigned long)mp_signed_bin_size(a)) { - return MP_VAL; - } - *outlen = mp_signed_bin_size(a); - return mp_to_signed_bin(a, b); -} -#endif - -/* End: bn_mp_to_signed_bin_n.c */ - -/* Start: bn_mp_to_unsigned_bin.c */ -#include -#ifdef BN_MP_TO_UNSIGNED_BIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* store in unsigned [big endian] format */ -int mp_to_unsigned_bin (mp_int * a, unsigned char *b) -{ - int x, res; - mp_int t; - - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - - x = 0; - while (mp_iszero (&t) == 0) { -#ifndef MP_8BIT - b[x++] = (unsigned char) (t.dp[0] & 255); -#else - b[x++] = (unsigned char) (t.dp[0] | ((t.dp[1] & 0x01) << 7)); -#endif - if ((res = mp_div_2d (&t, 8, &t, NULL)) != MP_OKAY) { - mp_clear (&t); - return res; - } - } - bn_reverse (b, x); - mp_clear (&t); - return MP_OKAY; -} -#endif - -/* End: bn_mp_to_unsigned_bin.c */ - -/* Start: bn_mp_to_unsigned_bin_n.c */ -#include -#ifdef BN_MP_TO_UNSIGNED_BIN_N_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* store in unsigned [big endian] format */ -int mp_to_unsigned_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen) -{ - if (*outlen < (unsigned long)mp_unsigned_bin_size(a)) { - return MP_VAL; - } - *outlen = mp_unsigned_bin_size(a); - return mp_to_unsigned_bin(a, b); -} -#endif - -/* End: bn_mp_to_unsigned_bin_n.c */ - -/* Start: bn_mp_toom_mul.c */ -#include -#ifdef BN_MP_TOOM_MUL_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* multiplication using the Toom-Cook 3-way algorithm - * - * Much more complicated than Karatsuba but has a lower - * asymptotic running time of O(N**1.464). This algorithm is - * only particularly useful on VERY large inputs - * (we're talking 1000s of digits here...). -*/ -int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c) -{ - mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2; - int res, B; - - /* init temps */ - if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, - &a0, &a1, &a2, &b0, &b1, - &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) { - return res; - } - - /* B */ - B = MIN(a->used, b->used) / 3; - - /* a = a2 * B**2 + a1 * B + a0 */ - if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_copy(a, &a1)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&a1, B); - mp_mod_2d(&a1, DIGIT_BIT * B, &a1); - - if ((res = mp_copy(a, &a2)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&a2, B*2); - - /* b = b2 * B**2 + b1 * B + b0 */ - if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_copy(b, &b1)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&b1, B); - mp_mod_2d(&b1, DIGIT_BIT * B, &b1); - - if ((res = mp_copy(b, &b2)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&b2, B*2); - - /* w0 = a0*b0 */ - if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) { - goto ERR; - } - - /* w4 = a2 * b2 */ - if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) { - goto ERR; - } - - /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */ - if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) { - goto ERR; - } - - /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */ - if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) { - goto ERR; - } - - - /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */ - if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) { - goto ERR; - } - - /* now solve the matrix - - 0 0 0 0 1 - 1 2 4 8 16 - 1 1 1 1 1 - 16 8 4 2 1 - 1 0 0 0 0 - - using 12 subtractions, 4 shifts, - 2 small divisions and 1 small multiplication - */ - - /* r1 - r4 */ - if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r0 */ - if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1/2 */ - if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3/2 */ - if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) { - goto ERR; - } - /* r2 - r0 - r4 */ - if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) { - goto ERR; - } - /* r1 - r2 */ - if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r2 */ - if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1 - 8r0 */ - if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - 8r4 */ - if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) { - goto ERR; - } - /* 3r2 - r1 - r3 */ - if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) { - goto ERR; - } - /* r1 - r2 */ - if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r2 */ - if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1/3 */ - if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) { - goto ERR; - } - /* r3/3 */ - if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) { - goto ERR; - } - - /* at this point shift W[n] by B*n */ - if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) { - goto ERR; - } - -ERR: - mp_clear_multi(&w0, &w1, &w2, &w3, &w4, - &a0, &a1, &a2, &b0, &b1, - &b2, &tmp1, &tmp2, NULL); - return res; -} - -#endif - -/* End: bn_mp_toom_mul.c */ - -/* Start: bn_mp_toom_sqr.c */ -#include -#ifdef BN_MP_TOOM_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* squaring using Toom-Cook 3-way algorithm */ -int -mp_toom_sqr(mp_int *a, mp_int *b) -{ - mp_int w0, w1, w2, w3, w4, tmp1, a0, a1, a2; - int res, B; - - /* init temps */ - if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL)) != MP_OKAY) { - return res; - } - - /* B */ - B = a->used / 3; - - /* a = a2 * B**2 + a1 * B + a0 */ - if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_copy(a, &a1)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&a1, B); - mp_mod_2d(&a1, DIGIT_BIT * B, &a1); - - if ((res = mp_copy(a, &a2)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&a2, B*2); - - /* w0 = a0*a0 */ - if ((res = mp_sqr(&a0, &w0)) != MP_OKAY) { - goto ERR; - } - - /* w4 = a2 * a2 */ - if ((res = mp_sqr(&a2, &w4)) != MP_OKAY) { - goto ERR; - } - - /* w1 = (a2 + 2(a1 + 2a0))**2 */ - if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_sqr(&tmp1, &w1)) != MP_OKAY) { - goto ERR; - } - - /* w3 = (a0 + 2(a1 + 2a2))**2 */ - if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_sqr(&tmp1, &w3)) != MP_OKAY) { - goto ERR; - } - - - /* w2 = (a2 + a1 + a0)**2 */ - if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sqr(&tmp1, &w2)) != MP_OKAY) { - goto ERR; - } - - /* now solve the matrix - - 0 0 0 0 1 - 1 2 4 8 16 - 1 1 1 1 1 - 16 8 4 2 1 - 1 0 0 0 0 - - using 12 subtractions, 4 shifts, 2 small divisions and 1 small multiplication. - */ - - /* r1 - r4 */ - if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r0 */ - if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1/2 */ - if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3/2 */ - if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) { - goto ERR; - } - /* r2 - r0 - r4 */ - if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) { - goto ERR; - } - /* r1 - r2 */ - if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r2 */ - if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1 - 8r0 */ - if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - 8r4 */ - if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) { - goto ERR; - } - /* 3r2 - r1 - r3 */ - if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) { - goto ERR; - } - /* r1 - r2 */ - if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r2 */ - if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1/3 */ - if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) { - goto ERR; - } - /* r3/3 */ - if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) { - goto ERR; - } - - /* at this point shift W[n] by B*n */ - if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_add(&w0, &w1, b)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, b, b)) != MP_OKAY) { - goto ERR; - } - -ERR: - mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL); - return res; -} - -#endif - -/* End: bn_mp_toom_sqr.c */ - -/* Start: bn_mp_toradix.c */ -#include -#ifdef BN_MP_TORADIX_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* stores a bignum as a ASCII string in a given radix (2..64) */ -int mp_toradix (mp_int * a, char *str, int radix) -{ - int res, digs; - mp_int t; - mp_digit d; - char *_s = str; - - /* check range of the radix */ - if (radix < 2 || radix > 64) { - return MP_VAL; - } - - /* quick out if its zero */ - if (mp_iszero(a) == 1) { - *str++ = '0'; - *str = '\0'; - return MP_OKAY; - } - - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - - /* if it is negative output a - */ - if (t.sign == MP_NEG) { - ++_s; - *str++ = '-'; - t.sign = MP_ZPOS; - } - - digs = 0; - while (mp_iszero (&t) == 0) { - if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { - mp_clear (&t); - return res; - } - *str++ = mp_s_rmap[d]; - ++digs; - } - - /* reverse the digits of the string. In this case _s points - * to the first digit [exluding the sign] of the number] - */ - bn_reverse ((unsigned char *)_s, digs); - - /* append a NULL so the string is properly terminated */ - *str = '\0'; - - mp_clear (&t); - return MP_OKAY; -} - -#endif - -/* End: bn_mp_toradix.c */ - -/* Start: bn_mp_toradix_n.c */ -#include -#ifdef BN_MP_TORADIX_N_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* stores a bignum as a ASCII string in a given radix (2..64) - * - * Stores upto maxlen-1 chars and always a NULL byte - */ -int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen) -{ - int res, digs; - mp_int t; - mp_digit d; - char *_s = str; - - /* check range of the maxlen, radix */ - if (maxlen < 2 || radix < 2 || radix > 64) { - return MP_VAL; - } - - /* quick out if its zero */ - if (mp_iszero(a) == MP_YES) { - *str++ = '0'; - *str = '\0'; - return MP_OKAY; - } - - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - - /* if it is negative output a - */ - if (t.sign == MP_NEG) { - /* we have to reverse our digits later... but not the - sign!! */ - ++_s; - - /* store the flag and mark the number as positive */ - *str++ = '-'; - t.sign = MP_ZPOS; - - /* subtract a char */ - --maxlen; - } - - digs = 0; - while (mp_iszero (&t) == 0) { - if (--maxlen < 1) { - /* no more room */ - break; - } - if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { - mp_clear (&t); - return res; - } - *str++ = mp_s_rmap[d]; - ++digs; - } - - /* reverse the digits of the string. In this case _s points - * to the first digit [exluding the sign] of the number - */ - bn_reverse ((unsigned char *)_s, digs); - - /* append a NULL so the string is properly terminated */ - *str = '\0'; - - mp_clear (&t); - return MP_OKAY; -} - -#endif - -/* End: bn_mp_toradix_n.c */ - -/* Start: bn_mp_unsigned_bin_size.c */ -#include -#ifdef BN_MP_UNSIGNED_BIN_SIZE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* get the size for an unsigned equivalent */ -int mp_unsigned_bin_size (mp_int * a) -{ - int size = mp_count_bits (a); - return (size / 8 + ((size & 7) != 0 ? 1 : 0)); -} -#endif - -/* End: bn_mp_unsigned_bin_size.c */ - -/* Start: bn_mp_xor.c */ -#include -#ifdef BN_MP_XOR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* XOR two ints together */ -int -mp_xor (mp_int * a, mp_int * b, mp_int * c) -{ - int res, ix, px; - mp_int t, *x; - - if (a->used > b->used) { - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - px = b->used; - x = b; - } else { - if ((res = mp_init_copy (&t, b)) != MP_OKAY) { - return res; - } - px = a->used; - x = a; - } - - for (ix = 0; ix < px; ix++) { - t.dp[ix] ^= x->dp[ix]; - } - mp_clamp (&t); - mp_exch (c, &t); - mp_clear (&t); - return MP_OKAY; -} -#endif - -/* End: bn_mp_xor.c */ - -/* Start: bn_mp_zero.c */ -#include -#ifdef BN_MP_ZERO_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* set to zero */ -void mp_zero (mp_int * a) -{ - int n; - mp_digit *tmp; - - a->sign = MP_ZPOS; - a->used = 0; - - tmp = a->dp; - for (n = 0; n < a->alloc; n++) { - *tmp++ = 0; - } -} -#endif - -/* End: bn_mp_zero.c */ - -/* Start: bn_prime_tab.c */ -#include -#ifdef BN_PRIME_TAB_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ -const mp_digit ltm_prime_tab[] = { - 0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013, - 0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035, - 0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059, - 0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F, -#ifndef MP_8BIT - 0x0083, - 0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD, - 0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF, - 0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107, - 0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137, - - 0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167, - 0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199, - 0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9, - 0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7, - 0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239, - 0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265, - 0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293, - 0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF, - - 0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301, - 0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B, - 0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371, - 0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD, - 0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5, - 0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419, - 0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449, - 0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B, - - 0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7, - 0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503, - 0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529, - 0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F, - 0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3, - 0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7, - 0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623, - 0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653 -#endif -}; -#endif - -/* End: bn_prime_tab.c */ - -/* Start: bn_reverse.c */ -#include -#ifdef BN_REVERSE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* reverse an array, used for radix code */ -void -bn_reverse (unsigned char *s, int len) -{ - int ix, iy; - unsigned char t; - - ix = 0; - iy = len - 1; - while (ix < iy) { - t = s[ix]; - s[ix] = s[iy]; - s[iy] = t; - ++ix; - --iy; - } -} -#endif - -/* End: bn_reverse.c */ - -/* Start: bn_s_mp_add.c */ -#include -#ifdef BN_S_MP_ADD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* low level addition, based on HAC pp.594, Algorithm 14.7 */ -int -s_mp_add (mp_int * a, mp_int * b, mp_int * c) -{ - mp_int *x; - int olduse, res, min, max; - - /* find sizes, we let |a| <= |b| which means we have to sort - * them. "x" will point to the input with the most digits - */ - if (a->used > b->used) { - min = b->used; - max = a->used; - x = a; - } else { - min = a->used; - max = b->used; - x = b; - } - - /* init result */ - if (c->alloc < max + 1) { - if ((res = mp_grow (c, max + 1)) != MP_OKAY) { - return res; - } - } - - /* get old used digit count and set new one */ - olduse = c->used; - c->used = max + 1; - - { - register mp_digit u, *tmpa, *tmpb, *tmpc; - register int i; - - /* alias for digit pointers */ - - /* first input */ - tmpa = a->dp; - - /* second input */ - tmpb = b->dp; - - /* destination */ - tmpc = c->dp; - - /* zero the carry */ - u = 0; - for (i = 0; i < min; i++) { - /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ - *tmpc = *tmpa++ + *tmpb++ + u; - - /* U = carry bit of T[i] */ - u = *tmpc >> ((mp_digit)DIGIT_BIT); - - /* take away carry bit from T[i] */ - *tmpc++ &= MP_MASK; - } - - /* now copy higher words if any, that is in A+B - * if A or B has more digits add those in - */ - if (min != max) { - for (; i < max; i++) { - /* T[i] = X[i] + U */ - *tmpc = x->dp[i] + u; - - /* U = carry bit of T[i] */ - u = *tmpc >> ((mp_digit)DIGIT_BIT); - - /* take away carry bit from T[i] */ - *tmpc++ &= MP_MASK; - } - } - - /* add carry */ - *tmpc++ = u; - - /* clear digits above oldused */ - for (i = c->used; i < olduse; i++) { - *tmpc++ = 0; - } - } - - mp_clamp (c); - return MP_OKAY; -} -#endif - -/* End: bn_s_mp_add.c */ - -/* Start: bn_s_mp_exptmod.c */ -#include -#ifdef BN_S_MP_EXPTMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ -#ifdef MP_LOW_MEM - #define TAB_SIZE 32 -#else - #define TAB_SIZE 256 -#endif - -int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode) -{ - mp_int M[TAB_SIZE], res, mu; - mp_digit buf; - int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; - int (*redux)(mp_int*,mp_int*,mp_int*); - - /* find window size */ - x = mp_count_bits (X); - if (x <= 7) { - winsize = 2; - } else if (x <= 36) { - winsize = 3; - } else if (x <= 140) { - winsize = 4; - } else if (x <= 450) { - winsize = 5; - } else if (x <= 1303) { - winsize = 6; - } else if (x <= 3529) { - winsize = 7; - } else { - winsize = 8; - } - -#ifdef MP_LOW_MEM - if (winsize > 5) { - winsize = 5; - } -#endif - - /* init M array */ - /* init first cell */ - if ((err = mp_init(&M[1])) != MP_OKAY) { - return err; - } - - /* now init the second half of the array */ - for (x = 1<<(winsize-1); x < (1 << winsize); x++) { - if ((err = mp_init(&M[x])) != MP_OKAY) { - for (y = 1<<(winsize-1); y < x; y++) { - mp_clear (&M[y]); - } - mp_clear(&M[1]); - return err; - } - } - - /* create mu, used for Barrett reduction */ - if ((err = mp_init (&mu)) != MP_OKAY) { - goto LBL_M; - } - - if (redmode == 0) { - if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) { - goto LBL_MU; - } - redux = mp_reduce; - } else { - if ((err = mp_reduce_2k_setup_l (P, &mu)) != MP_OKAY) { - goto LBL_MU; - } - redux = mp_reduce_2k_l; - } - - /* create M table - * - * The M table contains powers of the base, - * e.g. M[x] = G**x mod P - * - * The first half of the table is not - * computed though accept for M[0] and M[1] - */ - if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) { - goto LBL_MU; - } - - /* compute the value at M[1<<(winsize-1)] by squaring - * M[1] (winsize-1) times - */ - if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { - goto LBL_MU; - } - - for (x = 0; x < (winsize - 1); x++) { - /* square it */ - if ((err = mp_sqr (&M[1 << (winsize - 1)], - &M[1 << (winsize - 1)])) != MP_OKAY) { - goto LBL_MU; - } - - /* reduce modulo P */ - if ((err = redux (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) { - goto LBL_MU; - } - } - - /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) - * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) - */ - for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { - if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { - goto LBL_MU; - } - if ((err = redux (&M[x], P, &mu)) != MP_OKAY) { - goto LBL_MU; - } - } - - /* setup result */ - if ((err = mp_init (&res)) != MP_OKAY) { - goto LBL_MU; - } - mp_set (&res, 1); - - /* set initial mode and bit cnt */ - mode = 0; - bitcnt = 1; - buf = 0; - digidx = X->used - 1; - bitcpy = 0; - bitbuf = 0; - - for (;;) { - /* grab next digit as required */ - if (--bitcnt == 0) { - /* if digidx == -1 we are out of digits */ - if (digidx == -1) { - break; - } - /* read next digit and reset the bitcnt */ - buf = X->dp[digidx--]; - bitcnt = (int) DIGIT_BIT; - } - - /* grab the next msb from the exponent */ - y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; - buf <<= (mp_digit)1; - - /* if the bit is zero and mode == 0 then we ignore it - * These represent the leading zero bits before the first 1 bit - * in the exponent. Technically this opt is not required but it - * does lower the # of trivial squaring/reductions used - */ - if (mode == 0 && y == 0) { - continue; - } - - /* if the bit is zero and mode == 1 then we square */ - if (mode == 1 && y == 0) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; - } - continue; - } - - /* else we add it to the window */ - bitbuf |= (y << (winsize - ++bitcpy)); - mode = 2; - - if (bitcpy == winsize) { - /* ok window is filled so square as required and multiply */ - /* square first */ - for (x = 0; x < winsize; x++) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; - } - } - - /* then multiply */ - if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; - } - - /* empty window and reset */ - bitcpy = 0; - bitbuf = 0; - mode = 1; - } - } - - /* if bits remain then square/multiply */ - if (mode == 2 && bitcpy > 0) { - /* square then multiply if the bit is set */ - for (x = 0; x < bitcpy; x++) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; - } - - bitbuf <<= 1; - if ((bitbuf & (1 << winsize)) != 0) { - /* then multiply */ - if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; - } - } - } - } - - mp_exch (&res, Y); - err = MP_OKAY; -LBL_RES:mp_clear (&res); -LBL_MU:mp_clear (&mu); -LBL_M: - mp_clear(&M[1]); - for (x = 1<<(winsize-1); x < (1 << winsize); x++) { - mp_clear (&M[x]); - } - return err; -} -#endif - -/* End: bn_s_mp_exptmod.c */ - -/* Start: bn_s_mp_mul_digs.c */ -#include -#ifdef BN_S_MP_MUL_DIGS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* multiplies |a| * |b| and only computes upto digs digits of result - * HAC pp. 595, Algorithm 14.12 Modified so you can control how - * many digits of output are created. - */ -int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -{ - mp_int t; - int res, pa, pb, ix, iy; - mp_digit u; - mp_word r; - mp_digit tmpx, *tmpt, *tmpy; - - /* can we use the fast multiplier? */ - if (((digs) < MP_WARRAY) && - MIN (a->used, b->used) < - (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - return fast_s_mp_mul_digs (a, b, c, digs); - } - - if ((res = mp_init_size (&t, digs)) != MP_OKAY) { - return res; - } - t.used = digs; - - /* compute the digits of the product directly */ - pa = a->used; - for (ix = 0; ix < pa; ix++) { - /* set the carry to zero */ - u = 0; - - /* limit ourselves to making digs digits of output */ - pb = MIN (b->used, digs - ix); - - /* setup some aliases */ - /* copy of the digit from a used within the nested loop */ - tmpx = a->dp[ix]; - - /* an alias for the destination shifted ix places */ - tmpt = t.dp + ix; - - /* an alias for the digits of b */ - tmpy = b->dp; - - /* compute the columns of the output and propagate the carry */ - for (iy = 0; iy < pb; iy++) { - /* compute the column as a mp_word */ - r = ((mp_word)*tmpt) + - ((mp_word)tmpx) * ((mp_word)*tmpy++) + - ((mp_word) u); - - /* the new column is the lower part of the result */ - *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); - - /* get the carry word from the result */ - u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); - } - /* set carry if it is placed below digs */ - if (ix + iy < digs) { - *tmpt = u; - } - } - - mp_clamp (&t); - mp_exch (&t, c); - - mp_clear (&t); - return MP_OKAY; -} -#endif - -/* End: bn_s_mp_mul_digs.c */ - -/* Start: bn_s_mp_mul_high_digs.c */ -#include -#ifdef BN_S_MP_MUL_HIGH_DIGS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* multiplies |a| * |b| and does not compute the lower digs digits - * [meant to get the higher part of the product] - */ -int -s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -{ - mp_int t; - int res, pa, pb, ix, iy; - mp_digit u; - mp_word r; - mp_digit tmpx, *tmpt, *tmpy; - - /* can we use the fast multiplier? */ -#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C - if (((a->used + b->used + 1) < MP_WARRAY) - && MIN (a->used, b->used) < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - return fast_s_mp_mul_high_digs (a, b, c, digs); - } -#endif - - if ((res = mp_init_size (&t, a->used + b->used + 1)) != MP_OKAY) { - return res; - } - t.used = a->used + b->used + 1; - - pa = a->used; - pb = b->used; - for (ix = 0; ix < pa; ix++) { - /* clear the carry */ - u = 0; - - /* left hand side of A[ix] * B[iy] */ - tmpx = a->dp[ix]; - - /* alias to the address of where the digits will be stored */ - tmpt = &(t.dp[digs]); - - /* alias for where to read the right hand side from */ - tmpy = b->dp + (digs - ix); - - for (iy = digs - ix; iy < pb; iy++) { - /* calculate the double precision result */ - r = ((mp_word)*tmpt) + - ((mp_word)tmpx) * ((mp_word)*tmpy++) + - ((mp_word) u); - - /* get the lower part */ - *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); - - /* carry the carry */ - u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); - } - *tmpt = u; - } - mp_clamp (&t); - mp_exch (&t, c); - mp_clear (&t); - return MP_OKAY; -} -#endif - -/* End: bn_s_mp_mul_high_digs.c */ - -/* Start: bn_s_mp_sqr.c */ -#include -#ifdef BN_S_MP_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ -int s_mp_sqr (mp_int * a, mp_int * b) -{ - mp_int t; - int res, ix, iy, pa; - mp_word r; - mp_digit u, tmpx, *tmpt; - - pa = a->used; - if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) { - return res; - } - - /* default used is maximum possible size */ - t.used = 2*pa + 1; - - for (ix = 0; ix < pa; ix++) { - /* first calculate the digit at 2*ix */ - /* calculate double precision result */ - r = ((mp_word) t.dp[2*ix]) + - ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]); - - /* store lower part in result */ - t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK)); - - /* get the carry */ - u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); - - /* left hand side of A[ix] * A[iy] */ - tmpx = a->dp[ix]; - - /* alias for where to store the results */ - tmpt = t.dp + (2*ix + 1); - - for (iy = ix + 1; iy < pa; iy++) { - /* first calculate the product */ - r = ((mp_word)tmpx) * ((mp_word)a->dp[iy]); - - /* now calculate the double precision result, note we use - * addition instead of *2 since it's easier to optimize - */ - r = ((mp_word) *tmpt) + r + r + ((mp_word) u); - - /* store lower part */ - *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); - - /* get carry */ - u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); - } - /* propagate upwards */ - while (u != ((mp_digit) 0)) { - r = ((mp_word) *tmpt) + ((mp_word) u); - *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); - u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); - } - } - - mp_clamp (&t); - mp_exch (&t, b); - mp_clear (&t); - return MP_OKAY; -} -#endif - -/* End: bn_s_mp_sqr.c */ - -/* Start: bn_s_mp_sub.c */ -#include -#ifdef BN_S_MP_SUB_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */ -int -s_mp_sub (mp_int * a, mp_int * b, mp_int * c) -{ - int olduse, res, min, max; - - /* find sizes */ - min = b->used; - max = a->used; - - /* init result */ - if (c->alloc < max) { - if ((res = mp_grow (c, max)) != MP_OKAY) { - return res; - } - } - olduse = c->used; - c->used = max; - - { - register mp_digit u, *tmpa, *tmpb, *tmpc; - register int i; - - /* alias for digit pointers */ - tmpa = a->dp; - tmpb = b->dp; - tmpc = c->dp; - - /* set carry to zero */ - u = 0; - for (i = 0; i < min; i++) { - /* T[i] = A[i] - B[i] - U */ - *tmpc = *tmpa++ - *tmpb++ - u; - - /* U = carry bit of T[i] - * Note this saves performing an AND operation since - * if a carry does occur it will propagate all the way to the - * MSB. As a result a single shift is enough to get the carry - */ - u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); - - /* Clear carry from T[i] */ - *tmpc++ &= MP_MASK; - } - - /* now copy higher words if any, e.g. if A has more digits than B */ - for (; i < max; i++) { - /* T[i] = A[i] - U */ - *tmpc = *tmpa++ - u; - - /* U = carry bit of T[i] */ - u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); - - /* Clear carry from T[i] */ - *tmpc++ &= MP_MASK; - } - - /* clear digits above used (since we may not have grown result above) */ - for (i = c->used; i < olduse; i++) { - *tmpc++ = 0; - } - } - - mp_clamp (c); - return MP_OKAY; -} - -#endif - -/* End: bn_s_mp_sub.c */ - -/* Start: bncore.c */ -#include -#ifdef BNCORE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ - -/* Known optimal configurations - - CPU /Compiler /MUL CUTOFF/SQR CUTOFF -------------------------------------------------------------- - Intel P4 Northwood /GCC v3.4.1 / 88/ 128/LTM 0.32 ;-) - AMD Athlon64 /GCC v3.4.4 / 80/ 120/LTM 0.35 - -*/ - -int KARATSUBA_MUL_CUTOFF = 80, /* Min. number of digits before Karatsuba multiplication is used. */ - KARATSUBA_SQR_CUTOFF = 120, /* Min. number of digits before Karatsuba squaring is used. */ - - TOOM_MUL_CUTOFF = 350, /* no optimal values of these are known yet so set em high */ - TOOM_SQR_CUTOFF = 400; -#endif - -/* End: bncore.c */ - - -/* EOF */ diff --git a/libtommath/tombc/grammar.txt b/libtommath/tombc/grammar.txt deleted file mode 100644 index a780e75..0000000 --- a/libtommath/tombc/grammar.txt +++ /dev/null @@ -1,35 +0,0 @@ -program := program statement | statement | empty -statement := { statement } | - identifier = numexpression; | - identifier[numexpression] = numexpression; | - function(expressionlist); | - for (identifer = numexpression; numexpression; identifier = numexpression) { statement } | - while (numexpression) { statement } | - if (numexpresion) { statement } elif | - break; | - continue; - -elif := else statement | empty -function := abs | countbits | exptmod | jacobi | print | isprime | nextprime | issquare | readinteger | exit -expressionlist := expressionlist, expression | expression - -// LR(1) !!!? -expression := string | numexpression -numexpression := cmpexpr && cmpexpr | cmpexpr \|\| cmpexpr | cmpexpr -cmpexpr := boolexpr < boolexpr | boolexpr > boolexpr | boolexpr == boolexpr | - boolexpr <= boolexpr | boolexpr >= boolexpr | boolexpr -boolexpr := shiftexpr & shiftexpr | shiftexpr ^ shiftexpr | shiftexpr \| shiftexpr | shiftexpr -shiftexpr := addsubexpr << addsubexpr | addsubexpr >> addsubexpr | addsubexpr -addsubexpr := mulexpr + mulexpr | mulexpr - mulexpr | mulexpr -mulexpr := expr * expr | expr / expr | expr % expr | expr -expr := -nexpr | nexpr -nexpr := integer | identifier | ( numexpression ) | identifier[numexpression] - -identifier := identifer digits | identifier alpha | alpha -alpha := a ... z | A ... Z -integer := hexnumber | digits -hexnumber := 0xhexdigits -hexdigits := hexdigits hexdigit | hexdigit -hexdigit := 0 ... 9 | a ... f | A ... F -digits := digits digit | digit -digit := 0 ... 9 diff --git a/libtommath/tommath.h b/libtommath/tommath.h index 4b3a76f..4e6b1a5 100644 --- a/libtommath/tommath.h +++ b/libtommath/tommath.h @@ -10,46 +10,27 @@ * The library is free for all purposes without any express * guarantee it works. * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com + * Tom St Denis, tstdenis82@gmail.com, http://math.libtomcrypt.com */ #ifndef BN_H_ #define BN_H_ #include -#include #include -#include +#include #include #include -#ifndef MIN -# define MIN(x,y) ((x)<(y)?(x):(y)) -#endif - -#ifndef MAX -# define MAX(x,y) ((x)>(y)?(x):(y)) -#endif - #ifdef __cplusplus extern "C" { - -/* C++ compilers don't like assigning void * to mp_digit * */ -#define OPT_CAST(x) (x *) - -#else - -/* C on the other hand doesn't care */ -#define OPT_CAST(x) - #endif - /* detect 64-bit mode if possible */ -#if defined(__x86_64__) -# if !(defined(MP_64BIT) && defined(MP_16BIT) && defined(MP_8BIT)) -# define MP_64BIT -# endif +#if defined(__x86_64__) + #if !(defined(MP_32BIT) || defined(MP_16BIT) || defined(MP_8BIT)) + #define MP_64BIT + #endif #endif /* some default configurations. @@ -61,70 +42,68 @@ extern "C" { * [any size beyond that is ok provided it doesn't overflow the data type] */ #ifdef MP_8BIT - typedef unsigned char mp_digit; - typedef unsigned short mp_word; + typedef uint8_t mp_digit; + typedef uint16_t mp_word; +#define MP_SIZEOF_MP_DIGIT 1 +#ifdef DIGIT_BIT +#error You must not define DIGIT_BIT when using MP_8BIT +#endif #elif defined(MP_16BIT) - typedef unsigned short mp_digit; - typedef unsigned long mp_word; + typedef uint16_t mp_digit; + typedef uint32_t mp_word; +#define MP_SIZEOF_MP_DIGIT 2 +#ifdef DIGIT_BIT +#error You must not define DIGIT_BIT when using MP_16BIT +#endif #elif defined(MP_64BIT) /* for GCC only on supported platforms */ -#ifndef CRYPT - typedef unsigned long long ulong64; - typedef signed long long long64; + typedef uint64_t mp_digit; +#if defined(_WIN32) + typedef unsigned __int128 mp_word; +#elif defined(__GNUC__) + typedef unsigned long mp_word __attribute__ ((mode(TI))); +#else + /* it seems you have a problem + * but we assume you can somewhere define your own uint128_t */ + typedef uint128_t mp_word; #endif - typedef unsigned long mp_digit; - typedef unsigned long mp_word __attribute__ ((mode(TI))); - -# define DIGIT_BIT 60 + #define DIGIT_BIT 60 #else /* this is the default case, 28-bit digits */ - - /* this is to make porting into LibTomCrypt easier :-) */ -#ifndef CRYPT -# if defined(_MSC_VER) || defined(__BORLANDC__) - typedef unsigned __int64 ulong64; - typedef signed __int64 long64; -# else - typedef unsigned long long ulong64; - typedef signed long long long64; -# endif -#endif - typedef unsigned long mp_digit; - typedef ulong64 mp_word; + /* this is to make porting into LibTomCrypt easier :-) */ + typedef uint32_t mp_digit; + typedef uint64_t mp_word; -#ifdef MP_31BIT +#ifdef MP_31BIT /* this is an extension that uses 31-bit digits */ -# define DIGIT_BIT 31 + #define DIGIT_BIT 31 #else /* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */ -# define DIGIT_BIT 28 -# define MP_28BIT -#endif + #define DIGIT_BIT 28 + #define MP_28BIT #endif - -/* define heap macros */ -#ifndef CRYPT - /* default to libc stuff */ -# ifndef XMALLOC -# define XMALLOC malloc -# define XFREE free -# define XREALLOC realloc -# define XCALLOC calloc -# else - /* prototypes for our heap functions */ - extern void *XMALLOC(size_t n); - extern void *XREALLOC(void *p, size_t n); - extern void *XCALLOC(size_t n, size_t s); - extern void XFREE(void *p); -# endif #endif - /* otherwise the bits per digit is calculated automatically from the size of a mp_digit */ #ifndef DIGIT_BIT -# define DIGIT_BIT ((int)((CHAR_BIT * sizeof(mp_digit) - 1))) /* bits per digit */ + #define DIGIT_BIT (((CHAR_BIT * MP_SIZEOF_MP_DIGIT) - 1)) /* bits per digit */ + typedef uint_least32_t mp_min_u32; +#else + typedef mp_digit mp_min_u32; +#endif + +/* platforms that can use a better rand function */ +#if defined(__FreeBSD__) || defined(__OpenBSD__) || defined(__NetBSD__) || defined(__DragonFly__) + #define MP_USE_ALT_RAND 1 +#endif + +/* use arc4random on platforms that support it */ +#ifdef MP_USE_ALT_RAND + #define MP_GEN_RANDOM() arc4random() +#else + #define MP_GEN_RANDOM() rand() #endif #define MP_DIGIT_BIT DIGIT_BIT @@ -165,15 +144,15 @@ extern int KARATSUBA_MUL_CUTOFF, /* default precision */ #ifndef MP_PREC -# ifndef MP_LOW_MEM -# define MP_PREC 32 /* default digits of precision */ -# else -# define MP_PREC 8 /* default digits of precision */ -# endif + #ifndef MP_LOW_MEM + #define MP_PREC 32 /* default digits of precision */ + #else + #define MP_PREC 8 /* default digits of precision */ + #endif #endif /* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER_DIGIT*2) */ -#define MP_WARRAY (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGIT_BIT + 1)) +#define MP_WARRAY (1 << (((sizeof(mp_word) * CHAR_BIT) - (2 * DIGIT_BIT)) + 1)) /* the infamous mp_int structure */ typedef struct { @@ -190,7 +169,7 @@ typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat); #define SIGN(m) ((m)->sign) /* error code to char* string */ -char *mp_error_to_string(int code); +const char *mp_error_to_string(int code); /* ---> init and deinit bignum functions <--- */ /* init a bignum */ @@ -219,8 +198,9 @@ int mp_init_size(mp_int *a, int size); /* ---> Basic Manipulations <--- */ #define mp_iszero(a) (((a)->used == 0) ? MP_YES : MP_NO) -#define mp_iseven(a) (((a)->used == 0 || (((a)->dp[0] & 1) == 0)) ? MP_YES : MP_NO) -#define mp_isodd(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? MP_YES : MP_NO) +#define mp_iseven(a) ((((a)->used == 0) || (((a)->dp[0] & 1u) == 0u)) ? MP_YES : MP_NO) +#define mp_isodd(a) ((((a)->used > 0) && (((a)->dp[0] & 1u) == 1u)) ? MP_YES : MP_NO) +#define mp_isneg(a) (((a)->sign != MP_ZPOS) ? MP_YES : MP_NO) /* set to zero */ void mp_zero(mp_int *a); @@ -231,9 +211,21 @@ void mp_set(mp_int *a, mp_digit b); /* set a 32-bit const */ int mp_set_int(mp_int *a, unsigned long b); +/* set a platform dependent unsigned long value */ +int mp_set_long(mp_int *a, unsigned long b); + +/* set a platform dependent unsigned long long value */ +int mp_set_long_long(mp_int *a, unsigned long long b); + /* get a 32-bit value */ unsigned long mp_get_int(mp_int * a); +/* get a platform dependent unsigned long value */ +unsigned long mp_get_long(mp_int * a); + +/* get a platform dependent unsigned long long value */ +unsigned long long mp_get_long_long(mp_int * a); + /* initialize and set a digit */ int mp_init_set (mp_int * a, mp_digit b); @@ -249,6 +241,12 @@ int mp_init_copy(mp_int *a, mp_int *b); /* trim unused digits */ void mp_clamp(mp_int *a); +/* import binary data */ +int mp_import(mp_int* rop, size_t count, int order, size_t size, int endian, size_t nails, const void* op); + +/* export binary data */ +int mp_export(void* rop, size_t* countp, int order, size_t size, int endian, size_t nails, mp_int* op); + /* ---> digit manipulation <--- */ /* right shift by "b" digits */ @@ -257,19 +255,19 @@ void mp_rshd(mp_int *a, int b); /* left shift by "b" digits */ int mp_lshd(mp_int *a, int b); -/* c = a / 2**b */ +/* c = a / 2**b, implemented as c = a >> b */ int mp_div_2d(const mp_int *a, int b, mp_int *c, mp_int *d); /* b = a/2 */ int mp_div_2(mp_int *a, mp_int *b); -/* c = a * 2**b */ +/* c = a * 2**b, implemented as c = a << b */ int mp_mul_2d(const mp_int *a, int b, mp_int *c); /* b = a*2 */ int mp_mul_2(mp_int *a, mp_int *b); -/* c = a mod 2**d */ +/* c = a mod 2**b */ int mp_mod_2d(const mp_int *a, int b, mp_int *c); /* computes a = 2**b */ @@ -347,6 +345,7 @@ int mp_div_3(mp_int *a, mp_int *c, mp_digit *d); /* c = a**b */ int mp_expt_d(mp_int *a, mp_digit b, mp_int *c); +int mp_expt_d_ex (mp_int * a, mp_digit b, mp_int * c, int fast); /* c = a mod b, 0 <= c < b */ int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c); @@ -382,10 +381,14 @@ int mp_lcm(mp_int *a, mp_int *b, mp_int *c); * returns error if a < 0 and b is even */ int mp_n_root(mp_int *a, mp_digit b, mp_int *c); +int mp_n_root_ex (mp_int * a, mp_digit b, mp_int * c, int fast); /* special sqrt algo */ int mp_sqrt(mp_int *arg, mp_int *ret); +/* special sqrt (mod prime) */ +int mp_sqrtmod_prime(mp_int *arg, mp_int *prime, mp_int *ret); + /* is number a square? */ int mp_is_square(mp_int *arg, int *ret); @@ -447,13 +450,13 @@ int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); /* number of primes */ #ifdef MP_8BIT -# define PRIME_SIZE 31 + #define PRIME_SIZE 31 #else -# define PRIME_SIZE 256 + #define PRIME_SIZE 256 #endif /* table of first PRIME_SIZE primes */ -extern const mp_digit ltm_prime_tab[]; +extern const mp_digit ltm_prime_tab[PRIME_SIZE]; /* result=1 if a is divisible by one of the first PRIME_SIZE primes */ int mp_prime_is_divisible(mp_int *a, int *result); @@ -469,7 +472,7 @@ int mp_prime_fermat(mp_int *a, mp_int *b, int *result); int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result); /* This gives [for a given bit size] the number of trials required - * such that Miller-Rabin gives a prob of failure lower than 2^-96 + * such that Miller-Rabin gives a prob of failure lower than 2^-96 */ int mp_prime_rabin_miller_trials(int size); @@ -490,7 +493,7 @@ int mp_prime_is_prime(mp_int *a, int t, int *result); int mp_prime_next_prime(mp_int *a, int t, int bbs_style); /* makes a truly random prime of a given size (bytes), - * call with bbs = 1 if you want it to be congruent to 3 mod 4 + * call with bbs = 1 if you want it to be congruent to 3 mod 4 * * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself @@ -503,10 +506,9 @@ int mp_prime_next_prime(mp_int *a, int t, int bbs_style); /* makes a truly random prime of a given size (bits), * * Flags are as follows: - * + * * LTM_PRIME_BBS - make prime congruent to 3 mod 4 * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS) - * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero * LTM_PRIME_2MSB_ON - make the 2nd highest bit one * * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can @@ -534,8 +536,10 @@ int mp_toradix(mp_int *a, char *str, int radix); int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen); int mp_radix_size(mp_int *a, int radix, int *size); +#ifndef LTM_NO_FILE int mp_fread(mp_int *a, int radix, FILE *stream); int mp_fwrite(mp_int *a, int radix, FILE *stream); +#endif #define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len)) #define mp_raw_size(mp) mp_signed_bin_size(mp) @@ -549,31 +553,13 @@ int mp_fwrite(mp_int *a, int radix, FILE *stream); #define mp_todecimal(M, S) mp_toradix((M), (S), 10) #define mp_tohex(M, S) mp_toradix((M), (S), 16) -/* lowlevel functions, do not call! */ -int s_mp_add(mp_int *a, mp_int *b, mp_int *c); -int s_mp_sub(mp_int *a, mp_int *b, mp_int *c); -#define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1) -int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); -int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); -int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); -int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); -int fast_s_mp_sqr(mp_int *a, mp_int *b); -int s_mp_sqr(mp_int *a, mp_int *b); -int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c); -int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c); -int mp_karatsuba_sqr(mp_int *a, mp_int *b); -int mp_toom_sqr(mp_int *a, mp_int *b); -int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c); -int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c); -int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); -int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int mode); -int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int mode); -void bn_reverse(unsigned char *s, int len); - -extern const char *mp_s_rmap; - #ifdef __cplusplus -} + } #endif #endif + + +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/libtommath/tommath.out b/libtommath/tommath.out deleted file mode 100644 index 9f62617..0000000 --- a/libtommath/tommath.out +++ /dev/null @@ -1,139 +0,0 @@ -\BOOKMARK [0][-]{chapter.1}{Introduction}{} -\BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1} -\BOOKMARK [2][-]{subsection.1.1.1}{What is Multiple Precision Arithmetic?}{section.1.1} -\BOOKMARK [2][-]{subsection.1.1.2}{The Need for Multiple Precision Arithmetic}{section.1.1} -\BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1} -\BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1} -\BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1} -\BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3} -\BOOKMARK [2][-]{subsection.1.3.2}{Precision Notation}{section.1.3} -\BOOKMARK [2][-]{subsection.1.3.3}{Algorithm Inputs and Outputs}{section.1.3} -\BOOKMARK [2][-]{subsection.1.3.4}{Mathematical Expressions}{section.1.3} -\BOOKMARK [2][-]{subsection.1.3.5}{Work Effort}{section.1.3} -\BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1} -\BOOKMARK [1][-]{section.1.5}{Introduction to LibTomMath}{chapter.1} -\BOOKMARK [2][-]{subsection.1.5.1}{What is LibTomMath?}{section.1.5} -\BOOKMARK [2][-]{subsection.1.5.2}{Goals of LibTomMath}{section.1.5} -\BOOKMARK [1][-]{section.1.6}{Choice of LibTomMath}{chapter.1} -\BOOKMARK [2][-]{subsection.1.6.1}{Code Base}{section.1.6} -\BOOKMARK [2][-]{subsection.1.6.2}{API Simplicity}{section.1.6} -\BOOKMARK [2][-]{subsection.1.6.3}{Optimizations}{section.1.6} -\BOOKMARK [2][-]{subsection.1.6.4}{Portability and Stability}{section.1.6} -\BOOKMARK [2][-]{subsection.1.6.5}{Choice}{section.1.6} -\BOOKMARK [0][-]{chapter.2}{Getting Started}{} -\BOOKMARK [1][-]{section.2.1}{Library Basics}{chapter.2} -\BOOKMARK [1][-]{section.2.2}{What is a Multiple Precision Integer?}{chapter.2} -\BOOKMARK [2][-]{subsection.2.2.1}{The mp\137int Structure}{section.2.2} -\BOOKMARK [1][-]{section.2.3}{Argument Passing}{chapter.2} -\BOOKMARK [1][-]{section.2.4}{Return Values}{chapter.2} -\BOOKMARK [1][-]{section.2.5}{Initialization and Clearing}{chapter.2} -\BOOKMARK [2][-]{subsection.2.5.1}{Initializing an mp\137int}{section.2.5} -\BOOKMARK [2][-]{subsection.2.5.2}{Clearing an mp\137int}{section.2.5} -\BOOKMARK [1][-]{section.2.6}{Maintenance Algorithms}{chapter.2} -\BOOKMARK [2][-]{subsection.2.6.1}{Augmenting an mp\137int's Precision}{section.2.6} -\BOOKMARK [2][-]{subsection.2.6.2}{Initializing Variable Precision mp\137ints}{section.2.6} -\BOOKMARK [2][-]{subsection.2.6.3}{Multiple Integer Initializations and Clearings}{section.2.6} -\BOOKMARK [2][-]{subsection.2.6.4}{Clamping Excess Digits}{section.2.6} -\BOOKMARK [0][-]{chapter.3}{Basic Operations}{} -\BOOKMARK [1][-]{section.3.1}{Introduction}{chapter.3} -\BOOKMARK [1][-]{section.3.2}{Assigning Values to mp\137int Structures}{chapter.3} -\BOOKMARK [2][-]{subsection.3.2.1}{Copying an mp\137int}{section.3.2} -\BOOKMARK [2][-]{subsection.3.2.2}{Creating a Clone}{section.3.2} -\BOOKMARK [1][-]{section.3.3}{Zeroing an Integer}{chapter.3} -\BOOKMARK [1][-]{section.3.4}{Sign Manipulation}{chapter.3} -\BOOKMARK [2][-]{subsection.3.4.1}{Absolute Value}{section.3.4} -\BOOKMARK [2][-]{subsection.3.4.2}{Integer Negation}{section.3.4} -\BOOKMARK [1][-]{section.3.5}{Small Constants}{chapter.3} -\BOOKMARK [2][-]{subsection.3.5.1}{Setting Small Constants}{section.3.5} -\BOOKMARK [2][-]{subsection.3.5.2}{Setting Large Constants}{section.3.5} -\BOOKMARK [1][-]{section.3.6}{Comparisons}{chapter.3} -\BOOKMARK [2][-]{subsection.3.6.1}{Unsigned Comparisions}{section.3.6} -\BOOKMARK [2][-]{subsection.3.6.2}{Signed Comparisons}{section.3.6} -\BOOKMARK [0][-]{chapter.4}{Basic Arithmetic}{} -\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4} -\BOOKMARK [1][-]{section.4.2}{Addition and Subtraction}{chapter.4} -\BOOKMARK [2][-]{subsection.4.2.1}{Low Level Addition}{section.4.2} -\BOOKMARK [2][-]{subsection.4.2.2}{Low Level Subtraction}{section.4.2} -\BOOKMARK [2][-]{subsection.4.2.3}{High Level Addition}{section.4.2} -\BOOKMARK [2][-]{subsection.4.2.4}{High Level Subtraction}{section.4.2} -\BOOKMARK [1][-]{section.4.3}{Bit and Digit Shifting}{chapter.4} -\BOOKMARK [2][-]{subsection.4.3.1}{Multiplication by Two}{section.4.3} -\BOOKMARK [2][-]{subsection.4.3.2}{Division by Two}{section.4.3} -\BOOKMARK [1][-]{section.4.4}{Polynomial Basis Operations}{chapter.4} -\BOOKMARK [2][-]{subsection.4.4.1}{Multiplication by x}{section.4.4} -\BOOKMARK [2][-]{subsection.4.4.2}{Division by x}{section.4.4} -\BOOKMARK [1][-]{section.4.5}{Powers of Two}{chapter.4} -\BOOKMARK [2][-]{subsection.4.5.1}{Multiplication by Power of Two}{section.4.5} -\BOOKMARK [2][-]{subsection.4.5.2}{Division by Power of Two}{section.4.5} -\BOOKMARK [2][-]{subsection.4.5.3}{Remainder of Division by Power of Two}{section.4.5} -\BOOKMARK [0][-]{chapter.5}{Multiplication and Squaring}{} -\BOOKMARK [1][-]{section.5.1}{The Multipliers}{chapter.5} -\BOOKMARK [1][-]{section.5.2}{Multiplication}{chapter.5} -\BOOKMARK [2][-]{subsection.5.2.1}{The Baseline Multiplication}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.2}{Faster Multiplication by the ``Comba'' Method}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.3}{Polynomial Basis Multiplication}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.4}{Karatsuba Multiplication}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.5}{Toom-Cook 3-Way Multiplication}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.6}{Signed Multiplication}{section.5.2} -\BOOKMARK [1][-]{section.5.3}{Squaring}{chapter.5} -\BOOKMARK [2][-]{subsection.5.3.1}{The Baseline Squaring Algorithm}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.2}{Faster Squaring by the ``Comba'' Method}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.3}{Polynomial Basis Squaring}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.4}{Karatsuba Squaring}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.5}{Toom-Cook Squaring}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.6}{High Level Squaring}{section.5.3} -\BOOKMARK [0][-]{chapter.6}{Modular Reduction}{} -\BOOKMARK [1][-]{section.6.1}{Basics of Modular Reduction}{chapter.6} -\BOOKMARK [1][-]{section.6.2}{The Barrett Reduction}{chapter.6} -\BOOKMARK [2][-]{subsection.6.2.1}{Fixed Point Arithmetic}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.2}{Choosing a Radix Point}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.3}{Trimming the Quotient}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.4}{Trimming the Residue}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.5}{The Barrett Algorithm}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.6}{The Barrett Setup Algorithm}{section.6.2} -\BOOKMARK [1][-]{section.6.3}{The Montgomery Reduction}{chapter.6} -\BOOKMARK [2][-]{subsection.6.3.1}{Digit Based Montgomery Reduction}{section.6.3} -\BOOKMARK [2][-]{subsection.6.3.2}{Baseline Montgomery Reduction}{section.6.3} -\BOOKMARK [2][-]{subsection.6.3.3}{Faster ``Comba'' Montgomery Reduction}{section.6.3} -\BOOKMARK [2][-]{subsection.6.3.4}{Montgomery Setup}{section.6.3} -\BOOKMARK [1][-]{section.6.4}{The Diminished Radix Algorithm}{chapter.6} -\BOOKMARK [2][-]{subsection.6.4.1}{Choice of Moduli}{section.6.4} -\BOOKMARK [2][-]{subsection.6.4.2}{Choice of k}{section.6.4} -\BOOKMARK [2][-]{subsection.6.4.3}{Restricted Diminished Radix Reduction}{section.6.4} -\BOOKMARK [2][-]{subsection.6.4.4}{Unrestricted Diminished Radix Reduction}{section.6.4} -\BOOKMARK [1][-]{section.6.5}{Algorithm Comparison}{chapter.6} -\BOOKMARK [0][-]{chapter.7}{Exponentiation}{} -\BOOKMARK [1][-]{section.7.1}{Exponentiation Basics}{chapter.7} -\BOOKMARK [2][-]{subsection.7.1.1}{Single Digit Exponentiation}{section.7.1} -\BOOKMARK [1][-]{section.7.2}{k-ary Exponentiation}{chapter.7} -\BOOKMARK [2][-]{subsection.7.2.1}{Optimal Values of k}{section.7.2} -\BOOKMARK [2][-]{subsection.7.2.2}{Sliding-Window Exponentiation}{section.7.2} -\BOOKMARK [1][-]{section.7.3}{Modular Exponentiation}{chapter.7} -\BOOKMARK [2][-]{subsection.7.3.1}{Barrett Modular Exponentiation}{section.7.3} -\BOOKMARK [1][-]{section.7.4}{Quick Power of Two}{chapter.7} -\BOOKMARK [0][-]{chapter.8}{Higher Level Algorithms}{} -\BOOKMARK [1][-]{section.8.1}{Integer Division with Remainder}{chapter.8} -\BOOKMARK [2][-]{subsection.8.1.1}{Quotient Estimation}{section.8.1} -\BOOKMARK [2][-]{subsection.8.1.2}{Normalized Integers}{section.8.1} -\BOOKMARK [2][-]{subsection.8.1.3}{Radix- Division with Remainder}{section.8.1} -\BOOKMARK [1][-]{section.8.2}{Single Digit Helpers}{chapter.8} -\BOOKMARK [2][-]{subsection.8.2.1}{Single Digit Addition and Subtraction}{section.8.2} -\BOOKMARK [2][-]{subsection.8.2.2}{Single Digit Multiplication}{section.8.2} -\BOOKMARK [2][-]{subsection.8.2.3}{Single Digit Division}{section.8.2} -\BOOKMARK [2][-]{subsection.8.2.4}{Single Digit Root Extraction}{section.8.2} -\BOOKMARK [1][-]{section.8.3}{Random Number Generation}{chapter.8} -\BOOKMARK [1][-]{section.8.4}{Formatted Representations}{chapter.8} -\BOOKMARK [2][-]{subsection.8.4.1}{Reading Radix-n Input}{section.8.4} -\BOOKMARK [2][-]{subsection.8.4.2}{Generating Radix-n Output}{section.8.4} -\BOOKMARK [0][-]{chapter.9}{Number Theoretic Algorithms}{} -\BOOKMARK [1][-]{section.9.1}{Greatest Common Divisor}{chapter.9} -\BOOKMARK [2][-]{subsection.9.1.1}{Complete Greatest Common Divisor}{section.9.1} -\BOOKMARK [1][-]{section.9.2}{Least Common Multiple}{chapter.9} -\BOOKMARK [1][-]{section.9.3}{Jacobi Symbol Computation}{chapter.9} -\BOOKMARK [2][-]{subsection.9.3.1}{Jacobi Symbol}{section.9.3} -\BOOKMARK [1][-]{section.9.4}{Modular Inverse}{chapter.9} -\BOOKMARK [2][-]{subsection.9.4.1}{General Case}{section.9.4} -\BOOKMARK [1][-]{section.9.5}{Primality Tests}{chapter.9} -\BOOKMARK [2][-]{subsection.9.5.1}{Trial Division}{section.9.5} -\BOOKMARK [2][-]{subsection.9.5.2}{The Fermat Test}{section.9.5} -\BOOKMARK [2][-]{subsection.9.5.3}{The Miller-Rabin Test}{section.9.5} diff --git a/libtommath/tommath.pdf b/libtommath/tommath.pdf deleted file mode 100644 index c9571d8..0000000 Binary files a/libtommath/tommath.pdf and /dev/null differ diff --git a/libtommath/tommath.src b/libtommath/tommath.src index 4065822..768ed10 100644 --- a/libtommath/tommath.src +++ b/libtommath/tommath.src @@ -66,31 +66,20 @@ QUALCOMM Australia \\ } } \maketitle -This text has been placed in the public domain. This text corresponds to the v0.39 release of the +This text has been placed in the public domain. This text corresponds to the v0.39 release of the LibTomMath project. -\begin{alltt} -Tom St Denis -111 Banning Rd -Ottawa, Ontario -K2L 1C3 -Canada - -Phone: 1-613-836-3160 -Email: tomstdenis@gmail.com -\end{alltt} - -This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} +This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} {\em book} macro package and the Perl {\em booker} package. \tableofcontents \listoffigures \chapter*{Prefaces} -When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. -They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' -Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which +When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. +They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' +Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps -others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give +others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give back to society in the form of tools and knowledge that can help others in their endeavours. I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source @@ -103,9 +92,9 @@ from relatively straightforward algebra and I hope that this book can be a valua This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora of kind people donating their time, resources and kind words to help support my work. Writing a text of significant length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old, -comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg -were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to -continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. +comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg +were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to +continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I honour your kind gestures with this project. @@ -115,22 +104,22 @@ Open Source. Open Academia. Open Minds. \begin{flushright} Tom St Denis \end{flushright} \newpage -I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also +I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also contribute to educate others facing the problem of having to handle big number mathematical calculations. -This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of -how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about +This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of +how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about the layout and language used. -I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the -practical aspects of cryptography. +I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the +practical aspects of cryptography. -Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a -great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up -multiple precision calculations is often very important since we deal with outdated machine architecture where modular +Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a +great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up +multiple precision calculations is often very important since we deal with outdated machine architecture where modular reductions, for example, become painfully slow. -This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks +This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' \begin{flushright} @@ -142,22 +131,22 @@ Brazil \end{flushright} \newpage -It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about -Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not +It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about +Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. -At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the +At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real -contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. +contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. -When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, -and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close -friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, +When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, +and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close +friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, and I'm pleased to be involved with it. \begin{flushright} -Greg Rose, Sydney, Australia, June 2003. +Greg Rose, Sydney, Australia, June 2003. \end{flushright} \mainmatter @@ -167,23 +156,23 @@ Greg Rose, Sydney, Australia, June 2003. \subsection{What is Multiple Precision Arithmetic?} When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively -raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can -reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. -Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple +raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can +reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. +Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} - of algorithms can be designed to accomodate them. + of algorithms can be designed to accomodate them. By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in the decimal system with fixed precision $6 \cdot 7 = 2$. Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in -schools to manually add, subtract, multiply and divide. +schools to manually add, subtract, multiply and divide. \subsection{The Need for Multiple Precision Arithmetic} The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation -of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require -integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a -typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and +of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require +integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a +typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. \begin{figure}[!here] @@ -201,27 +190,27 @@ Java \cite{JAVA} only provide instrinsic support for integers which are relative \label{fig:ISOC} \end{figure} -The largest data type guaranteed to be provided by the ISO C programming -language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they -see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is -insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be -trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, -rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by +The largest data type guaranteed to be provided by the ISO C programming +language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they +see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is +insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be +trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, +rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by extending the range of representable integers while using single precision data types. -Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic -primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in -various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several -major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and +Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic +primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in +various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several +major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and deployment of efficient algorithms. -However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. -Another auxiliary use of multiple precision integers is high precision floating point data types. -The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. -Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE -floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small +However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. +Another auxiliary use of multiple precision integers is high precision floating point data types. +The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. +Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE +floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small (\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create -a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where +a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where scientific applications must minimize the total output error over long calculations. Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). @@ -229,152 +218,152 @@ In fact the library discussed within this text has already been used to form a p \subsection{Benefits of Multiple Precision Arithmetic} \index{precision} -The benefit of multiple precision representations over single or fixed precision representations is that -no precision is lost while representing the result of an operation which requires excess precision. For example, -the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple -precision algorithm would augment the precision of the destination to accomodate the result while a single precision system +The benefit of multiple precision representations over single or fixed precision representations is that +no precision is lost while representing the result of an operation which requires excess precision. For example, +the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple +precision algorithm would augment the precision of the destination to accomodate the result while a single precision system would truncate excess bits to maintain a fixed level of precision. It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic -curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum -size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the -integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard +curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum +size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the +integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. -Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the +Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved -platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the -inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input -without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to +platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the +inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input +without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to be written and tested once. \section{Purpose of This Text} -The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. -That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' -elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} -give considerably detailed explanations of the theoretical aspects of algorithms and often very little information -regarding the practical implementation aspects. - -In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For -example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple -algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning +The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. +That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' +elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} +give considerably detailed explanations of the theoretical aspects of algorithms and often very little information +regarding the practical implementation aspects. + +In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For +example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple +algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple -as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not +as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). -Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers -and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve -any form of useful performance in non-trivial applications. +Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers +and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve +any form of useful performance in non-trivial applications. To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer -package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used -to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field -tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text +package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used +to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field +tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text discusses a very large portion of the inner workings of the library. -The algorithms that are presented will always include at least one ``pseudo-code'' description followed -by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same -algorithm in other programming languages as the reader sees fit. +The algorithms that are presented will always include at least one ``pseudo-code'' description followed +by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same +algorithm in other programming languages as the reader sees fit. This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing -the reader how the algorithms fit together as well as where to start on various taskings. +the reader how the algorithms fit together as well as where to start on various taskings. \section{Discussion and Notation} \subsection{Notation} A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent -the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits -of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer -$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. +the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits +of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer +$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. \index{mp\_int} -The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well -as auxilary data required to manipulate the data. These additional members are discussed further in section -\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be -synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members -are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the -member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would -evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that -$a.length = 5$. +The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well +as auxilary data required to manipulate the data. These additional members are discussed further in section +\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be +synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members +are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the +member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would +evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that +$a.length = 5$. For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used -to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is -a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to -mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These +to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is +a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to +mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple -precision algorithm to solve the same problem. +precision algorithm to solve the same problem. \subsection{Precision Notation} -The variable $\beta$ represents the radix of a single digit of a multiple precision integer and -must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in -the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range -$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the +The variable $\beta$ represents the radix of a single digit of a multiple precision integer and +must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in +the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range +$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the carry. Since all modern computers are binary, it is assumed that $q$ is two. \index{mp\_digit} \index{mp\_word} -Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent -a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In -several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. -For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to +Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent +a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In +several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. +For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision -variable it is assumed that all single precision variables are promoted to double precision during the evaluation. +variable it is assumed that all single precision variables are promoted to double precision during the evaluation. Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single precision data type. -For example, if $\beta = 10^2$ a single precision data type may represent a value in the +For example, if $\beta = 10^2$ a single precision data type may represent a value in the range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let $a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. -In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit -in a single precision data type and as a result $c \ne \hat c$. +In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit +in a single precision data type and as a result $c \ne \hat c$. \subsection{Algorithm Inputs and Outputs} Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision -as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This -distinction is important as scalars are often used as array indicies and various other counters. +as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This +distinction is important as scalars are often used as array indicies and various other counters. \subsection{Mathematical Expressions} -The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression +The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression -rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when -the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, -$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a -fraction a real value division is implied, for example ${5 \over 2} = 2.5$. +rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when +the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, +$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a +fraction a real value division is implied, for example ${5 \over 2} = 2.5$. The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation -of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. +of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. \subsection{Work Effort} \index{big-Oh} -To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all -single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. -That is a single precision addition, multiplication and division are assumed to take the same time to +To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all +single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. +That is a single precision addition, multiplication and division are assumed to take the same time to complete. While this is generally not true in practice, it will simplify the discussions considerably. -Some algorithms have slight advantages over others which is why some constants will not be removed in -the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a -baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these -would both be said to be equivalent to $O(n^2)$. However, -in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a +Some algorithms have slight advantages over others which is why some constants will not be removed in +the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a +baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these +would both be said to be equivalent to $O(n^2)$. However, +in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a result small constant factors in the work effort will make an observable difference in algorithm efficiency. -All of the algorithms presented in this text have a polynomial time work level. That is, of the form -$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how +All of the algorithms presented in this text have a polynomial time work level. That is, of the form +$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how various optimizations will help pay off in the long run. \section{Exercises} Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to -the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought -provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent -chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the -subject material. +the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought +provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent +chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the +subject material. That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular are encouraged to verify they can answer the problems correctly before moving on. Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of -the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these -exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the +the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these +exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the scoring system used. \begin{figure}[here] @@ -404,21 +393,21 @@ scoring system used. \end{figure} Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or -devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level +devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These -two levels are essentially entry level questions. +two levels are essentially entry level questions. -Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often -fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always +Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often +fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can answer these questions will feel comfortable with the concepts behind the topic at hand. -Problems at the fourth level are meant to be similar to those of the level three questions except they will require -additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide -the exact details of the answer until a subsequent chapter. +Problems at the fourth level are meant to be similar to those of the level three questions except they will require +additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide +the exact details of the answer until a subsequent chapter. -Problems at the fifth level are meant to be the hardest -problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a +Problems at the fifth level are meant to be the hardest +problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a mastery of the subject matter at hand. Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader @@ -427,43 +416,43 @@ is encouraged to answer the follow-up problems and try to draw the relevance of \section{Introduction to LibTomMath} \subsection{What is LibTomMath?} -LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it -is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on -any given platform. +LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it +is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on +any given platform. The library has been successfully tested under numerous operating systems including Unix\footnote{All of these -trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such -as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such +trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such +as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such as public key cryptosystems and still maintain a relatively small footprint. \subsection{Goals of LibTomMath} -Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, -even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the -library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM -processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window -exponentiation and Montgomery reduction have been provided to make the library more efficient. +Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, +even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the +library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM +processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window +exponentiation and Montgomery reduction have been provided to make the library more efficient. -Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface -(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized -algorithms automatically without the developer's specific attention. One such example is the generic multiplication -algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication -based on the magnitude of the inputs and the configuration of the library. +Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface +(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized +algorithms automatically without the developer's specific attention. One such example is the generic multiplication +algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication +based on the magnitude of the inputs and the configuration of the library. -Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should +Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should be source compatible with another popular library which makes it more attractive for developers to use. In this case the -MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits -in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument +MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits +in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument passing conventions, it has been written from scratch by Tom St Denis. -The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' -library exists which can be used to teach computer science students how to perform fast and reliable multiple precision -integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. +The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' +library exists which can be used to teach computer science students how to perform fast and reliable multiple precision +integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. \section{Choice of LibTomMath} LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but -for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL -\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for +for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL +\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for reasons that will be explained in the following sub-sections. \subsection{Code Base} @@ -472,115 +461,115 @@ segments of code littered throughout the source. This clean and uncluttered app developer can more readily discern the true intent of a given section of source code without trying to keep track of what conditional code will be used. -The code base of LibTomMath is well organized. Each function is in its own separate source code file +The code base of LibTomMath is well organized. Each function is in its own separate source code file which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing -very hard. GMP has many conditional code segments which also hinder tracing. +very hard. GMP has many conditional code segments which also hinder tracing. When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} - which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about + which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about $50$KiB) but LibTomMath is also much faster and more complete than MPI. \subsection{API Simplicity} -LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build -with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the -functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided -which is an extremely valuable benefit for the student and developer alike. +LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build +with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the +functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided +which is an extremely valuable benefit for the student and developer alike. -The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to -illegible short hand. LibTomMath does not share this characteristic. +The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to +illegible short hand. LibTomMath does not share this characteristic. The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In -effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely +effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely undersireable in many situations. \subsection{Optimizations} While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does -feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP +feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP -only had Barrett and Montgomery modular reduction algorithms.}. +only had Barrett and Montgomery modular reduction algorithms.}. LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular -exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually +exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually slower than the best libraries such as GMP and OpenSSL by only a small factor. \subsection{Portability and Stability} -LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler -(\textit{GCC}). This means that without changes the library will build without configuration or setting up any -variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of -MPI has recently stopped working on his library and LIP has long since been discontinued. +LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler +(\textit{GCC}). This means that without changes the library will build without configuration or setting up any +variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of +MPI has recently stopped working on his library and LIP has long since been discontinued. GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active development and are very stable across a variety of platforms. \subsection{Choice} LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for -the case study of this text. Various source files from the LibTomMath project will be included within the text. However, -the reader is encouraged to download their own copy of the library to actually be able to work with the library. +the case study of this text. Various source files from the LibTomMath project will be included within the text. However, +the reader is encouraged to download their own copy of the library to actually be able to work with the library. \chapter{Getting Started} \section{Library Basics} -The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, -a problem along with allowable solution parameters should be identified and analyzed. In this particular case the +The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, +a problem along with allowable solution parameters should be identified and analyzed. In this particular case the inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written as portable source code that is reasonably efficient across several different computer platforms. -After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. -That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, +After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. +That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. -By building outwards from a base foundation instead of using a parallel design methodology the resulting project is +By building outwards from a base foundation instead of using a parallel design methodology the resulting project is highly modular. Being highly modular is a desirable property of any project as it often means the resulting product -has a small footprint and updates are easy to perform. +has a small footprint and updates are easy to perform. -Usually when I start a project I will begin with the header files. I define the data types I think I will need and -prototype the initial functions that are not dependent on other functions (within the library). After I +Usually when I start a project I will begin with the header files. I define the data types I think I will need and +prototype the initial functions that are not dependent on other functions (within the library). After I implement these base functions I prototype more dependent functions and implement them. The process repeats until -I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as -mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to -why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the -dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the -mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development +I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as +mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to +why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the +dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the +mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. FIGU,design_process,Design Flow of the First Few Original LibTomMath Functions. Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing -the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. +the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. -It only makes sense to begin the text with the preliminary data types and support algorithms required as well. +It only makes sense to begin the text with the preliminary data types and support algorithms required as well. This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. \section{What is a Multiple Precision Integer?} -Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot -be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is -to use fixed precision data types to create and manipulate multiple precision integers which may represent values -that are very large. +Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot +be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is +to use fixed precision data types to create and manipulate multiple precision integers which may represent values +that are very large. As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system -the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits -(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds -column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based -multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed +the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits +(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds +column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based +multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed precision computer words with the exception that a different radix is used. -What most people probably do not think about explicitly are the various other attributes that describe a multiple precision -integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, -that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in -its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper -arithmetic. The third property is how many digits placeholders are available to hold the integer. +What most people probably do not think about explicitly are the various other attributes that describe a multiple precision +integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, +that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in +its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper +arithmetic. The third property is how many digits placeholders are available to hold the integer. The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, -if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. +if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer -will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision -integer or mp\_int for short. +will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision +integer or mp\_int for short. \subsection{The mp\_int Structure} \label{sec:MPINT} -The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for -any such data type but it does provide for making composite data types known as structures. The following is the structure definition +The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for +any such data type but it does provide for making composite data types known as structures. The following is the structure definition used within LibTomMath. \index{mp\_int} @@ -607,46 +596,46 @@ The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. \begin{enumerate} \item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent -a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. +a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. -\item The \textbf{alloc} parameter denotes how -many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count -of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the -array to accommodate the precision of the result. +\item The \textbf{alloc} parameter denotes how +many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count +of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the +array to accommodate the precision of the result. -\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple -precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least +\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple +precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored -first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, -if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then -it would represent the integer $a + b\beta + c\beta^2 + \ldots$ +first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, +if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then +it would represent the integer $a + b\beta + c\beta^2 + \ldots$ \index{MP\_ZPOS} \index{MP\_NEG} -\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). +\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). \end{enumerate} \subsubsection{Valid mp\_int Structures} -Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. +Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). \begin{enumerate} \item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated array of digits. \item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. -\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, +\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, leading zero digits in the most significant positions must be trimmed. \begin{enumerate} \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. \end{enumerate} -\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; +\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; this represents the mp\_int value of zero. \end{enumerate} \section{Argument Passing} -A convention of argument passing must be adopted early on in the development of any library. Making the function -prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. -In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int -structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. +A convention of argument passing must be adopted early on in the development of any library. Making the function +prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. +In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int +structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. Consider the following examples. \begin{verbatim} @@ -659,25 +648,25 @@ The left to right order is a fairly natural way to implement the functions since functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order -of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In -truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been -adopted. - -Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a -destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important -feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. -However, to implement this feature specific care has to be given to ensure the destination is not modified before the +of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In +truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been +adopted. + +Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a +destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important +feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. +However, to implement this feature specific care has to be given to ensure the destination is not modified before the source is fully read. \section{Return Values} -A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them -to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end +A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them +to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may fault by dereferencing memory not owned by the application. -In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for -instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor -will it check pointers for validity. Any function that can cause a runtime error will return an error code as an +In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for +instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor +will it check pointers for validity. Any function that can cause a runtime error will return an error code as an \textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). \index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} @@ -696,7 +685,7 @@ will it check pointers for validity. Any function that can cause a runtime erro \end{figure} When an error is detected within a function it should free any memory it allocated, often during the initialization of -temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the +temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the function was called. Error checking with this style of API is fairly simple. \begin{verbatim} @@ -707,19 +696,19 @@ function was called. Error checking with this style of API is fairly simple. } \end{verbatim} -The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal +The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. \section{Initialization and Clearing} -The logical starting point when actually writing multiple precision integer functions is the initialization and +The logical starting point when actually writing multiple precision integer functions is the initialization and clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate -and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste -memory and become unmanageable. +and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste +memory and become unmanageable. If the memory for the digits has been successfully allocated then the rest of the members of the structure must be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set @@ -754,16 +743,16 @@ structure are set to valid values. The mp\_init algorithm will perform such an \textbf{Algorithm mp\_init.} The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly -a valid assumption if the input resides on the stack. +a valid assumption if the input resides on the stack. Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for -the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} -name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} +the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} +name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest precision number you'll be working with. Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow -heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack +heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack memory and the number of heap operations will be trivial. Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and @@ -775,14 +764,14 @@ This function introduces the idiosyncrasy that all iterative loops, commonly ini when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured -the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate +the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate decrementally. EXAM,bn_mp_init.c -One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It -is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The -call to mp\_init() is used only to initialize the members of the structure to a known default state. +One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It +is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The +call to mp\_init() is used only to initialize the members of the structure to a known default state. Here we see (line @23,XMALLOC@) the memory allocation is performed first. This allows us to exit cleanly and quickly if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there @@ -791,17 +780,17 @@ but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to mall memory allocation routine. In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been -accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a +accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a portable fashion you have to actually assign the value. The for loop (line @28,for@) performs this required operation. -After the memory has been successfully initialized the remainder of the members are initialized +After the memory has been successfully initialized the remainder of the members are initialized (lines @29,used@ through @31,sign@) to their respective default states. At this point the algorithm has succeeded and -a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the -mp\_int structure has been properly initialized and is safe to use with other functions within the library. +a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the +mp\_int structure has been properly initialized and is safe to use with other functions within the library. \subsection{Clearing an mp\_int} -When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be +When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be returned to the application's memory pool with the mp\_clear algorithm. \begin{figure}[here] @@ -826,12 +815,12 @@ returned to the application's memory pool with the mp\_clear algorithm. \end{figure} \textbf{Algorithm mp\_clear.} -This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that +This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal is to free the allocated memory. The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this -algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid +algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid digit pointer \textbf{dp} setting. Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm @@ -844,11 +833,11 @@ checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is \textbf{NULL} in which case the if statement will evaluate to true. The digits of the mp\_int are cleared by the for loop (line @25,for@) which assigns a zero to every digit. Similar to mp\_init() -the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. +the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer -still has to be reset to \textbf{NULL} manually (line @33,NULL@). +still has to be reset to \textbf{NULL} manually (line @33,NULL@). Now that the digits have been cleared and deallocated the other members are set to their final values (lines @34,= 0@ and @35,ZPOS@). @@ -856,16 +845,16 @@ Now that the digits have been cleared and deallocated the other members are set The previous sections describes how to initialize and clear an mp\_int structure. To further support operations that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be -able to augment the precision of an mp\_int and -initialize mp\_ints with differing initial conditions. +able to augment the precision of an mp\_int and +initialize mp\_ints with differing initial conditions. These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level algorithms such as addition, multiplication and modular exponentiation. \subsection{Augmenting an mp\_int's Precision} -When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire -result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member -is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it +When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire +result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member +is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. \newpage\begin{figure}[here] @@ -891,14 +880,14 @@ must be re-sized appropriately to accomodate the result. The mp\_grow algorithm \end{figure} \textbf{Algorithm mp\_grow.} -It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to -prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. +It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to +prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. -The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). -This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. +The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). +This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. -It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much -akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are +It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much +akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are assumed to contain undefined values they are initially set to zero. EXAM,bn_mp_grow.c @@ -915,12 +904,12 @@ the re-allocation. All that is left is to clear the newly allocated digits and Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would -result in a memory leak if XREALLOC ever failed. +result in a memory leak if XREALLOC ever failed. \subsection{Initializing Variable Precision mp\_ints} -Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size -of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it -will allocate \textit{at least} a specified number of digits. +Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size +of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it +will allocate \textit{at least} a specified number of digits. \begin{figure}[here] \begin{small} @@ -947,30 +936,30 @@ will allocate \textit{at least} a specified number of digits. \end{figure} \textbf{Algorithm mp\_init\_size.} -This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of -digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a -multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial +This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of +digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a +multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial allocations from becoming a bottleneck in the rest of the algorithms. -Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This +Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is correct no further memory re-allocations are required to work with the mp\_int. EXAM,bn_mp_init_size.c -The number of digits $b$ requested is padded (line @22,MP_PREC@) by first augmenting it to the next multiple of -\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the -mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be -returned (line @27,return@). +The number of digits $b$ requested is padded (line @22,MP_PREC@) by first augmenting it to the next multiple of +\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the +mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be +returned (line @27,return@). -The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The -\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set -to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines @29,used@, @30,alloc@ and @31,sign@). If the function -returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the +The digits are allocated with the malloc() function (line @27,XMALLOC@) and set to zero afterwards (line @38,for@). The +\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set +to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines @29,used@, @30,alloc@ and @31,sign@). If the function +returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the functions to work with. \subsection{Multiple Integer Initializations and Clearings} -Occasionally a function will require a series of mp\_int data types to be made available simultaneously. +Occasionally a function will require a series of mp\_int data types to be made available simultaneously. The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single statement. It is essentially a shortcut to multiple initializations. @@ -995,42 +984,42 @@ statement. It is essentially a shortcut to multiple initializations. \end{figure} \textbf{Algorithm mp\_init\_multi.} -The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected -(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' +The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected +(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' initialization which allows for quick recovery from runtime errors. EXAM,bn_mp_init_multi.c This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int -structures in an actual C array they are simply passed as arguments to the function. This function makes use of the -``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument -appended on the right. +structures in an actual C array they are simply passed as arguments to the function. This function makes use of the +``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument +appended on the right. The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count $n$ of succesfully initialized mp\_int structures is maintained (line @47,n++@) such that if a failure does occur, -the algorithm can backtrack and free the previously initialized structures (lines @27,if@ to @46,}@). +the algorithm can backtrack and free the previously initialized structures (lines @27,if@ to @46,}@). \subsection{Clamping Excess Digits} -When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of -the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a -$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ -though, with no final carry into the last position. However, suppose the destination had to be first expanded -(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. +When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of +the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a +$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ +though, with no final carry into the last position. However, suppose the destination had to be first expanded +(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. That would be a considerable waste of time since heap operations are relatively slow. The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked -there would be an excess high order zero digit. +there would be an excess high order zero digit. -For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit +For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would -accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very -low the representation is excessively large. +accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very +low the representation is excessively large. -The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the -\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a -positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to +The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the +\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a +positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to \textbf{MP\_ZPOS}. \begin{figure}[here] @@ -1052,16 +1041,16 @@ positive number which means that if the \textbf{used} count is decremented to ze \textbf{Algorithm mp\_clamp.} As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at -the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for +the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for when all of the digits are zero to ensure that the mp\_int is valid at all times. EXAM,bn_mp_clamp.c Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming -language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is -important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously +language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is +important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously undesirable. The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not -the pointer ``a''. +the pointer ``a''. \section*{Exercises} \begin{tabular}{cl} @@ -1087,19 +1076,19 @@ $\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy mig \section{Introduction} In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining -mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low +mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they work before proceeding since these algorithms will be used almost intrinsically in the following chapters. The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures -represent. +represent. \section{Assigning Values to mp\_int Structures} \subsection{Copying an mp\_int} Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same -value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. +value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. \newpage\begin{figure}[here] \begin{center} @@ -1124,40 +1113,40 @@ value as the mp\_int it was copied from. The mp\_copy algorithm provides this f \textbf{Algorithm mp\_copy.} This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will -represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the +represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. -If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow +If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of $b$. \textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the -text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in -step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is +text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in +step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return -the error code itself. However, the C code presented will demonstrate all of the error handling logic required to +the error code itself. However, the C code presented will demonstrate all of the error handling logic required to implement the pseudo-code. EXAM,bn_mp_copy.c Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output -mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without -copying digits (line @24,a == b@). +mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without +copying digits (line @24,a == b@). The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than $a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines @29,alloc@ to @33,}@). In order to simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits of the mp\_ints $a$ and $b$ respectively. These aliases (lines @42,tmpa@ and @45,tmpb@) allow the compiler to access the digits without first dereferencing the -mp\_int pointers and then subsequently the pointer to the digits. +mp\_int pointers and then subsequently the pointer to the digits. -After the aliases are established the digits from $a$ are copied into $b$ (lines @48,for@ to @50,}@) and then the excess -digits of $b$ are set to zero (lines @53,for@ to @55,}@). Both ``for'' loops make use of the pointer aliases and in -fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization +After the aliases are established the digits from $a$ are copied into $b$ (lines @48,for@ to @50,}@) and then the excess +digits of $b$ are set to zero (lines @53,for@ to @55,}@). Both ``for'' loops make use of the pointer aliases and in +fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization allows the alias to stay in a machine register fairly easy between the two loops. \textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will -be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the +be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the number of pointer dereferencing operations required to access data. For example, a for loop may resemble \begin{alltt} @@ -1166,7 +1155,7 @@ for (x = 0; x < 100; x++) \{ \} \end{alltt} -This could be re-written using aliases as +This could be re-written using aliases as \begin{alltt} mp_digit *tmpa; @@ -1176,17 +1165,17 @@ for (x = 0; x < 100; x++) \{ \} \end{alltt} -In this case an alias is used to access the -array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required +In this case an alias is used to access the +array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases. -The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations -may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may -work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer -aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code +The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations +may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may +work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer +aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code stands a better chance of being faster. -The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' +The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' loop of the function mp\_copy() re-written to not use pointer aliases. \begin{alltt} @@ -1196,13 +1185,13 @@ loop of the function mp\_copy() re-written to not use pointer aliases. \} \end{alltt} -Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more +Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more complicated as there are four variables within the statement instead of just two. \subsubsection{Nested Statements} Another commonly used technique in the source routines is that certain sections of code are nested. This is used in particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six) -will typically have three different phases. First the temporaries are initialized, then the columns calculated and +will typically have three different phases. First the temporaries are initialized, then the columns calculated and finally the carries are propagated. In this example the middle column production phase will typically be nested as it uses temporary variables and aliases the most. @@ -1211,9 +1200,9 @@ the various temporary variables required do not propagate into other sections of \subsection{Creating a Clone} -Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int -and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is -useful within functions that need to modify an argument but do not wish to actually modify the original copy. The +Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int +and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is +useful within functions that need to modify an argument but do not wish to actually modify the original copy. The mp\_init\_copy algorithm has been designed to help perform this task. \begin{figure}[here] @@ -1233,14 +1222,14 @@ mp\_init\_copy algorithm has been designed to help perform this task. \end{figure} \textbf{Algorithm mp\_init\_copy.} -This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As -such this algorithm will perform two operations in one step. +This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As +such this algorithm will perform two operations in one step. EXAM,bn_mp_init_copy.c -This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that +This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that \textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call -and \textbf{a} will be left intact. +and \textbf{a} will be left intact. \section{Zeroing an Integer} Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to @@ -1264,11 +1253,11 @@ perform this task. \end{figure} \textbf{Algorithm mp\_zero.} -This algorithm simply resets a mp\_int to the default state. +This algorithm simply resets a mp\_int to the default state. EXAM,bn_mp_zero.c -After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the +After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the \textbf{sign} variable is set to \textbf{MP\_ZPOS}. \section{Sign Manipulation} @@ -1296,7 +1285,7 @@ the absolute value of an mp\_int. \textbf{Algorithm mp\_abs.} This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, -for instance, the developer to pass the same mp\_int as the source and destination to this function without addition +for instance, the developer to pass the same mp\_int as the source and destination to this function without addition logic to handle it. EXAM,bn_mp_abs.c @@ -1331,7 +1320,7 @@ the negative of an mp\_int input. \textbf{Algorithm mp\_neg.} This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then -the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if +the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if $a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return zero as negative. @@ -1356,9 +1345,9 @@ Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For 2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ 3. $a.used \leftarrow \left \lbrace \begin{array}{ll} 1 & \mbox{if }a_0 > 0 \\ - 0 & \mbox{if }a_0 = 0 + 0 & \mbox{if }a_0 = 0 \end{array} \right .$ \\ -\hline +\hline \end{tabular} \end{center} \caption{Algorithm mp\_set} @@ -1370,16 +1359,16 @@ single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adj EXAM,bn_mp_set.c -First we zero (line @21,mp_zero@) the mp\_int to make sure that the other members are initialized for a +First we zero (line @21,mp_zero@) the mp\_int to make sure that the other members are initialized for a small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count -is zero. Next we set the digit and reduce it modulo $\beta$ (line @22,MP_MASK@). After this step we have to +is zero. Next we set the digit and reduce it modulo $\beta$ (line @22,MP_MASK@). After this step we have to check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise to zero. -We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with +We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with $2^k - 1$ will perform the same operation. -One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses +One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses this function should take that into account. Only trivially small constants can be set using this function. \subsection{Setting Large Constants} @@ -1407,9 +1396,9 @@ data type as input and will always treat it as a 32-bit integer. \end{figure} \textbf{Algorithm mp\_set\_int.} -The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the +The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the -next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is +next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have zero digits used and the newly added four bits would be ignored. @@ -1418,23 +1407,23 @@ Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorith EXAM,bn_mp_set_int.c This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird -addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits. While it may not -seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@ -as well as the call to mp\_clamp() on line @40,mp_clamp@. Both functions will clamp excess leading digits which keeps +addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits. While it may not +seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@ +as well as the call to mp\_clamp() on line @40,mp_clamp@. Both functions will clamp excess leading digits which keeps the number of used digits low. \section{Comparisons} \subsection{Unsigned Comparisions} Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ -to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude -positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. +to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude +positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two -mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the +mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the signs are known to agree in advance. -To facilitate working with the results of the comparison functions three constants are required. +To facilitate working with the results of the comparison functions three constants are required. \begin{figure}[here] \begin{center} @@ -1470,24 +1459,24 @@ To facilitate working with the results of the comparison functions three constan \textbf{Algorithm mp\_cmp\_mag.} By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return -\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. -Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. -If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. +\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. +Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. +If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. EXAM,bn_mp_cmp_mag.c -The two if statements (lines @24,if@ and @28,if@) compare the number of digits in the two inputs. These two are -performed before all of the digits are compared since it is a very cheap test to perform and can potentially save -considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be +The two if statements (lines @24,if@ and @28,if@) compare the number of digits in the two inputs. These two are +performed before all of the digits are compared since it is a very cheap test to perform and can potentially save +considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits. \subsection{Signed Comparisons} -Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude +Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude comparison a trivial signed comparison algorithm can be written. \begin{figure}[here] @@ -1510,16 +1499,16 @@ comparison a trivial signed comparison algorithm can be written. \end{figure} \textbf{Algorithm mp\_cmp.} -The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate -comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step -three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then +The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate +comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step +three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then $\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. EXAM,bn_mp_cmp.c The two if statements (lines @22,if@ and @26,if@) perform the initial sign comparison. If the signs are not the equal then which ever -has the positive sign is larger. The inputs are compared (line @30,if@) based on magnitudes. If the signs were both -negative then the unsigned comparison is performed in the opposite direction (line @31,mp_cmp_mag@). Otherwise, the signs are assumed to +has the positive sign is larger. The inputs are compared (line @30,if@) based on magnitudes. If the signs were both +negative then the unsigned comparison is performed in the opposite direction (line @31,mp_cmp_mag@). Otherwise, the signs are assumed to be both positive and a forward direction unsigned comparison is performed. \section*{Exercises} @@ -1536,38 +1525,38 @@ $\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of \chapter{Basic Arithmetic} \section{Introduction} -At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been -established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These -algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important -that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms -which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. +At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been +established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These +algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important +that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms +which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. MARK,SHIFTS -All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right -logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real -number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). -Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. +All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right +logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real +number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). +Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed -from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the -result is $110_2$. +from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the +result is $110_2$. \section{Addition and Subtraction} In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers -$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. +$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. As a result subtraction can be performed with a trivial series of logical operations and an addition. However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the -sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or +sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or subtraction algorithms with the sign fixed up appropriately. The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of the integers respectively. \subsection{Low Level Addition} -An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the -trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. +An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the +trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. \newpage @@ -1614,18 +1603,18 @@ Historically that convention stems from the MPI library where ``s\_'' stood for \end{figure} \textbf{Algorithm s\_mp\_add.} -This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. -Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the +This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. +Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic will simply add all of the smallest input to the largest input and store that first part of the result in the destination. Then it will apply a simpler addition loop to excess digits of the larger input. -The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two +The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the -same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum -of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. +same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum +of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. At this point the first addition loop will go through as many digit positions that both inputs have. The carry variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce @@ -1644,32 +1633,32 @@ EXAM,bn_s_mp_add.c We first sort (lines @27,if@ to @35,}@) the inputs based on magnitude and determine the $min$ and $max$ variables. Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we -grow the destination (@37,init@ to @42,}@) ensure that it can accomodate the result of the addition. +grow the destination (@37,init@ to @42,}@) ensure that it can accomodate the result of the addition. -Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on +Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ represent the two inputs and destination variables respectively. These aliases are used to ensure the compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. -The initial carry $u$ will be cleared (line @65,u = 0@), note that $u$ is of type mp\_digit which ensures type +The initial carry $u$ will be cleared (line @65,u = 0@), note that $u$ is of type mp\_digit which ensures type compatibility within the implementation. The initial addition (line @66,for@ to @75,}@) adds digits from both inputs until the smallest input runs out of digits. Similarly the conditional addition loop -(line @81,for@ to @90,}@) adds the remaining digits from the larger of the two inputs. The addition is finished +(line @81,for@ to @90,}@) adds the remaining digits from the larger of the two inputs. The addition is finished with the final carry being stored in $tmpc$ (line @94,tmpc++@). Note the ``++'' operator within the same expression. After line @94,tmpc++@, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful for the next loop (line @97,for@ to @99,}@) which set any old upper digits to zero. \subsection{Low Level Subtraction} The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the -unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must -be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. +unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must +be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. MARK,GAMMA For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent -the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For -this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a -mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). +the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For +this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a +mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$. @@ -1685,7 +1674,7 @@ data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case 1. $min \leftarrow b.used$ \\ 2. $max \leftarrow a.used$ \\ 3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ +4. $oldused \leftarrow c.used$ \\ 5. $c.used \leftarrow max$ \\ 6. $u \leftarrow 0$ \\ 7. for $n$ from $0$ to $min - 1$ do \\ @@ -1715,54 +1704,54 @@ passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \ver algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. -The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 -set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at -most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and +The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 +set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at +most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and set to the maximal count for the operation. -The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision -subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction -loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. +The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision +subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction +loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. -For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to -the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the -third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the -way to the most significant bit. +For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to +the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the +third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the +way to the most significant bit. -Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most +Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that -is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the -carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. +is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the +carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step 10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. EXAM,bn_s_mp_sub.c -Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded -(lines @24,min@ and @25,max@). In reality the $min$ and $max$ variables are only aliases and are only -used to make the source code easier to read. Again the pointer alias optimization is used +Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded +(lines @24,min@ and @25,max@). In reality the $min$ and $max$ variables are only aliases and are only +used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized (lines @42,tmpa@, @43,tmpb@ and @44,tmpc@) for $a$, $b$ and $c$ respectively. The first subtraction loop (lines @47,u = 0@ through @61,}@) subtract digits from both inputs until the smaller of -the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' -method of extracting the carry (line @57, >>@). The traditional method for extracting the carry would be to shift -by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of -the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry -extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the -most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This +the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' +method of extracting the carry (line @57, >>@). The traditional method for extracting the carry would be to shift +by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of +the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry +extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the +most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on twos compliment machines which is a safe assumption to make. -If $a$ has a larger magnitude than $b$ an additional loop (lines @64,for@ through @73,}@) is required to propagate -the carry through $a$ and copy the result to $c$. +If $a$ has a larger magnitude than $b$ an additional loop (lines @64,for@ through @73,}@) is required to propagate +the carry through $a$ and copy the result to $c$. \subsection{High Level Addition} Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be -established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data -types. +established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data +types. -Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} +Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. \begin{figure}[!here] @@ -1790,8 +1779,8 @@ flag. A high level addition is actually performed as a series of eight separate \end{figure} \textbf{Algorithm mp\_add.} -This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from -either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly +This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from +either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly straightforward but restricted since subtraction can only produce positive results. \begin{figure}[here] @@ -1821,9 +1810,9 @@ straightforward but restricted since subtraction can only produce positive resul \label{fig:AddChart} \end{figure} -Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three -specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are -forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best +Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three +specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are +forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best follows how the implementation actually was achieved. Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms @@ -1831,8 +1820,8 @@ s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would -produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp -within algorithm s\_mp\_add will force $-0$ to become $0$. +produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp +within algorithm s\_mp\_add will force $-0$ to become $0$. EXAM,bn_mp_add.c @@ -1842,7 +1831,7 @@ explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observ level functions do so. Returning their return code is sufficient. \subsection{High Level Subtraction} -The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. +The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. \newpage\begin{figure}[!here] \begin{center} @@ -1872,7 +1861,7 @@ The high level signed subtraction algorithm is essentially the same as the high \end{figure} \textbf{Algorithm mp\_sub.} -This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or +This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or \cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and the operations required. @@ -1899,28 +1888,28 @@ the operations required. \label{fig:SubChart} \end{figure} -Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the -algorithm from producing $-a - -a = -0$ as a result. +Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the +algorithm from producing $-a - -a = -0$ as a result. EXAM,bn_mp_sub.c Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations -and forward it to the end of the function. On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a -``greater than or equal to'' comparison. +and forward it to the end of the function. On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a +``greater than or equal to'' comparison. \section{Bit and Digit Shifting} MARK,POLY -It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. -This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. +It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. +This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations -are on radix-$\beta$ digits. +are on radix-$\beta$ digits. \subsection{Multiplication by Two} -In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient -operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. +In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient +operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. \newpage\begin{figure}[!here] \begin{small} @@ -1954,26 +1943,26 @@ operation to perform. A single precision logical shift left is sufficient to mu \end{figure} \textbf{Algorithm mp\_mul\_2.} -This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such -an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since -it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. +This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such +an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since +it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. -Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together +Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus -the previous carry. Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with +the previous carry. Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with forwarding the carry to the next iteration. -Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. +Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. EXAM,bn_mp_mul_2.c This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference -is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling. +is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling. \subsection{Division by Two} A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. @@ -2014,26 +2003,26 @@ core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent reading past the end of the array of digits. -Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the -least significant bit not the most significant bit. +Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the +least significant bit not the most significant bit. EXAM,bn_mp_div_2.c \section{Polynomial Basis Operations} Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as -the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single +the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer -division and Karatsuba multiplication. +division and Karatsuba multiplication. Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that $y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the -polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. +polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. \subsection{Multiplication by $x$} -Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one +Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to -multiplying by the integer $\beta$. +multiplying by the integer $\beta$. \newpage\begin{figure}[!here] \begin{small} @@ -2064,16 +2053,16 @@ multiplying by the integer $\beta$. \end{figure} \textbf{Algorithm mp\_lshd.} -This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs +This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is -typically used on values where the original value is no longer required. The algorithm will return success immediately if -$b \le 0$ since the rest of algorithm is only valid when $b > 0$. +typically used on values where the original value is no longer required. The algorithm will return success immediately if +$b \le 0$ since the rest of algorithm is only valid when $b > 0$. First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over -the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). -The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on +the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). +The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on step 8 sets the lower $b$ digits to zero. \newpage @@ -2082,14 +2071,14 @@ FIGU,sliding_window,Sliding Window Movement EXAM,bn_mp_lshd.c The if statement (line @24,if@) ensures that the $b$ variable is greater than zero since we do not interpret negative -shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates +shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $top$ (line @42,top@) is an alias -for the leading digit while $bottom$ (line @45,bottom@) is an alias for the trailing edge. The aliases form a -window of exactly $b$ digits over the input. +for the leading digit while $bottom$ (line @45,bottom@) is an alias for the trailing edge. The aliases form a +window of exactly $b$ digits over the input. \subsection{Division by $x$} -Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. +Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. \newpage\begin{figure}[!here] \begin{small} @@ -2122,13 +2111,13 @@ Division by powers of $x$ is easily achieved by shifting the digits right and re \textbf{Algorithm mp\_rshd.} This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since -it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. +it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal to the shift count $b$ then it will simply zero the input and return. After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that -is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. +is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. Also the digits are copied from the leading to the trailing edge. Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. @@ -2141,9 +2130,9 @@ the upper digits of the input to make sure the result is correct. \section{Powers of Two} -Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For +Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single -shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. +shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. \subsection{Multiplication by Power of Two} @@ -2184,29 +2173,29 @@ shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole d This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to quickly compute the product. -First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than -$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ +First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than +$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ left. -After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts -required. If it is non-zero a modified shift loop is used to calculate the remaining product. +After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts +required. If it is non-zero a modified shift loop is used to calculate the remaining product. Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ -variable is used to extract the upper $d$ bits to form the carry for the next iteration. +variable is used to extract the upper $d$ bits to form the carry for the next iteration. -This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to +This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. EXAM,bn_mp_mul_2d.c -The shifting is performed in--place which means the first step (line @24,a != c@) is to copy the input to the +The shifting is performed in--place which means the first step (line @24,a != c@) is to copy the input to the destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then has to be grown (line @31,grow@) to accomodate the result. -If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples -of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift +If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples +of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift loop (lines @45,if@ to @76,}@) we make use of pre--computed values $shift$ and $mask$. These are used to -extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a -chain between consecutive iterations to propagate the carry. +extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a +chain between consecutive iterations to propagate the carry. \subsection{Division by Power of Two} @@ -2244,14 +2233,14 @@ chain between consecutive iterations to propagate the carry. \end{figure} \textbf{Algorithm mp\_div\_2d.} -This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm +This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division by using algorithm mp\_mod\_2d. EXAM,bn_mp_div_2d.c -The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally -ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the +The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally +ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before the quotient is obtained. @@ -2261,7 +2250,7 @@ the direction of the shifts. \subsection{Remainder of Division by Power of Two} The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This -algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. +algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. \begin{figure}[!here] \begin{small} @@ -2293,8 +2282,8 @@ algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (m \end{figure} \textbf{Algorithm mp\_mod\_2d.} -This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the -result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ +This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the +result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. EXAM,bn_mp_mod_2d.c @@ -2303,8 +2292,8 @@ We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in su than the input we just mp\_copy() the input and return right away. After this point we know we must actually perform some work to produce the remainder. -Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce -the number. First we zero any digits above the last digit in $2^b$ (line @41,for@). Next we reduce the +Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce +the number. First we zero any digits above the last digit in $2^b$ (line @41,for@). Next we reduce the leading digit of both (line @45,&=@) and then mp\_clamp(). \section*{Exercises} @@ -2338,40 +2327,40 @@ $\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another \chapter{Multiplication and Squaring} \section{The Multipliers} -For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of -algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction -where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication -and squaring, leaving modular reductions for the subsequent chapter. +For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of +algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction +where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication +and squaring, leaving modular reductions for the subsequent chapter. -The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular +The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular -exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, -35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision +exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, +35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision multiplications. -For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied -against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the -overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in -1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. -This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. +For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied +against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the +overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in +1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. +This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. \section{Multiplication} \subsection{The Baseline Multiplication} \label{sec:basemult} \index{baseline multiplication} Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication -algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision -multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To -simplify most discussions, it will be assumed that the inputs have comparable number of digits. - -The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be -used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important -facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this -modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product -will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. - -Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to -include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The +algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision +multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To +simplify most discussions, it will be assumed that the inputs have comparable number of digits. + +The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be +used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important +facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this +modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product +will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. + +Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to +include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}). \newpage\begin{figure}[!here] @@ -2415,20 +2404,20 @@ Compute the product. \\ \textbf{Algorithm s\_mp\_mul\_digs.} This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem -a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent -algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. -Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the +a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent +algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. +Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the inputs. The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either -input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A -temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to -compute products when either $a = c$ or $b = c$ without overwriting the inputs. +input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A +temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to +compute products when either $a = c$ or $b = c$ without overwriting the inputs. All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm -will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the -innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. +will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the +innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best visualized in the following table. @@ -2442,20 +2431,20 @@ visualized in the following table. && & 5 & 7 & 6 & $10^0(1)(576)$ \\ &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ -\hline +\hline \end{tabular} \end{center} \caption{Long-Hand Multiplication Diagram} \end{figure} -Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate +Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step -5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit -$t_{ix+iy}$ and the result would be lost. +5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit +$t_{ix+iy}$ and the result would be lost. At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result @@ -2463,53 +2452,53 @@ exceed the precision requested. EXAM,bn_s_mp_mul_digs.c -First we determine (line @30,if@) if the Comba method can be used first since it's faster. The conditions for -sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than -\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is +First we determine (line @30,if@) if the Comba method can be used first since it's faster. The conditions for +sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than +\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium. If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int -$t$ (line @36,init@) to the exact size of the output to avoid further re--allocations. At this point we now +$t$ (line @36,init@) to the exact size of the output to avoid further re--allocations. At this point we now begin the $O(n^2)$ loop. This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of -digits as output. In each iteration of the outer loop the $pb$ variable is set (line @48,MIN@) to the maximum -number of inner loop iterations. +digits as output. In each iteration of the outer loop the $pb$ variable is set (line @48,MIN@) to the maximum +number of inner loop iterations. Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the -carry from the previous iteration. A particularly important observation is that most modern optimizing -C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that +carry from the previous iteration. A particularly important observation is that most modern optimizing +C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that is required for the product. In x86 terms for example, this means using the MUL instruction. -Each digit of the product is stored in turn (line @68,tmpt@) and the carry propagated (line @71,>>@) to the +Each digit of the product is stored in turn (line @68,tmpt@) and the carry propagated (line @71,>>@) to the next iteration. \subsection{Faster Multiplication by the ``Comba'' Method} MARK,COMBA -One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be -computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement -in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. -Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an -interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written +One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be +computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement +in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. +Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an +interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written five years before. -At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight -twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products -are produced then added together to form the final result. In the baseline algorithm the columns are added together -after each iteration to get the result instantaneously. +At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight +twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products +are produced then added together to form the final result. In the baseline algorithm the columns are added together +after each iteration to get the result instantaneously. -In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at -the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated -after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute -the product vector $\vec x$ as follows. +In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at +the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated +after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute +the product vector $\vec x$ as follows. \begin{equation} \vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace \end{equation} Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication -of $576$ and $241$. +of $576$ and $241$. \newpage\begin{figure}[here] \begin{small} @@ -2520,15 +2509,15 @@ of $576$ and $241$. \hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ -\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ -\hline +\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ +\hline \end{tabular} \end{center} \end{small} \caption{Comba Multiplication Diagram} \end{figure} -At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. +At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is congruent to adding a leading zero digit. @@ -2551,16 +2540,16 @@ congruent to adding a leading zero digit. \caption{Algorithm Comba Fixup} \end{figure} -With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case +With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case $241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more efficient than the baseline algorithm why not simply always use this algorithm? \subsubsection{Column Weight.} -At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output +At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then -an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is +an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is min$(m, n)$ which is fairly obvious. The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall @@ -2571,7 +2560,7 @@ two quantities we must not violate the following k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} \end{equation} -Which reduces to +Which reduces to \begin{equation} k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} @@ -2584,9 +2573,9 @@ found. k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} \end{equation} -The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration -the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since -$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. +The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration +the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since +$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. \newpage\begin{figure}[!here] \begin{small} @@ -2632,74 +2621,74 @@ Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the s \textbf{Algorithm fast\_s\_mp\_mul\_digs.} This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. -The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the +The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration. The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than $b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable -$ix$ is. This is used for the immediately subsequent statement where we find $iy$. +$ix$ is. This is used for the immediately subsequent statement where we find $iy$. The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each -pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to -move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until +pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to +move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until $tx \ge a.used$ or $ty < 0$ occurs. After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator into the next round by dividing $\_ \hat W$ by $\beta$. -To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the -cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require -$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, +To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the +cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require +$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply -and addition operations in the nested loop in parallel. +and addition operations in the nested loop in parallel. EXAM,bn_fast_s_mp_mul_digs.c As per the pseudo--code we first calculate $pa$ (line @47,MIN@) as the number of digits to output. Next we begin the outer loop to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines @61,tmpx@, @62,tmpy@) to point -inside the two multiplicands quickly. - -The inner loop (lines @70,for@ to @72,}@) of this implementation is where the tradeoff come into play. Originally this comba -implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix -the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write -one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth -is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often -slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the +inside the two multiplicands quickly. + +The inner loop (lines @70,for@ to @72,}@) of this implementation is where the tradeoff come into play. Originally this comba +implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix +the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write +one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth +is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often +slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the compiler has aliased $\_ \hat W$ to a CPU register. -After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines @75,W[ix]@, @78,>>@) to forward it as -a carry for the next pass. After the outer loop we use the final carry (line @82,W[ix]@) as the last digit of the product. +After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines @75,W[ix]@, @78,>>@) to forward it as +a carry for the next pass. After the outer loop we use the final carry (line @82,W[ix]@) as the last digit of the product. \subsection{Polynomial Basis Multiplication} To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms -the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and +the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and $g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. - + The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients requires $O(n^2)$ time and would in practice be slower than the Comba technique. -However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown -coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with -Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in -effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. +However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown +coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with +Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in +effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. -The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since -$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the -fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required +The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since +$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the +fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term -is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product +is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product $W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather -simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. -The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the +simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. +The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. -If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that +If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} +$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that $\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. @@ -2709,10 +2698,10 @@ example, when $n = 2$ and $q = 1$ then following two equations are equivalent to \end{eqnarray} Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the -polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. +polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. -As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of -multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is +As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of +multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is $O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} summarizes the exponents for various values of $n$. @@ -2737,23 +2726,23 @@ summarizes the exponents for various values of $n$. At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large -numbers. +numbers. \subsubsection{Cutoff Point} -The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, +The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the polynomial basis approach more costly to use with small inputs. -Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a -point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and -when $m > y$ the Comba methods are slower than the polynomial basis algorithms. +Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a +point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and +when $m > y$ the Comba methods are slower than the polynomial basis algorithms. The exact location of $y$ depends on several key architectural elements of the computer platform in question. \begin{enumerate} \item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower -the cutoff point $y$ will be. +the cutoff point $y$ will be. \item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This @@ -2766,11 +2755,11 @@ influence over the cutoff point. A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when -a high resolution timer is available. +a high resolution timer is available. \subsection{Karatsuba Multiplication} Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for -general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with +general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. \begin{equation} @@ -2778,8 +2767,8 @@ f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd \end{equation} Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying -this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns -out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points +this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns +out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points $\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$. Consider the resultant system of equations. \begin{center} @@ -2792,7 +2781,7 @@ $\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} -making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. +making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. \newpage\begin{figure}[!here] \begin{small} @@ -2830,7 +2819,7 @@ Calculate the final product. \\ 18. $c \leftarrow t1 + x1y1$ \\ 19. Clear all of the temporary variables. \\ 20. Return(\textit{MP\_OKAY}).\\ -\hline +\hline \end{tabular} \end{center} \end{small} @@ -2839,13 +2828,13 @@ Calculate the final product. \\ \textbf{Algorithm mp\_karatsuba\_mul.} This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description -from Knuth \cite[pp. 294-295]{TAOCPV2}. +from Knuth \cite[pp. 294-295]{TAOCPV2}. \index{radix point} In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must -be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the -smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 -compute the lower halves. Step 6 and 7 computer the upper halves. +be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the +smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 +compute the lower halves. Step 6 and 7 computer the upper halves. After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products $x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed. By using $x0$ instead @@ -2857,29 +2846,29 @@ EXAM,bn_mp_karatsuba_mul.c The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense -to handle error recovery with a single piece of code. Lines @61,if@ to @75,if@ handle initializing all of the temporary variables +to handle error recovery with a single piece of code. Lines @61,if@ to @75,if@ handle initializing all of the temporary variables required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only the temporaries that have been successfully allocated so far. -The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the +The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective number of digits for the next section of code. The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd -to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and -\textbf{sign} members are copied first. The first for loop on line @98,for@ copies the lower halves. Since they are both the same magnitude it -is simpler to calculate both lower halves in a single loop. The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and +to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and +\textbf{sign} members are copied first. The first for loop on line @98,for@ copies the lower halves. Since they are both the same magnitude it +is simpler to calculate both lower halves in a single loop. The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and $y1$ respectively. By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs. When line @152,err@ is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that -the same code that handles errors can be used to clear the temporary variables and return. +the same code that handles errors can be used to clear the temporary variables and return. \subsection{Toom-Cook $3$-Way Multiplication} -Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are -chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, -$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients +Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are +chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, +$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients of the $W(x)$. With the five relations that Toom-Cook specifies, the following system of equations is formed. @@ -2897,7 +2886,7 @@ $\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point -(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. +(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. \begin{figure}[!here] \begin{small} @@ -2967,7 +2956,7 @@ Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ \end{figure} \textbf{Algorithm mp\_toom\_mul.} -This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this +This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across any given step. @@ -2979,34 +2968,34 @@ The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\ to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find $f(y)$ and $g(y)$ which significantly speeds up the algorithm. -After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients +After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients $w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates -that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. +that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. -Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer +Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer result $a \cdot b$ is produced. EXAM,bn_mp_toom_mul.c -The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very +The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this algorithm is not practical as Karatsuba has a much lower cutoff point. -First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines @40,mod@ to @69,rshd@) with +First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines @40,mod@ to @69,rshd@) with combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly -for $b$. +for $b$. Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so we get those out of the way first (lines @72,mul@ and @77,mul@). Next we compute $w1, w2$ and $w3$ using Horners method. After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively -straight forward. +straight forward. \subsection{Signed Multiplication} Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all -of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. +of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. \begin{figure}[!here] \begin{small} @@ -3040,24 +3029,24 @@ of the multiplication algorithms have been unsigned multiplications which leaves \end{figure} \textbf{Algorithm mp\_mul.} -This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms +This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm -s\_mp\_mul\_digs will clear it. +s\_mp\_mul\_digs will clear it. EXAM,bn_mp_mul.c -The implementation is rather simplistic and is not particularly noteworthy. Line @22,?@ computes the sign of the result using the ``?'' -operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. +The implementation is rather simplistic and is not particularly noteworthy. Line @22,?@ computes the sign of the result using the ``?'' +operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. \section{Squaring} \label{sec:basesquare} Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications -performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider -the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, -$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ -and $3 \cdot 1 = 1 \cdot 3$. +performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider +the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, +$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ +and $3 \cdot 1 = 1 \cdot 3$. For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ required for multiplication. The following diagram gives an example of the operations required. @@ -3077,19 +3066,19 @@ $\times$ &&1&2&3&\\ MARK,SQUARE Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ -represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. +represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will -appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double -products and at most one square (\textit{see the exercise section}). +appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double +products and at most one square (\textit{see the exercise section}). -The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, -occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. +The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, +occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. Column two of row one is a square and column three is the first unique column. \subsection{The Baseline Squaring Algorithm} The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines -will not handle. +will not handle. \begin{figure}[!here] \begin{small} @@ -3131,30 +3120,30 @@ will not handle. \textbf{Algorithm s\_mp\_sqr.} This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC -\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the +\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the destination mp\_int to be the same as the source mp\_int. The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate -the carry and compute the double products. +the carry and compute the double products. The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that when it is multiplied by two, it can be properly represented by a mp\_word. -Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial -results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. +Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial +results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. EXAM,bn_s_mp_sqr.c Inside the outer loop (line @32,for@) the square term is calculated on line @35,r =@. The carry (line @42,>>@) has been -extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized +extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized (lines @45,tmpx@ and @48,tmpt@) to simplify the inner loop. The doubling is performed using two -additions (line @57,r + r@) since it is usually faster than shifting, if not at least as fast. +additions (line @57,r + r@) since it is usually faster than shifting, if not at least as fast. The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to -square a number. +square a number. \subsection{Faster Squaring by the ``Comba'' Method} A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional @@ -3164,10 +3153,10 @@ performance hazards. The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, -$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. +$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. -However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two -mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and +However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two +mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind. \newpage\begin{figure}[!here] @@ -3205,7 +3194,7 @@ Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ 9. for $ix$ from $pa$ to $oldused - 1$ do \\ \hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\ 10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ -11. Return(\textit{MP\_OKAY}). \\ +11. Return(\textit{MP\_OKAY}). \\ \hline \end{tabular} \end{center} @@ -3214,8 +3203,8 @@ Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ \end{figure} \textbf{Algorithm fast\_s\_mp\_sqr.} -This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm -s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. +This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm +s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of. First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop @@ -3230,35 +3219,35 @@ only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rf EXAM,bn_fast_s_mp_sqr.c -This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for -the special case of squaring. +This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for +the special case of squaring. \subsection{Polynomial Basis Squaring} The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ -multiplications to find the $\zeta$ relations, squaring operations are performed instead. +multiplications to find the $\zeta$ relations, squaring operations are performed instead. \subsection{Karatsuba Squaring} -Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. -Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a +Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. +Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a number with the following equation. \begin{equation} h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2 \end{equation} -Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$. As in -Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of +Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$. As in +Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of $O \left ( n^{lg(3)} \right )$. -If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm -instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the -time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff -point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. +If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm +instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the +time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff +point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. -Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. +Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication -were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. +were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. \newpage\begin{figure}[!here] \begin{small} @@ -3312,7 +3301,7 @@ Now if $5n$ single precision additions and a squaring of $n$-digits is faster th this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality. Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or -machine clock cycles.}. +machine clock cycles.}. \begin{equation} 5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2 @@ -3330,27 +3319,27 @@ ${13 \over 9}$ & $<$ & $n$ \\ This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a -ratio of 1:7. } than simpler operations such as addition. +ratio of 1:7. } than simpler operations such as addition. EXAM,bn_mp_karatsuba_sqr.c -This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and +This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and shift the input into the two halves. The loop from line @54,{@ to line @70,}@ has been modified since only one input exists. The \textbf{used} count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents -to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. +to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 it is actually below the Comba limit (\textit{at 110 digits}). -This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are -redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and +This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are +redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and mp\_clears are executed normally. \subsection{Toom-Cook Squaring} The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used -instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to -derive their own Toom-Cook squaring algorithm. +instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to +derive their own Toom-Cook squaring algorithm. \subsection{High Level Squaring} \newpage\begin{figure}[!here] @@ -3383,7 +3372,7 @@ derive their own Toom-Cook squaring algorithm. \textbf{Algorithm mp\_sqr.} This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least \textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If -neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. +neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. EXAM,bn_mp_sqr.c @@ -3394,11 +3383,11 @@ $\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix & \\ $\left [ 2 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\ & of double products and at most one square is stated. Prove this statement. \\ - & \\ + & \\ $\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ & \\ $\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ - & \\ + & \\ $\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ & required for equation $6.7$ to be true. \\ & \\ @@ -3416,61 +3405,61 @@ $\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and T MARK,REDUCTION \section{Basics of Modular Reduction} \index{modular residue} -Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, +Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} -modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered +modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered in~\ref{sec:division}. -Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result -$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the +Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result +$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the ``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and -other forms of residues. +other forms of residues. -Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions -is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the -RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in -elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular -exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the -range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check -algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. +Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions +is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the +RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in +elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular +exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the +range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check +algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. \section{The Barrett Reduction} The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate -division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to +division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to \begin{equation} c = a - b \cdot \lfloor a/b \rfloor \end{equation} -Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper -targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, -DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. +Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper +targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, +DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. It would take another common optimization to optimize the algorithm. \subsection{Fixed Point Arithmetic} The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed -point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were -fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit -integer and a $q$-bit fraction part (\textit{where $p+q = k$}). +point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were +fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit +integer and a $q$-bit fraction part (\textit{where $p+q = k$}). In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the -value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by -moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted -to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the -fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. +value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by +moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted +to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the +fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication -of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is -equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer +of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is +equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer $a$ by another integer $b$ can be achieved with the following expression. \begin{equation} \lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor \end{equation} -The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with +The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations -are considerably faster than division on most processors. +are considerably faster than division on most processors. Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and @@ -3491,11 +3480,11 @@ c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough -precision. +precision. -Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and -another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to -reduce the number. +Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and +another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to +reduce the number. For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing $a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. @@ -3503,19 +3492,19 @@ By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (m \subsection{Choosing a Radix Point} Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best -that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. +that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of -the initial multiplication that finds the quotient. +the initial multiplication that finds the quotient. Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent -the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if -two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the +the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if +two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the $m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to -express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then +express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then ${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient is bound by $0 \le {a' \over b} < 1$. -Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits +Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits ``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation @@ -3523,52 +3512,52 @@ with the irrelevant digits trimmed. Now the modular reduction is trimmed to the c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor \end{equation} -Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the -exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor -would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient +Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the +exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor +would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient $\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient -can be off by an additional value of one for a total of at most two. This implies that -$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting +can be off by an additional value of one for a total of at most two. This implies that +$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting $b$ once or twice the residue is found. The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single -precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. +precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. This is considerably faster than the original attempt. -For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ -represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. -With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ -is found. +For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ +represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. +With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then +$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ +is found. \subsection{Trimming the Quotient} -So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As +So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for -optimization. +optimization. After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower -half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision -multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. -In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. +half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision +multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. +In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number -of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. +of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. \subsection{Trimming the Residue} After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small -multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the +multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are -implicitly zero. +implicitly zero. The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full $O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can -be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces -only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. +be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces +only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which -is considerably faster than the straightforward $3m^2$ method. +is considerably faster than the straightforward $3m^2$ method. \subsection{The Barrett Algorithm} \newpage\begin{figure}[!here] @@ -3612,26 +3601,26 @@ Now subtract the modulus if the residue is too large (e.g. quotient too small). \textbf{Algorithm mp\_reduce.} This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC -\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must +\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must be adhered to for the algorithm to work. First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. -Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this -algorithm and is assumed to be calculated and stored before the algorithm is used. +Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this +algorithm and is assumed to be calculated and stored before the algorithm is used. -Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called +Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called $s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number -of digits in $b$ is very much smaller than $\beta$. +of digits in $b$ is very much smaller than $\beta$. -While it is known that -$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied -``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be -fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. +While it is known that +$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied +``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be +fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. -The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is +The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. EXAM,bn_mp_reduce.c @@ -3639,11 +3628,11 @@ EXAM,bn_mp_reduce.c The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits in the modulus. In the source code this is evaluated on lines @36,if@ to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is -safe to do so. +safe to do so. \subsection{The Barrett Setup Algorithm} In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for -future use so that the Barrett algorithm can be used without delay. +future use so that the Barrett algorithm can be used without delay. \newpage\begin{figure}[!here] \begin{small} @@ -3670,24 +3659,24 @@ is equivalent and much faster. The final value is computed by taking the intege EXAM,bn_mp_reduce_setup.c This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable -which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the -remainder to be passed as NULL meaning to ignore the value. +which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the +remainder to be passed as NULL meaning to ignore the value. \section{The Montgomery Reduction} -Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting -form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a -residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. +Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting +form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a +residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of $n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. \textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way -to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. +to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. \textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually -this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to -multiplication by $k^{-1}$ modulo $n$. +this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to +multiplication by $k^{-1}$ modulo $n$. From these two simple facts the following simple algorithm can be derived. @@ -3713,8 +3702,8 @@ From these two simple facts the following simple algorithm can be derived. The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since -$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the -final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to +$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the +final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to $0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. \begin{figure}[here] @@ -3739,12 +3728,12 @@ $0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction \label{fig:MONT1} \end{figure} -Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$). The result of -the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^9$ modulo $257$ the correct residue -$r \equiv 158$ is produced. +Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$). The result of +the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^9$ modulo $257$ the correct residue +$r \equiv 158$ is produced. Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts -and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. +and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. Fortunately there exists an alternative representation of the algorithm. \begin{figure}[!here] @@ -3793,10 +3782,10 @@ precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a s \label{fig:MONT2} \end{figure} -Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. -With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the -loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is -zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. +Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. +With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the +loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is +zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. \subsection{Digit Based Montgomery Reduction} Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the @@ -3820,9 +3809,9 @@ previous algorithm re-written to compute the Montgomery reduction in this new fa \caption{Algorithm Montgomery Reduction (modified II)} \end{figure} -The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of +The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This -problem breaks down to solving the following congruency. +problem breaks down to solving the following congruency. \begin{center} \begin{tabular}{rcl} @@ -3832,10 +3821,10 @@ $\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ \end{tabular} \end{center} -In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used -extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. +In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used +extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. -For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ +For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ represent the value to reduce. \newpage\begin{figure} @@ -3851,14 +3840,14 @@ represent the value to reduce. \caption{Example of Montgomery Reduction} \end{figure} -The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ +The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and -the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. +the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. \subsection{Baseline Montgomery Reduction} -The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for -Montgomery reductions. +The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for +Montgomery reductions. \newpage\begin{figure}[!here] \begin{small} @@ -3906,9 +3895,9 @@ Divide by $\beta^k$ and fix up as required. \\ \textbf{Algorithm mp\_montgomery\_reduce.} This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The -restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as +restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in -advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. +advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on the size of the input. This algorithm is discussed in ~COMBARED~. @@ -3917,17 +3906,17 @@ Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calcu calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. -Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications +Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision -multiplications. +multiplications. EXAM,bn_mp_montgomery_reduce.c This is the baseline implementation of the Montgomery reduction algorithm. Lines @30,digs@ to @35,}@ determine if the Comba based -routine can be used instead. Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop. +routine can be used instead. Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop. The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and -the alias $tmpn$ refers to the modulus $n$. +the alias $tmpn$ refers to the modulus $n$. \subsection{Faster ``Comba'' Montgomery Reduction} MARK,COMBARED @@ -3935,14 +3924,14 @@ MARK,COMBARED The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates -a $k \times 1$ product $k$ times. +a $k \times 1$ product $k$ times. -The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the -carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. -Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. +The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the +carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. +Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases -the speed of the algorithm. +the speed of the algorithm. \newpage\begin{figure}[!here] \begin{small} @@ -3991,9 +3980,9 @@ Zero excess digits and fixup $x$. \\ \textbf{Algorithm fast\_mp\_montgomery\_reduce.} This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions -on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the +on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo -a modulus of at most $3,556$ bits in length. +a modulus of at most $3,556$ bits in length. As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step @@ -4007,23 +3996,23 @@ how the upper bits of those same words are not reduced modulo $\beta$. This is point. Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are -stored in the destination $x$. +stored in the destination $x$. EXAM,bn_fast_mp_montgomery_reduce.c The $\hat W$ array is first filled with digits of $x$ on line @49,for@ then the rest of the digits are zeroed on line @54,for@. Both loops share -the same alias variables to make the code easier to read. +the same alias variables to make the code easier to read. The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This -forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line @101,>>@ fixes the carry +forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line @101,>>@ fixes the carry for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. The for loop on line @113,for@ propagates the rest of the carries upwards through the columns. The for loop on line @126,for@ reduces the columns modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th -digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. +digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. \subsection{Montgomery Setup} -To calculate the variable $\rho$ a relatively simple algorithm will be required. +To calculate the variable $\rho$ a relatively simple algorithm will be required. \begin{figure}[!here] \begin{small} @@ -4044,17 +4033,17 @@ To calculate the variable $\rho$ a relatively simple algorithm will be required. \end{tabular} \end{center} \end{small} -\caption{Algorithm mp\_montgomery\_setup} +\caption{Algorithm mp\_montgomery\_setup} \end{figure} \textbf{Algorithm mp\_montgomery\_setup.} -This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick -to calculate $1/n_0$ when $\beta$ is a power of two. +This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick +to calculate $1/n_0$ when $\beta$ is a power of two. EXAM,bn_mp_montgomery_setup.c This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess -multiplications when $\beta$ is not the default 28-bits. +multiplications when $\beta$ is not the default 28-bits. \section{The Diminished Radix Algorithm} The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett @@ -4064,7 +4053,7 @@ or Montgomery methods for certain forms of moduli. The technique is based on th (x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} \end{equation} -This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that +This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof of the above equation is very simple. First write $x$ in the product form. @@ -4078,7 +4067,7 @@ Now reduce both sides modulo $(n - k)$. x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} \end{equation} -The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ +The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. \begin{figure}[!here] @@ -4108,7 +4097,7 @@ into the equation the original congruence is reproduced, thus concluding the pro This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. -\begin{equation} +\begin{equation} 0 \le x < n^2 + k^2 - 2nk \end{equation} @@ -4119,15 +4108,15 @@ q < n - 2k - k^2/n \end{equation} Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as -$0 \le x < n$. By step four the sum $x + q$ is bounded by +$0 \le x < n$. By step four the sum $x + q$ is bounded by \begin{equation} 0 \le q + x < (k + 1)n - 2k^2 - 1 \end{equation} With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the -sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the -range $0 \le x < (n - k - 1)^2$. +sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the +range $0 \le x < (n - k - 1)^2$. \begin{figure} \begin{small} @@ -4140,13 +4129,13 @@ $q \leftarrow q*k = 1446759$ \\ $x \leftarrow x \mbox{ mod } n = 21$ \\ $x \leftarrow x + q = 1446780$ \\ $x \leftarrow x - (n - k) = 1446527$ \\ -\hline +\hline $q \leftarrow \lfloor x/n \rfloor = 5650$ \\ $q \leftarrow q*k = 16950$ \\ $x \leftarrow x \mbox{ mod } n = 127$ \\ $x \leftarrow x + q = 17077$ \\ $x \leftarrow x - (n - k) = 16824$ \\ -\hline +\hline $q \leftarrow \lfloor x/n \rfloor = 65$ \\ $q \leftarrow q*k = 195$ \\ $x \leftarrow x \mbox{ mod } n = 184$ \\ @@ -4169,29 +4158,29 @@ three passes were required to find the residue $x \equiv 126$. On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. -Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. -Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division -by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ -which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. +Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. +Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division +by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ +which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. -However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be -performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. -Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. +However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be +performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. +Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted -modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the -$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. +modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the +$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. \subsection{Choice of $k$} Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might -as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. +as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. \subsection{Restricted Diminished Radix Reduction} -The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce +The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation -of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition -of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular +of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition +of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular exponentiations are performed. \newpage\begin{figure}[!here] @@ -4227,31 +4216,31 @@ exponentiations are performed. \textbf{Algorithm mp\_dr\_reduce.} This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction -with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. +with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th -digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to -$x$ before the addition of the multiple of the upper half. +digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to +$x$ before the addition of the multiple of the upper half. At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes -at step 3. +at step 3. EXAM,bn_mp_dr_reduce.c The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line @49,top:@ is where the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of -the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. +the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits a division by $\beta^m$ can be simulated virtually for free. The loop on line @61,for@ performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) in this algorithm. -By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line @71,for@ the -same pointer will point to the $m+1$'th digit where the zeroes will be placed. +By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line @71,for@ the +same pointer will point to the $m+1$'th digit where the zeroes will be placed. -Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. +Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. With the same logic at line @82,sub@ the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code does not need to be checked. @@ -4342,20 +4331,20 @@ algorithm is much faster than either Montgomery or Barrett reduction when the mo \textbf{Algorithm mp\_reduce\_2k.} This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right -shift which makes the algorithm fairly inexpensive to use. +shift which makes the algorithm fairly inexpensive to use. EXAM,bn_mp_reduce_2k.c The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d on line @31,mp_div_2d@ calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without -any multiplications. +any multiplications. -The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are -positive. By using the unsigned versions the overhead is kept to a minimum. +The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are +positive. By using the unsigned versions the overhead is kept to a minimum. \subsubsection{Unrestricted Setup} -To setup this reduction algorithm the value of $k = 2^p - n$ is required. +To setup this reduction algorithm the value of $k = 2^p - n$ is required. \begin{figure}[!here] \begin{small} @@ -4379,7 +4368,7 @@ To setup this reduction algorithm the value of $k = 2^p - n$ is required. \textbf{Algorithm mp\_reduce\_2k\_setup.} This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction -is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. +is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. EXAM,bn_mp_reduce_2k_setup.c @@ -4418,7 +4407,7 @@ significant bit. The resulting sum will be a power of two. \end{figure} \textbf{Algorithm mp\_reduce\_is\_2k.} -This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. +This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. EXAM,bn_mp_reduce_is_2k.c @@ -4427,7 +4416,7 @@ EXAM,bn_mp_reduce_is_2k.c \section{Algorithm Comparison} So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since -all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. +all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. \begin{center} \begin{small} @@ -4463,12 +4452,12 @@ $\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix R & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ & terminate within $1 \le k \le 10$ iterations. \\ & \\ -\end{tabular} +\end{tabular} \chapter{Exponentiation} Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed -in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key +in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any such cryptosystem and many methods have been sought to speed it up. @@ -4478,7 +4467,7 @@ the number of multiplications becomes prohibitive. Imagine what would happen if with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which -are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least +are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least significant bit. If $b$ is a $k$-bit integer than the following equation is true. \begin{equation} @@ -4495,7 +4484,7 @@ The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term si $a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average $k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. -While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to +While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to be computed in an auxilary variable. Consider the following equivalent algorithm. \begin{figure}[!here] @@ -4521,7 +4510,7 @@ be computed in an auxilary variable. Consider the following equivalent algorith This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the -product. +product. For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. @@ -4542,13 +4531,13 @@ For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrate \caption{Example of Left to Right Exponentiation} \end{figure} -When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is -called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. +When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is +called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. \subsection{Single Digit Exponentiation} -The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended -to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of -$b$ that are greater than three. +The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended +to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of +$b$ that are greater than three. \newpage\begin{figure}[!here] \begin{small} @@ -4576,10 +4565,10 @@ $b$ that are greater than three. \textbf{Algorithm mp\_expt\_d.} This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to -quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the -exponent is a fixed width. +quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the +exponent is a fixed width. -A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of +A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of $1$ in the subsequent step. Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared @@ -4587,12 +4576,12 @@ on step 3.1. In the following step if the most significant bit of $b$ is one th of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. -EXAM,bn_mp_expt_d.c +EXAM,bn_mp_expt_d_ex.c -Line @29,mp_set@ sets the initial value of the result to $1$. Next the loop on line @31,for@ steps through each bit of the exponent starting from -the most significant down towards the least significant. The invariant squaring operation placed on line @333,mp_sqr@ is performed first. After +This describes only the algorithm that is used when the parameter $fast$ is $0$. Line @31,mp_set@ sets the initial value of the result to $1$. Next the loop on line @54,for@ steps through each bit of the exponent starting from +the most significant down towards the least significant. The invariant squaring operation placed on line @333,mp_sqr@ is performed first. After the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line -@47,<<@ moves all of the bits of the exponent upwards towards the most significant location. +@69,<<@ moves all of the bits of the exponent upwards towards the most significant location. \section{$k$-ary Exponentiation} When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor @@ -4625,7 +4614,7 @@ portion of the entire exponent. Consider the following modification to the basi The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and -$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. +$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The @@ -4635,7 +4624,7 @@ has increased slightly but the number of multiplications has nearly halved. \subsection{Optimal Values of $k$} An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ -for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. +for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. \begin{figure}[here] \begin{center} @@ -4661,10 +4650,10 @@ for various exponent sizes and compares the number of multiplication and squarin \subsection{Sliding-Window Exponentiation} A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially -this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the -algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. +this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the +algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. -Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. +Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm {\ref{fig:KARY}}. \begin{figure}[here] \begin{center} @@ -4715,29 +4704,29 @@ Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half -the size as the previous table. +the size as the previous table. -Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as -the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the +Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as +the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where -a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ -squarings. The second method requires $8$ multiplications and $18$ squarings. +a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ +squarings. The second method requires $8$ multiplications and $18$ squarings. -In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. +In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. \section{Modular Exponentiation} -Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing -$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it -modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. +Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing +$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it +modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using -one of the algorithms presented in ~REDUCTION~. +one of the algorithms presented in ~REDUCTION~. Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm -terminates with an error. +terminates with an error. \begin{figure}[!here] \begin{small} @@ -4764,10 +4753,10 @@ terminates with an error. \end{figure} \textbf{Algorithm mp\_exptmod.} -The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm -which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation +The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm +which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation -algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). +algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). EXAM,bn_mp_exptmod.c @@ -4776,7 +4765,7 @@ negative the algorithm tries to perform a modular exponentiation with the modula the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive exponent. -If the exponent is positive the algorithm resumes the exponentiation. Line @63,dr_@ determines if the modulus is of the restricted Diminished Radix +If the exponent is positive the algorithm resumes the exponentiation. Line @63,dr_@ determines if the modulus is of the restricted Diminished Radix form. If it is not line @65,reduce@ attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one of three values. @@ -4787,7 +4776,7 @@ of three values. \end{enumerate} Line @69,if@ determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, -the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. +the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. \subsection{Barrett Modular Exponentiation} @@ -4892,9 +4881,9 @@ No more windows left. Check for residual bits of exponent. \\ This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction algorithm to keep the product small throughout the algorithm. -The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the +The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This -table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. +table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ @@ -4902,49 +4891,49 @@ times. The rest of the table elements are found by multiplying the previous ele Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. \begin{enumerate} -\item The variable $mode$ dictates how the bits of the exponent are interpreted. +\item The variable $mode$ dictates how the bits of the exponent are interpreted. \begin{enumerate} - \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply - $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. - \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits - are read and a single squaring is performed. If a non-zero bit is read a new window is created. + \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply + $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. + \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits + are read and a single squaring is performed. If a non-zero bit is read a new window is created. \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit downwards. \end{enumerate} \item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit is fetched from the exponent. -\item The variable $buf$ holds the currently read digit of the exponent. +\item The variable $buf$ holds the currently read digit of the exponent. \item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. \item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and the appropriate operations performed. -\item The variable $bitbuf$ holds the current bits of the window being formed. +\item The variable $bitbuf$ holds the current bits of the window being formed. \end{enumerate} All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is -read and if there are no digits left than the loop terminates. +read and if there are no digits left than the loop terminates. After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit -upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to +upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to trailing edges the entire exponent is read from most significant bit to least significant bit. -At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the +At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle -the two cases of $mode = 1$ and $mode = 2$ respectively. +the two cases of $mode = 1$ and $mode = 2$ respectively. FIGU,expt_state,Sliding Window State Diagram -By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then -a Left-to-Right algorithm is used to process the remaining few bits. +By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then +a Left-to-Right algorithm is used to process the remaining few bits. EXAM,bn_s_mp_exptmod.c Lines @31,if@ through @45,}@ determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted -from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement -on line @37,if@ the value of $x$ is already known to be greater than $140$. +from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement +on line @37,if@ the value of $x$ is already known to be greater than $140$. The conditional piece of code beginning on line @42,ifdef@ allows the window size to be restricted to five bits. This logic is used to ensure -the table of precomputed powers of $G$ remains relatively small. +the table of precomputed powers of $G$ remains relatively small. The for loop on line @60,for@ initializes the $M$ array while lines @71,mp_init@ and @75,mp_reduce@ through @85,}@ initialize the reduction function that will be used for this modulus. @@ -4982,11 +4971,11 @@ EXAM,bn_mp_2expt.c \chapter{Higher Level Algorithms} This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These -routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. +routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic -for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. -These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate +for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. +These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate various representations of integers. For example, converting from an mp\_int to a string of character. \section{Integer Division with Remainder} @@ -4994,7 +4983,7 @@ various representations of integers. For example, converting from an mp\_int to Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables -will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and +will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. \newpage\begin{figure}[!here] @@ -5024,42 +5013,42 @@ let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The followin As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. -To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and +To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading -digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly -arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. +digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly +arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. -Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder +Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder $y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the -remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since -$237 \cdot 23 + 20 = 5471$ is true. +remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since +$237 \cdot 23 + 20 = 5471$ is true. \subsection{Quotient Estimation} \label{sec:divest} As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the -dividend and divisor are zero. +dividend and divisor are zero. The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} -of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate -using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ +of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate +using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ represent the most significant digits of the dividend and divisor respectively. -\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to +\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to $k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } -The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other +The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility -$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of +$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of inequalities will prove the hypothesis. \begin{equation} y - \hat k x \le y - \hat k x_s\beta^s \end{equation} -This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. +This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. \begin{equation} y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) @@ -5084,13 +5073,13 @@ Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which co For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both $x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will -lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. +lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. -\begin{equation} -{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} +\begin{equation} +{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} \end{equation} -At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. +At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. \subsection{Radix-$\beta$ Division with Remainder} \newpage\begin{figure}[!here] @@ -5188,23 +5177,23 @@ Finalize the result. \\ This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed division and will produce a fully qualified quotient and remainder. -First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly -zero and the remainder is the dividend. +First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly +zero and the remainder is the dividend. After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are -positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. -This is performed by shifting both to the left by enough bits to get the desired normalization. +positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. +This is performed by shifting both to the left by enough bits to get the desired normalization. -At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is +At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is $2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted -to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the +to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two -times to produce the desired leading digit of the quotient. +times to produce the desired leading digit of the quotient. Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by -induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. +induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher @@ -5212,51 +5201,51 @@ order approximation to adjust the quotient digit. After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of -algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. +algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. -Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the +Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} -is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie +is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should -respectively be replaced with a zero. +respectively be replaced with a zero. EXAM,bn_mp_div.c The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division -algorithm with only the quotient is +algorithm with only the quotient is \begin{verbatim} mp_div(&a, &b, &c, NULL); /* c = [a/b] */ \end{verbatim} -Lines @108,if@ and @113,if@ handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor -respectively. After the two trivial cases all of the temporary variables are initialized. Line @147,neg@ determines the sign of -the quotient and line @148,sign@ ensures that both $x$ and $y$ are positive. +Lines @108,if@ and @113,if@ handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor +respectively. After the two trivial cases all of the temporary variables are initialized. Line @147,neg@ determines the sign of +the quotient and line @148,sign@ ensures that both $x$ and $y$ are positive. The number of bits in the leading digit is calculated on line @151,norm@. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting them to the left by $lg(\beta) - 1 - k$ bits. -Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the +Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the leading digit of the quotient. The loop beginning on line @184,for@ will produce the remainder of the quotient digits. The conditional ``continue'' on line @186,continue@ is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits -above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. +above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. -Lines @214,t1@, @216,t1@ and @222,t2@ through @225,t2@ manually construct the high accuracy estimations by setting the digits of the two mp\_int -variables directly. +Lines @214,t1@, @216,t1@ and @222,t2@ through @225,t2@ manually construct the high accuracy estimations by setting the digits of the two mp\_int +variables directly. \section{Single Digit Helpers} -This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of +This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of the helper functions assume the single digit input is positive and will treat them as such. \subsection{Single Digit Addition and Subtraction} -Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction +Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. \newpage\begin{figure}[!here] @@ -5322,17 +5311,17 @@ only has one digit. \caption{Algorithm mp\_mul\_d} \end{figure} \textbf{Algorithm mp\_mul\_d.} -This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. -Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. +This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. +Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. EXAM,bn_mp_mul_d.c -In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is -read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. +In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is +read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. \subsection{Single Digit Division} Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the -divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. +divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. \newpage\begin{figure}[!here] \begin{small} @@ -5369,35 +5358,35 @@ divisor is only a single digit a specialized variant of the division algorithm c \textbf{Algorithm mp\_div\_d.} This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ -after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. +after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction -from chapter seven. +from chapter seven. EXAM,bn_mp_div_d.c Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. -The division and remainder on lines @44,/@ and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based -processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC -compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. +The division and remainder on lines @90,/@ and @91,-@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based +processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC +compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. \subsection{Single Digit Root Extraction} -Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation -(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. +Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation +(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. \begin{equation} x_{i+1} = x_i - {f(x_i) \over f'(x_i)} \label{eqn:newton} \end{equation} -In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is +In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain -such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the -algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. +such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the +algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. \newpage\begin{figure}[!here] \begin{small} @@ -5438,19 +5427,19 @@ algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$ \textbf{Algorithm mp\_n\_root.} This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator. That is at first the denominator is calculated by finding -$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ -multiplications by t$1$ inside the loop. +$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ +multiplications by t$1$ inside the loop. The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the -root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. +root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. EXAM,bn_mp_n_root.c \section{Random Number Generation} -Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho +Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented -is solely for simulations and not intended for cryptographic use. +is solely for simulations and not intended for cryptographic use. \newpage\begin{figure}[!here] \begin{small} @@ -5478,7 +5467,7 @@ is solely for simulations and not intended for cryptographic use. \textbf{Algorithm mp\_rand.} This algorithm produces a pseudo-random integer of $b$ digits. By ensuring that the first digit is non-zero the algorithm also guarantees that the final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of -the integers from $0$ to $\beta - 1$. +the integers from $0$ to $\beta - 1$. EXAM,bn_mp_rand.c @@ -5488,7 +5477,7 @@ be given a string of characters such as ``114585'' and turn it into the radix-$\ into a program. \subsection{Reading Radix-n Input} -For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to +For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary @@ -5498,7 +5487,7 @@ mediums. \begin{center} \begin{tabular}{cc|cc|cc|cc} \hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\ -\hline +\hline 0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\ 4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ 8 & 8 & 9 & 9 & 10 & A & 11 & B \\ @@ -5552,7 +5541,7 @@ mediums. \caption{Algorithm mp\_read\_radix} \end{figure} \textbf{Algorithm mp\_read\_radix.} -This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the +This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the string to indicate the value is negative, otherwise it is assumed to be positive. The algorithm will read up to $sn$ characters from the input and will stop when it reads a character it cannot map the algorithm stops reading characters from the string. This allows numbers to be embedded as part of larger input without any significant problem. @@ -5560,7 +5549,7 @@ as part of larger input without any significant problem. EXAM,bn_mp_read_radix.c \subsection{Generating Radix-$n$ Output} -Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. +Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. \newpage\begin{figure}[!here] \begin{small} @@ -5594,10 +5583,10 @@ Generating radix-$n$ output is fairly trivial with a division and remainder algo \caption{Algorithm mp\_toradix} \end{figure} \textbf{Algorithm mp\_toradix.} -This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing +This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$. Note that instead of actually dividing by $r^k$ in each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration. As a result a series of trivial $n \times 1$ divisions -are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order +are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order (see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''. \begin{figure} @@ -5619,7 +5608,7 @@ are required instead of a series of $n \times k$ divisions. One design flaw of EXAM,bn_mp_toradix.c \chapter{Number Theoretic Algorithms} -This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi +This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and various Sieve based factoring algorithms. @@ -5629,7 +5618,7 @@ both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mb simultaneously. The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then -$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. +$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. \newpage\begin{figure}[!here] \begin{small} @@ -5653,9 +5642,9 @@ $r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \rig \end{figure} This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are -relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of -greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. -In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. +relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of +greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. +In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. \begin{figure}[!here] \begin{small} @@ -5679,17 +5668,17 @@ In particular, we would like $a - b$ to decrease in magnitude which implies that \textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.} The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$. In other -words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always -divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the +words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always +divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}. -As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that +As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that $b - a$ is still very much larger than $a$. A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does not divide the greatest common divisor but will divide $b - a$. In this case ${b - a} \over p$ is also an integer and still divisible by the greatest common divisor. -However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first. -Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. +However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first. +Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. \begin{figure}[!here] \begin{small} @@ -5722,14 +5711,14 @@ Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can b \label{fig:gcd3} \end{figure} -This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ +This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ decreases more rapidly. The first loop on step two removes powers of $p$ that are in common. A count, $k$, is kept which will present a common -divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely -divided out of the difference $b - a$ so long as the division leaves no remainder. +divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely +divided out of the difference $b - a$ so long as the division leaves no remainder. In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often. It also helps that division by $p$ be easy to compute. The ideal choice of $p$ is two since division by two amounts to a right logical shift. Another important observation is that by -step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the +step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the largest of the pair. \subsection{Complete Greatest Common Divisor} @@ -5777,15 +5766,15 @@ and will produce the greatest common divisor. \textbf{Algorithm mp\_gcd.} This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as -Algorithm B and in practice this appears to be true. +Algorithm B and in practice this appears to be true. -The first two steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the -largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of +The first two steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the +largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of $a$ and $b$ respectively and the algorithm will proceed to reduce the pair. Step five will divide out any common factors of two and keep track of the count in the variable $k$. After this step, two is no longer a -factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step -six and seven ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while--loops will iterate since +factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step +six and seven ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while--loops will iterate since they cannot both be even. By step eight both of $u$ and $v$ are odd which is required for the inner logic. First the pair are swapped such that $v$ is equal to @@ -5793,22 +5782,22 @@ or greater than $u$. This ensures that the subtraction on step 8.2 will always factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd. After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six. The result -must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. +must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. EXAM,bn_mp_gcd.c -This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the +This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise -it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three +it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three trivial cases of inputs are handled on lines @23,zero@ through @29,}@. After those lines the inputs are assumed to be non-zero. -Lines @32,if@ and @36,if@ make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two +Lines @32,if@ and @36,if@ make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two must be divided out of the two inputs. The block starting at line @43,common@ removes common factors of two by first counting the number of trailing -zero bits in both. The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values. It is assumed that -the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than -entries than are accessible by an ``int'' so this is not a limitation.}. +zero bits in both. The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values. It is assumed that +the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than +entries than are accessible by an ``int'' so this is not a limitation.}. -At this point there are no more common factors of two in the two values. The divisions by a power of two on lines @60,div_2d@ and @67,div_2d@ remove +At this point there are no more common factors of two in the two values. The divisions by a power of two on lines @60,div_2d@ and @67,div_2d@ remove any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm. The while loop on line @72, while@ performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative. @@ -5819,8 +5808,8 @@ least common multiple is normally denoted as $[ a, b ]$ and numerically equivale and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$. The least common multiple arises often in coding theory as well as number theory. If two functions have periods of $a$ and $b$ respectively they will -collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on -Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). +collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on +Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $. \begin{figure}[!here] @@ -5848,7 +5837,7 @@ dividing the product of the two inputs by their greatest common divisor. EXAM,bn_mp_lcm.c \section{Jacobi Symbol Computation} -To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is +To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is equivalent to equation \ref{eqn:legendre}. @@ -5858,9 +5847,9 @@ equivalent to equation \ref{eqn:legendre}. a^{(p-1)/2} \equiv \begin{array}{rl} -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ 0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\ - 1 & \mbox{if }a\mbox{ is a quadratic residue}. + 1 & \mbox{if }a\mbox{ is a quadratic residue}. \end{array} \mbox{ (mod }p\mbox{)} -\label{eqn:legendre} +\label{eqn:legendre} \end{equation} \textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.} @@ -5884,7 +5873,7 @@ then the quantity in the braces must be zero. By reduction, \begin{eqnarray} \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\ \left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\ -x^2 \equiv a \mbox{ (mod }p\mbox{)} +x^2 \equiv a \mbox{ (mod }p\mbox{)} \end{eqnarray} As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue. If $a$ does not divide $p$ and $a$ @@ -5904,47 +5893,47 @@ the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equ By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function. The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for further details.} will be used to derive an efficient Jacobi symbol algorithm. Where $p$ is an odd integer greater than two and $a, b \in \Z$ the -following are true. +following are true. \begin{enumerate} -\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. +\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. \item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$. \item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$. \item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$. -\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically -$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. +\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically +$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. \end{enumerate} Using these facts if $a = 2^k \cdot a'$ then \begin{eqnarray} \left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\ - = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) + = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) \label{eqn:jacobi} \end{eqnarray} -By fact five, +By fact five, \begin{equation} -\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} \end{equation} -Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then +Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then \begin{equation} -\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} +\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} \end{equation} By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed. \begin{equation} -\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4} +\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4} \end{equation} -The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of -$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the -factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the -Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. +The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of +$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the +factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the +Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. \newpage\begin{figure}[!here] \begin{small} @@ -5988,12 +5977,12 @@ Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$ \end{figure} \textbf{Algorithm mp\_jacobi.} This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm -is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. +is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively. Step five determines the number of two factors in the -input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one -if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled -the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$ +input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one +if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled +the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$ are congruent to one modulo four, otherwise it evaluates to negative one. By step nine if $a'$ does not equal one a recursion is required. Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute @@ -6001,22 +5990,22 @@ $\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi EXAM,bn_mp_jacobi.c -As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C -variable name character. +As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C +variable name character. The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm. If the input is non-trivial the algorithm has to proceed compute the Jacobi. The variable $s$ is used to hold the current Jacobi product. Note that $s$ is merely a C ``int'' data type since -the values it may obtain are merely $-1$, $0$ and $1$. +the values it may obtain are merely $-1$, $0$ and $1$. After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$. Technically only the least significant -bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same +bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same processor requirements and neither is faster than the other. Line @59, if@ through @70, }@ determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than $k$ is even and the value is one. Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight. The value of -$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines @73, if@ through @75, }@. +$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines @73, if@ through @75, }@. -Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. +Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. \textit{-- Comment about default $s$ and such...} @@ -6024,31 +6013,31 @@ Finally, if $a1$ does not equal one the algorithm must recurse and compute $\lef \label{sec:modinv} The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$. The integer $b$ is called the multiplicative inverse of $a$ which is -denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and +denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and fields of integers. However, the former will be the matter of discussion. -The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the +The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial. \begin{equation} ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)} \end{equation} -However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite -requires all of the prime factors. This approach also is very slow as the size of $p$ grows. +However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite +requires all of the prime factors. This approach also is very slow as the size of $p$ grows. -A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear +A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation. \begin{equation} ab + pq = 1 \end{equation} -Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of -$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. +Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of +$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The -binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine -equation. +binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine +equation. \subsection{General Case} \newpage\begin{figure}[!here] @@ -6100,12 +6089,12 @@ equation. \end{small} \end{figure} \textbf{Algorithm mp\_invmod.} -This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the +This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete -Diophantine solution. +Diophantine solution. If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative -inverse for $a$ and the error is reported. +inverse for $a$ and the error is reported. The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd. In this case the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is @@ -6115,8 +6104,8 @@ Ca + Db = v \end{equation} If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists. Otherwise, $C$ -is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie -within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$ +is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie +within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$ then only a couple of additions or subtractions will be required to adjust the inverse. EXAM,bn_mp_invmod.c @@ -6124,22 +6113,22 @@ EXAM,bn_mp_invmod.c \subsubsection{Odd Moduli} When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse. In particular by attempting to solve -the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. +the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. -The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This +The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This optimization will halve the time required to compute the modular inverse. \section{Primality Tests} -A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime -since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. +A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime +since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is -prime the algorithm may be incorrect. +prime the algorithm may be incorrect. -As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as +As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question. \subsection{Trial Division} @@ -6153,13 +6142,13 @@ of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by -$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range -$3 \le q \le 100$. +$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range +$3 \le q \le 100$. -At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to -be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate -approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The -array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. +At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to +be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate +approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The +array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. \begin{figure}[!here] \begin{small} @@ -6183,27 +6172,27 @@ array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers \caption{Algorithm mp\_prime\_is\_divisible} \end{figure} \textbf{Algorithm mp\_prime\_is\_divisible.} -This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. +This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. EXAM,bn_mp_prime_is_divisible.c -The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a +The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a mp\_digit. The table \_\_prime\_tab is defined in the following file. EXAM,bn_prime_tab.c Note that there are two possible tables. When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes -upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. +upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. \subsection{The Fermat Test} -The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in +The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of -the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to -$a^1 = a$. +the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to +$a^1 = a$. -If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case +If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$. However, this test is not absolute as it is possible that the order -of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several +of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows in size. @@ -6229,13 +6218,13 @@ in size. \end{figure} \textbf{Algorithm mp\_prime\_fermat.} This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not. It uses a single modular exponentiation to -determine the result. +determine the result. EXAM,bn_mp_prime_fermat.c \subsection{The Miller-Rabin Test} -The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen -candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the +The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen +candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime. @@ -6271,11 +6260,11 @@ some value not congruent to $\pm 1$ when squared equals one which cannot occur i \end{figure} \textbf{Algorithm mp\_prime\_miller\_rabin.} This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$. It will set $c = 1$ if the algorithm cannot determine -if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. +if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not. Otherwise, the algorithm will square $y$ upto $s - 1$ times stopping only when $y \equiv -1$. If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$ -is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably +is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably composite then it is \textit{probably} prime. EXAM,bn_mp_prime_miller_rabin.c diff --git a/libtommath/tommath.tex b/libtommath/tommath.tex deleted file mode 100644 index c79a537..0000000 --- a/libtommath/tommath.tex +++ /dev/null @@ -1,6691 +0,0 @@ -\documentclass[b5paper]{book} -\usepackage{hyperref} -\usepackage{makeidx} -\usepackage{amssymb} -\usepackage{color} -\usepackage{alltt} -\usepackage{graphicx} -\usepackage{layout} -\def\union{\cup} -\def\intersect{\cap} -\def\getsrandom{\stackrel{\rm R}{\gets}} -\def\cross{\times} -\def\cat{\hspace{0.5em} \| \hspace{0.5em}} -\def\catn{$\|$} -\def\divides{\hspace{0.3em} | \hspace{0.3em}} -\def\nequiv{\not\equiv} -\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} -\def\lcm{{\rm lcm}} -\def\gcd{{\rm gcd}} -\def\log{{\rm log}} -\def\ord{{\rm ord}} -\def\abs{{\mathit abs}} -\def\rep{{\mathit rep}} -\def\mod{{\mathit\ mod\ }} -\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} -\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} -\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} -\def\Or{{\rm\ or\ }} -\def\And{{\rm\ and\ }} -\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} -\def\implies{\Rightarrow} -\def\undefined{{\rm ``undefined"}} -\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} -\let\oldphi\phi -\def\phi{\varphi} -\def\Pr{{\rm Pr}} -\newcommand{\str}[1]{{\mathbf{#1}}} -\def\F{{\mathbb F}} -\def\N{{\mathbb N}} -\def\Z{{\mathbb Z}} -\def\R{{\mathbb R}} -\def\C{{\mathbb C}} -\def\Q{{\mathbb Q}} -\definecolor{DGray}{gray}{0.5} -\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} -\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} -\def\gap{\vspace{0.5ex}} -\makeindex -\begin{document} -\frontmatter -\pagestyle{empty} -\title{Multi--Precision Math} -\author{\mbox{ -%\begin{small} -\begin{tabular}{c} -Tom St Denis \\ -Algonquin College \\ -\\ -Mads Rasmussen \\ -Open Communications Security \\ -\\ -Greg Rose \\ -QUALCOMM Australia \\ -\end{tabular} -%\end{small} -} -} -\maketitle -This text has been placed in the public domain. This text corresponds to the v0.39 release of the -LibTomMath project. - -\begin{alltt} -Tom St Denis -111 Banning Rd -Ottawa, Ontario -K2L 1C3 -Canada - -Phone: 1-613-836-3160 -Email: tomstdenis@gmail.com -\end{alltt} - -This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} -{\em book} macro package and the Perl {\em booker} package. - -\tableofcontents -\listoffigures -\chapter*{Prefaces} -When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. -They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' -Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which -perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps -others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give -back to society in the form of tools and knowledge that can help others in their endeavours. - -I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source -code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not -explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works -itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality -of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far -from relatively straightforward algebra and I hope that this book can be a valuable learning asset. - -This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora -of kind people donating their time, resources and kind words to help support my work. Writing a text of significant -length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old, -comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg -were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to -continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. - -To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I -honour your kind gestures with this project. - -Open Source. Open Academia. Open Minds. - -\begin{flushright} Tom St Denis \end{flushright} - -\newpage -I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also -contribute to educate others facing the problem of having to handle big number mathematical calculations. - -This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of -how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about -the layout and language used. - -I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the -practical aspects of cryptography. - -Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a -great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up -multiple precision calculations is often very important since we deal with outdated machine architecture where modular -reductions, for example, become painfully slow. - -This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks -themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' - -\begin{flushright} -Mads Rasmussen - -S\~{a}o Paulo - SP - -Brazil -\end{flushright} - -\newpage -It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about -Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not -really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. - -At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the -sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real -contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. -Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. - -When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, -and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close -friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, -and I'm pleased to be involved with it. - -\begin{flushright} -Greg Rose, Sydney, Australia, June 2003. -\end{flushright} - -\mainmatter -\pagestyle{headings} -\chapter{Introduction} -\section{Multiple Precision Arithmetic} - -\subsection{What is Multiple Precision Arithmetic?} -When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively -raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can -reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. -Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple -precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} - of algorithms can be designed to accomodate them. - -By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in -the decimal system with fixed precision $6 \cdot 7 = 2$. - -Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in -schools to manually add, subtract, multiply and divide. - -\subsection{The Need for Multiple Precision Arithmetic} -The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation -of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require -integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a -typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and -Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. - -\begin{figure}[!here] -\begin{center} -\begin{tabular}{|r|c|} -\hline \textbf{Data Type} & \textbf{Range} \\ -\hline char & $-128 \ldots 127$ \\ -\hline short & $-32768 \ldots 32767$ \\ -\hline long & $-2147483648 \ldots 2147483647$ \\ -\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\ -\hline -\end{tabular} -\end{center} -\caption{Typical Data Types for the C Programming Language} -\label{fig:ISOC} -\end{figure} - -The largest data type guaranteed to be provided by the ISO C programming -language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they -see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is -insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be -trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, -rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by -extending the range of representable integers while using single precision data types. - -Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic -primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in -various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several -major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and -deployment of efficient algorithms. - -However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. -Another auxiliary use of multiple precision integers is high precision floating point data types. -The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. -Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE -floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small -(\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create -a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where -scientific applications must minimize the total output error over long calculations. - -Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). -In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}. - -\subsection{Benefits of Multiple Precision Arithmetic} -\index{precision} -The benefit of multiple precision representations over single or fixed precision representations is that -no precision is lost while representing the result of an operation which requires excess precision. For example, -the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple -precision algorithm would augment the precision of the destination to accomodate the result while a single precision system -would truncate excess bits to maintain a fixed level of precision. - -It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic -curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum -size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the -integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard -processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not -normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. - -Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the -overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved -platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the -inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input -without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to -be written and tested once. - -\section{Purpose of This Text} -The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. -That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' -elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} -give considerably detailed explanations of the theoretical aspects of algorithms and often very little information -regarding the practical implementation aspects. - -In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For -example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple -algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning -the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple -as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not -discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). - -Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers -and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve -any form of useful performance in non-trivial applications. - -To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer -package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used -to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field -tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text -discusses a very large portion of the inner workings of the library. - -The algorithms that are presented will always include at least one ``pseudo-code'' description followed -by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same -algorithm in other programming languages as the reader sees fit. - -This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing -the reader how the algorithms fit together as well as where to start on various taskings. - -\section{Discussion and Notation} -\subsection{Notation} -A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent -the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits -of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer -$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. - -\index{mp\_int} -The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well -as auxilary data required to manipulate the data. These additional members are discussed further in section -\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be -synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members -are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the -member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would -evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that -$a.length = 5$. - -For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used -to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is -a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to -mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These -algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple -precision algorithm to solve the same problem. - -\subsection{Precision Notation} -The variable $\beta$ represents the radix of a single digit of a multiple precision integer and -must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in -the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range -$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the -carry. Since all modern computers are binary, it is assumed that $q$ is two. - -\index{mp\_digit} \index{mp\_word} -Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent -a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In -several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. -For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to -the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision -variable it is assumed that all single precision variables are promoted to double precision during the evaluation. -Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single -precision data type. - -For example, if $\beta = 10^2$ a single precision data type may represent a value in the -range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let -$a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written -as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. -In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit -in a single precision data type and as a result $c \ne \hat c$. - -\subsection{Algorithm Inputs and Outputs} -Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision -as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This -distinction is important as scalars are often used as array indicies and various other counters. - -\subsection{Mathematical Expressions} -The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression -itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression -rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when -the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, -$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a -fraction a real value division is implied, for example ${5 \over 2} = 2.5$. - -The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation -of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. - -\subsection{Work Effort} -\index{big-Oh} -To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all -single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. -That is a single precision addition, multiplication and division are assumed to take the same time to -complete. While this is generally not true in practice, it will simplify the discussions considerably. - -Some algorithms have slight advantages over others which is why some constants will not be removed in -the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a -baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these -would both be said to be equivalent to $O(n^2)$. However, -in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a -result small constant factors in the work effort will make an observable difference in algorithm efficiency. - -All of the algorithms presented in this text have a polynomial time work level. That is, of the form -$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how -various optimizations will help pay off in the long run. - -\section{Exercises} -Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to -the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought -provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent -chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the -subject material. - -That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular -are encouraged to verify they can answer the problems correctly before moving on. - -Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of -the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these -exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the -scoring system used. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|l|} -\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ - & minutes to solve. Usually does not involve much computer time \\ - & to solve. \\ -\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ - & time usage. Usually requires a program to be written to \\ - & solve the problem. \\ -\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ - & of work. Usually involves trivial research and development of \\ - & new theory from the perspective of a student. \\ -\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ - & of work and research, the solution to which will demonstrate \\ - & a higher mastery of the subject matter. \\ -\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ - & novice to solve. Solutions to these problems will demonstrate a \\ - & complete mastery of the given subject. \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Exercise Scoring System} -\end{figure} - -Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or -devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level -are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These -two levels are essentially entry level questions. - -Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often -fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always -involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can -answer these questions will feel comfortable with the concepts behind the topic at hand. - -Problems at the fourth level are meant to be similar to those of the level three questions except they will require -additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide -the exact details of the answer until a subsequent chapter. - -Problems at the fifth level are meant to be the hardest -problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a -mastery of the subject matter at hand. - -Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader -is encouraged to answer the follow-up problems and try to draw the relevance of problems. - -\section{Introduction to LibTomMath} - -\subsection{What is LibTomMath?} -LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it -is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on -any given platform. - -The library has been successfully tested under numerous operating systems including Unix\footnote{All of these -trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such -as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such -as public key cryptosystems and still maintain a relatively small footprint. - -\subsection{Goals of LibTomMath} - -Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, -even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the -library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM -processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window -exponentiation and Montgomery reduction have been provided to make the library more efficient. - -Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface -(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized -algorithms automatically without the developer's specific attention. One such example is the generic multiplication -algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication -based on the magnitude of the inputs and the configuration of the library. - -Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should -be source compatible with another popular library which makes it more attractive for developers to use. In this case the -MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits -in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument -passing conventions, it has been written from scratch by Tom St Denis. - -The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' -library exists which can be used to teach computer science students how to perform fast and reliable multiple precision -integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. - -\section{Choice of LibTomMath} -LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but -for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL -\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for -reasons that will be explained in the following sub-sections. - -\subsection{Code Base} -The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional -segments of code littered throughout the source. This clean and uncluttered approach to the library means that a -developer can more readily discern the true intent of a given section of source code without trying to keep track of -what conditional code will be used. - -The code base of LibTomMath is well organized. Each function is in its own separate source code file -which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source -file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing -very hard. GMP has many conditional code segments which also hinder tracing. - -When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} - which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about -$50$KiB) but LibTomMath is also much faster and more complete than MPI. - -\subsection{API Simplicity} -LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build -with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the -functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided -which is an extremely valuable benefit for the student and developer alike. - -The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to -illegible short hand. LibTomMath does not share this characteristic. - -The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors -are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In -effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely -undersireable in many situations. - -\subsection{Optimizations} -While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does -feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP -and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few -of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP -only had Barrett and Montgomery modular reduction algorithms.}. - -LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular -exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually -slower than the best libraries such as GMP and OpenSSL by only a small factor. - -\subsection{Portability and Stability} -LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler -(\textit{GCC}). This means that without changes the library will build without configuration or setting up any -variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of -MPI has recently stopped working on his library and LIP has long since been discontinued. - -GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active -development and are very stable across a variety of platforms. - -\subsection{Choice} -LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for -the case study of this text. Various source files from the LibTomMath project will be included within the text. However, -the reader is encouraged to download their own copy of the library to actually be able to work with the library. - -\chapter{Getting Started} -\section{Library Basics} -The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, -a problem along with allowable solution parameters should be identified and analyzed. In this particular case the -inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written -as portable source code that is reasonably efficient across several different computer platforms. - -After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. -That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, -before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. -By building outwards from a base foundation instead of using a parallel design methodology the resulting project is -highly modular. Being highly modular is a desirable property of any project as it often means the resulting product -has a small footprint and updates are easy to perform. - -Usually when I start a project I will begin with the header files. I define the data types I think I will need and -prototype the initial functions that are not dependent on other functions (within the library). After I -implement these base functions I prototype more dependent functions and implement them. The process repeats until -I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as -mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to -why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the -dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the -mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development -for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. - -\begin{center} -\begin{figure}[here] -\includegraphics{pics/design_process.ps} -\caption{Design Flow of the First Few Original LibTomMath Functions.} -\label{pic:design_process} -\end{figure} -\end{center} - -Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing -the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. - -It only makes sense to begin the text with the preliminary data types and support algorithms required as well. -This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. - -\section{What is a Multiple Precision Integer?} -Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot -be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is -to use fixed precision data types to create and manipulate multiple precision integers which may represent values -that are very large. - -As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system -the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits -(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds -column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based -multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed -precision computer words with the exception that a different radix is used. - -What most people probably do not think about explicitly are the various other attributes that describe a multiple precision -integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, -that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in -its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper -arithmetic. The third property is how many digits placeholders are available to hold the integer. - -The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, -if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. -Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer -will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision -integer or mp\_int for short. - -\subsection{The mp\_int Structure} -\label{sec:MPINT} -The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for -any such data type but it does provide for making composite data types known as structures. The following is the structure definition -used within LibTomMath. - -\index{mp\_int} -\begin{figure}[here] -\begin{center} -\begin{small} -%\begin{verbatim} -\begin{tabular}{|l|} -\hline -typedef struct \{ \\ -\hspace{3mm}int used, alloc, sign;\\ -\hspace{3mm}mp\_digit *dp;\\ -\} \textbf{mp\_int}; \\ -\hline -\end{tabular} -%\end{verbatim} -\end{small} -\caption{The mp\_int Structure} -\label{fig:mpint} -\end{center} -\end{figure} - -The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. - -\begin{enumerate} -\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent -a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. - -\item The \textbf{alloc} parameter denotes how -many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count -of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the -array to accommodate the precision of the result. - -\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple -precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least -significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored -first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, -if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then -it would represent the integer $a + b\beta + c\beta^2 + \ldots$ - -\index{MP\_ZPOS} \index{MP\_NEG} -\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). -\end{enumerate} - -\subsubsection{Valid mp\_int Structures} -Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. -The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). - -\begin{enumerate} -\item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated -array of digits. -\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. -\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, -leading zero digits in the most significant positions must be trimmed. - \begin{enumerate} - \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. - \end{enumerate} -\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; -this represents the mp\_int value of zero. -\end{enumerate} - -\section{Argument Passing} -A convention of argument passing must be adopted early on in the development of any library. Making the function -prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. -In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int -structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. -Consider the following examples. - -\begin{verbatim} - mp_mul(&a, &b, &c); /* c = a * b */ - mp_add(&a, &b, &a); /* a = a + b */ - mp_sqr(&a, &b); /* b = a * a */ -\end{verbatim} - -The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the -functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. - -Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order -of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In -truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been -adopted. - -Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a -destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important -feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. -However, to implement this feature specific care has to be given to ensure the destination is not modified before the -source is fully read. - -\section{Return Values} -A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them -to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end -developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may -fault by dereferencing memory not owned by the application. - -In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for -instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor -will it check pointers for validity. Any function that can cause a runtime error will return an error code as an -\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). - -\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} -\begin{figure}[here] -\begin{center} -\begin{tabular}{|l|l|} -\hline \textbf{Value} & \textbf{Meaning} \\ -\hline \textbf{MP\_OKAY} & The function was successful \\ -\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ -\hline \textbf{MP\_MEM} & The function ran out of heap memory \\ -\hline -\end{tabular} -\end{center} -\caption{LibTomMath Error Codes} -\label{fig:errcodes} -\end{figure} - -When an error is detected within a function it should free any memory it allocated, often during the initialization of -temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the -function was called. Error checking with this style of API is fairly simple. - -\begin{verbatim} - int err; - if ((err = mp_add(&a, &b, &c)) != MP_OKAY) { - printf("Error: %s\n", mp_error_to_string(err)); - exit(EXIT_FAILURE); - } -\end{verbatim} - -The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal -and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. - -\section{Initialization and Clearing} -The logical starting point when actually writing multiple precision integer functions is the initialization and -clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. - -Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of -the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though -the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations -would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate -and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste -memory and become unmanageable. - -If the memory for the digits has been successfully allocated then the rest of the members of the structure must -be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set -to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}. - -\subsection{Initializing an mp\_int} -An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the -structure are set to valid values. The mp\_init algorithm will perform such an action. - -\index{mp\_init} -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\ -\hline \\ -1. Allocate memory for \textbf{MP\_PREC} digits. \\ -2. If the allocation failed return(\textit{MP\_MEM}) \\ -3. for $n$ from $0$ to $MP\_PREC - 1$ do \\ -\hspace{3mm}3.1 $a_n \leftarrow 0$\\ -4. $a.sign \leftarrow MP\_ZPOS$\\ -5. $a.used \leftarrow 0$\\ -6. $a.alloc \leftarrow MP\_PREC$\\ -7. Return(\textit{MP\_OKAY})\\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init} -\end{figure} - -\textbf{Algorithm mp\_init.} -The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly -manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly -a valid assumption if the input resides on the stack. - -Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for -the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} -name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} -used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest -precision number you'll be working with. - -Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow -heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack -memory and the number of heap operations will be trivial. - -Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and -\textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless -of the original condition of the input. - -\textbf{Remark.} -This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally -when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that -a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each -iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured -the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate -decrementally. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It -is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The -call to mp\_init() is used only to initialize the members of the structure to a known default state. - -Here we see (line 24) the memory allocation is performed first. This allows us to exit cleanly and quickly -if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there -was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function -but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in -memory allocation routine. - -In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been -accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a -portable fashion you have to actually assign the value. The for loop (line 30) performs this required -operation. - -After the memory has been successfully initialized the remainder of the members are initialized -(lines 34 through 35) to their respective default states. At this point the algorithm has succeeded and -a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the -mp\_int structure has been properly initialized and is safe to use with other functions within the library. - -\subsection{Clearing an mp\_int} -When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be -returned to the application's memory pool with the mp\_clear algorithm. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_clear}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. The memory for $a$ shall be deallocated. \\ -\hline \\ -1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\ -2. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}2.1 $a_n \leftarrow 0$ \\ -3. Free the memory allocated for the digits of $a$. \\ -4. $a.used \leftarrow 0$ \\ -5. $a.alloc \leftarrow 0$ \\ -6. $a.sign \leftarrow MP\_ZPOS$ \\ -7. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_clear} -\end{figure} - -\textbf{Algorithm mp\_clear.} -This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that -if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal -is to free the allocated memory. - -The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this -algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid -digit pointer \textbf{dp} setting. - -Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm -with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line 25) -checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be -\textbf{NULL} in which case the if statement will evaluate to true. - -The digits of the mp\_int are cleared by the for loop (line 27) which assigns a zero to every digit. Similar to mp\_init() -the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. - -The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to -a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer -still has to be reset to \textbf{NULL} manually (line 35). - -Now that the digits have been cleared and deallocated the other members are set to their final values (lines 36 and 37). - -\section{Maintenance Algorithms} - -The previous sections describes how to initialize and clear an mp\_int structure. To further support operations -that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be -able to augment the precision of an mp\_int and -initialize mp\_ints with differing initial conditions. - -These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level -algorithms such as addition, multiplication and modular exponentiation. - -\subsection{Augmenting an mp\_int's Precision} -When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire -result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member -is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it -must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_grow}. \\ -\textbf{Input}. An mp\_int $a$ and an integer $b$. \\ -\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\ -\hline \\ -1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\ -2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ -3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ -4. Re-allocate the array of digits $a$ to size $v$ \\ -5. If the allocation failed then return(\textit{MP\_MEM}). \\ -6. for n from a.alloc to $v - 1$ do \\ -\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\ -7. $a.alloc \leftarrow v$ \\ -8. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_grow} -\end{figure} - -\textbf{Algorithm mp\_grow.} -It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to -prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. - -The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). -This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. - -It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much -akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are -assumed to contain undefined values they are initially set to zero. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line 24) checks -if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc} -the function skips the re-allocation part thus saving time. - -When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is -padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 25). The XREALLOC function is used -to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc -function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before -the re-allocation. All that is left is to clear the newly allocated digits and return. - -Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return -an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would -result in a memory leak if XREALLOC ever failed. - -\subsection{Initializing Variable Precision mp\_ints} -Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size -of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it -will allocate \textit{at least} a specified number of digits. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_size}. \\ -\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\ -\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\ -\hline \\ -1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\ -2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ -3. Allocate $v$ digits. \\ -4. for $n$ from $0$ to $v - 1$ do \\ -\hspace{3mm}4.1 $a_n \leftarrow 0$ \\ -5. $a.sign \leftarrow MP\_ZPOS$\\ -6. $a.used \leftarrow 0$\\ -7. $a.alloc \leftarrow v$\\ -8. Return(\textit{MP\_OKAY})\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_init\_size} -\end{figure} - -\textbf{Algorithm mp\_init\_size.} -This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of -digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a -multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial -allocations from becoming a bottleneck in the rest of the algorithms. - -Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This -particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is -correct no further memory re-allocations are required to work with the mp\_int. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The number of digits $b$ requested is padded (line 24) by first augmenting it to the next multiple of -\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the -mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be -returned (line 29). - -The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The -\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set -to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 33, 34 and 35). If the function -returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the -functions to work with. - -\subsection{Multiple Integer Initializations and Clearings} -Occasionally a function will require a series of mp\_int data types to be made available simultaneously. -The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single -statement. It is essentially a shortcut to multiple initializations. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_multi}. \\ -\textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\ -\textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\ -\hline \\ -1. for $n$ from 0 to $k - 1$ do \\ -\hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\ -\hspace{+3mm}1.2. If initialization failed then do \\ -\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\ -\hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\ -\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\ -2. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init\_multi} -\end{figure} - -\textbf{Algorithm mp\_init\_multi.} -The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected -(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' -initialization which allows for quick recovery from runtime errors. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int -structures in an actual C array they are simply passed as arguments to the function. This function makes use of the -``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument -appended on the right. - -The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count -$n$ of succesfully initialized mp\_int structures is maintained (line 48) such that if a failure does occur, -the algorithm can backtrack and free the previously initialized structures (lines 28 to 47). - - -\subsection{Clamping Excess Digits} -When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of -the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a -$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ -though, with no final carry into the last position. However, suppose the destination had to be first expanded -(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. -That would be a considerable waste of time since heap operations are relatively slow. - -The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function -terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked -there would be an excess high order zero digit. - -For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit -will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would -accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very -low the representation is excessively large. - -The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the -\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a -positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to -\textbf{MP\_ZPOS}. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_clamp}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Any excess leading zero digits of $a$ are removed \\ -\hline \\ -1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\ -\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\ -2. if $a.used = 0$ then do \\ -\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\ -\hline \\ -\end{tabular} -\end{center} -\caption{Algorithm mp\_clamp} -\end{figure} - -\textbf{Algorithm mp\_clamp.} -As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at -the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for -when all of the digits are zero to ensure that the mp\_int is valid at all times. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Note on line 28 how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming -language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is -important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously -undesirable. The parenthesis on line 31 is used to make sure the \textbf{used} count is decremented and not -the pointer ``a''. - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ - & \\ -$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ - & \\ -$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ - & encryption when $\beta = 2^{28}$. \\ - & \\ -$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ - & \\ -$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ - & \\ -\end{tabular} - - -%%% -% CHAPTER FOUR -%%% - -\chapter{Basic Operations} - -\section{Introduction} -In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining -mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low -level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they -work before proceeding since these algorithms will be used almost intrinsically in the following chapters. - -The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of -mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures -represent. - -\section{Assigning Values to mp\_int Structures} -\subsection{Copying an mp\_int} -Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making -a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same -value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_copy}. \\ -\textbf{Input}. An mp\_int $a$ and $b$. \\ -\textbf{Output}. Store a copy of $a$ in $b$. \\ -\hline \\ -1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\ -2. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\ -3. for $n$ from $a.used$ to $b.used - 1$ do \\ -\hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\ -4. $b.used \leftarrow a.used$ \\ -5. $b.sign \leftarrow a.sign$ \\ -6. return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_copy} -\end{figure} - -\textbf{Algorithm mp\_copy.} -This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will -represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the -mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. - -If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow -algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two -and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of -$b$. - -\textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the -text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in -step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is -limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return -the error code itself. However, the C code presented will demonstrate all of the error handling logic required to -implement the pseudo-code. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output -mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without -copying digits (line 25). - -The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than -$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 30 to 33). In order to -simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits -of the mp\_ints $a$ and $b$ respectively. These aliases (lines 43 and 46) allow the compiler to access the digits without first dereferencing the -mp\_int pointers and then subsequently the pointer to the digits. - -After the aliases are established the digits from $a$ are copied into $b$ (lines 49 to 51) and then the excess -digits of $b$ are set to zero (lines 54 to 56). Both ``for'' loops make use of the pointer aliases and in -fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization -allows the alias to stay in a machine register fairly easy between the two loops. - -\textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will -be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the -number of pointer dereferencing operations required to access data. For example, a for loop may resemble - -\begin{alltt} -for (x = 0; x < 100; x++) \{ - a->num[4]->dp[x] = 0; -\} -\end{alltt} - -This could be re-written using aliases as - -\begin{alltt} -mp_digit *tmpa; -a = a->num[4]->dp; -for (x = 0; x < 100; x++) \{ - *a++ = 0; -\} -\end{alltt} - -In this case an alias is used to access the -array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required -as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases. - -The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations -may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may -work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer -aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code -stands a better chance of being faster. - -The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' -loop of the function mp\_copy() re-written to not use pointer aliases. - -\begin{alltt} - /* copy all the digits */ - for (n = 0; n < a->used; n++) \{ - b->dp[n] = a->dp[n]; - \} -\end{alltt} - -Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more -complicated as there are four variables within the statement instead of just two. - -\subsubsection{Nested Statements} -Another commonly used technique in the source routines is that certain sections of code are nested. This is used in -particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six) -will typically have three different phases. First the temporaries are initialized, then the columns calculated and -finally the carries are propagated. In this example the middle column production phase will typically be nested as it -uses temporary variables and aliases the most. - -The nesting also simplies the source code as variables that are nested are only valid for their scope. As a result -the various temporary variables required do not propagate into other sections of code. - - -\subsection{Creating a Clone} -Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int -and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is -useful within functions that need to modify an argument but do not wish to actually modify the original copy. The -mp\_init\_copy algorithm has been designed to help perform this task. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_copy}. \\ -\textbf{Input}. An mp\_int $a$ and $b$\\ -\textbf{Output}. $a$ is initialized to be a copy of $b$. \\ -\hline \\ -1. Init $a$. (\textit{mp\_init}) \\ -2. Copy $b$ to $a$. (\textit{mp\_copy}) \\ -3. Return the status of the copy operation. \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init\_copy} -\end{figure} - -\textbf{Algorithm mp\_init\_copy.} -This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As -such this algorithm will perform two operations in one step. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that -\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call -and \textbf{a} will be left intact. - -\section{Zeroing an Integer} -Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to -perform this task. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_zero}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Zero the contents of $a$ \\ -\hline \\ -1. $a.used \leftarrow 0$ \\ -2. $a.sign \leftarrow$ MP\_ZPOS \\ -3. for $n$ from 0 to $a.alloc - 1$ do \\ -\hspace{3mm}3.1 $a_n \leftarrow 0$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_zero} -\end{figure} - -\textbf{Algorithm mp\_zero.} -This algorithm simply resets a mp\_int to the default state. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the -\textbf{sign} variable is set to \textbf{MP\_ZPOS}. - -\section{Sign Manipulation} -\subsection{Absolute Value} -With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute -the absolute value of an mp\_int. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_abs}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Computes $b = \vert a \vert$ \\ -\hline \\ -1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ -2. If the copy failed return(\textit{MP\_MEM}). \\ -3. $b.sign \leftarrow MP\_ZPOS$ \\ -4. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_abs} -\end{figure} - -\textbf{Algorithm mp\_abs.} -This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an -algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, -for instance, the developer to pass the same mp\_int as the source and destination to this function without addition -logic to handle it. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This fairly trivial algorithm first eliminates non--required duplications (line 28) and then sets the -\textbf{sign} flag to \textbf{MP\_ZPOS}. - -\subsection{Integer Negation} -With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute -the negative of an mp\_int input. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_neg}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Computes $b = -a$ \\ -\hline \\ -1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ -2. If the copy failed return(\textit{MP\_MEM}). \\ -3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\ -4. If $a.sign = MP\_ZPOS$ then do \\ -\hspace{3mm}4.1 $b.sign = MP\_NEG$. \\ -5. else do \\ -\hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\ -6. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_neg} -\end{figure} - -\textbf{Algorithm mp\_neg.} -This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then -the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if -$a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return -zero as negative. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Like mp\_abs() this function avoids non--required duplications (line 22) and then sets the sign. We -have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero -than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}. - -\section{Small Constants} -\subsection{Setting Small Constants} -Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_set}. \\ -\textbf{Input}. An mp\_int $a$ and a digit $b$ \\ -\textbf{Output}. Make $a$ equivalent to $b$ \\ -\hline \\ -1. Zero $a$ (\textit{mp\_zero}). \\ -2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ -3. $a.used \leftarrow \left \lbrace \begin{array}{ll} - 1 & \mbox{if }a_0 > 0 \\ - 0 & \mbox{if }a_0 = 0 - \end{array} \right .$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_set} -\end{figure} - -\textbf{Algorithm mp\_set.} -This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The -single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -First we zero (line 21) the mp\_int to make sure that the other members are initialized for a -small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count -is zero. Next we set the digit and reduce it modulo $\beta$ (line 22). After this step we have to -check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise -to zero. - -We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with -$2^k - 1$ will perform the same operation. - -One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses -this function should take that into account. Only trivially small constants can be set using this function. - -\subsection{Setting Large Constants} -To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long'' -data type as input and will always treat it as a 32-bit integer. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_set\_int}. \\ -\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\ -\textbf{Output}. Make $a$ equivalent to $b$ \\ -\hline \\ -1. Zero $a$ (\textit{mp\_zero}) \\ -2. for $n$ from 0 to 7 do \\ -\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\ -\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ -\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ -\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\ -3. Clamp excess used digits (\textit{mp\_clamp}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_set\_int} -\end{figure} - -\textbf{Algorithm mp\_set\_int.} -The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the -mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the -next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is -incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have -zero digits used and the newly added four bits would be ignored. - -Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird -addition on line 39 ensures that the newly added in bits are added to the number of digits. While it may not -seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 28 -as well as the call to mp\_clamp() on line 41. Both functions will clamp excess leading digits which keeps -the number of used digits low. - -\section{Comparisons} -\subsection{Unsigned Comparisions} -Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, -to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ -to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude -positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. - -The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two -mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the -signs are known to agree in advance. - -To facilitate working with the results of the comparison functions three constants are required. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{|r|l|} -\hline \textbf{Constant} & \textbf{Meaning} \\ -\hline \textbf{MP\_GT} & Greater Than \\ -\hline \textbf{MP\_EQ} & Equal To \\ -\hline \textbf{MP\_LT} & Less Than \\ -\hline -\end{tabular} -\end{center} -\caption{Comparison Return Codes} -\end{figure} - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_cmp\_mag}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$. \\ -\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ -\hline \\ -1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ -2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ -3. for n from $a.used - 1$ to 0 do \\ -\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ -\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ -4. Return(\textit{MP\_EQ}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_cmp\_mag} -\end{figure} - -\textbf{Algorithm mp\_cmp\_mag.} -By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return -\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. -Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. -If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. - -By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to -the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The two if statements (lines 25 and 29) compare the number of digits in the two inputs. These two are -performed before all of the digits are compared since it is a very cheap test to perform and can potentially save -considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be -smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits. - - - -\subsection{Signed Comparisons} -Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude -comparison a trivial signed comparison algorithm can be written. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_cmp}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ -\hline \\ -1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ -2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ -3. if $a.sign = MP\_NEG$ then \\ -\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\ -4 Otherwise \\ -\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_cmp} -\end{figure} - -\textbf{Algorithm mp\_cmp.} -The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate -comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step -three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then -$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The two if statements (lines 23 and 24) perform the initial sign comparison. If the signs are not the equal then which ever -has the positive sign is larger. The inputs are compared (line 32) based on magnitudes. If the signs were both -negative then the unsigned comparison is performed in the opposite direction (line 34). Otherwise, the signs are assumed to -be both positive and a forward direction unsigned comparison is performed. - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ - & \\ -$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ - & of two random digits (of equal magnitude) before a difference is found. \\ - & \\ -$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ - & on the observations made in the previous problem. \\ - & -\end{tabular} - -\chapter{Basic Arithmetic} -\section{Introduction} -At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been -established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These -algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important -that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms -which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. - -All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right -logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real -number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). -Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. -For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. - -One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed -from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the -result is $110_2$. - -\section{Addition and Subtraction} -In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers -$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. -As a result subtraction can be performed with a trivial series of logical operations and an addition. - -However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the -sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or -subtraction algorithms with the sign fixed up appropriately. - -The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of -the integers respectively. - -\subsection{Low Level Addition} -An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the -trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. -Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. - -\newpage -\begin{figure}[!here] -\begin{center} -\begin{small} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_add}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\ -\hline \\ -1. if $a.used > b.used$ then \\ -\hspace{+3mm}1.1 $min \leftarrow b.used$ \\ -\hspace{+3mm}1.2 $max \leftarrow a.used$ \\ -\hspace{+3mm}1.3 $x \leftarrow a$ \\ -2. else \\ -\hspace{+3mm}2.1 $min \leftarrow a.used$ \\ -\hspace{+3mm}2.2 $max \leftarrow b.used$ \\ -\hspace{+3mm}2.3 $x \leftarrow b$ \\ -3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ -5. $c.used \leftarrow max + 1$ \\ -6. $u \leftarrow 0$ \\ -7. for $n$ from $0$ to $min - 1$ do \\ -\hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\ -\hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\ -\hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -8. if $min \ne max$ then do \\ -\hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ -\hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\ -\hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\ -\hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -9. $c_{max} \leftarrow u$ \\ -10. if $olduse > max$ then \\ -\hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\ -\hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\ -11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\ -12. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Algorithm s\_mp\_add} -\end{figure} - -\textbf{Algorithm s\_mp\_add.} -This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. -Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the -MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. - -The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic -will simply add all of the smallest input to the largest input and store that first part of the result in the -destination. Then it will apply a simpler addition loop to excess digits of the larger input. - -The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two -inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the -same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum -of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. - -At this point the first addition loop will go through as many digit positions that both inputs have. The carry -variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce -one digit of the summand. First -two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored -in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$. - -Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias -for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits -and the carry to the destination. - -The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition. - - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -We first sort (lines 28 to 36) the inputs based on magnitude and determine the $min$ and $max$ variables. -Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we -grow the destination (38 to 42) ensure that it can accomodate the result of the addition. - -Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on -lines 56, 59 and 62 represent the two inputs and destination variables respectively. These aliases are used to ensure the -compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. - -The initial carry $u$ will be cleared (line 65), note that $u$ is of type mp\_digit which ensures type -compatibility within the implementation. The initial addition (line 66 to 75) adds digits from -both inputs until the smallest input runs out of digits. Similarly the conditional addition loop -(line 81 to 90) adds the remaining digits from the larger of the two inputs. The addition is finished -with the final carry being stored in $tmpc$ (line 94). Note the ``++'' operator within the same expression. -After line 94, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful -for the next loop (line 97 to 99) which set any old upper digits to zero. - -\subsection{Low Level Subtraction} -The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the -unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must -be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. -This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. - - -For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent -the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For -this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a -mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). - -For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' -data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$. - -\newpage\begin{figure}[!here] -\begin{center} -\begin{small} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_sub}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ -\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ -\hline \\ -1. $min \leftarrow b.used$ \\ -2. $max \leftarrow a.used$ \\ -3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ -5. $c.used \leftarrow max$ \\ -6. $u \leftarrow 0$ \\ -7. for $n$ from $0$ to $min - 1$ do \\ -\hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\ -\hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ -\hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -8. if $min < max$ then do \\ -\hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ -\hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\ -\hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ -\hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -9. if $oldused > max$ then do \\ -\hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\ -\hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\ -10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Algorithm s\_mp\_sub} -\end{figure} - -\textbf{Algorithm s\_mp\_sub.} -This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when -passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This -algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case -of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. - -The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 -set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at -most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and -set to the maximal count for the operation. - -The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision -subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction -loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. - -For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to -the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the -third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the -way to the most significant bit. - -Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most -significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that -is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the -carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. - -If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step -10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded -(lines 25 and 26). In reality the $min$ and $max$ variables are only aliases and are only -used to make the source code easier to read. Again the pointer alias optimization is used -within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized -(lines 42, 43 and 44) for $a$, $b$ and $c$ respectively. - -The first subtraction loop (lines 47 through 61) subtract digits from both inputs until the smaller of -the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' -method of extracting the carry (line 57). The traditional method for extracting the carry would be to shift -by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of -the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry -extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the -most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This -optimization only works on twos compliment machines which is a safe assumption to make. - -If $a$ has a larger magnitude than $b$ an additional loop (lines 64 through 73) is required to propagate -the carry through $a$ and copy the result to $c$. - -\subsection{High Level Addition} -Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be -established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data -types. - -Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} -flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. - -\begin{figure}[!here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_add}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The signed addition $c = a + b$. \\ -\hline \\ -1. if $a.sign = b.sign$ then do \\ -\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\ -2. else do \\ -\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ -\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ -\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ -\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_add} -\end{figure} - -\textbf{Algorithm mp\_add.} -This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from -either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly -straightforward but restricted since subtraction can only produce positive results. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|} -\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ -\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ -\hline &&&&\\ - -\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ -\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ - -\hline &&&&\\ - -\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ - -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Addition Guide Chart} -\label{fig:AddChart} -\end{figure} - -Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three -specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are -forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best -follows how the implementation actually was achieved. - -Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms -s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} -to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. - -For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would -produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp -within algorithm s\_mp\_add will force $-0$ to become $0$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which -is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without -explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower -level functions do so. Returning their return code is sufficient. - -\subsection{High Level Subtraction} -The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. - -\newpage\begin{figure}[!here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_sub}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The signed subtraction $c = a - b$. \\ -\hline \\ -1. if $a.sign \ne b.sign$ then do \\ -\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ -2. else do \\ -\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ -\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} - MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ - MP\_NEG & \mbox{otherwise} \\ - \end{array} \right .$ \\ -\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_sub} -\end{figure} - -\textbf{Algorithm mp\_sub.} -This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or -\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and -the operations required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|} -\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ -\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\ -\hline &&&& \\ -\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline &&&& \\ -\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ -\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Subtraction Guide Chart} -\label{fig:SubChart} -\end{figure} - -Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the -algorithm from producing $-a - -a = -0$ as a result. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations -and forward it to the end of the function. On line 39 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a -``greater than or equal to'' comparison. - -\section{Bit and Digit Shifting} -It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. -This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. - -In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift -the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations -are on radix-$\beta$ digits. - -\subsection{Multiplication by Two} - -In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient -operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_2}. \\ -\textbf{Input}. One mp\_int $a$ \\ -\textbf{Output}. $b = 2a$. \\ -\hline \\ -1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\ -2. $oldused \leftarrow b.used$ \\ -3. $b.used \leftarrow a.used$ \\ -4. $r \leftarrow 0$ \\ -5. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\ -\hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}5.3 $r \leftarrow rr$ \\ -6. If $r \ne 0$ then do \\ -\hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\ -\hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\ -7. If $b.used < oldused - 1$ then do \\ -\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ -\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ -8. $b.sign \leftarrow a.sign$ \\ -9. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_2} -\end{figure} - -\textbf{Algorithm mp\_mul\_2.} -This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such -an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since -it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. - -Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count -is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. - -Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together -are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to -obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus -the previous carry. Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with -forwarding the carry to the next iteration. - -Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. -Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference -is the use of the logical shift operator on line 52 to perform a single precision doubling. - -\subsection{Division by Two} -A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_2}. \\ -\textbf{Input}. One mp\_int $a$ \\ -\textbf{Output}. $b = a/2$. \\ -\hline \\ -1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\ -2. If the reallocation failed return(\textit{MP\_MEM}). \\ -3. $oldused \leftarrow b.used$ \\ -4. $b.used \leftarrow a.used$ \\ -5. $r \leftarrow 0$ \\ -6. for $n$ from $b.used - 1$ to $0$ do \\ -\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ -\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}6.3 $r \leftarrow rr$ \\ -7. If $b.used < oldused - 1$ then do \\ -\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ -\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ -8. $b.sign \leftarrow a.sign$ \\ -9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\ -10. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_2} -\end{figure} - -\textbf{Algorithm mp\_div\_2.} -This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition -core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm -could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent -reading past the end of the array of digits. - -Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the -least significant bit not the most significant bit. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section{Polynomial Basis Operations} -Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as -the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single -place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer -division and Karatsuba multiplication. - -Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that -$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the -polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. - -\subsection{Multiplication by $x$} - -Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one -degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to -multiplying by the integer $\beta$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_lshd}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\ -\hline \\ -1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ -2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\ -3. If the reallocation failed return(\textit{MP\_MEM}). \\ -4. $a.used \leftarrow a.used + b$ \\ -5. $i \leftarrow a.used - 1$ \\ -6. $j \leftarrow a.used - 1 - b$ \\ -7. for $n$ from $a.used - 1$ to $b$ do \\ -\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ -\hspace{3mm}7.2 $i \leftarrow i - 1$ \\ -\hspace{3mm}7.3 $j \leftarrow j - 1$ \\ -8. for $n$ from 0 to $b - 1$ do \\ -\hspace{3mm}8.1 $a_n \leftarrow 0$ \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_lshd} -\end{figure} - -\textbf{Algorithm mp\_lshd.} -This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs -from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The -motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally -different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is -typically used on values where the original value is no longer required. The algorithm will return success immediately if -$b \le 0$ since the rest of algorithm is only valid when $b > 0$. - -First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over -the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). -The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on -step 8 sets the lower $b$ digits to zero. - -\newpage -\begin{center} -\begin{figure}[here] -\includegraphics{pics/sliding_window.ps} -\caption{Sliding Window Movement} -\label{pic:sliding_window} -\end{figure} -\end{center} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The if statement (line 24) ensures that the $b$ variable is greater than zero since we do not interpret negative -shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates -the need for an additional variable in the for loop. The variable $top$ (line 42) is an alias -for the leading digit while $bottom$ (line 45) is an alias for the trailing edge. The aliases form a -window of exactly $b$ digits over the input. - -\subsection{Division by $x$} - -Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_rshd}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ -\hline \\ -1. If $b \le 0$ then return. \\ -2. If $a.used \le b$ then do \\ -\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\ -\hspace{3mm}2.2 Return. \\ -3. $i \leftarrow 0$ \\ -4. $j \leftarrow b$ \\ -5. for $n$ from 0 to $a.used - b - 1$ do \\ -\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ -\hspace{3mm}5.2 $i \leftarrow i + 1$ \\ -\hspace{3mm}5.3 $j \leftarrow j + 1$ \\ -6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ -\hspace{3mm}6.1 $a_n \leftarrow 0$ \\ -7. $a.used \leftarrow a.used - b$ \\ -8. Return. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_rshd} -\end{figure} - -\textbf{Algorithm mp\_rshd.} -This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since -it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. - -If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal -to the shift count $b$ then it will simply zero the input and return. - -After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that -is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. -Also the digits are copied from the leading to the trailing edge. - -Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we -form a sliding window except we copy in the other direction. After the window (line 60) we then zero -the upper digits of the input to make sure the result is correct. - -\section{Powers of Two} - -Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For -example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single -shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. - -\subsection{Multiplication by Power of Two} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ -\hline \\ -1. $c \leftarrow a$. (\textit{mp\_copy}) \\ -2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ -3. If the reallocation failed return(\textit{MP\_MEM}). \\ -4. If $b \ge lg(\beta)$ then \\ -\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ -\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ -5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -6. If $d \ne 0$ then do \\ -\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ -\hspace{3mm}6.2 $r \leftarrow 0$ \\ -\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ -\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ -\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ -\hspace{3mm}6.4 If $r > 0$ then do \\ -\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ -\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ -7. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_2d} -\end{figure} - -\textbf{Algorithm mp\_mul\_2d.} -This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to -quickly compute the product. - -First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than -$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ -left. - -After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts -required. If it is non-zero a modified shift loop is used to calculate the remaining product. -Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ -variable is used to extract the upper $d$ bits to form the carry for the next iteration. - -This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to -complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The shifting is performed in--place which means the first step (line 25) is to copy the input to the -destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then -has to be grown (line 32) to accomodate the result. - -If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples -of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift -loop (lines 46 to 76) we make use of pre--computed values $shift$ and $mask$. These are used to -extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a -chain between consecutive iterations to propagate the carry. - -\subsection{Division by Power of Two} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then do \\ -\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ -\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ -\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ -2. $c \leftarrow a$ \\ -3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -4. If $b \ge lg(\beta)$ then do \\ -\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ -5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -6. If $k \ne 0$ then do \\ -\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ -\hspace{3mm}6.2 $r \leftarrow 0$ \\ -\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ -\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ -\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ -\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ -7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_2d} -\end{figure} - -\textbf{Algorithm mp\_div\_2d.} -This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm -mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division -by using algorithm mp\_mod\_2d. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally -ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the -result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before -the quotient is obtained. - -The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. The only significant difference is -the direction of the shifts. - -\subsection{Remainder of Division by Power of Two} - -The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This -algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mod\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then do \\ -\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $b > a.used \cdot lg(\beta)$ then do \\ -\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ -\hspace{3mm}2.2 Return the result of step 2.1. \\ -3. $c \leftarrow a$ \\ -4. If step 3 failed return(\textit{MP\_MEM}). \\ -5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ -\hspace{3mm}5.1 $c_n \leftarrow 0$ \\ -6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ -8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mod\_2d} -\end{figure} - -\textbf{Algorithm mp\_mod\_2d.} -This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the -result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ -is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger -than the input we just mp\_copy() the input and return right away. After this point we know we must actually -perform some work to produce the remainder. - -Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce -the number. First we zero any digits above the last digit in $2^b$ (line 42). Next we reduce the -leading digit of both (line 46) and then mp\_clamp(). - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ - & in $O(n)$ time. \\ - &\\ -$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ - & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ - & upto $64$ with a hamming weight less than three. \\ - &\\ -$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ - & $2^k - 1$ as well. \\ - &\\ -$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ - & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ - & any $n$-bit input. Note that the time of addition is ignored in the \\ - & calculation. \\ - & \\ -$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ - & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ - & the cost of addition. \\ - & \\ -$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ - & for $n = 64 \ldots 1024$ in steps of $64$. \\ - & \\ -$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ - & calculating the result of a signed comparison. \\ - & -\end{tabular} - -\chapter{Multiplication and Squaring} -\section{The Multipliers} -For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of -algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction -where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication -and squaring, leaving modular reductions for the subsequent chapter. - -The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular -exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular -exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, -35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision -multiplications. - -For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied -against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the -overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in -1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. -This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. - -\section{Multiplication} -\subsection{The Baseline Multiplication} -\label{sec:basemult} -\index{baseline multiplication} -Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication -algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision -multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To -simplify most discussions, it will be assumed that the inputs have comparable number of digits. - -The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be -used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important -facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this -modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product -will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. - -Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to -include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The -constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}). - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ -\hline \\ -1. If min$(a.used, b.used) < \delta$ then do \\ -\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\ -\hspace{3mm}1.2 Return the result of step 1.1 \\ -\\ -Allocate and initialize a temporary mp\_int. \\ -2. Init $t$ to be of size $digs$ \\ -3. If step 2 failed return(\textit{MP\_MEM}). \\ -4. $t.used \leftarrow digs$ \\ -\\ -Compute the product. \\ -5. for $ix$ from $0$ to $a.used - 1$ do \\ -\hspace{3mm}5.1 $u \leftarrow 0$ \\ -\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ -\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\ -\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\ -\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\ -\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}5.5 if $ix + pb < digs$ then do \\ -\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\ -6. Clamp excess digits of $t$. \\ -7. Swap $c$ with $t$ \\ -8. Clear $t$ \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_mul\_digs} -\end{figure} - -\textbf{Algorithm s\_mp\_mul\_digs.} -This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem -a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent -algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. -Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the -inputs. - -The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either -input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A -temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to -compute products when either $a = c$ or $b = c$ without overwriting the inputs. - -All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable -is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm -will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the -innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. - -For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best -visualized in the following table. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|l|} -\hline && & 5 & 7 & 6 & \\ -\hline $\times$&& & 2 & 4 & 1 & \\ -\hline &&&&&&\\ - && & 5 & 7 & 6 & $10^0(1)(576)$ \\ - &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ - 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ -\hline -\end{tabular} -\end{center} -\caption{Long-Hand Multiplication Diagram} -\end{figure} - -Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate -count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. - -Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step -is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a -double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step -5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit -$t_{ix+iy}$ and the result would be lost. - -At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th -digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result -exceed the precision requested. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -First we determine (line 31) if the Comba method can be used first since it's faster. The conditions for -sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than -\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is -set to $\delta$ but can be reduced when memory is at a premium. - -If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int -$t$ (line 37) to the exact size of the output to avoid further re--allocations. At this point we now -begin the $O(n^2)$ loop. - -This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of -digits as output. In each iteration of the outer loop the $pb$ variable is set (line 49) to the maximum -number of inner loop iterations. - -Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the -carry from the previous iteration. A particularly important observation is that most modern optimizing -C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that -is required for the product. In x86 terms for example, this means using the MUL instruction. - -Each digit of the product is stored in turn (line 69) and the carry propagated (line 72) to the -next iteration. - -\subsection{Faster Multiplication by the ``Comba'' Method} - -One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be -computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement -in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. -Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an -interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written -five years before. - -At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight -twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products -are produced then added together to form the final result. In the baseline algorithm the columns are added together -after each iteration to get the result instantaneously. - -In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at -the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated -after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute -the product vector $\vec x$ as follows. - -\begin{equation} -\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace -\end{equation} - -Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication -of $576$ and $241$. - -\newpage\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|} - \hline & & 5 & 7 & 6 & First Input\\ - \hline $\times$ & & 2 & 4 & 1 & Second Input\\ -\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ - & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ - $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ -\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Comba Multiplication Diagram} -\end{figure} - -At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. -Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is -congruent to adding a leading zero digit. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Comba Fixup}. \\ -\textbf{Input}. Vector $\vec x$ of dimension $k$ \\ -\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\ -\hline \\ -1. for $n$ from $0$ to $k - 1$ do \\ -\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\ -\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\ -2. Return($\vec x$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Comba Fixup} -\end{figure} - -With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case -$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more -efficient than the baseline algorithm why not simply always use this algorithm? - -\subsubsection{Column Weight.} -At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output -independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix -the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of -three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then -an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is -min$(m, n)$ which is fairly obvious. - -The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall -from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these -two quantities we must not violate the following - -\begin{equation} -k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} -\end{equation} - -Which reduces to - -\begin{equation} -k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} -\end{equation} - -Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is -found. - -\begin{equation} -k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} -\end{equation} - -The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration -the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since -$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the stack. \\ -1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ -2. If step 1 failed return(\textit{MP\_MEM}).\\ -\\ -3. $pa \leftarrow \mbox{MIN}(digs, a.used + b.used)$ \\ -\\ -4. $\_ \hat W \leftarrow 0$ \\ -5. for $ix$ from 0 to $pa - 1$ do \\ -\hspace{3mm}5.1 $ty \leftarrow \mbox{MIN}(b.used - 1, ix)$ \\ -\hspace{3mm}5.2 $tx \leftarrow ix - ty$ \\ -\hspace{3mm}5.3 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ -\hspace{3mm}5.4 for $iz$ from 0 to $iy - 1$ do \\ -\hspace{6mm}5.4.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx+iy}b_{ty-iy}$ \\ -\hspace{3mm}5.5 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$\\ -\hspace{3mm}5.6 $\_ \hat W \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ -\\ -6. $oldused \leftarrow c.used$ \\ -7. $c.used \leftarrow digs$ \\ -8. for $ix$ from $0$ to $pa$ do \\ -\hspace{3mm}8.1 $c_{ix} \leftarrow W_{ix}$ \\ -9. for $ix$ from $pa + 1$ to $oldused - 1$ do \\ -\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ -\\ -10. Clamp $c$. \\ -11. Return MP\_OKAY. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_s\_mp\_mul\_digs} -\label{fig:COMBAMULT} -\end{figure} - -\textbf{Algorithm fast\_s\_mp\_mul\_digs.} -This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. - -The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the -loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and -reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration. - -The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than -$b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable -$ix$ is. This is used for the immediately subsequent statement where we find $iy$. - -The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time -means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each -pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to -move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until -$tx \ge a.used$ or $ty < 0$ occurs. - -After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator -into the next round by dividing $\_ \hat W$ by $\beta$. - -To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the -cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require -$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, -the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply -and addition operations in the nested loop in parallel. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -As per the pseudo--code we first calculate $pa$ (line 48) as the number of digits to output. Next we begin the outer loop -to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines 62, 63) to point -inside the two multiplicands quickly. - -The inner loop (lines 71 to 74) of this implementation is where the tradeoff come into play. Originally this comba -implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix -the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write -one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth -is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often -slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the -compiler has aliased $\_ \hat W$ to a CPU register. - -After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 77, 80) to forward it as -a carry for the next pass. After the outer loop we use the final carry (line 77) as the last digit of the product. - -\subsection{Polynomial Basis Multiplication} -To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms -the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and -$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. - -The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will -directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients -requires $O(n^2)$ time and would in practice be slower than the Comba technique. - -However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown -coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with -Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in -effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. - -The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since -$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the -fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required -by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. - -When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term -is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product -$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather -simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. -The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the -points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. - -If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For -example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. - -\begin{eqnarray} -\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\ -16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0) -\end{eqnarray} - -Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the -polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. - -As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of -multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is -$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} -summarizes the exponents for various values of $n$. - -\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\ -\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\ -\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\ -\hline $4$ & $1.403677461$ &\\ -\hline $5$ & $1.365212389$ &\\ -\hline $10$ & $1.278753601$ &\\ -\hline $100$ & $1.149426538$ &\\ -\hline $1000$ & $1.100270931$ &\\ -\hline $10000$ & $1.075252070$ &\\ -\hline -\end{tabular} -\end{center} -\caption{Asymptotic Running Time of Polynomial Basis Multiplication} -\label{fig:exponent} -\end{figure} - -At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead -of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large -numbers. - -\subsubsection{Cutoff Point} -The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, -the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the -polynomial basis approach more costly to use with small inputs. - -Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a -point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and -when $m > y$ the Comba methods are slower than the polynomial basis algorithms. - -The exact location of $y$ depends on several key architectural elements of the computer platform in question. - -\begin{enumerate} -\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example -on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower -the cutoff point $y$ will be. - -\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits -grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This -directly reflects on the ratio previous mentioned. - -\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an -influence over the cutoff point. - -\end{enumerate} - -A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point -is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when -a high resolution timer is available. - -\subsection{Karatsuba Multiplication} -Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for -general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with -light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. - -\begin{equation} -f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd -\end{equation} - -Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying -this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns -out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points -$\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$. Consider the resultant system of equations. - -\begin{center} -\begin{tabular}{rcrcrcrc} -$\zeta_{0}$ & $=$ & & & & & $w_0$ \\ -$\zeta_{1}$ & $=$ & $w_2$ & $+$ & $w_1$ & $+$ & $w_0$ \\ -$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ -\end{tabular} -\end{center} - -By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity -of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} -making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ -\hline \\ -1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ -2. If step 2 failed then return(\textit{MP\_MEM}). \\ -\\ -Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ -3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ -4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ -6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\ -7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ -\\ -Calculate the three products. \\ -8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\ -9. $x1y1 \leftarrow x1 \cdot y1$ \\ -10. $t1 \leftarrow x1 + x0$ (\textit{mp\_add}) \\ -11. $x0 \leftarrow y1 + y0$ \\ -12. $t1 \leftarrow t1 \cdot x0$ \\ -\\ -Calculate the middle term. \\ -13. $x0 \leftarrow x0y0 + x1y1$ \\ -14. $t1 \leftarrow t1 - x0$ (\textit{s\_mp\_sub}) \\ -\\ -Calculate the final product. \\ -15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\ -16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ -17. $t1 \leftarrow x0y0 + t1$ \\ -18. $c \leftarrow t1 + x1y1$ \\ -19. Clear all of the temporary variables. \\ -20. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_karatsuba\_mul} -\end{figure} - -\textbf{Algorithm mp\_karatsuba\_mul.} -This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description -from Knuth \cite[pp. 294-295]{TAOCPV2}. - -\index{radix point} -In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must -be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the -smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 -compute the lower halves. Step 6 and 7 computer the upper halves. - -After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products -$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed. By using $x0$ instead -of an additional temporary variable, the algorithm can avoid an addition memory allocation operation. - -The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional -wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense -to handle error recovery with a single piece of code. Lines 62 to 76 handle initializing all of the temporary variables -required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only -the temporaries that have been successfully allocated so far. - -The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the -additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective -number of digits for the next section of code. - -The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd -to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and -\textbf{sign} members are copied first. The first for loop on line 96 copies the lower halves. Since they are both the same magnitude it -is simpler to calculate both lower halves in a single loop. The for loop on lines 102 and 107 calculate the upper halves $x1$ and -$y1$ respectively. - -By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs. - -When line 151 is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that -the same code that handles errors can be used to clear the temporary variables and return. - -\subsection{Toom-Cook $3$-Way Multiplication} -Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are -chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, -$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients -of the $W(x)$. - -With the five relations that Toom-Cook specifies, the following system of equations is formed. - -\begin{center} -\begin{tabular}{rcrcrcrcrcr} -$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\ -$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\ -$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\ -$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\ -$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\ -\end{tabular} -\end{center} - -A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power -of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that -the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point -(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toom\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b $ \\ -\hline \\ -Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\ -1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\ -2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -\\ -Find the five equations for $w_0, w_1, ..., w_4$. \\ -8. $w_0 \leftarrow a_0 \cdot b_0$ \\ -9. $w_4 \leftarrow a_2 \cdot b_2$ \\ -10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\ -11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ -12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\ -13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\ -14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\ -15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\ -16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ -17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\ -\\ -Continued on the next page.\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toom\_mul} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b $ \\ -\hline \\ -Now solve the system of equations. \\ -18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\ -19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\ -20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\ -21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ -22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\ -23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\ -24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ -25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\ -\\ -Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ -26. for $n$ from $1$ to $4$ do \\ -\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\ -27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\ -28. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toom\_mul (continued)} -\end{figure} - -\textbf{Algorithm mp\_toom\_mul.} -This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this -algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this -description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across -any given step. - -The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller -integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required. - -The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond -to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find -$f(y)$ and $g(y)$ which significantly speeds up the algorithm. - -After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients -$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of -the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates -that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. - -Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer -result $a \cdot b$ is produced. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very -large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with -Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this -algorithm is not practical as Karatsuba has a much lower cutoff point. - -First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines 41 to 70) with -combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly -for $b$. - -Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so -we get those out of the way first (lines 73 and 78). Next we compute $w1, w2$ and $w3$ using Horners method. - -After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively -straight forward. - -\subsection{Signed Multiplication} -Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all -of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b$ \\ -\hline \\ -1. If $a.sign = b.sign$ then \\ -\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ -2. else \\ -\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ -3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ -\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ -4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ -\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ -5. else \\ -\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ -\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ -\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ -\hspace{3mm}5.3 else \\ -\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ -6. $c.sign \leftarrow sign$ \\ -7. Return the result of the unsigned multiplication performed. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul} -\end{figure} - -\textbf{Algorithm mp\_mul.} -This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms -available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm -s\_mp\_mul\_digs will clear it. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The implementation is rather simplistic and is not particularly noteworthy. Line 22 computes the sign of the result using the ``?'' -operator from the C programming language. Line 48 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. - -\section{Squaring} -\label{sec:basesquare} - -Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization -available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications -performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider -the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, -$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ -and $3 \cdot 1 = 1 \cdot 3$. - -For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ -required for multiplication. The following diagram gives an example of the operations required. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{ccccc|c} -&&1&2&3&\\ -$\times$ &&1&2&3&\\ -\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\ - & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\ - $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\ -\end{tabular} -\end{center} -\caption{Squaring Optimization Diagram} -\end{figure} - -Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ -represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. - -The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will -appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double -products and at most one square (\textit{see the exercise section}). - -The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, -occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. -Column two of row one is a square and column three is the first unique column. - -\subsection{The Baseline Squaring Algorithm} -The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines -will not handle. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\ -2. If step 1 failed return(\textit{MP\_MEM}) \\ -3. $t.used \leftarrow 2 \cdot a.used + 1$ \\ -4. For $ix$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}Calculate the square. \\ -\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\ -\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}Calculate the double products after the square. \\ -\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\ -\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\ -\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}Set the last carry. \\ -\hspace{3mm}4.5 While $u > 0$ do \\ -\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\ -\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\ -\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\ -6. Exchange $b$ and $t$. \\ -7. Clear $t$ (\textit{mp\_clear}) \\ -8. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_sqr} -\end{figure} - -\textbf{Algorithm s\_mp\_sqr.} -This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC -\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the -destination mp\_int to be the same as the source mp\_int. - -The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while -the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate -the carry and compute the double products. - -The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this -very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that -when it is multiplied by two, it can be properly represented by a mp\_word. - -Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial -results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Inside the outer loop (line 34) the square term is calculated on line 37. The carry (line 44) has been -extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized -(lines 47 and 50) to simplify the inner loop. The doubling is performed using two -additions (line 59) since it is usually faster than shifting, if not at least as fast. - -The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops -get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to -square a number. - -\subsection{Faster Squaring by the ``Comba'' Method} -A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional -drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these -performance hazards. - -The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry -propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact -that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, -$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. - -However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two -mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and -carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ -1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ -2. If step 1 failed return(\textit{MP\_MEM}). \\ -\\ -3. $pa \leftarrow 2 \cdot a.used$ \\ -4. $\hat W1 \leftarrow 0$ \\ -5. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}5.1 $\_ \hat W \leftarrow 0$ \\ -\hspace{3mm}5.2 $ty \leftarrow \mbox{MIN}(a.used - 1, ix)$ \\ -\hspace{3mm}5.3 $tx \leftarrow ix - ty$ \\ -\hspace{3mm}5.4 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ -\hspace{3mm}5.5 $iy \leftarrow \mbox{MIN}(iy, \lfloor \left (ty - tx + 1 \right )/2 \rfloor)$ \\ -\hspace{3mm}5.6 for $iz$ from $0$ to $iz - 1$ do \\ -\hspace{6mm}5.6.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx + iz}a_{ty - iz}$ \\ -\hspace{3mm}5.7 $\_ \hat W \leftarrow 2 \cdot \_ \hat W + \hat W1$ \\ -\hspace{3mm}5.8 if $ix$ is even then \\ -\hspace{6mm}5.8.1 $\_ \hat W \leftarrow \_ \hat W + \left ( a_{\lfloor ix/2 \rfloor}\right )^2$ \\ -\hspace{3mm}5.9 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ -\hspace{3mm}5.10 $\hat W1 \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ -\\ -6. $oldused \leftarrow b.used$ \\ -7. $b.used \leftarrow 2 \cdot a.used$ \\ -8. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}8.1 $b_{ix} \leftarrow W_{ix}$ \\ -9. for $ix$ from $pa$ to $oldused - 1$ do \\ -\hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\ -10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_s\_mp\_sqr} -\end{figure} - -\textbf{Algorithm fast\_s\_mp\_sqr.} -This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm -s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. -This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of. - -First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop -products are to be doubled. If we had added the previous carry in we would be doubling too much. Next we perform an -addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits. For example, $a_3 \cdot a_5$ is equal -$a_5 \cdot a_3$. Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum -of the products just outside the inner loop we have to avoid doing this. This is also a good thing since we perform -fewer multiplications and the routine ends up being faster. - -Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8). We add in the square -only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for -the special case of squaring. - -\subsection{Polynomial Basis Squaring} -The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception -is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ -multiplications to find the $\zeta$ relations, squaring operations are performed instead. - -\subsection{Karatsuba Squaring} -Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. -Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a -number with the following equation. - -\begin{equation} -h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2 -\end{equation} - -Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$. As in -Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of -$O \left ( n^{lg(3)} \right )$. - -If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm -instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the -time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff -point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. - -Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. -The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication -were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\ -2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\ -\\ -Split the input. e.g. $a = x1\beta^B + x0$ \\ -3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\ -4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\ -\\ -Calculate the three squares. \\ -6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\ -7. $x1x1 \leftarrow x1^2$ \\ -8. $t1 \leftarrow x1 + x0$ (\textit{s\_mp\_add}) \\ -9. $t1 \leftarrow t1^2$ \\ -\\ -Compute the middle term. \\ -10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\ -11. $t1 \leftarrow t1 - t2$ \\ -\\ -Compute final product. \\ -12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\ -13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\ -14. $t1 \leftarrow t1 + x0x0$ \\ -15. $b \leftarrow t1 + x1x1$ \\ -16. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_karatsuba\_sqr} -\end{figure} - -\textbf{Algorithm mp\_karatsuba\_sqr.} -This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based -multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings. - -The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is -placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$ -as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form. - -By expanding $\left (x1 + x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $(x0 - x1)^2 - (x1^2 + x0^2) = 2 \cdot x0 \cdot x1$. -Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then -this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality. - -Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or -machine clock cycles.}. - -\begin{equation} -5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2 -\end{equation} - -For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold. -\begin{center} -\begin{tabular}{rcl} -${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\ -${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\ -${13 \over 9}$ & $<$ & $n$ \\ -\end{tabular} -\end{center} - -This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors -where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On -the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a -ratio of 1:7. } than simpler operations such as addition. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and -shift the input into the two halves. The loop from line 54 to line 70 has been modified since only one input exists. The \textbf{used} -count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents -to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. - -By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point -is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 -it is actually below the Comba limit (\textit{at 110 digits}). - -This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are -redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and -mp\_clears are executed normally. - -\subsection{Toom-Cook Squaring} -The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used -instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to -derive their own Toom-Cook squaring algorithm. - -\subsection{High Level Squaring} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\ -\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\ -2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\ -\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\ -3. else \\ -\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\ -\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\ -\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\ -\hspace{3mm}3.3 else \\ -\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\ -4. $b.sign \leftarrow MP\_ZPOS$ \\ -5. Return the result of the unsigned squaring performed. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_sqr} -\end{figure} - -\textbf{Algorithm mp\_sqr.} -This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least -\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If -neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ - & that have different number of digits in Karatsuba multiplication. \\ - & \\ -$\left [ 2 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\ - & of double products and at most one square is stated. Prove this statement. \\ - & \\ -$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ - & \\ -$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ - & \\ -$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ - & required for equation $6.7$ to be true. \\ - & \\ -$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\ - & compute subsets of the columns in each thread. Determine a cutoff point where \\ - & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\ - &\\ -$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook. You must \\ - & increase the throughput of mp\_exptmod() for random odd moduli in the range \\ - & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\ - & \\ -\end{tabular} - -\chapter{Modular Reduction} -\section{Basics of Modular Reduction} -\index{modular residue} -Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, -such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} -modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered -in~\ref{sec:division}. - -Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result -$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the -``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and -other forms of residues. - -Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions -is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the -RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in -elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular -exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the -range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check -algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. - -\section{The Barrett Reduction} -The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate -division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to - -\begin{equation} -c = a - b \cdot \lfloor a/b \rfloor -\end{equation} - -Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper -targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, -DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. -It would take another common optimization to optimize the algorithm. - -\subsection{Fixed Point Arithmetic} -The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed -point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were -fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit -integer and a $q$-bit fraction part (\textit{where $p+q = k$}). - -In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the -value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by -moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted -to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the -fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. - -This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication -of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is -equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer -$a$ by another integer $b$ can be achieved with the following expression. - -\begin{equation} -\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor -\end{equation} - -The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with -modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations -are considerably faster than division on most processors. - -Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which -leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and -the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. The value of $2^q$ must be close to or ideally -larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach -to work correctly. Plugging this form of divison into the original equation the following modular residue equation arises. - -\begin{equation} -c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor -\end{equation} - -Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$ -variable also helps re-inforce the idea that it is meant to be computed once and re-used. - -\begin{equation} -c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor -\end{equation} - -Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett -reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough -precision. - -Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and -another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to -reduce the number. - -For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing -$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. -By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found. - -\subsection{Choosing a Radix Point} -Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best -that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. -See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of -the initial multiplication that finds the quotient. - -Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent -the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if -two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the -$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to -express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then -${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient -is bound by $0 \le {a' \over b} < 1$. - -Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits -``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input -with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation - -\begin{equation} -c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor -\end{equation} - -Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the -exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor -would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off -by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient -can be off by an additional value of one for a total of at most two. This implies that -$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting -$b$ once or twice the residue is found. - -The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single -precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. -This is considerably faster than the original attempt. - -For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ -represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. -With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ -is found. - -\subsection{Trimming the Quotient} -So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As -it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for -optimization. - -After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower -half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision -multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. -In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. - -The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision -multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number -of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. - -\subsection{Trimming the Residue} -After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small -multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the -result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are -implicitly zero. - -The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full -$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can -be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces -only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. - -With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which -is considerably faster than the straightforward $3m^2$ method. - -\subsection{The Barrett Algorithm} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\ -\textbf{Output}. $a \mbox{ (mod }b\mbox{)}$ \\ -\hline \\ -Let $m$ represent the number of digits in $b$. \\ -1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\ -2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\ -\\ -Produce the quotient. \\ -3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\ -4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\ -\\ -Subtract the multiple of modulus from the input. \\ -5. $a \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\ -7. $a \leftarrow a - q$ (\textit{mp\_sub}) \\ -\\ -Add $\beta^{m+1}$ if a carry occured. \\ -8. If $a < 0$ then (\textit{mp\_cmp\_d}) \\ -\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\ -\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\ -\hspace{3mm}8.3 $a \leftarrow a + q$ \\ -\\ -Now subtract the modulus if the residue is too large (e.g. quotient too small). \\ -9. While $a \ge b$ do (\textit{mp\_cmp}) \\ -\hspace{3mm}9.1 $c \leftarrow a - b$ \\ -10. Clear $q$. \\ -11. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce} -\end{figure} - -\textbf{Algorithm mp\_reduce.} -This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC -\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must -be adhered to for the algorithm to work. - -First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting -a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order -for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. -Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this -algorithm and is assumed to be calculated and stored before the algorithm is used. - -Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called -$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that -instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number -of digits in $b$ is very much smaller than $\beta$. - -While it is known that -$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied -``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be -fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. - -The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is -performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves -the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits -in the modulus. In the source code this is evaluated on lines 36 to 44 where algorithm s\_mp\_mul\_high\_digs is used when it is -safe to do so. - -\subsection{The Barrett Setup Algorithm} -In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for -future use so that the Barrett algorithm can be used without delay. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_setup}. \\ -\textbf{Input}. mp\_int $a$ ($a > 1$) \\ -\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ -\hline \\ -1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ -2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ -3. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_setup} -\end{figure} - -\textbf{Algorithm mp\_reduce\_setup.} -This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which -is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable -which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the -remainder to be passed as NULL meaning to ignore the value. - -\section{The Montgomery Reduction} -Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting -form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a -residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. - -Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of -$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input -is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. - -\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way -to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. - -\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually -this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to -multiplication by $k^{-1}$ modulo $n$. - -From these two simple facts the following simple algorithm can be derived. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction}. \\ -\textbf{Input}. Integer $x$, $n$ and $k$ \\ -\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $1$ to $k$ do \\ -\hspace{3mm}1.1 If $x$ is odd then \\ -\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\ -\hspace{3mm}1.2 $x \leftarrow x/2$ \\ -2. Return $x$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction} -\end{figure} - -The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is -added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since -$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the -final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to -$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|l|} -\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\ -\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\ -\hline $2$ & $x/2 = 1453$ \\ -\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\ -\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\ -\hline $5$ & $x/2 = 278$ \\ -\hline $6$ & $x/2 = 139$ \\ -\hline $7$ & $x + n = 396$, $x/2 = 198$ \\ -\hline $8$ & $x/2 = 99$ \\ -\hline $9$ & $x + n = 356$, $x/2 = 178$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example of Montgomery Reduction (I)} -\label{fig:MONT1} -\end{figure} - -Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$). The result of -the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^9$ modulo $257$ the correct residue -$r \equiv 158$ is produced. - -Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts -and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. -Fortunately there exists an alternative representation of the algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\ -\textbf{Input}. Integer $x$, $n$ and $k$ ($2^k > n$) \\ -\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $1$ to $k$ do \\ -\hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\ -\hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\ -2. Return $x/2^k$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction (modified I)} -\end{figure} - -This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single -precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|l|r|} -\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} & \textbf{Result ($x$) in Binary} \\ -\hline -- & $5555$ & $1010110110011$ \\ -\hline $1$ & $x + 2^{0}n = 5812$ & $1011010110100$ \\ -\hline $2$ & $5812$ & $1011010110100$ \\ -\hline $3$ & $x + 2^{2}n = 6840$ & $1101010111000$ \\ -\hline $4$ & $x + 2^{3}n = 8896$ & $10001011000000$ \\ -\hline $5$ & $8896$ & $10001011000000$ \\ -\hline $6$ & $8896$ & $10001011000000$ \\ -\hline $7$ & $x + 2^{6}n = 25344$ & $110001100000000$ \\ -\hline $8$ & $25344$ & $110001100000000$ \\ -\hline $9$ & $x + 2^{7}n = 91136$ & $10110010000000000$ \\ -\hline -- & $x/2^k = 178$ & \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example of Montgomery Reduction (II)} -\label{fig:MONT2} -\end{figure} - -Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. -With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the -loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is -zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. - -\subsection{Digit Based Montgomery Reduction} -Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the -previous algorithm re-written to compute the Montgomery reduction in this new fashion. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\ -\textbf{Input}. Integer $x$, $n$ and $k$ ($\beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $0$ to $k - 1$ do \\ -\hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\ -2. Return $x/\beta^k$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction (modified II)} -\end{figure} - -The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of -the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This -problem breaks down to solving the following congruency. - -\begin{center} -\begin{tabular}{rcl} -$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\ -$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\ -$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ -\end{tabular} -\end{center} - -In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used -extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. - -For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ -represent the value to reduce. - -\newpage\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\ -\hline -- & $33$ & --\\ -\hline $0$ & $33 + \mu n = 50$ & $1$ \\ -\hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Montgomery Reduction} -\end{figure} - -The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ -which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in -the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and -the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. - -\subsection{Baseline Montgomery Reduction} -The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for -Montgomery reductions. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\ -\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ -\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. $digs \leftarrow 2n.used + 1$ \\ -2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\ -\hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\ -\\ -Setup $x$ for the reduction. \\ -3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\ -4. $x.used \leftarrow digs$ \\ -\\ -Eliminate the lower $k$ digits. \\ -5. For $ix$ from $0$ to $k - 1$ do \\ -\hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}5.2 $u \leftarrow 0$ \\ -\hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\ -\hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\ -\hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}5.4 While $u > 0$ do \\ -\hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\ -\hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\ -\hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\ -\hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\ -\\ -Divide by $\beta^k$ and fix up as required. \\ -6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\ -7. If $x \ge n$ then \\ -\hspace{3mm}7.1 $x \leftarrow x - n$ \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_montgomery\_reduce} -\end{figure} - -\textbf{Algorithm mp\_montgomery\_reduce.} -This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based -on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The -restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as -for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in -advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. - -Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on -the size of the input. This algorithm is discussed in sub-section 6.3.3. - -Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop -calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and -multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. - -Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications -in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision -multiplications. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This is the baseline implementation of the Montgomery reduction algorithm. Lines 31 to 36 determine if the Comba based -routine can be used instead. Line 47 computes the value of $\mu$ for that particular iteration of the outer loop. - -The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and -the alias $tmpn$ refers to the modulus $n$. - -\subsection{Faster ``Comba'' Montgomery Reduction} - -The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial -nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba -technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates -a $k \times 1$ product $k$ times. - -The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the -carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. -Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. - -With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases -the speed of the algorithm. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ -\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ -\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ -1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ -Copy the digits of $x$ into the array $\hat W$ \\ -2. For $ix$ from $0$ to $x.used - 1$ do \\ -\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ -3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ -\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ -Elimiate the lower $k$ digits. \\ -4. for $ix$ from $0$ to $n.used - 1$ do \\ -\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ -\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ -\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ -Propagate carries upwards. \\ -5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ -\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ -Shift right and reduce modulo $\beta$ simultaneously. \\ -6. for $ix$ from $0$ to $n.used + 1$ do \\ -\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ -Zero excess digits and fixup $x$. \\ -7. if $x.used > n.used + 1$ then do \\ -\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ -\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ -8. $x.used \leftarrow n.used + 1$ \\ -9. Clamp excessive digits of $x$. \\ -10. If $x \ge n$ then \\ -\hspace{3mm}10.1 $x \leftarrow x - n$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_mp\_montgomery\_reduce} -\end{figure} - -\textbf{Algorithm fast\_mp\_montgomery\_reduce.} -This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly -faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions -on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the -the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo -a modulus of at most $3,556$ bits in length. - -As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the -contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step -4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such -as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing -a single precision multiplication instead half the amount of time is spent. - -Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step -4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note -how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no -point. - -Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are -stored in the destination $x$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The $\hat W$ array is first filled with digits of $x$ on line 48 then the rest of the digits are zeroed on line 55. Both loops share -the same alias variables to make the code easier to read. - -The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This -forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line 110 fixes the carry -for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. - -The for loop on line 109 propagates the rest of the carries upwards through the columns. The for loop on line 126 reduces the columns -modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th -digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. - -\subsection{Montgomery Setup} -To calculate the variable $\rho$ a relatively simple algorithm will be required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_montgomery\_setup}. \\ -\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\ -\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ -\hline \\ -1. $b \leftarrow n_0$ \\ -2. If $b$ is even return(\textit{MP\_VAL}) \\ -3. $x \leftarrow (((b + 2) \mbox{ AND } 4) << 1) + b$ \\ -4. for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\ -\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\ -5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\ -6. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_montgomery\_setup} -\end{figure} - -\textbf{Algorithm mp\_montgomery\_setup.} -This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick -to calculate $1/n_0$ when $\beta$ is a power of two. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess -multiplications when $\beta$ is not the default 28-bits. - -\section{The Diminished Radix Algorithm} -The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett -or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence. - -\begin{equation} -(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} -\end{equation} - -This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that -then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof -of the above equation is very simple. First write $x$ in the product form. - -\begin{equation} -x = qn + r -\end{equation} - -Now reduce both sides modulo $(n - k)$. - -\begin{equation} -x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} -\end{equation} - -The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ -into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Diminished Radix Reduction}. \\ -\textbf{Input}. Integer $x$, $n$, $k$ \\ -\textbf{Output}. $x \mbox{ mod } (n - k)$ \\ -\hline \\ -1. $q \leftarrow \lfloor x / n \rfloor$ \\ -2. $q \leftarrow k \cdot q$ \\ -3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\ -4. $x \leftarrow x + q$ \\ -5. If $x \ge (n - k)$ then \\ -\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\ -\hspace{3mm}5.2 Goto step 1. \\ -6. Return $x$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Diminished Radix Reduction} -\label{fig:DR} -\end{figure} - -This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always -once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. - -\begin{equation} -0 \le x < n^2 + k^2 - 2nk -\end{equation} - -The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following. - -\begin{equation} -q < n - 2k - k^2/n -\end{equation} - -Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as -$0 \le x < n$. By step four the sum $x + q$ is bounded by - -\begin{equation} -0 \le q + x < (k + 1)n - 2k^2 - 1 -\end{equation} - -With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the -sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the -range $0 \le x < (n - k - 1)^2$. - -\begin{figure} -\begin{small} -\begin{center} -\begin{tabular}{|l|} -\hline -$x = 123456789, n = 256, k = 3$ \\ -\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\ -$q \leftarrow q*k = 1446759$ \\ -$x \leftarrow x \mbox{ mod } n = 21$ \\ -$x \leftarrow x + q = 1446780$ \\ -$x \leftarrow x - (n - k) = 1446527$ \\ -\hline -$q \leftarrow \lfloor x/n \rfloor = 5650$ \\ -$q \leftarrow q*k = 16950$ \\ -$x \leftarrow x \mbox{ mod } n = 127$ \\ -$x \leftarrow x + q = 17077$ \\ -$x \leftarrow x - (n - k) = 16824$ \\ -\hline -$q \leftarrow \lfloor x/n \rfloor = 65$ \\ -$q \leftarrow q*k = 195$ \\ -$x \leftarrow x \mbox{ mod } n = 184$ \\ -$x \leftarrow x + q = 379$ \\ -$x \leftarrow x - (n - k) = 126$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example Diminished Radix Reduction} -\label{fig:EXDR} -\end{figure} - -Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$ -is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only -three passes were required to find the residue $x \equiv 126$. - - -\subsection{Choice of Moduli} -On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other -modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. - -Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. -Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division -by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ -which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. - -However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be -performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. -Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. - -Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted -modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the -$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. - -\subsection{Choice of $k$} -Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ -in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might -as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. - -\subsection{Restricted Diminished Radix Reduction} -The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce -an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation -of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition -of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular -exponentiations are performed. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_reduce}. \\ -\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\ -\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\ -\textbf{Output}. $x \mbox{ mod } n$ \\ -\hline \\ -1. $m \leftarrow n.used$ \\ -2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\ -3. $\mu \leftarrow 0$ \\ -4. for $i$ from $0$ to $m - 1$ do \\ -\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\ -\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -5. $x_{m} \leftarrow \mu$ \\ -6. for $i$ from $m + 1$ to $x.used - 1$ do \\ -\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\ -7. Clamp excess digits of $x$. \\ -8. If $x \ge n$ then \\ -\hspace{3mm}8.1 $x \leftarrow x - n$ \\ -\hspace{3mm}8.2 Goto step 3. \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_reduce} -\end{figure} - -\textbf{Algorithm mp\_dr\_reduce.} -This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction -with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. - -This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ -and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing -the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th -digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to -$x$ before the addition of the multiple of the upper half. - -At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes -at step 3. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line 52 is where -the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of -the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. - -The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits -a division by $\beta^m$ can be simulated virtually for free. The loop on line 64 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) -in this algorithm. - -By line 67 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line 74 the -same pointer will point to the $m+1$'th digit where the zeroes will be placed. - -Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. -With the same logic at line 81 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used -as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code -does not need to be checked. - -\subsubsection{Setup} -To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for -completeness. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_setup}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $k = \beta - n_0$ \\ -\hline \\ -1. $k \leftarrow \beta - n_0$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_setup} -\end{figure} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsubsection{Modulus Detection} -Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be -of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ -\hline -1. If $n.used < 2$ then return($0$). \\ -2. for $ix$ from $1$ to $n.used - 1$ do \\ -\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ -3. Return($1$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_is\_modulus} -\end{figure} - -\textbf{Algorithm mp\_dr\_is\_modulus.} -This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are -in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to -step 3 then $n$ must be of Diminished Radix form. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsection{Unrestricted Diminished Radix Reduction} -The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm -is a straightforward adaptation of algorithm~\ref{fig:DR}. - -In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new -algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_2k}. \\ -\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ -\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ -\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ -\hline -1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -2. While $a \ge n$ do \\ -\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\ -\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\ -\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.5 If $a \ge n$ then do \\ -\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_2k} -\end{figure} - -\textbf{Algorithm mp\_reduce\_2k.} -This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right -shift which makes the algorithm fairly inexpensive to use. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d -on line 31 calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size -is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without -any multiplications. - -The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are -positive. By using the unsigned versions the overhead is kept to a minimum. - -\subsubsection{Unrestricted Setup} -To setup this reduction algorithm the value of $k = 2^p - n$ is required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $k = 2^p - n$ \\ -\hline -1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\ -3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\ -4. $k \leftarrow x_0$ \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_2k\_setup} -\end{figure} - -\textbf{Algorithm mp\_reduce\_2k\_setup.} -This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction -is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsubsection{Unrestricted Detection} -An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. - -\begin{enumerate} -\item The number has only one digit. -\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. -\end{enumerate} - -If either condition is true than there is a power of two $2^p$ such that $0 < 2^p - n < \beta$. If the input is only -one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact -that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most -significant bit. The resulting sum will be a power of two. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $1$ if of proper form, $0$ otherwise \\ -\hline -1. If $n.used = 0$ then return($0$). \\ -2. If $n.used = 1$ then return($1$). \\ -3. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -4. for $x$ from $lg(\beta)$ to $p$ do \\ -\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\ -5. Return($1$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_is\_2k} -\end{figure} - -\textbf{Algorithm mp\_reduce\_is\_2k.} -This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - - - -\section{Algorithm Comparison} -So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses -that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since -all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. - -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\ -\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\ -\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\ -\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\ -\hline -\end{tabular} -\end{small} -\end{center} - -In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery -reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of -calling the half precision multipliers, addition and division by $\beta$ algorithms. - -For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly -shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms -primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in -modular exponentiation to greatly speed up the operation. - - - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\ - & calculates the correct value of $\rho$. \\ - & \\ -$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\ - & \\ -$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\ - & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ - & terminate within $1 \le k \le 10$ iterations. \\ - & \\ -\end{tabular} - - -\chapter{Exponentiation} -Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed -in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key -cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any -such cryptosystem and many methods have been sought to speed it up. - -\section{Exponentiation Basics} -A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size -the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature -with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. - -Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which -are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least -significant bit. If $b$ is a $k$-bit integer than the following equation is true. - -\begin{equation} -a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i} -\end{equation} - -By taking the base $a$ logarithm of both sides of the equation the following equation is the result. - -\begin{equation} -b = \sum_{i=0}^{k-1}2^i \cdot b_i -\end{equation} - -The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to -$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average -$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. - -While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to -be computed in an auxilary variable. Consider the following equivalent algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Left to Right Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$ and $k$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $k - 1$ to $0$ do \\ -\hspace{3mm}2.1 $c \leftarrow c^2$ \\ -\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Left to Right Exponentiation} -\label{fig:LTOR} -\end{figure} - -This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is -multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the -product. - -For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. - -\newpage\begin{figure} -\begin{center} -\begin{tabular}{|c|c|} -\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\ -\hline - & $1$ \\ -\hline $5$ & $a$ \\ -\hline $4$ & $a^2$ \\ -\hline $3$ & $a^4 \cdot a$ \\ -\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\ -\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\ -\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Left to Right Exponentiation} -\end{figure} - -When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is -called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. - -\subsection{Single Digit Exponentiation} -The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended -to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of -$b$ that are greater than three. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_expt\_d}. \\ -\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\ -2. $c \leftarrow 1$ (\textit{mp\_set}) \\ -3. for $x$ from 1 to $lg(\beta)$ do \\ -\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\ -\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\ -\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\ -\hspace{3mm}3.3 $b \leftarrow b << 1$ \\ -4. Clear $g$. \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_expt\_d} -\end{figure} - -\textbf{Algorithm mp\_expt\_d.} -This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to -quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the -exponent is a fixed width. - -A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of -$1$ in the subsequent step. - -Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared -on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value -of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each -iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Line 29 sets the initial value of the result to $1$. Next the loop on line 31 steps through each bit of the exponent starting from -the most significant down towards the least significant. The invariant squaring operation placed on line 33 is performed first. After -the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line -47 moves all of the bits of the exponent upwards towards the most significant location. - -\section{$k$-ary Exponentiation} -When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor -slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to -the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY} -computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a -portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{$k$-ary Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $t - 1$ to $0$ do \\ -\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\ -\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\ -\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{$k$-ary Exponentiation} -\label{fig:KARY} -\end{figure} - -The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been -precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and -$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. -However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. - -Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The -original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings -has increased slightly but the number of multiplications has nearly halved. - -\subsection{Optimal Values of $k$} -An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest -approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ -for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\ -\hline $16$ & $2$ & $27$ & $24$ \\ -\hline $32$ & $3$ & $49$ & $48$ \\ -\hline $64$ & $3$ & $92$ & $96$ \\ -\hline $128$ & $4$ & $175$ & $192$ \\ -\hline $256$ & $4$ & $335$ & $384$ \\ -\hline $512$ & $5$ & $645$ & $768$ \\ -\hline $1024$ & $6$ & $1257$ & $1536$ \\ -\hline $2048$ & $6$ & $2452$ & $3072$ \\ -\hline $4096$ & $7$ & $4808$ & $6144$ \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Optimal Values of $k$ for $k$-ary Exponentiation} -\label{fig:OPTK} -\end{figure} - -\subsection{Sliding-Window Exponentiation} -A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially -this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the -algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. - -Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\ -\hline $16$ & $3$ & $24$ & $27$ \\ -\hline $32$ & $3$ & $45$ & $49$ \\ -\hline $64$ & $4$ & $87$ & $92$ \\ -\hline $128$ & $4$ & $167$ & $175$ \\ -\hline $256$ & $5$ & $322$ & $335$ \\ -\hline $512$ & $6$ & $628$ & $645$ \\ -\hline $1024$ & $6$ & $1225$ & $1257$ \\ -\hline $2048$ & $7$ & $2403$ & $2452$ \\ -\hline $4096$ & $8$ & $4735$ & $4808$ \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Optimal Values of $k$ for Sliding Window Exponentiation} -\label{fig:OPTK2} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $t - 1$ to $0$ do \\ -\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\ -\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\ -\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\ -\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\ -\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Sliding Window $k$-ary Exponentiation} -\end{figure} - -Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this -algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half -the size as the previous table. - -Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as -the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the -exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where -a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ -squarings. The second method requires $8$ multiplications and $18$ squarings. - -In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. - -\section{Modular Exponentiation} - -Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing -$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it -modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. - -This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using -one of the algorithms presented in chapter six. - -Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm -will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The -value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm -terminates with an error. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_exptmod}. \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ -2. If $b.sign = MP\_NEG$ then \\ -\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\ -\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\ -\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\ -3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\ -\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\ -4. else \\ -\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_exptmod} -\end{figure} - -\textbf{Algorithm mp\_exptmod.} -The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm -which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation -except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation -algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -In order to keep the algorithms in a known state the first step on line 29 is to reject any negative modulus as input. If the exponent is -negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned -the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive -exponent. - -If the exponent is positive the algorithm resumes the exponentiation. Line 77 determines if the modulus is of the restricted Diminished Radix -form. If it is not line 70 attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one -of three values. - -\begin{enumerate} -\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form. -\item $dr = 1$ means that the modulus is of restricted Diminished Radix form. -\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form. -\end{enumerate} - -Line 69 determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, -the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. - -\subsection{Barrett Modular Exponentiation} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_exptmod}. \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -1. $k \leftarrow lg(x)$ \\ -2. $winsize \leftarrow \left \lbrace \begin{array}{ll} - 2 & \mbox{if }k \le 7 \\ - 3 & \mbox{if }7 < k \le 36 \\ - 4 & \mbox{if }36 < k \le 140 \\ - 5 & \mbox{if }140 < k \le 450 \\ - 6 & \mbox{if }450 < k \le 1303 \\ - 7 & \mbox{if }1303 < k \le 3529 \\ - 8 & \mbox{if }3529 < k \\ - \end{array} \right .$ \\ -3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\ -4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\ -5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\ -\\ -Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\ -6. $k \leftarrow 2^{winsize - 1}$ \\ -7. $M_{k} \leftarrow M_1$ \\ -8. for $ix$ from 0 to $winsize - 2$ do \\ -\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\ -\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ -9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\ -\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\ -\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ -10. $res \leftarrow 1$ \\ -\\ -Start Sliding Window. \\ -11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\ -12. Loop \\ -\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\ -\hspace{3mm}12.2 If $bitcnt = 0$ then do \\ -\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\ -\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\ -\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\ -\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\ -Continued on next page. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_exptmod} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\ -\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\ -\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\ -\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\ -\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\ -\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}12.6.3 Goto step 12. \\ -\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\ -\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\ -\hspace{3mm}12.9 $mode \leftarrow 2$ \\ -\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\ -\hspace{6mm}Window is full so perform the squarings and single multiplication. \\ -\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\ -\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\ -\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\ -\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}Reset the window. \\ -\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\ -\\ -No more windows left. Check for residual bits of exponent. \\ -13. If $mode = 2$ and $bitcpy > 0$ then do \\ -\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\ -\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\ -\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\ -\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\ -\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\ -\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -14. $y \leftarrow res$ \\ -15. Clear $res$, $mu$ and the $M$ array. \\ -16. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_exptmod (continued)} -\end{figure} - -\textbf{Algorithm s\_mp\_exptmod.} -This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction -algorithm to keep the product small throughout the algorithm. - -The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the -larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This -table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. - -After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make -the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ -times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$. - -Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. -\begin{enumerate} -\item The variable $mode$ dictates how the bits of the exponent are interpreted. -\begin{enumerate} - \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply - $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. - \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits - are read and a single squaring is performed. If a non-zero bit is read a new window is created. - \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit - downwards. -\end{enumerate} -\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit - is fetched from the exponent. -\item The variable $buf$ holds the currently read digit of the exponent. -\item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. -\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and - the appropriate operations performed. -\item The variable $bitbuf$ holds the current bits of the window being formed. -\end{enumerate} - -All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step -inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is -read and if there are no digits left than the loop terminates. - -After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit -upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to -trailing edges the entire exponent is read from most significant bit to least significant bit. - -At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the -algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle -the two cases of $mode = 1$ and $mode = 2$ respectively. - -\begin{center} -\begin{figure}[here] -\includegraphics{pics/expt_state.ps} -\caption{Sliding Window State Diagram} -\label{pic:expt_state} -\end{figure} -\end{center} - -By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then -a Left-to-Right algorithm is used to process the remaining few bits. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Lines 32 through 46 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted -from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement -on line 38 the value of $x$ is already known to be greater than $140$. - -The conditional piece of code beginning on line 48 allows the window size to be restricted to five bits. This logic is used to ensure -the table of precomputed powers of $G$ remains relatively small. - -The for loop on line 61 initializes the $M$ array while lines 72 and 77 through 86 initialize the reduction -function that will be used for this modulus. - --- More later. - -\section{Quick Power of Two} -Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is -equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_2expt}. \\ -\textbf{Input}. integer $b$ \\ -\textbf{Output}. $a \leftarrow 2^b$ \\ -\hline \\ -1. $a \leftarrow 0$ \\ -2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\ -3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\ -4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_2expt} -\end{figure} - -\textbf{Algorithm mp\_2expt.} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\chapter{Higher Level Algorithms} - -This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These -routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. - -The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic -for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. -These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate -various representations of integers. For example, converting from an mp\_int to a string of character. - -\section{Integer Division with Remainder} -\label{sec:division} - -Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication -the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables -will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and -let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\ -\textbf{Input}. integer $x$ and $y$ \\ -\textbf{Output}. $q = \lfloor y/x\rfloor, r = y - xq$ \\ -\hline \\ -1. $q \leftarrow 0$ \\ -2. $n \leftarrow \vert \vert y \vert \vert - \vert \vert x \vert \vert$ \\ -3. for $t$ from $n$ down to $0$ do \\ -\hspace{3mm}3.1 Maximize $k$ such that $kx\beta^t$ is less than or equal to $y$ and $(k + 1)x\beta^t$ is greater. \\ -\hspace{3mm}3.2 $q \leftarrow q + k\beta^t$ \\ -\hspace{3mm}3.3 $y \leftarrow y - kx\beta^t$ \\ -4. $r \leftarrow y$ \\ -5. Return($q, r$) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Radix-$\beta$ Integer Division} -\label{fig:raddiv} -\end{figure} - -As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which -their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. - -To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and -simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method -used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading -digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly -arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. -As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. - -Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder -$y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the -remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since -$237 \cdot 23 + 20 = 5471$ is true. - -\subsection{Quotient Estimation} -\label{sec:divest} -As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading -digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically -speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the -dividend and divisor are zero. - -The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} -of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate -using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ -represent the most significant digits of the dividend and divisor respectively. - -\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to -$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } -The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other -cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility -$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of -inequalities will prove the hypothesis. - -\begin{equation} -y - \hat k x \le y - \hat k x_s\beta^s -\end{equation} - -This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. - -\begin{equation} -y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) -\end{equation} - -By simplifying the previous inequality the following inequality is formed. - -\begin{equation} -y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s -\end{equation} - -Subsequently, - -\begin{equation} -y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x -\end{equation} - -Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED} - - -\subsection{Normalized Integers} -For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both -$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original -remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will -lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. - -\begin{equation} -{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} -\end{equation} - -At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. - -\subsection{Radix-$\beta$ Division with Remainder} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div}. \\ -\textbf{Input}. mp\_int $a, b$ \\ -\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ -\hline \\ -1. If $b = 0$ return(\textit{MP\_VAL}). \\ -2. If $\vert a \vert < \vert b \vert$ then do \\ -\hspace{3mm}2.1 $d \leftarrow a$ \\ -\hspace{3mm}2.2 $c \leftarrow 0$ \\ -\hspace{3mm}2.3 Return(\textit{MP\_OKAY}). \\ -\\ -Setup the quotient to receive the digits. \\ -3. Grow $q$ to $a.used + 2$ digits. \\ -4. $q \leftarrow 0$ \\ -5. $x \leftarrow \vert a \vert , y \leftarrow \vert b \vert$ \\ -6. $sign \leftarrow \left \lbrace \begin{array}{ll} - MP\_ZPOS & \mbox{if }a.sign = b.sign \\ - MP\_NEG & \mbox{otherwise} \\ - \end{array} \right .$ \\ -\\ -Normalize the inputs such that the leading digit of $y$ is greater than or equal to $\beta / 2$. \\ -7. $norm \leftarrow (lg(\beta) - 1) - (\lceil lg(y) \rceil \mbox{ (mod }lg(\beta)\mbox{)})$ \\ -8. $x \leftarrow x \cdot 2^{norm}, y \leftarrow y \cdot 2^{norm}$ \\ -\\ -Find the leading digit of the quotient. \\ -9. $n \leftarrow x.used - 1, t \leftarrow y.used - 1$ \\ -10. $y \leftarrow y \cdot \beta^{n - t}$ \\ -11. While ($x \ge y$) do \\ -\hspace{3mm}11.1 $q_{n - t} \leftarrow q_{n - t} + 1$ \\ -\hspace{3mm}11.2 $x \leftarrow x - y$ \\ -12. $y \leftarrow \lfloor y / \beta^{n-t} \rfloor$ \\ -\\ -Continued on the next page. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div} (continued). \\ -\textbf{Input}. mp\_int $a, b$ \\ -\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ -\hline \\ -Now find the remainder fo the digits. \\ -13. for $i$ from $n$ down to $(t + 1)$ do \\ -\hspace{3mm}13.1 If $i > x.used$ then jump to the next iteration of this loop. \\ -\hspace{3mm}13.2 If $x_{i} = y_{t}$ then \\ -\hspace{6mm}13.2.1 $q_{i - t - 1} \leftarrow \beta - 1$ \\ -\hspace{3mm}13.3 else \\ -\hspace{6mm}13.3.1 $\hat r \leftarrow x_{i} \cdot \beta + x_{i - 1}$ \\ -\hspace{6mm}13.3.2 $\hat r \leftarrow \lfloor \hat r / y_{t} \rfloor$ \\ -\hspace{6mm}13.3.3 $q_{i - t - 1} \leftarrow \hat r$ \\ -\hspace{3mm}13.4 $q_{i - t - 1} \leftarrow q_{i - t - 1} + 1$ \\ -\\ -Fixup quotient estimation. \\ -\hspace{3mm}13.5 Loop \\ -\hspace{6mm}13.5.1 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ -\hspace{6mm}13.5.2 t$1 \leftarrow 0$ \\ -\hspace{6mm}13.5.3 t$1_0 \leftarrow y_{t - 1}, $ t$1_1 \leftarrow y_t,$ t$1.used \leftarrow 2$ \\ -\hspace{6mm}13.5.4 $t1 \leftarrow t1 \cdot q_{i - t - 1}$ \\ -\hspace{6mm}13.5.5 t$2_0 \leftarrow x_{i - 2}, $ t$2_1 \leftarrow x_{i - 1}, $ t$2_2 \leftarrow x_i, $ t$2.used \leftarrow 3$ \\ -\hspace{6mm}13.5.6 If $\vert t1 \vert > \vert t2 \vert$ then goto step 13.5. \\ -\hspace{3mm}13.6 t$1 \leftarrow y \cdot q_{i - t - 1}$ \\ -\hspace{3mm}13.7 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ -\hspace{3mm}13.8 $x \leftarrow x - $ t$1$ \\ -\hspace{3mm}13.9 If $x.sign = MP\_NEG$ then \\ -\hspace{6mm}13.10 t$1 \leftarrow y$ \\ -\hspace{6mm}13.11 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ -\hspace{6mm}13.12 $x \leftarrow x + $ t$1$ \\ -\hspace{6mm}13.13 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ -\\ -Finalize the result. \\ -14. Clamp excess digits of $q$ \\ -15. $c \leftarrow q, c.sign \leftarrow sign$ \\ -16. $x.sign \leftarrow a.sign$ \\ -17. $d \leftarrow \lfloor x / 2^{norm} \rfloor$ \\ -18. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div (continued)} -\end{figure} -\textbf{Algorithm mp\_div.} -This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed -division and will produce a fully qualified quotient and remainder. - -First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly -zero and the remainder is the dividend. - -After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the -divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are -positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. -This is performed by shifting both to the left by enough bits to get the desired normalization. - -At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is -$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted -to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the -shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two -times to produce the desired leading digit of the quotient. - -Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly -accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by -induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. - -Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is -to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher -order approximation to adjust the quotient digit. - -After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced -by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of -algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. - -Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the -remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} -is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie -outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should -respectively be replaced with a zero. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or -remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division -algorithm with only the quotient is - -\begin{verbatim} -mp_div(&a, &b, &c, NULL); /* c = [a/b] */ -\end{verbatim} - -Lines 109 and 113 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor -respectively. After the two trivial cases all of the temporary variables are initialized. Line 148 determines the sign of -the quotient and line 148 ensures that both $x$ and $y$ are positive. - -The number of bits in the leading digit is calculated on line 151. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits -of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is -exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting -them to the left by $lg(\beta) - 1 - k$ bits. - -Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the -leading digit of the quotient. The loop beginning on line 184 will produce the remainder of the quotient digits. - -The conditional ``continue'' on line 187 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the -algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits -above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. - -Lines 214, 216 and 223 through 225 manually construct the high accuracy estimations by setting the digits of the two mp\_int -variables directly. - -\section{Single Digit Helpers} - -This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of -the helper functions assume the single digit input is positive and will treat them as such. - -\subsection{Single Digit Addition and Subtraction} - -Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction -algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_add\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = a + b$ \\ -\hline \\ -1. $t \leftarrow b$ (\textit{mp\_set}) \\ -2. $c \leftarrow a + t$ \\ -3. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_add\_d} -\end{figure} - -\textbf{Algorithm mp\_add\_d.} -This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Clever use of the letter 't'. - -\subsubsection{Subtraction} -The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int. - -\subsection{Single Digit Multiplication} -Single digit multiplication arises enough in division and radix conversion that it ought to be implement as a special case of the baseline -multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands -only has one digit. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = ab$ \\ -\hline \\ -1. $pa \leftarrow a.used$ \\ -2. Grow $c$ to at least $pa + 1$ digits. \\ -3. $oldused \leftarrow c.used$ \\ -4. $c.used \leftarrow pa + 1$ \\ -5. $c.sign \leftarrow a.sign$ \\ -6. $\mu \leftarrow 0$ \\ -7. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}7.1 $\hat r \leftarrow \mu + a_{ix}b$ \\ -\hspace{3mm}7.2 $c_{ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}7.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -8. $c_{pa} \leftarrow \mu$ \\ -9. for $ix$ from $pa + 1$ to $oldused$ do \\ -\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ -10. Clamp excess digits of $c$. \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_d} -\end{figure} -\textbf{Algorithm mp\_mul\_d.} -This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. -Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is -read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. - -\subsection{Single Digit Division} -Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the -divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = \lfloor a / b \rfloor, d = a - cb$ \\ -\hline \\ -1. If $b = 0$ then return(\textit{MP\_VAL}).\\ -2. If $b = 3$ then use algorithm mp\_div\_3 instead. \\ -3. Init $q$ to $a.used$ digits. \\ -4. $q.used \leftarrow a.used$ \\ -5. $q.sign \leftarrow a.sign$ \\ -6. $\hat w \leftarrow 0$ \\ -7. for $ix$ from $a.used - 1$ down to $0$ do \\ -\hspace{3mm}7.1 $\hat w \leftarrow \hat w \beta + a_{ix}$ \\ -\hspace{3mm}7.2 If $\hat w \ge b$ then \\ -\hspace{6mm}7.2.1 $t \leftarrow \lfloor \hat w / b \rfloor$ \\ -\hspace{6mm}7.2.2 $\hat w \leftarrow \hat w \mbox{ (mod }b\mbox{)}$ \\ -\hspace{3mm}7.3 else\\ -\hspace{6mm}7.3.1 $t \leftarrow 0$ \\ -\hspace{3mm}7.4 $q_{ix} \leftarrow t$ \\ -8. $d \leftarrow \hat w$ \\ -9. Clamp excess digits of $q$. \\ -10. $c \leftarrow q$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_d} -\end{figure} -\textbf{Algorithm mp\_div\_d.} -This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the -algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ -after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. - -If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with -a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction -from chapter seven. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to -indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. - -The division and remainder on lines 44 and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based -processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC -compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. - -\subsection{Single Digit Root Extraction} - -Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation -(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. - -\begin{equation} -x_{i+1} = x_i - {f(x_i) \over f'(x_i)} -\label{eqn:newton} -\end{equation} - -In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is -simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain -such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the -algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_n\_root}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c^b \le a$ \\ -\hline \\ -1. If $b$ is even and $a.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ -2. $sign \leftarrow a.sign$ \\ -3. $a.sign \leftarrow MP\_ZPOS$ \\ -4. t$2 \leftarrow 2$ \\ -5. Loop \\ -\hspace{3mm}5.1 t$1 \leftarrow $ t$2$ \\ -\hspace{3mm}5.2 t$3 \leftarrow $ t$1^{b - 1}$ \\ -\hspace{3mm}5.3 t$2 \leftarrow $ t$3 $ $\cdot$ t$1$ \\ -\hspace{3mm}5.4 t$2 \leftarrow $ t$2 - a$ \\ -\hspace{3mm}5.5 t$3 \leftarrow $ t$3 \cdot b$ \\ -\hspace{3mm}5.6 t$3 \leftarrow \lfloor $t$2 / $t$3 \rfloor$ \\ -\hspace{3mm}5.7 t$2 \leftarrow $ t$1 - $ t$3$ \\ -\hspace{3mm}5.8 If t$1 \ne $ t$2$ then goto step 5. \\ -6. Loop \\ -\hspace{3mm}6.1 t$2 \leftarrow $ t$1^b$ \\ -\hspace{3mm}6.2 If t$2 > a$ then \\ -\hspace{6mm}6.2.1 t$1 \leftarrow $ t$1 - 1$ \\ -\hspace{6mm}6.2.2 Goto step 6. \\ -7. $a.sign \leftarrow sign$ \\ -8. $c \leftarrow $ t$1$ \\ -9. $c.sign \leftarrow sign$ \\ -10. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_n\_root} -\end{figure} -\textbf{Algorithm mp\_n\_root.} -This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation -that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator. That is at first the denominator is calculated by finding -$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ -multiplications by t$1$ inside the loop. - -The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the -root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_n\_root.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section{Random Number Generation} - -Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho -factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented -is solely for simulations and not intended for cryptographic use. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_rand}. \\ -\textbf{Input}. An integer $b$ \\ -\textbf{Output}. A pseudo-random number of $b$ digits \\ -\hline \\ -1. $a \leftarrow 0$ \\ -2. If $b \le 0$ return(\textit{MP\_OKAY}) \\ -3. Pick a non-zero random digit $d$. \\ -4. $a \leftarrow a + d$ \\ -5. for $ix$ from 1 to $d - 1$ do \\ -\hspace{3mm}5.1 $a \leftarrow a \cdot \beta$ \\ -\hspace{3mm}5.2 Pick a random digit $d$. \\ -\hspace{3mm}5.3 $a \leftarrow a + d$ \\ -6. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_rand} -\end{figure} -\textbf{Algorithm mp\_rand.} -This algorithm produces a pseudo-random integer of $b$ digits. By ensuring that the first digit is non-zero the algorithm also guarantees that the -final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of -the integers from $0$ to $\beta - 1$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_rand.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section{Formatted Representations} -The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties. For example, the ability to -be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers -into a program. - -\subsection{Reading Radix-n Input} -For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to -printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the -map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen -such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary -mediums. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{cc|cc|cc|cc} -\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\ -\hline -0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\ -4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ -8 & 8 & 9 & 9 & 10 & A & 11 & B \\ -12 & C & 13 & D & 14 & E & 15 & F \\ -16 & G & 17 & H & 18 & I & 19 & J \\ -20 & K & 21 & L & 22 & M & 23 & N \\ -24 & O & 25 & P & 26 & Q & 27 & R \\ -28 & S & 29 & T & 30 & U & 31 & V \\ -32 & W & 33 & X & 34 & Y & 35 & Z \\ -36 & a & 37 & b & 38 & c & 39 & d \\ -40 & e & 41 & f & 42 & g & 43 & h \\ -44 & i & 45 & j & 46 & k & 47 & l \\ -48 & m & 49 & n & 50 & o & 51 & p \\ -52 & q & 53 & r & 54 & s & 55 & t \\ -56 & u & 57 & v & 58 & w & 59 & x \\ -60 & y & 61 & z & 62 & $+$ & 63 & $/$ \\ -\hline -\end{tabular} -\end{center} -\caption{Lower ASCII Map} -\label{fig:ASC} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_read\_radix}. \\ -\textbf{Input}. A string $str$ of length $sn$ and radix $r$. \\ -\textbf{Output}. The radix-$\beta$ equivalent mp\_int. \\ -\hline \\ -1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ -2. $ix \leftarrow 0$ \\ -3. If $str_0 =$ ``-'' then do \\ -\hspace{3mm}3.1 $ix \leftarrow ix + 1$ \\ -\hspace{3mm}3.2 $sign \leftarrow MP\_NEG$ \\ -4. else \\ -\hspace{3mm}4.1 $sign \leftarrow MP\_ZPOS$ \\ -5. $a \leftarrow 0$ \\ -6. for $iy$ from $ix$ to $sn - 1$ do \\ -\hspace{3mm}6.1 Let $y$ denote the position in the map of $str_{iy}$. \\ -\hspace{3mm}6.2 If $str_{iy}$ is not in the map or $y \ge r$ then goto step 7. \\ -\hspace{3mm}6.3 $a \leftarrow a \cdot r$ \\ -\hspace{3mm}6.4 $a \leftarrow a + y$ \\ -7. If $a \ne 0$ then $a.sign \leftarrow sign$ \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_read\_radix} -\end{figure} -\textbf{Algorithm mp\_read\_radix.} -This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the -string to indicate the value is negative, otherwise it is assumed to be positive. The algorithm will read up to $sn$ characters from the input -and will stop when it reads a character it cannot map the algorithm stops reading characters from the string. This allows numbers to be embedded -as part of larger input without any significant problem. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_read\_radix.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsection{Generating Radix-$n$ Output} -Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toradix}. \\ -\textbf{Input}. A mp\_int $a$ and an integer $r$\\ -\textbf{Output}. The radix-$r$ representation of $a$ \\ -\hline \\ -1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ -2. If $a = 0$ then $str = $ ``$0$'' and return(\textit{MP\_OKAY}). \\ -3. $t \leftarrow a$ \\ -4. $str \leftarrow$ ``'' \\ -5. if $t.sign = MP\_NEG$ then \\ -\hspace{3mm}5.1 $str \leftarrow str + $ ``-'' \\ -\hspace{3mm}5.2 $t.sign = MP\_ZPOS$ \\ -6. While ($t \ne 0$) do \\ -\hspace{3mm}6.1 $d \leftarrow t \mbox{ (mod }r\mbox{)}$ \\ -\hspace{3mm}6.2 $t \leftarrow \lfloor t / r \rfloor$ \\ -\hspace{3mm}6.3 Look up $d$ in the map and store the equivalent character in $y$. \\ -\hspace{3mm}6.4 $str \leftarrow str + y$ \\ -7. If $str_0 = $``$-$'' then \\ -\hspace{3mm}7.1 Reverse the digits $str_1, str_2, \ldots str_n$. \\ -8. Otherwise \\ -\hspace{3mm}8.1 Reverse the digits $str_0, str_1, \ldots str_n$. \\ -9. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toradix} -\end{figure} -\textbf{Algorithm mp\_toradix.} -This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing -successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$. Note that instead of actually dividing by $r^k$ in -each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration. As a result a series of trivial $n \times 1$ divisions -are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order -(see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''. - -\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\ -\hline $1234$ & -- & -- \\ -\hline $123$ & $4$ & ``4'' \\ -\hline $12$ & $3$ & ``43'' \\ -\hline $1$ & $2$ & ``432'' \\ -\hline $0$ & $1$ & ``4321'' \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Algorithm mp\_toradix.} -\label{fig:mpradix} -\end{figure} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_toradix.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\chapter{Number Theoretic Algorithms} -This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi -symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and -various Sieve based factoring algorithms. - -\section{Greatest Common Divisor} -The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of -both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur -simultaneously. - -The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then -$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\ -\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ -\textbf{Output}. The greatest common divisor $(a, b)$. \\ -\hline \\ -1. While ($b > 0$) do \\ -\hspace{3mm}1.1 $r \leftarrow a \mbox{ (mod }b\mbox{)}$ \\ -\hspace{3mm}1.2 $a \leftarrow b$ \\ -\hspace{3mm}1.3 $b \leftarrow r$ \\ -2. Return($a$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Greatest Common Divisor (I)} -\label{fig:gcd1} -\end{figure} - -This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are -relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of -greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. -In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\ -\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ -\textbf{Output}. The greatest common divisor $(a, b)$. \\ -\hline \\ -1. While ($b > 0$) do \\ -\hspace{3mm}1.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ -\hspace{3mm}1.2 $b \leftarrow b - a$ \\ -2. Return($a$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Greatest Common Divisor (II)} -\label{fig:gcd2} -\end{figure} - -\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.} -The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$. In other -words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always -divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the -second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}. - -As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that -$b - a$ is still very much larger than $a$. A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does -not divide the greatest common divisor but will divide $b - a$. In this case ${b - a} \over p$ is also an integer and still divisible by -the greatest common divisor. - -However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first. -Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\ -\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ -\textbf{Output}. The greatest common divisor $(a, b)$. \\ -\hline \\ -1. $k \leftarrow 0$ \\ -2. While $a$ and $b$ are both divisible by $p$ do \\ -\hspace{3mm}2.1 $a \leftarrow \lfloor a / p \rfloor$ \\ -\hspace{3mm}2.2 $b \leftarrow \lfloor b / p \rfloor$ \\ -\hspace{3mm}2.3 $k \leftarrow k + 1$ \\ -3. While $a$ is divisible by $p$ do \\ -\hspace{3mm}3.1 $a \leftarrow \lfloor a / p \rfloor$ \\ -4. While $b$ is divisible by $p$ do \\ -\hspace{3mm}4.1 $b \leftarrow \lfloor b / p \rfloor$ \\ -5. While ($b > 0$) do \\ -\hspace{3mm}5.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ -\hspace{3mm}5.2 $b \leftarrow b - a$ \\ -\hspace{3mm}5.3 While $b$ is divisible by $p$ do \\ -\hspace{6mm}5.3.1 $b \leftarrow \lfloor b / p \rfloor$ \\ -6. Return($a \cdot p^k$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Greatest Common Divisor (III)} -\label{fig:gcd3} -\end{figure} - -This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ -decreases more rapidly. The first loop on step two removes powers of $p$ that are in common. A count, $k$, is kept which will present a common -divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely -divided out of the difference $b - a$ so long as the division leaves no remainder. - -In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often. It also helps that division by $p$ be easy -to compute. The ideal choice of $p$ is two since division by two amounts to a right logical shift. Another important observation is that by -step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the -largest of the pair. - -\subsection{Complete Greatest Common Divisor} -The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly -and will produce the greatest common divisor. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_gcd}. \\ -\textbf{Input}. mp\_int $a$ and $b$ \\ -\textbf{Output}. The greatest common divisor $c = (a, b)$. \\ -\hline \\ -1. If $a = 0$ then \\ -\hspace{3mm}1.1 $c \leftarrow \vert b \vert $ \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $b = 0$ then \\ -\hspace{3mm}2.1 $c \leftarrow \vert a \vert $ \\ -\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ -3. $u \leftarrow \vert a \vert, v \leftarrow \vert b \vert$ \\ -4. $k \leftarrow 0$ \\ -5. While $u.used > 0$ and $v.used > 0$ and $u_0 \equiv v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ -\hspace{3mm}5.2 $u \leftarrow \lfloor u / 2 \rfloor$ \\ -\hspace{3mm}5.3 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ -7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -8. While $v.used > 0$ \\ -\hspace{3mm}8.1 If $\vert u \vert > \vert v \vert$ then \\ -\hspace{6mm}8.1.1 Swap $u$ and $v$. \\ -\hspace{3mm}8.2 $v \leftarrow \vert v \vert - \vert u \vert$ \\ -\hspace{3mm}8.3 While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{6mm}8.3.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -9. $c \leftarrow u \cdot 2^k$ \\ -10. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_gcd} -\end{figure} -\textbf{Algorithm mp\_gcd.} -This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of -Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as -Algorithm B and in practice this appears to be true. - -The first two steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the -largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of -$a$ and $b$ respectively and the algorithm will proceed to reduce the pair. - -Step five will divide out any common factors of two and keep track of the count in the variable $k$. After this step, two is no longer a -factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step -six and seven ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while--loops will iterate since -they cannot both be even. - -By step eight both of $u$ and $v$ are odd which is required for the inner logic. First the pair are swapped such that $v$ is equal to -or greater than $u$. This ensures that the subtraction on step 8.2 will always produce a positive and even result. Step 8.3 removes any -factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd. - -After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six. The result -must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_gcd.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the -integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise -it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three -trivial cases of inputs are handled on lines 24 through 30. After those lines the inputs are assumed to be non-zero. - -Lines 32 and 37 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two -must be divided out of the two inputs. The block starting at line 44 removes common factors of two by first counting the number of trailing -zero bits in both. The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values. It is assumed that -the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than -entries than are accessible by an ``int'' so this is not a limitation.}. - -At this point there are no more common factors of two in the two values. The divisions by a power of two on lines 62 and 68 remove -any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm. The while loop -on line 73 performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in -place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative. - -\section{Least Common Multiple} -The least common multiple of a pair of integers is their product divided by their greatest common divisor. For two integers $a$ and $b$ the -least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$. For example, if $a = 2 \cdot 2 \cdot 3 = 12$ -and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$. - -The least common multiple arises often in coding theory as well as number theory. If two functions have periods of $a$ and $b$ respectively they will -collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on -Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). -Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_lcm}. \\ -\textbf{Input}. mp\_int $a$ and $b$ \\ -\textbf{Output}. The least common multiple $c = [a, b]$. \\ -\hline \\ -1. $c \leftarrow (a, b)$ \\ -2. $t \leftarrow a \cdot b$ \\ -3. $c \leftarrow \lfloor t / c \rfloor$ \\ -4. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_lcm} -\end{figure} -\textbf{Algorithm mp\_lcm.} -This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$. It computes the least common multiple directly by -dividing the product of the two inputs by their greatest common divisor. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_lcm.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section{Jacobi Symbol Computation} -To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is -defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is -equivalent to equation \ref{eqn:legendre}. - -\textit{-- Tom, don't be an ass, cite your source here...!} - -\begin{equation} -a^{(p-1)/2} \equiv \begin{array}{rl} - -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ - 0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\ - 1 & \mbox{if }a\mbox{ is a quadratic residue}. - \end{array} \mbox{ (mod }p\mbox{)} -\label{eqn:legendre} -\end{equation} - -\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.} -An integer $a$ is a quadratic residue if the following equation has a solution. - -\begin{equation} -x^2 \equiv a \mbox{ (mod }p\mbox{)} -\label{eqn:root} -\end{equation} - -Consider the following equation. - -\begin{equation} -0 \equiv x^{p-1} - 1 \equiv \left \lbrace \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \right \rbrace + \left ( a^{(p-1)/2} - 1 \right ) \mbox{ (mod }p\mbox{)} -\label{eqn:rooti} -\end{equation} - -Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true. If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$ -then the quantity in the braces must be zero. By reduction, - -\begin{eqnarray} -\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\ -\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\ -x^2 \equiv a \mbox{ (mod }p\mbox{)} -\end{eqnarray} - -As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue. If $a$ does not divide $p$ and $a$ -is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since -\begin{equation} -0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)} -\end{equation} -One of the terms on the right hand side must be zero. \textbf{QED} - -\subsection{Jacobi Symbol} -The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2. If $p = \prod_{i=0}^n p_i$ then -the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation. - -\begin{equation} -\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right ) -\end{equation} - -By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function. The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for -further details.} will be used to derive an efficient Jacobi symbol algorithm. Where $p$ is an odd integer greater than two and $a, b \in \Z$ the -following are true. - -\begin{enumerate} -\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. -\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$. -\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$. -\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$. -\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically -$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. -\end{enumerate} - -Using these facts if $a = 2^k \cdot a'$ then - -\begin{eqnarray} -\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\ - = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) -\label{eqn:jacobi} -\end{eqnarray} - -By fact five, - -\begin{equation} -\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} -\end{equation} - -Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then - -\begin{equation} -\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} -\end{equation} - -By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed. - -\begin{equation} -\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4} -\end{equation} - -The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of -$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the -factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the -Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_jacobi}. \\ -\textbf{Input}. mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\ -\textbf{Output}. The Jacobi symbol $c = \left ( {a \over p } \right )$. \\ -\hline \\ -1. If $a = 0$ then \\ -\hspace{3mm}1.1 $c \leftarrow 0$ \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $a = 1$ then \\ -\hspace{3mm}2.1 $c \leftarrow 1$ \\ -\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ -3. $a' \leftarrow a$ \\ -4. $k \leftarrow 0$ \\ -5. While $a'.used > 0$ and $a'_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ -\hspace{3mm}5.2 $a' \leftarrow \lfloor a' / 2 \rfloor$ \\ -6. If $k \equiv 0 \mbox{ (mod }2\mbox{)}$ then \\ -\hspace{3mm}6.1 $s \leftarrow 1$ \\ -7. else \\ -\hspace{3mm}7.1 $r \leftarrow p_0 \mbox{ (mod }8\mbox{)}$ \\ -\hspace{3mm}7.2 If $r = 1$ or $r = 7$ then \\ -\hspace{6mm}7.2.1 $s \leftarrow 1$ \\ -\hspace{3mm}7.3 else \\ -\hspace{6mm}7.3.1 $s \leftarrow -1$ \\ -8. If $p_0 \equiv a'_0 \equiv 3 \mbox{ (mod }4\mbox{)}$ then \\ -\hspace{3mm}8.1 $s \leftarrow -s$ \\ -9. If $a' \ne 1$ then \\ -\hspace{3mm}9.1 $p' \leftarrow p \mbox{ (mod }a'\mbox{)}$ \\ -\hspace{3mm}9.2 $s \leftarrow s \cdot \mbox{mp\_jacobi}(p', a')$ \\ -10. $c \leftarrow s$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_jacobi} -\end{figure} -\textbf{Algorithm mp\_jacobi.} -This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm -is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. - -Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively. Step five determines the number of two factors in the -input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one -if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled -the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$ -are congruent to one modulo four, otherwise it evaluates to negative one. - -By step nine if $a'$ does not equal one a recursion is required. Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute -$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_jacobi.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C -variable name character. - -The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm. If the input is non-trivial the algorithm -has to proceed compute the Jacobi. The variable $s$ is used to hold the current Jacobi product. Note that $s$ is merely a C ``int'' data type since -the values it may obtain are merely $-1$, $0$ and $1$. - -After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$. Technically only the least significant -bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same -processor requirements and neither is faster than the other. - -Line 58 through 71 determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than -$k$ is even and the value is one. Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight. The value of -$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 71 through 74. - -Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. - -\textit{-- Comment about default $s$ and such...} - -\section{Modular Inverse} -\label{sec:modinv} -The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there -exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$. The integer $b$ is called the multiplicative inverse of $a$ which is -denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and -fields of integers. However, the former will be the matter of discussion. - -The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the -order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial. - -\begin{equation} -ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)} -\end{equation} - -However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite -requires all of the prime factors. This approach also is very slow as the size of $p$ grows. - -A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear -Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation. - -\begin{equation} -ab + pq = 1 -\end{equation} - -Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of -$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. -However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The -binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine -equation. - -\subsection{General Case} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_invmod}. \\ -\textbf{Input}. mp\_int $a$ and $b$, $(a, b) = 1$, $p \ge 2$, $0 < a < p$. \\ -\textbf{Output}. The modular inverse $c \equiv a^{-1} \mbox{ (mod }b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then return(\textit{MP\_VAL}). \\ -2. If $b_0 \equiv 1 \mbox{ (mod }2\mbox{)}$ then use algorithm fast\_mp\_invmod. \\ -3. $x \leftarrow \vert a \vert, y \leftarrow b$ \\ -4. If $x_0 \equiv y_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ then return(\textit{MP\_VAL}). \\ -5. $B \leftarrow 0, C \leftarrow 0, A \leftarrow 1, D \leftarrow 1$ \\ -6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ -\hspace{3mm}6.2 If ($A.used > 0$ and $A_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($B.used > 0$ and $B_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ -\hspace{6mm}6.2.1 $A \leftarrow A + y$ \\ -\hspace{6mm}6.2.2 $B \leftarrow B - x$ \\ -\hspace{3mm}6.3 $A \leftarrow \lfloor A / 2 \rfloor$ \\ -\hspace{3mm}6.4 $B \leftarrow \lfloor B / 2 \rfloor$ \\ -7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -\hspace{3mm}7.2 If ($C.used > 0$ and $C_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($D.used > 0$ and $D_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ -\hspace{6mm}7.2.1 $C \leftarrow C + y$ \\ -\hspace{6mm}7.2.2 $D \leftarrow D - x$ \\ -\hspace{3mm}7.3 $C \leftarrow \lfloor C / 2 \rfloor$ \\ -\hspace{3mm}7.4 $D \leftarrow \lfloor D / 2 \rfloor$ \\ -8. If $u \ge v$ then \\ -\hspace{3mm}8.1 $u \leftarrow u - v$ \\ -\hspace{3mm}8.2 $A \leftarrow A - C$ \\ -\hspace{3mm}8.3 $B \leftarrow B - D$ \\ -9. else \\ -\hspace{3mm}9.1 $v \leftarrow v - u$ \\ -\hspace{3mm}9.2 $C \leftarrow C - A$ \\ -\hspace{3mm}9.3 $D \leftarrow D - B$ \\ -10. If $u \ne 0$ goto step 6. \\ -11. If $v \ne 1$ return(\textit{MP\_VAL}). \\ -12. While $C \le 0$ do \\ -\hspace{3mm}12.1 $C \leftarrow C + b$ \\ -13. While $C \ge b$ do \\ -\hspace{3mm}13.1 $C \leftarrow C - b$ \\ -14. $c \leftarrow C$ \\ -15. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\end{figure} -\textbf{Algorithm mp\_invmod.} -This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the -extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete -Diophantine solution. - -If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative -inverse for $a$ and the error is reported. - -The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd. In this case -the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is - -\begin{equation} -Ca + Db = v -\end{equation} - -If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists. Otherwise, $C$ -is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie -within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$ -then only a couple of additions or subtractions will be required to adjust the inverse. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_invmod.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsubsection{Odd Moduli} - -When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse. In particular by attempting to solve -the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. - -The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This -optimization will halve the time required to compute the modular inverse. - -\section{Primality Tests} - -A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime -since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. - -Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or -not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all -probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is -prime the algorithm may be incorrect. - -As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as -well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question. - -\subsection{Trial Division} - -Trial division means to attempt to evenly divide a candidate integer by small prime integers. If the candidate can be evenly divided it obviously -cannot be prime. By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime. However, such a test -would require a prohibitive amount of time as $n$ grows. - -Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead. By performing trial division with only a subset -of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime. However, often it can prove a candidate is not prime. - -The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be -discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by -$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range -$3 \le q \le 100$. - -At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to -be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate -approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The -array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $c = 1$ if $n$ is divisible by a small prime, otherwise $c = 0$. \\ -\hline \\ -1. for $ix$ from $0$ to $PRIME\_SIZE$ do \\ -\hspace{3mm}1.1 $d \leftarrow n \mbox{ (mod }\_\_prime\_tab_{ix}\mbox{)}$ \\ -\hspace{3mm}1.2 If $d = 0$ then \\ -\hspace{6mm}1.2.1 $c \leftarrow 1$ \\ -\hspace{6mm}1.2.2 Return(\textit{MP\_OKAY}). \\ -2. $c \leftarrow 0$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_prime\_is\_divisible} -\end{figure} -\textbf{Algorithm mp\_prime\_is\_divisible.} -This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_is\_divisible.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a -mp\_digit. The table \_\_prime\_tab is defined in the following file. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_prime\_tab.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Note that there are two possible tables. When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes -upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. - -\subsection{The Fermat Test} -The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in -fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of -the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to -$a^1 = a$. - -If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case -it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$. However, this test is not absolute as it is possible that the order -of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several -integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows -in size. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_prime\_fermat}. \\ -\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ -\textbf{Output}. $c = 1$ if $b^a \equiv b \mbox{ (mod }a\mbox{)}$, otherwise $c = 0$. \\ -\hline \\ -1. $t \leftarrow b^a \mbox{ (mod }a\mbox{)}$ \\ -2. If $t = b$ then \\ -\hspace{3mm}2.1 $c = 1$ \\ -3. else \\ -\hspace{3mm}3.1 $c = 0$ \\ -4. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_prime\_fermat} -\end{figure} -\textbf{Algorithm mp\_prime\_fermat.} -This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not. It uses a single modular exponentiation to -determine the result. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_fermat.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsection{The Miller-Rabin Test} -The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen -candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the -value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that -some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_prime\_miller\_rabin}. \\ -\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ -\textbf{Output}. $c = 1$ if $a$ is a Miller-Rabin prime to the base $a$, otherwise $c = 0$. \\ -\hline -1. $a' \leftarrow a - 1$ \\ -2. $r \leftarrow n1$ \\ -3. $c \leftarrow 0, s \leftarrow 0$ \\ -4. While $r.used > 0$ and $r_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}4.1 $s \leftarrow s + 1$ \\ -\hspace{3mm}4.2 $r \leftarrow \lfloor r / 2 \rfloor$ \\ -5. $y \leftarrow b^r \mbox{ (mod }a\mbox{)}$ \\ -6. If $y \nequiv \pm 1$ then \\ -\hspace{3mm}6.1 $j \leftarrow 1$ \\ -\hspace{3mm}6.2 While $j \le (s - 1)$ and $y \nequiv a'$ \\ -\hspace{6mm}6.2.1 $y \leftarrow y^2 \mbox{ (mod }a\mbox{)}$ \\ -\hspace{6mm}6.2.2 If $y = 1$ then goto step 8. \\ -\hspace{6mm}6.2.3 $j \leftarrow j + 1$ \\ -\hspace{3mm}6.3 If $y \nequiv a'$ goto step 8. \\ -7. $c \leftarrow 1$\\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_prime\_miller\_rabin} -\end{figure} -\textbf{Algorithm mp\_prime\_miller\_rabin.} -This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$. It will set $c = 1$ if the algorithm cannot determine -if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. - -If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not. Otherwise, the algorithm will -square $y$ upto $s - 1$ times stopping only when $y \equiv -1$. If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$ -is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably -composite then it is \textit{probably} prime. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_miller\_rabin.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - - - - -\backmatter -\appendix -\begin{thebibliography}{ABCDEF} -\bibitem[1]{TAOCPV2} -Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998 - -\bibitem[2]{HAC} -A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996 - -\bibitem[3]{ROSE} -Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999 - -\bibitem[4]{COMBA} -Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990) - -\bibitem[5]{KARA} -A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294 - -\bibitem[6]{KARAP} -Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002 - -\bibitem[7]{BARRETT} -Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag. - -\bibitem[8]{MONT} -P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985. - -\bibitem[9]{DRMET} -Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories - -\bibitem[10]{MMB} -J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89 - -\bibitem[11]{RSAREF} -R.L. Rivest, A. Shamir, L. Adleman, \textit{A Method for Obtaining Digital Signatures and Public-Key Cryptosystems} - -\bibitem[12]{DHREF} -Whitfield Diffie, Martin E. Hellman, \textit{New Directions in Cryptography}, IEEE Transactions on Information Theory, 1976 - -\bibitem[13]{IEEE} -IEEE Standard for Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985) - -\bibitem[14]{GMP} -GNU Multiple Precision (GMP), \url{http://www.swox.com/gmp/} - -\bibitem[15]{MPI} -Multiple Precision Integer Library (MPI), Michael Fromberger, \url{http://thayer.dartmouth.edu/~sting/mpi/} - -\bibitem[16]{OPENSSL} -OpenSSL Cryptographic Toolkit, \url{http://openssl.org} - -\bibitem[17]{LIP} -Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip} - -\bibitem[18]{ISOC} -JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.'' - -\bibitem[19]{JAVA} -The Sun Java Website, \url{http://java.sun.com/} - -\end{thebibliography} - -\input{tommath.ind} - -\end{document} diff --git a/libtommath/tommath_class.h b/libtommath/tommath_class.h index b9cc902..2085521 100644 --- a/libtommath/tommath_class.h +++ b/libtommath/tommath_class.h @@ -38,7 +38,9 @@ #define BN_MP_DR_REDUCE_C #define BN_MP_DR_SETUP_C #define BN_MP_EXCH_C +#define BN_MP_EXPORT_C #define BN_MP_EXPT_D_C +#define BN_MP_EXPT_D_EX_C #define BN_MP_EXPTMOD_C #define BN_MP_EXPTMOD_FAST_C #define BN_MP_EXTEUCLID_C @@ -46,7 +48,10 @@ #define BN_MP_FWRITE_C #define BN_MP_GCD_C #define BN_MP_GET_INT_C +#define BN_MP_GET_LONG_C +#define BN_MP_GET_LONG_LONG_C #define BN_MP_GROW_C +#define BN_MP_IMPORT_C #define BN_MP_INIT_C #define BN_MP_INIT_COPY_C #define BN_MP_INIT_MULTI_C @@ -73,6 +78,7 @@ #define BN_MP_MUL_D_C #define BN_MP_MULMOD_C #define BN_MP_N_ROOT_C +#define BN_MP_N_ROOT_EX_C #define BN_MP_NEG_C #define BN_MP_OR_C #define BN_MP_PRIME_FERMAT_C @@ -99,11 +105,14 @@ #define BN_MP_RSHD_C #define BN_MP_SET_C #define BN_MP_SET_INT_C +#define BN_MP_SET_LONG_C +#define BN_MP_SET_LONG_LONG_C #define BN_MP_SHRINK_C #define BN_MP_SIGNED_BIN_SIZE_C #define BN_MP_SQR_C #define BN_MP_SQRMOD_C #define BN_MP_SQRT_C +#define BN_MP_SQRTMOD_PRIME_C #define BN_MP_SUB_C #define BN_MP_SUB_D_C #define BN_MP_SUBMOD_C @@ -315,12 +324,23 @@ #if defined(BN_MP_EXCH_C) #endif +#if defined(BN_MP_EXPORT_C) + #define BN_MP_INIT_COPY_C + #define BN_MP_COUNT_BITS_C + #define BN_MP_DIV_2D_C + #define BN_MP_CLEAR_C +#endif + #if defined(BN_MP_EXPT_D_C) + #define BN_MP_EXPT_D_EX_C +#endif + +#if defined(BN_MP_EXPT_D_EX_C) #define BN_MP_INIT_COPY_C #define BN_MP_SET_C - #define BN_MP_SQR_C - #define BN_MP_CLEAR_C #define BN_MP_MUL_C + #define BN_MP_CLEAR_C + #define BN_MP_SQR_C #endif #if defined(BN_MP_EXPTMOD_C) @@ -387,7 +407,6 @@ #if defined(BN_MP_GCD_C) #define BN_MP_ISZERO_C #define BN_MP_ABS_C - #define BN_MP_ZERO_C #define BN_MP_INIT_COPY_C #define BN_MP_CNT_LSB_C #define BN_MP_DIV_2D_C @@ -401,13 +420,26 @@ #if defined(BN_MP_GET_INT_C) #endif +#if defined(BN_MP_GET_LONG_C) +#endif + +#if defined(BN_MP_GET_LONG_LONG_C) +#endif + #if defined(BN_MP_GROW_C) #endif +#if defined(BN_MP_IMPORT_C) + #define BN_MP_ZERO_C + #define BN_MP_MUL_2D_C + #define BN_MP_CLAMP_C +#endif + #if defined(BN_MP_INIT_C) #endif #if defined(BN_MP_INIT_COPY_C) + #define BN_MP_INIT_SIZE_C #define BN_MP_COPY_C #endif @@ -481,8 +513,9 @@ #define BN_MP_MUL_C #define BN_MP_INIT_SIZE_C #define BN_MP_CLAMP_C - #define BN_MP_SUB_C + #define BN_S_MP_ADD_C #define BN_MP_ADD_C + #define BN_S_MP_SUB_C #define BN_MP_LSHD_C #define BN_MP_CLEAR_C #endif @@ -491,8 +524,8 @@ #define BN_MP_INIT_SIZE_C #define BN_MP_CLAMP_C #define BN_MP_SQR_C - #define BN_MP_SUB_C #define BN_S_MP_ADD_C + #define BN_S_MP_SUB_C #define BN_MP_LSHD_C #define BN_MP_ADD_C #define BN_MP_CLEAR_C @@ -516,8 +549,9 @@ #define BN_MP_INIT_C #define BN_MP_DIV_C #define BN_MP_CLEAR_C - #define BN_MP_ADD_C + #define BN_MP_ISZERO_C #define BN_MP_EXCH_C + #define BN_MP_ADD_C #endif #if defined(BN_MP_MOD_2D_C) @@ -583,10 +617,14 @@ #endif #if defined(BN_MP_N_ROOT_C) + #define BN_MP_N_ROOT_EX_C +#endif + +#if defined(BN_MP_N_ROOT_EX_C) #define BN_MP_INIT_C #define BN_MP_SET_C #define BN_MP_COPY_C - #define BN_MP_EXPT_D_C + #define BN_MP_EXPT_D_EX_C #define BN_MP_MUL_C #define BN_MP_SUB_C #define BN_MP_MUL_D_C @@ -667,9 +705,9 @@ #endif #if defined(BN_MP_RADIX_SIZE_C) + #define BN_MP_ISZERO_C #define BN_MP_COUNT_BITS_C #define BN_MP_INIT_COPY_C - #define BN_MP_ISZERO_C #define BN_MP_DIV_D_C #define BN_MP_CLEAR_C #endif @@ -687,7 +725,6 @@ #if defined(BN_MP_READ_RADIX_C) #define BN_MP_ZERO_C #define BN_MP_S_RMAP_C - #define BN_MP_RADIX_SMAP_C #define BN_MP_MUL_D_C #define BN_MP_ADD_D_C #define BN_MP_ISZERO_C @@ -788,6 +825,12 @@ #define BN_MP_CLAMP_C #endif +#if defined(BN_MP_SET_LONG_C) +#endif + +#if defined(BN_MP_SET_LONG_LONG_C) +#endif + #if defined(BN_MP_SHRINK_C) #endif @@ -823,6 +866,25 @@ #define BN_MP_CLEAR_C #endif +#if defined(BN_MP_SQRTMOD_PRIME_C) + #define BN_MP_CMP_D_C + #define BN_MP_ZERO_C + #define BN_MP_JACOBI_C + #define BN_MP_INIT_MULTI_C + #define BN_MP_MOD_D_C + #define BN_MP_ADD_D_C + #define BN_MP_DIV_2_C + #define BN_MP_EXPTMOD_C + #define BN_MP_COPY_C + #define BN_MP_SUB_D_C + #define BN_MP_ISEVEN_C + #define BN_MP_SET_INT_C + #define BN_MP_SQRMOD_C + #define BN_MP_MULMOD_C + #define BN_MP_SET_C + #define BN_MP_CLEAR_MULTI_C +#endif + #if defined(BN_MP_SUB_C) #define BN_S_MP_ADD_C #define BN_MP_CMP_MAG_C diff --git a/libtommath/tommath_superclass.h b/libtommath/tommath_superclass.h index e3926df..1b26841 100644 --- a/libtommath/tommath_superclass.h +++ b/libtommath/tommath_superclass.h @@ -70,3 +70,7 @@ #endif #endif + +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/unix/Makefile.in b/unix/Makefile.in index 570bce0..70e7c9f3f2 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -320,8 +320,8 @@ TOMMATH_OBJS = bncore.o bn_reverse.o bn_fast_s_mp_mul_digs.o \ bn_mp_cnt_lsb.o bn_mp_copy.o \ bn_mp_count_bits.o bn_mp_div.o bn_mp_div_d.o bn_mp_div_2.o \ bn_mp_div_2d.o bn_mp_div_3.o \ - bn_mp_exch.o bn_mp_expt_d.o bn_mp_grow.o bn_mp_init.o \ - bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \ + bn_mp_exch.o bn_mp_expt_d_ex.o bn_mp_expt_d.o bn_mp_grow.o \ + bn_mp_init.o bn_mp_init_copy.o bn_mp_init_multi.o bn_mp_init_set.o \ bn_mp_init_set_int.o bn_mp_init_size.o bn_mp_karatsuba_mul.o \ bn_mp_karatsuba_sqr.o \ bn_mp_lshd.o bn_mp_mod.o bn_mp_mod_2d.o bn_mp_mul.o bn_mp_mul_2.o \ @@ -502,6 +502,7 @@ TOMMATH_SRCS = \ $(TOMMATH_DIR)/bn_mp_div_2d.c \ $(TOMMATH_DIR)/bn_mp_div_3.c \ $(TOMMATH_DIR)/bn_mp_exch.c \ + $(TOMMATH_DIR)/bn_mp_expt_d_ex.c \ $(TOMMATH_DIR)/bn_mp_expt_d.c \ $(TOMMATH_DIR)/bn_mp_grow.c \ $(TOMMATH_DIR)/bn_mp_init.c \ @@ -1417,6 +1418,9 @@ bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS) bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_exch.c +bn_mp_expt_d_ex.o: $(TOMMATH_DIR)/bn_mp_expt_d_ex.c $(MATHHDRS) + $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d_ex.c + bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d.c -- cgit v0.12 From 8377ed833a9566731442ef744b419425638d5040 Mon Sep 17 00:00:00 2001 From: gahr Date: Sun, 13 Mar 2016 17:23:31 +0000 Subject: [e6f27aa56f] Add files I missed in my previous commit --- libtommath/bn_mp_export.c | 88 ++++++++++++++++++++++++++ libtommath/bn_mp_expt_d_ex.c | 83 ++++++++++++++++++++++++ libtommath/bn_mp_get_long.c | 41 ++++++++++++ libtommath/bn_mp_get_long_long.c | 41 ++++++++++++ libtommath/bn_mp_import.c | 73 ++++++++++++++++++++++ libtommath/bn_mp_n_root_ex.c | 132 +++++++++++++++++++++++++++++++++++++++ libtommath/bn_mp_set_long.c | 24 +++++++ libtommath/bn_mp_set_long_long.c | 24 +++++++ libtommath/bn_mp_sqrtmod_prime.c | 124 ++++++++++++++++++++++++++++++++++++ libtommath/tommath_private.h | 125 ++++++++++++++++++++++++++++++++++++ 10 files changed, 755 insertions(+) create mode 100644 libtommath/bn_mp_export.c create mode 100644 libtommath/bn_mp_expt_d_ex.c create mode 100644 libtommath/bn_mp_get_long.c create mode 100644 libtommath/bn_mp_get_long_long.c create mode 100644 libtommath/bn_mp_import.c create mode 100644 libtommath/bn_mp_n_root_ex.c create mode 100644 libtommath/bn_mp_set_long.c create mode 100644 libtommath/bn_mp_set_long_long.c create mode 100644 libtommath/bn_mp_sqrtmod_prime.c create mode 100644 libtommath/tommath_private.h diff --git a/libtommath/bn_mp_export.c b/libtommath/bn_mp_export.c new file mode 100644 index 0000000..ac4c2f9 --- /dev/null +++ b/libtommath/bn_mp_export.c @@ -0,0 +1,88 @@ +#include +#ifdef BN_MP_EXPORT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tstdenis82@gmail.com, http://libtom.org + */ + +/* based on gmp's mpz_export. + * see http://gmplib.org/manual/Integer-Import-and-Export.html + */ +int mp_export(void* rop, size_t* countp, int order, size_t size, + int endian, size_t nails, mp_int* op) { + int result; + size_t odd_nails, nail_bytes, i, j, bits, count; + unsigned char odd_nail_mask; + + mp_int t; + + if ((result = mp_init_copy(&t, op)) != MP_OKAY) { + return result; + } + + if (endian == 0) { + union { + unsigned int i; + char c[4]; + } lint; + lint.i = 0x01020304; + + endian = (lint.c[0] == 4) ? -1 : 1; + } + + odd_nails = (nails % 8); + odd_nail_mask = 0xff; + for (i = 0; i < odd_nails; ++i) { + odd_nail_mask ^= (1 << (7 - i)); + } + nail_bytes = nails / 8; + + bits = mp_count_bits(&t); + count = (bits / ((size * 8) - nails)) + (((bits % ((size * 8) - nails)) != 0) ? 1 : 0); + + for (i = 0; i < count; ++i) { + for (j = 0; j < size; ++j) { + unsigned char* byte = ( + (unsigned char*)rop + + (((order == -1) ? i : ((count - 1) - i)) * size) + + ((endian == -1) ? j : ((size - 1) - j)) + ); + + if (j >= (size - nail_bytes)) { + *byte = 0; + continue; + } + + *byte = (unsigned char)((j == ((size - nail_bytes) - 1)) ? (t.dp[0] & odd_nail_mask) : (t.dp[0] & 0xFF)); + + if ((result = mp_div_2d(&t, ((j == ((size - nail_bytes) - 1)) ? (8 - odd_nails) : 8), &t, NULL)) != MP_OKAY) { + mp_clear(&t); + return result; + } + } + } + + mp_clear(&t); + + if (countp != NULL) { + *countp = count; + } + + return MP_OKAY; +} + +#endif + +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/libtommath/bn_mp_expt_d_ex.c b/libtommath/bn_mp_expt_d_ex.c new file mode 100644 index 0000000..649d224 --- /dev/null +++ b/libtommath/bn_mp_expt_d_ex.c @@ -0,0 +1,83 @@ +#include +#ifdef BN_MP_EXPT_D_EX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tstdenis82@gmail.com, http://libtom.org + */ + +/* calculate c = a**b using a square-multiply algorithm */ +int mp_expt_d_ex (mp_int * a, mp_digit b, mp_int * c, int fast) +{ + int res; + unsigned int x; + + mp_int g; + + if ((res = mp_init_copy (&g, a)) != MP_OKAY) { + return res; + } + + /* set initial result */ + mp_set (c, 1); + + if (fast != 0) { + while (b > 0) { + /* if the bit is set multiply */ + if ((b & 1) != 0) { + if ((res = mp_mul (c, &g, c)) != MP_OKAY) { + mp_clear (&g); + return res; + } + } + + /* square */ + if (b > 1) { + if ((res = mp_sqr (&g, &g)) != MP_OKAY) { + mp_clear (&g); + return res; + } + } + + /* shift to next bit */ + b >>= 1; + } + } + else { + for (x = 0; x < DIGIT_BIT; x++) { + /* square */ + if ((res = mp_sqr (c, c)) != MP_OKAY) { + mp_clear (&g); + return res; + } + + /* if the bit is set multiply */ + if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) { + if ((res = mp_mul (c, &g, c)) != MP_OKAY) { + mp_clear (&g); + return res; + } + } + + /* shift to next bit */ + b <<= 1; + } + } /* if ... else */ + + mp_clear (&g); + return MP_OKAY; +} +#endif + +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/libtommath/bn_mp_get_long.c b/libtommath/bn_mp_get_long.c new file mode 100644 index 0000000..7c3d0fe --- /dev/null +++ b/libtommath/bn_mp_get_long.c @@ -0,0 +1,41 @@ +#include +#ifdef BN_MP_GET_LONG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tstdenis82@gmail.com, http://libtom.org + */ + +/* get the lower unsigned long of an mp_int, platform dependent */ +unsigned long mp_get_long(mp_int * a) +{ + int i; + unsigned long res; + + if (a->used == 0) { + return 0; + } + + /* get number of digits of the lsb we have to read */ + i = MIN(a->used,(int)(((sizeof(unsigned long) * CHAR_BIT) + DIGIT_BIT - 1) / DIGIT_BIT)) - 1; + + /* get most significant digit of result */ + res = DIGIT(a,i); + +#if (ULONG_MAX != 0xffffffffuL) || (DIGIT_BIT < 32) + while (--i >= 0) { + res = (res << DIGIT_BIT) | DIGIT(a,i); + } +#endif + return res; +} +#endif diff --git a/libtommath/bn_mp_get_long_long.c b/libtommath/bn_mp_get_long_long.c new file mode 100644 index 0000000..4b959e6 --- /dev/null +++ b/libtommath/bn_mp_get_long_long.c @@ -0,0 +1,41 @@ +#include +#ifdef BN_MP_GET_LONG_LONG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tstdenis82@gmail.com, http://libtom.org + */ + +/* get the lower unsigned long long of an mp_int, platform dependent */ +unsigned long long mp_get_long_long (mp_int * a) +{ + int i; + unsigned long long res; + + if (a->used == 0) { + return 0; + } + + /* get number of digits of the lsb we have to read */ + i = MIN(a->used,(int)(((sizeof(unsigned long long) * CHAR_BIT) + DIGIT_BIT - 1) / DIGIT_BIT)) - 1; + + /* get most significant digit of result */ + res = DIGIT(a,i); + +#if DIGIT_BIT < 64 + while (--i >= 0) { + res = (res << DIGIT_BIT) | DIGIT(a,i); + } +#endif + return res; +} +#endif diff --git a/libtommath/bn_mp_import.c b/libtommath/bn_mp_import.c new file mode 100644 index 0000000..dd4b8e6 --- /dev/null +++ b/libtommath/bn_mp_import.c @@ -0,0 +1,73 @@ +#include +#ifdef BN_MP_IMPORT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tstdenis82@gmail.com, http://libtom.org + */ + +/* based on gmp's mpz_import. + * see http://gmplib.org/manual/Integer-Import-and-Export.html + */ +int mp_import(mp_int* rop, size_t count, int order, size_t size, + int endian, size_t nails, const void* op) { + int result; + size_t odd_nails, nail_bytes, i, j; + unsigned char odd_nail_mask; + + mp_zero(rop); + + if (endian == 0) { + union { + unsigned int i; + char c[4]; + } lint; + lint.i = 0x01020304; + + endian = (lint.c[0] == 4) ? -1 : 1; + } + + odd_nails = (nails % 8); + odd_nail_mask = 0xff; + for (i = 0; i < odd_nails; ++i) { + odd_nail_mask ^= (1 << (7 - i)); + } + nail_bytes = nails / 8; + + for (i = 0; i < count; ++i) { + for (j = 0; j < (size - nail_bytes); ++j) { + unsigned char byte = *( + (unsigned char*)op + + (((order == 1) ? i : ((count - 1) - i)) * size) + + ((endian == 1) ? (j + nail_bytes) : (((size - 1) - j) - nail_bytes)) + ); + + if ( + (result = mp_mul_2d(rop, ((j == 0) ? (8 - odd_nails) : 8), rop)) != MP_OKAY) { + return result; + } + + rop->dp[0] |= (j == 0) ? (byte & odd_nail_mask) : byte; + rop->used += 1; + } + } + + mp_clamp(rop); + + return MP_OKAY; +} + +#endif + +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/libtommath/bn_mp_n_root_ex.c b/libtommath/bn_mp_n_root_ex.c new file mode 100644 index 0000000..79d1dfb --- /dev/null +++ b/libtommath/bn_mp_n_root_ex.c @@ -0,0 +1,132 @@ +#include +#ifdef BN_MP_N_ROOT_EX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tstdenis82@gmail.com, http://libtom.org + */ + +/* find the n'th root of an integer + * + * Result found such that (c)**b <= a and (c+1)**b > a + * + * This algorithm uses Newton's approximation + * x[i+1] = x[i] - f(x[i])/f'(x[i]) + * which will find the root in log(N) time where + * each step involves a fair bit. This is not meant to + * find huge roots [square and cube, etc]. + */ +int mp_n_root_ex (mp_int * a, mp_digit b, mp_int * c, int fast) +{ + mp_int t1, t2, t3; + int res, neg; + + /* input must be positive if b is even */ + if (((b & 1) == 0) && (a->sign == MP_NEG)) { + return MP_VAL; + } + + if ((res = mp_init (&t1)) != MP_OKAY) { + return res; + } + + if ((res = mp_init (&t2)) != MP_OKAY) { + goto LBL_T1; + } + + if ((res = mp_init (&t3)) != MP_OKAY) { + goto LBL_T2; + } + + /* if a is negative fudge the sign but keep track */ + neg = a->sign; + a->sign = MP_ZPOS; + + /* t2 = 2 */ + mp_set (&t2, 2); + + do { + /* t1 = t2 */ + if ((res = mp_copy (&t2, &t1)) != MP_OKAY) { + goto LBL_T3; + } + + /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */ + + /* t3 = t1**(b-1) */ + if ((res = mp_expt_d_ex (&t1, b - 1, &t3, fast)) != MP_OKAY) { + goto LBL_T3; + } + + /* numerator */ + /* t2 = t1**b */ + if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) { + goto LBL_T3; + } + + /* t2 = t1**b - a */ + if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) { + goto LBL_T3; + } + + /* denominator */ + /* t3 = t1**(b-1) * b */ + if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) { + goto LBL_T3; + } + + /* t3 = (t1**b - a)/(b * t1**(b-1)) */ + if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) { + goto LBL_T3; + } + + if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) { + goto LBL_T3; + } + } while (mp_cmp (&t1, &t2) != MP_EQ); + + /* result can be off by a few so check */ + for (;;) { + if ((res = mp_expt_d_ex (&t1, b, &t2, fast)) != MP_OKAY) { + goto LBL_T3; + } + + if (mp_cmp (&t2, a) == MP_GT) { + if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) { + goto LBL_T3; + } + } else { + break; + } + } + + /* reset the sign of a first */ + a->sign = neg; + + /* set the result */ + mp_exch (&t1, c); + + /* set the sign of the result */ + c->sign = neg; + + res = MP_OKAY; + +LBL_T3:mp_clear (&t3); +LBL_T2:mp_clear (&t2); +LBL_T1:mp_clear (&t1); + return res; +} +#endif + +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/libtommath/bn_mp_set_long.c b/libtommath/bn_mp_set_long.c new file mode 100644 index 0000000..281fce7 --- /dev/null +++ b/libtommath/bn_mp_set_long.c @@ -0,0 +1,24 @@ +#include +#ifdef BN_MP_SET_LONG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tstdenis82@gmail.com, http://libtom.org + */ + +/* set a platform dependent unsigned long int */ +MP_SET_XLONG(mp_set_long, unsigned long) +#endif + +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/libtommath/bn_mp_set_long_long.c b/libtommath/bn_mp_set_long_long.c new file mode 100644 index 0000000..3c4b01a --- /dev/null +++ b/libtommath/bn_mp_set_long_long.c @@ -0,0 +1,24 @@ +#include +#ifdef BN_MP_SET_LONG_LONG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tstdenis82@gmail.com, http://libtom.org + */ + +/* set a platform dependent unsigned long long int */ +MP_SET_XLONG(mp_set_long_long, unsigned long long) +#endif + +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/libtommath/bn_mp_sqrtmod_prime.c b/libtommath/bn_mp_sqrtmod_prime.c new file mode 100644 index 0000000..968729e --- /dev/null +++ b/libtommath/bn_mp_sqrtmod_prime.c @@ -0,0 +1,124 @@ +#include +#ifdef BN_MP_SQRTMOD_PRIME_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library is free for all purposes without any express + * guarantee it works. + */ + +/* Tonelli-Shanks algorithm + * https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm + * https://gmplib.org/list-archives/gmp-discuss/2013-April/005300.html + * + */ + +int mp_sqrtmod_prime(mp_int *n, mp_int *prime, mp_int *ret) +{ + int res, legendre; + mp_int t1, C, Q, S, Z, M, T, R, two; + mp_digit i; + + /* first handle the simple cases */ + if (mp_cmp_d(n, 0) == MP_EQ) { + mp_zero(ret); + return MP_OKAY; + } + if (mp_cmp_d(prime, 2) == MP_EQ) return MP_VAL; /* prime must be odd */ + if ((res = mp_jacobi(n, prime, &legendre)) != MP_OKAY) return res; + if (legendre == -1) return MP_VAL; /* quadratic non-residue mod prime */ + + if ((res = mp_init_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL)) != MP_OKAY) { + return res; + } + + /* SPECIAL CASE: if prime mod 4 == 3 + * compute directly: res = n^(prime+1)/4 mod prime + * Handbook of Applied Cryptography algorithm 3.36 + */ + if ((res = mp_mod_d(prime, 4, &i)) != MP_OKAY) goto cleanup; + if (i == 3) { + if ((res = mp_add_d(prime, 1, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_exptmod(n, &t1, prime, ret)) != MP_OKAY) goto cleanup; + res = MP_OKAY; + goto cleanup; + } + + /* NOW: Tonelli-Shanks algorithm */ + + /* factor out powers of 2 from prime-1, defining Q and S as: prime-1 = Q*2^S */ + if ((res = mp_copy(prime, &Q)) != MP_OKAY) goto cleanup; + if ((res = mp_sub_d(&Q, 1, &Q)) != MP_OKAY) goto cleanup; + /* Q = prime - 1 */ + mp_zero(&S); + /* S = 0 */ + while (mp_iseven(&Q) != MP_NO) { + if ((res = mp_div_2(&Q, &Q)) != MP_OKAY) goto cleanup; + /* Q = Q / 2 */ + if ((res = mp_add_d(&S, 1, &S)) != MP_OKAY) goto cleanup; + /* S = S + 1 */ + } + + /* find a Z such that the Legendre symbol (Z|prime) == -1 */ + if ((res = mp_set_int(&Z, 2)) != MP_OKAY) goto cleanup; + /* Z = 2 */ + while(1) { + if ((res = mp_jacobi(&Z, prime, &legendre)) != MP_OKAY) goto cleanup; + if (legendre == -1) break; + if ((res = mp_add_d(&Z, 1, &Z)) != MP_OKAY) goto cleanup; + /* Z = Z + 1 */ + } + + if ((res = mp_exptmod(&Z, &Q, prime, &C)) != MP_OKAY) goto cleanup; + /* C = Z ^ Q mod prime */ + if ((res = mp_add_d(&Q, 1, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + /* t1 = (Q + 1) / 2 */ + if ((res = mp_exptmod(n, &t1, prime, &R)) != MP_OKAY) goto cleanup; + /* R = n ^ ((Q + 1) / 2) mod prime */ + if ((res = mp_exptmod(n, &Q, prime, &T)) != MP_OKAY) goto cleanup; + /* T = n ^ Q mod prime */ + if ((res = mp_copy(&S, &M)) != MP_OKAY) goto cleanup; + /* M = S */ + if ((res = mp_set_int(&two, 2)) != MP_OKAY) goto cleanup; + + res = MP_VAL; + while (1) { + if ((res = mp_copy(&T, &t1)) != MP_OKAY) goto cleanup; + i = 0; + while (1) { + if (mp_cmp_d(&t1, 1) == MP_EQ) break; + if ((res = mp_exptmod(&t1, &two, prime, &t1)) != MP_OKAY) goto cleanup; + i++; + } + if (i == 0) { + if ((res = mp_copy(&R, ret)) != MP_OKAY) goto cleanup; + res = MP_OKAY; + goto cleanup; + } + if ((res = mp_sub_d(&M, i, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_sub_d(&t1, 1, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_exptmod(&two, &t1, prime, &t1)) != MP_OKAY) goto cleanup; + /* t1 = 2 ^ (M - i - 1) */ + if ((res = mp_exptmod(&C, &t1, prime, &t1)) != MP_OKAY) goto cleanup; + /* t1 = C ^ (2 ^ (M - i - 1)) mod prime */ + if ((res = mp_sqrmod(&t1, prime, &C)) != MP_OKAY) goto cleanup; + /* C = (t1 * t1) mod prime */ + if ((res = mp_mulmod(&R, &t1, prime, &R)) != MP_OKAY) goto cleanup; + /* R = (R * t1) mod prime */ + if ((res = mp_mulmod(&T, &C, prime, &T)) != MP_OKAY) goto cleanup; + /* T = (T * C) mod prime */ + mp_set(&M, i); + /* M = i */ + } + +cleanup: + mp_clear_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL); + return res; +} + +#endif diff --git a/libtommath/tommath_private.h b/libtommath/tommath_private.h new file mode 100644 index 0000000..d23c333 --- /dev/null +++ b/libtommath/tommath_private.h @@ -0,0 +1,125 @@ +/* LibTomMath, multiple-precision integer library -- Tom St Denis + * + * LibTomMath is a library that provides multiple-precision + * integer arithmetic as well as number theoretic functionality. + * + * The library was designed directly after the MPI library by + * Michael Fromberger but has been written from scratch with + * additional optimizations in place. + * + * The library is free for all purposes without any express + * guarantee it works. + * + * Tom St Denis, tstdenis82@gmail.com, http://math.libtomcrypt.com + */ +#ifndef TOMMATH_PRIV_H_ +#define TOMMATH_PRIV_H_ + +#include +#include + +#ifndef MIN +#define MIN(x,y) (((x) < (y)) ? (x) : (y)) +#endif + +#ifndef MAX +#define MAX(x,y) (((x) > (y)) ? (x) : (y)) +#endif + +#ifdef __cplusplus +extern "C" { + +/* C++ compilers don't like assigning void * to mp_digit * */ +#define OPT_CAST(x) (x *) + +#else + +/* C on the other hand doesn't care */ +#define OPT_CAST(x) + +#endif + +/* define heap macros */ +#if 0 +#ifndef XMALLOC + /* default to libc stuff */ + #define XMALLOC malloc + #define XFREE free + #define XREALLOC realloc + #define XCALLOC calloc +#else + /* prototypes for our heap functions */ + extern void *XMALLOC(size_t n); + extern void *XREALLOC(void *p, size_t n); + extern void *XCALLOC(size_t n, size_t s); + extern void XFREE(void *p); +#endif +#endif + +/* lowlevel functions, do not call! */ +int s_mp_add(mp_int *a, mp_int *b, mp_int *c); +int s_mp_sub(mp_int *a, mp_int *b, mp_int *c); +#define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1) +int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); +int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); +int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); +int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); +int fast_s_mp_sqr(mp_int *a, mp_int *b); +int s_mp_sqr(mp_int *a, mp_int *b); +int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c); +int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c); +int mp_karatsuba_sqr(mp_int *a, mp_int *b); +int mp_toom_sqr(mp_int *a, mp_int *b); +int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c); +int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c); +int fast_mp_montgomery_reduce(mp_int *x, mp_int *n, mp_digit rho); +int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int redmode); +int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode); +void bn_reverse(unsigned char *s, int len); + +extern const char *mp_s_rmap; + +/* Fancy macro to set an MPI from another type. + * There are several things assumed: + * x is the counter and unsigned + * a is the pointer to the MPI + * b is the original value that should be set in the MPI. + */ +#define MP_SET_XLONG(func_name, type) \ +int func_name (mp_int * a, type b) \ +{ \ + unsigned int x; \ + int res; \ + \ + mp_zero (a); \ + \ + /* set four bits at a time */ \ + for (x = 0; x < (sizeof(type) * 2u); x++) { \ + /* shift the number up four bits */ \ + if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) { \ + return res; \ + } \ + \ + /* OR in the top four bits of the source */ \ + a->dp[0] |= (b >> ((sizeof(type) * 8u) - 4u)) & 15u; \ + \ + /* shift the source up to the next four bits */ \ + b <<= 4; \ + \ + /* ensure that digits are not clamped off */ \ + a->used += 1; \ + } \ + mp_clamp (a); \ + return MP_OKAY; \ +} + +#ifdef __cplusplus + } +#endif + +#endif + + +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ -- cgit v0.12 From 6b63d42886b68cbfdbd19c6fee04b836320a646d Mon Sep 17 00:00:00 2001 From: venkat Date: Wed, 16 Mar 2016 08:58:27 +0000 Subject: Update tzdata to 2016b from ietf.org --- library/tzdata/America/Cayman | 177 +-------------------- library/tzdata/America/Metlakatla | 169 ++++++++++++++++++++ library/tzdata/America/Port-au-Prince | 168 -------------------- library/tzdata/America/Santa_Isabel | 285 +--------------------------------- library/tzdata/Asia/Barnaul | 71 +++++++++ library/tzdata/Asia/Chita | 1 + library/tzdata/Asia/Gaza | 168 ++++++++++---------- library/tzdata/Asia/Hebron | 168 ++++++++++---------- library/tzdata/Asia/Karachi | 4 +- library/tzdata/Asia/Sakhalin | 1 + library/tzdata/Asia/Tehran | 124 +++++++++++++++ library/tzdata/Europe/Astrakhan | 70 +++++++++ library/tzdata/Europe/Chisinau | 6 +- library/tzdata/Europe/Samara | 2 +- library/tzdata/Europe/Ulyanovsk | 73 +++++++++ 15 files changed, 689 insertions(+), 798 deletions(-) create mode 100644 library/tzdata/Asia/Barnaul create mode 100644 library/tzdata/Europe/Astrakhan create mode 100644 library/tzdata/Europe/Ulyanovsk diff --git a/library/tzdata/America/Cayman b/library/tzdata/America/Cayman index 5231ca9..92ce5e2 100644 --- a/library/tzdata/America/Cayman +++ b/library/tzdata/America/Cayman @@ -1,176 +1,5 @@ # created by tools/tclZIC.tcl - do not edit - -set TZData(:America/Cayman) { - {-9223372036854775808 -19532 0 LMT} - {-2524502068 -18431 0 KMT} - {-1827687169 -18000 0 EST} - {1451624400 -18000 0 EST} - {1457852400 -14400 1 EDT} - {1478412000 -18000 0 EST} - {1489302000 -14400 1 EDT} - {1509861600 -18000 0 EST} - {1520751600 -14400 1 EDT} - {1541311200 -18000 0 EST} - {1552201200 -14400 1 EDT} - {1572760800 -18000 0 EST} - {1583650800 -14400 1 EDT} - {1604210400 -18000 0 EST} - {1615705200 -14400 1 EDT} - {1636264800 -18000 0 EST} - {1647154800 -14400 1 EDT} - {1667714400 -18000 0 EST} - {1678604400 -14400 1 EDT} - {1699164000 -18000 0 EST} - {1710054000 -14400 1 EDT} - {1730613600 -18000 0 EST} - {1741503600 -14400 1 EDT} - {1762063200 -18000 0 EST} - {1772953200 -14400 1 EDT} - {1793512800 -18000 0 EST} - {1805007600 -14400 1 EDT} - {1825567200 -18000 0 EST} - {1836457200 -14400 1 EDT} - {1857016800 -18000 0 EST} - {1867906800 -14400 1 EDT} - {1888466400 -18000 0 EST} - {1899356400 -14400 1 EDT} - {1919916000 -18000 0 EST} - {1930806000 -14400 1 EDT} - {1951365600 -18000 0 EST} - {1962860400 -14400 1 EDT} - {1983420000 -18000 0 EST} - {1994310000 -14400 1 EDT} - {2014869600 -18000 0 EST} - {2025759600 -14400 1 EDT} - {2046319200 -18000 0 EST} - {2057209200 -14400 1 EDT} - {2077768800 -18000 0 EST} - {2088658800 -14400 1 EDT} - {2109218400 -18000 0 EST} - {2120108400 -14400 1 EDT} - {2140668000 -18000 0 EST} - {2152162800 -14400 1 EDT} - {2172722400 -18000 0 EST} - {2183612400 -14400 1 EDT} - {2204172000 -18000 0 EST} - {2215062000 -14400 1 EDT} - {2235621600 -18000 0 EST} - {2246511600 -14400 1 EDT} - {2267071200 -18000 0 EST} - {2277961200 -14400 1 EDT} - {2298520800 -18000 0 EST} - {2309410800 -14400 1 EDT} - {2329970400 -18000 0 EST} - {2341465200 -14400 1 EDT} - {2362024800 -18000 0 EST} - {2372914800 -14400 1 EDT} - {2393474400 -18000 0 EST} - {2404364400 -14400 1 EDT} - {2424924000 -18000 0 EST} - {2435814000 -14400 1 EDT} - {2456373600 -18000 0 EST} - {2467263600 -14400 1 EDT} - {2487823200 -18000 0 EST} - {2499318000 -14400 1 EDT} - {2519877600 -18000 0 EST} - {2530767600 -14400 1 EDT} - {2551327200 -18000 0 EST} - {2562217200 -14400 1 EDT} - {2582776800 -18000 0 EST} - {2593666800 -14400 1 EDT} - {2614226400 -18000 0 EST} - {2625116400 -14400 1 EDT} - {2645676000 -18000 0 EST} - {2656566000 -14400 1 EDT} - {2677125600 -18000 0 EST} - {2688620400 -14400 1 EDT} - {2709180000 -18000 0 EST} - {2720070000 -14400 1 EDT} - {2740629600 -18000 0 EST} - {2751519600 -14400 1 EDT} - {2772079200 -18000 0 EST} - {2782969200 -14400 1 EDT} - {2803528800 -18000 0 EST} - {2814418800 -14400 1 EDT} - {2834978400 -18000 0 EST} - {2846473200 -14400 1 EDT} - {2867032800 -18000 0 EST} - {2877922800 -14400 1 EDT} - {2898482400 -18000 0 EST} - {2909372400 -14400 1 EDT} - {2929932000 -18000 0 EST} - {2940822000 -14400 1 EDT} - {2961381600 -18000 0 EST} - {2972271600 -14400 1 EDT} - {2992831200 -18000 0 EST} - {3003721200 -14400 1 EDT} - {3024280800 -18000 0 EST} - {3035775600 -14400 1 EDT} - {3056335200 -18000 0 EST} - {3067225200 -14400 1 EDT} - {3087784800 -18000 0 EST} - {3098674800 -14400 1 EDT} - {3119234400 -18000 0 EST} - {3130124400 -14400 1 EDT} - {3150684000 -18000 0 EST} - {3161574000 -14400 1 EDT} - {3182133600 -18000 0 EST} - {3193023600 -14400 1 EDT} - {3213583200 -18000 0 EST} - {3225078000 -14400 1 EDT} - {3245637600 -18000 0 EST} - {3256527600 -14400 1 EDT} - {3277087200 -18000 0 EST} - {3287977200 -14400 1 EDT} - {3308536800 -18000 0 EST} - {3319426800 -14400 1 EDT} - {3339986400 -18000 0 EST} - {3350876400 -14400 1 EDT} - {3371436000 -18000 0 EST} - {3382930800 -14400 1 EDT} - {3403490400 -18000 0 EST} - {3414380400 -14400 1 EDT} - {3434940000 -18000 0 EST} - {3445830000 -14400 1 EDT} - {3466389600 -18000 0 EST} - {3477279600 -14400 1 EDT} - {3497839200 -18000 0 EST} - {3508729200 -14400 1 EDT} - {3529288800 -18000 0 EST} - {3540178800 -14400 1 EDT} - {3560738400 -18000 0 EST} - {3572233200 -14400 1 EDT} - {3592792800 -18000 0 EST} - {3603682800 -14400 1 EDT} - {3624242400 -18000 0 EST} - {3635132400 -14400 1 EDT} - {3655692000 -18000 0 EST} - {3666582000 -14400 1 EDT} - {3687141600 -18000 0 EST} - {3698031600 -14400 1 EDT} - {3718591200 -18000 0 EST} - {3730086000 -14400 1 EDT} - {3750645600 -18000 0 EST} - {3761535600 -14400 1 EDT} - {3782095200 -18000 0 EST} - {3792985200 -14400 1 EDT} - {3813544800 -18000 0 EST} - {3824434800 -14400 1 EDT} - {3844994400 -18000 0 EST} - {3855884400 -14400 1 EDT} - {3876444000 -18000 0 EST} - {3887334000 -14400 1 EDT} - {3907893600 -18000 0 EST} - {3919388400 -14400 1 EDT} - {3939948000 -18000 0 EST} - {3950838000 -14400 1 EDT} - {3971397600 -18000 0 EST} - {3982287600 -14400 1 EDT} - {4002847200 -18000 0 EST} - {4013737200 -14400 1 EDT} - {4034296800 -18000 0 EST} - {4045186800 -14400 1 EDT} - {4065746400 -18000 0 EST} - {4076636400 -14400 1 EDT} - {4097196000 -18000 0 EST} +if {![info exists TZData(America/Panama)]} { + LoadTimeZoneFile America/Panama } +set TZData(:America/Cayman) $TZData(:America/Panama) diff --git a/library/tzdata/America/Metlakatla b/library/tzdata/America/Metlakatla index 8ea80fa..407948d 100644 --- a/library/tzdata/America/Metlakatla +++ b/library/tzdata/America/Metlakatla @@ -40,4 +40,173 @@ set TZData(:America/Metlakatla) { {404902800 -28800 0 PST} {420026400 -25200 1 PDT} {436356000 -28800 0 PST} + {1446372000 -32400 0 AKST} + {1457866800 -28800 1 AKDT} + {1478426400 -32400 0 AKST} + {1489316400 -28800 1 AKDT} + {1509876000 -32400 0 AKST} + {1520766000 -28800 1 AKDT} + {1541325600 -32400 0 AKST} + {1552215600 -28800 1 AKDT} + {1572775200 -32400 0 AKST} + {1583665200 -28800 1 AKDT} + {1604224800 -32400 0 AKST} + {1615719600 -28800 1 AKDT} + {1636279200 -32400 0 AKST} + {1647169200 -28800 1 AKDT} + {1667728800 -32400 0 AKST} + {1678618800 -28800 1 AKDT} + {1699178400 -32400 0 AKST} + {1710068400 -28800 1 AKDT} + {1730628000 -32400 0 AKST} + {1741518000 -28800 1 AKDT} + {1762077600 -32400 0 AKST} + {1772967600 -28800 1 AKDT} + {1793527200 -32400 0 AKST} + {1805022000 -28800 1 AKDT} + {1825581600 -32400 0 AKST} + {1836471600 -28800 1 AKDT} + {1857031200 -32400 0 AKST} + {1867921200 -28800 1 AKDT} + {1888480800 -32400 0 AKST} + {1899370800 -28800 1 AKDT} + {1919930400 -32400 0 AKST} + {1930820400 -28800 1 AKDT} + {1951380000 -32400 0 AKST} + {1962874800 -28800 1 AKDT} + {1983434400 -32400 0 AKST} + {1994324400 -28800 1 AKDT} + {2014884000 -32400 0 AKST} + {2025774000 -28800 1 AKDT} + {2046333600 -32400 0 AKST} + {2057223600 -28800 1 AKDT} + {2077783200 -32400 0 AKST} + {2088673200 -28800 1 AKDT} + {2109232800 -32400 0 AKST} + {2120122800 -28800 1 AKDT} + {2140682400 -32400 0 AKST} + {2152177200 -28800 1 AKDT} + {2172736800 -32400 0 AKST} + {2183626800 -28800 1 AKDT} + {2204186400 -32400 0 AKST} + {2215076400 -28800 1 AKDT} + {2235636000 -32400 0 AKST} + {2246526000 -28800 1 AKDT} + {2267085600 -32400 0 AKST} + {2277975600 -28800 1 AKDT} + {2298535200 -32400 0 AKST} + {2309425200 -28800 1 AKDT} + {2329984800 -32400 0 AKST} + {2341479600 -28800 1 AKDT} + {2362039200 -32400 0 AKST} + {2372929200 -28800 1 AKDT} + {2393488800 -32400 0 AKST} + {2404378800 -28800 1 AKDT} + {2424938400 -32400 0 AKST} + {2435828400 -28800 1 AKDT} + {2456388000 -32400 0 AKST} + {2467278000 -28800 1 AKDT} + {2487837600 -32400 0 AKST} + {2499332400 -28800 1 AKDT} + {2519892000 -32400 0 AKST} + {2530782000 -28800 1 AKDT} + {2551341600 -32400 0 AKST} + {2562231600 -28800 1 AKDT} + {2582791200 -32400 0 AKST} + {2593681200 -28800 1 AKDT} + {2614240800 -32400 0 AKST} + {2625130800 -28800 1 AKDT} + {2645690400 -32400 0 AKST} + {2656580400 -28800 1 AKDT} + {2677140000 -32400 0 AKST} + {2688634800 -28800 1 AKDT} + {2709194400 -32400 0 AKST} + {2720084400 -28800 1 AKDT} + {2740644000 -32400 0 AKST} + {2751534000 -28800 1 AKDT} + {2772093600 -32400 0 AKST} + {2782983600 -28800 1 AKDT} + {2803543200 -32400 0 AKST} + {2814433200 -28800 1 AKDT} + {2834992800 -32400 0 AKST} + {2846487600 -28800 1 AKDT} + {2867047200 -32400 0 AKST} + {2877937200 -28800 1 AKDT} + {2898496800 -32400 0 AKST} + {2909386800 -28800 1 AKDT} + {2929946400 -32400 0 AKST} + {2940836400 -28800 1 AKDT} + {2961396000 -32400 0 AKST} + {2972286000 -28800 1 AKDT} + {2992845600 -32400 0 AKST} + {3003735600 -28800 1 AKDT} + {3024295200 -32400 0 AKST} + {3035790000 -28800 1 AKDT} + {3056349600 -32400 0 AKST} + {3067239600 -28800 1 AKDT} + {3087799200 -32400 0 AKST} + {3098689200 -28800 1 AKDT} + {3119248800 -32400 0 AKST} + {3130138800 -28800 1 AKDT} + {3150698400 -32400 0 AKST} + {3161588400 -28800 1 AKDT} + {3182148000 -32400 0 AKST} + {3193038000 -28800 1 AKDT} + {3213597600 -32400 0 AKST} + {3225092400 -28800 1 AKDT} + {3245652000 -32400 0 AKST} + {3256542000 -28800 1 AKDT} + {3277101600 -32400 0 AKST} + {3287991600 -28800 1 AKDT} + {3308551200 -32400 0 AKST} + {3319441200 -28800 1 AKDT} + {3340000800 -32400 0 AKST} + {3350890800 -28800 1 AKDT} + {3371450400 -32400 0 AKST} + {3382945200 -28800 1 AKDT} + {3403504800 -32400 0 AKST} + {3414394800 -28800 1 AKDT} + {3434954400 -32400 0 AKST} + {3445844400 -28800 1 AKDT} + {3466404000 -32400 0 AKST} + {3477294000 -28800 1 AKDT} + {3497853600 -32400 0 AKST} + {3508743600 -28800 1 AKDT} + {3529303200 -32400 0 AKST} + {3540193200 -28800 1 AKDT} + {3560752800 -32400 0 AKST} + {3572247600 -28800 1 AKDT} + {3592807200 -32400 0 AKST} + {3603697200 -28800 1 AKDT} + {3624256800 -32400 0 AKST} + {3635146800 -28800 1 AKDT} + {3655706400 -32400 0 AKST} + {3666596400 -28800 1 AKDT} + {3687156000 -32400 0 AKST} + {3698046000 -28800 1 AKDT} + {3718605600 -32400 0 AKST} + {3730100400 -28800 1 AKDT} + {3750660000 -32400 0 AKST} + {3761550000 -28800 1 AKDT} + {3782109600 -32400 0 AKST} + {3792999600 -28800 1 AKDT} + {3813559200 -32400 0 AKST} + {3824449200 -28800 1 AKDT} + {3845008800 -32400 0 AKST} + {3855898800 -28800 1 AKDT} + {3876458400 -32400 0 AKST} + {3887348400 -28800 1 AKDT} + {3907908000 -32400 0 AKST} + {3919402800 -28800 1 AKDT} + {3939962400 -32400 0 AKST} + {3950852400 -28800 1 AKDT} + {3971412000 -32400 0 AKST} + {3982302000 -28800 1 AKDT} + {4002861600 -32400 0 AKST} + {4013751600 -28800 1 AKDT} + {4034311200 -32400 0 AKST} + {4045201200 -28800 1 AKDT} + {4065760800 -32400 0 AKST} + {4076650800 -28800 1 AKDT} + {4097210400 -32400 0 AKST} } diff --git a/library/tzdata/America/Port-au-Prince b/library/tzdata/America/Port-au-Prince index f1d7fc4..b8b60d6 100644 --- a/library/tzdata/America/Port-au-Prince +++ b/library/tzdata/America/Port-au-Prince @@ -46,172 +46,4 @@ set TZData(:America/Port-au-Prince) { {1414908000 -18000 0 EST} {1425798000 -14400 1 EDT} {1446357600 -18000 0 EST} - {1457852400 -14400 1 EDT} - {1478412000 -18000 0 EST} - {1489302000 -14400 1 EDT} - {1509861600 -18000 0 EST} - {1520751600 -14400 1 EDT} - {1541311200 -18000 0 EST} - {1552201200 -14400 1 EDT} - {1572760800 -18000 0 EST} - {1583650800 -14400 1 EDT} - {1604210400 -18000 0 EST} - {1615705200 -14400 1 EDT} - {1636264800 -18000 0 EST} - {1647154800 -14400 1 EDT} - {1667714400 -18000 0 EST} - {1678604400 -14400 1 EDT} - {1699164000 -18000 0 EST} - {1710054000 -14400 1 EDT} - {1730613600 -18000 0 EST} - {1741503600 -14400 1 EDT} - {1762063200 -18000 0 EST} - {1772953200 -14400 1 EDT} - {1793512800 -18000 0 EST} - {1805007600 -14400 1 EDT} - {1825567200 -18000 0 EST} - {1836457200 -14400 1 EDT} - {1857016800 -18000 0 EST} - {1867906800 -14400 1 EDT} - {1888466400 -18000 0 EST} - {1899356400 -14400 1 EDT} - {1919916000 -18000 0 EST} - {1930806000 -14400 1 EDT} - {1951365600 -18000 0 EST} - {1962860400 -14400 1 EDT} - {1983420000 -18000 0 EST} - {1994310000 -14400 1 EDT} - {2014869600 -18000 0 EST} - {2025759600 -14400 1 EDT} - {2046319200 -18000 0 EST} - {2057209200 -14400 1 EDT} - {2077768800 -18000 0 EST} - {2088658800 -14400 1 EDT} - {2109218400 -18000 0 EST} - {2120108400 -14400 1 EDT} - {2140668000 -18000 0 EST} - {2152162800 -14400 1 EDT} - {2172722400 -18000 0 EST} - {2183612400 -14400 1 EDT} - {2204172000 -18000 0 EST} - {2215062000 -14400 1 EDT} - {2235621600 -18000 0 EST} - {2246511600 -14400 1 EDT} - {2267071200 -18000 0 EST} - {2277961200 -14400 1 EDT} - {2298520800 -18000 0 EST} - {2309410800 -14400 1 EDT} - {2329970400 -18000 0 EST} - {2341465200 -14400 1 EDT} - {2362024800 -18000 0 EST} - {2372914800 -14400 1 EDT} - {2393474400 -18000 0 EST} - {2404364400 -14400 1 EDT} - {2424924000 -18000 0 EST} - {2435814000 -14400 1 EDT} - {2456373600 -18000 0 EST} - {2467263600 -14400 1 EDT} - {2487823200 -18000 0 EST} - {2499318000 -14400 1 EDT} - {2519877600 -18000 0 EST} - {2530767600 -14400 1 EDT} - {2551327200 -18000 0 EST} - {2562217200 -14400 1 EDT} - {2582776800 -18000 0 EST} - {2593666800 -14400 1 EDT} - {2614226400 -18000 0 EST} - {2625116400 -14400 1 EDT} - {2645676000 -18000 0 EST} - {2656566000 -14400 1 EDT} - {2677125600 -18000 0 EST} - {2688620400 -14400 1 EDT} - {2709180000 -18000 0 EST} - {2720070000 -14400 1 EDT} - {2740629600 -18000 0 EST} - {2751519600 -14400 1 EDT} - {2772079200 -18000 0 EST} - {2782969200 -14400 1 EDT} - {2803528800 -18000 0 EST} - {2814418800 -14400 1 EDT} - {2834978400 -18000 0 EST} - {2846473200 -14400 1 EDT} - {2867032800 -18000 0 EST} - {2877922800 -14400 1 EDT} - {2898482400 -18000 0 EST} - {2909372400 -14400 1 EDT} - {2929932000 -18000 0 EST} - {2940822000 -14400 1 EDT} - {2961381600 -18000 0 EST} - {2972271600 -14400 1 EDT} - {2992831200 -18000 0 EST} - {3003721200 -14400 1 EDT} - {3024280800 -18000 0 EST} - {3035775600 -14400 1 EDT} - {3056335200 -18000 0 EST} - {3067225200 -14400 1 EDT} - {3087784800 -18000 0 EST} - {3098674800 -14400 1 EDT} - {3119234400 -18000 0 EST} - {3130124400 -14400 1 EDT} - {3150684000 -18000 0 EST} - {3161574000 -14400 1 EDT} - {3182133600 -18000 0 EST} - {3193023600 -14400 1 EDT} - {3213583200 -18000 0 EST} - {3225078000 -14400 1 EDT} - {3245637600 -18000 0 EST} - {3256527600 -14400 1 EDT} - {3277087200 -18000 0 EST} - {3287977200 -14400 1 EDT} - {3308536800 -18000 0 EST} - {3319426800 -14400 1 EDT} - {3339986400 -18000 0 EST} - {3350876400 -14400 1 EDT} - {3371436000 -18000 0 EST} - {3382930800 -14400 1 EDT} - {3403490400 -18000 0 EST} - {3414380400 -14400 1 EDT} - {3434940000 -18000 0 EST} - {3445830000 -14400 1 EDT} - {3466389600 -18000 0 EST} - {3477279600 -14400 1 EDT} - {3497839200 -18000 0 EST} - {3508729200 -14400 1 EDT} - {3529288800 -18000 0 EST} - {3540178800 -14400 1 EDT} - {3560738400 -18000 0 EST} - {3572233200 -14400 1 EDT} - {3592792800 -18000 0 EST} - {3603682800 -14400 1 EDT} - {3624242400 -18000 0 EST} - {3635132400 -14400 1 EDT} - {3655692000 -18000 0 EST} - {3666582000 -14400 1 EDT} - {3687141600 -18000 0 EST} - {3698031600 -14400 1 EDT} - {3718591200 -18000 0 EST} - {3730086000 -14400 1 EDT} - {3750645600 -18000 0 EST} - {3761535600 -14400 1 EDT} - {3782095200 -18000 0 EST} - {3792985200 -14400 1 EDT} - {3813544800 -18000 0 EST} - {3824434800 -14400 1 EDT} - {3844994400 -18000 0 EST} - {3855884400 -14400 1 EDT} - {3876444000 -18000 0 EST} - {3887334000 -14400 1 EDT} - {3907893600 -18000 0 EST} - {3919388400 -14400 1 EDT} - {3939948000 -18000 0 EST} - {3950838000 -14400 1 EDT} - {3971397600 -18000 0 EST} - {3982287600 -14400 1 EDT} - {4002847200 -18000 0 EST} - {4013737200 -14400 1 EDT} - {4034296800 -18000 0 EST} - {4045186800 -14400 1 EDT} - {4065746400 -18000 0 EST} - {4076636400 -14400 1 EDT} - {4097196000 -18000 0 EST} } diff --git a/library/tzdata/America/Santa_Isabel b/library/tzdata/America/Santa_Isabel index 87cb5a8..a3a3b4f 100644 --- a/library/tzdata/America/Santa_Isabel +++ b/library/tzdata/America/Santa_Isabel @@ -1,284 +1,5 @@ # created by tools/tclZIC.tcl - do not edit - -set TZData(:America/Santa_Isabel) { - {-9223372036854775808 -27568 0 LMT} - {-1514736000 -25200 0 MST} - {-1451667600 -28800 0 PST} - {-1343062800 -25200 0 MST} - {-1234803600 -28800 0 PST} - {-1222963200 -25200 1 PDT} - {-1207242000 -28800 0 PST} - {-873820800 -25200 1 PWT} - {-769395600 -25200 1 PPT} - {-761677200 -28800 0 PST} - {-686073600 -25200 1 PDT} - {-661539600 -28800 0 PST} - {-504892800 -28800 0 PST} - {-495036000 -25200 1 PDT} - {-481734000 -28800 0 PST} - {-463586400 -25200 1 PDT} - {-450284400 -28800 0 PST} - {-431532000 -25200 1 PDT} - {-418230000 -28800 0 PST} - {-400082400 -25200 1 PDT} - {-386780400 -28800 0 PST} - {-368632800 -25200 1 PDT} - {-355330800 -28800 0 PST} - {-337183200 -25200 1 PDT} - {-323881200 -28800 0 PST} - {-305733600 -25200 1 PDT} - {-292431600 -28800 0 PST} - {-283968000 -28800 0 PST} - {189331200 -28800 0 PST} - {199274400 -25200 1 PDT} - {215600400 -28800 0 PST} - {230724000 -25200 1 PDT} - {247050000 -28800 0 PST} - {262778400 -25200 1 PDT} - {278499600 -28800 0 PST} - {294228000 -25200 1 PDT} - {309949200 -28800 0 PST} - {325677600 -25200 1 PDT} - {341398800 -28800 0 PST} - {357127200 -25200 1 PDT} - {372848400 -28800 0 PST} - {388576800 -25200 1 PDT} - {404902800 -28800 0 PST} - {420026400 -25200 1 PDT} - {436352400 -28800 0 PST} - {452080800 -25200 1 PDT} - {467802000 -28800 0 PST} - {483530400 -25200 1 PDT} - {499251600 -28800 0 PST} - {514980000 -25200 1 PDT} - {530701200 -28800 0 PST} - {544615200 -25200 1 PDT} - {562150800 -28800 0 PST} - {576064800 -25200 1 PDT} - {594205200 -28800 0 PST} - {607514400 -25200 1 PDT} - {625654800 -28800 0 PST} - {638964000 -25200 1 PDT} - {657104400 -28800 0 PST} - {671018400 -25200 1 PDT} - {688554000 -28800 0 PST} - {702468000 -25200 1 PDT} - {720003600 -28800 0 PST} - {733917600 -25200 1 PDT} - {752058000 -28800 0 PST} - {765367200 -25200 1 PDT} - {783507600 -28800 0 PST} - {796816800 -25200 1 PDT} - {814957200 -28800 0 PST} - {820483200 -28800 0 PST} - {828871200 -25200 1 PDT} - {846406800 -28800 0 PST} - {860320800 -25200 1 PDT} - {877856400 -28800 0 PST} - {891770400 -25200 1 PDT} - {909306000 -28800 0 PST} - {923220000 -25200 1 PDT} - {941360400 -28800 0 PST} - {954669600 -25200 1 PDT} - {972810000 -28800 0 PST} - {978336000 -28800 0 PST} - {986119200 -25200 1 PDT} - {1004259600 -28800 0 PST} - {1014192000 -28800 0 PST} - {1018173600 -25200 1 PDT} - {1035709200 -28800 0 PST} - {1049623200 -25200 1 PDT} - {1067158800 -28800 0 PST} - {1081072800 -25200 1 PDT} - {1099213200 -28800 0 PST} - {1112522400 -25200 1 PDT} - {1130662800 -28800 0 PST} - {1143972000 -25200 1 PDT} - {1162112400 -28800 0 PST} - {1175421600 -25200 1 PDT} - {1193562000 -28800 0 PST} - {1207476000 -25200 1 PDT} - {1225011600 -28800 0 PST} - {1238925600 -25200 1 PDT} - {1256461200 -28800 0 PST} - {1270375200 -25200 1 PDT} - {1288515600 -28800 0 PST} - {1301824800 -25200 1 PDT} - {1319965200 -28800 0 PST} - {1333274400 -25200 1 PDT} - {1351414800 -28800 0 PST} - {1365328800 -25200 1 PDT} - {1382864400 -28800 0 PST} - {1396778400 -25200 1 PDT} - {1414314000 -28800 0 PST} - {1428228000 -25200 1 PDT} - {1445763600 -28800 0 PST} - {1459677600 -25200 1 PDT} - {1477818000 -28800 0 PST} - {1491127200 -25200 1 PDT} - {1509267600 -28800 0 PST} - {1522576800 -25200 1 PDT} - {1540717200 -28800 0 PST} - {1554631200 -25200 1 PDT} - {1572166800 -28800 0 PST} - {1586080800 -25200 1 PDT} - {1603616400 -28800 0 PST} - {1617530400 -25200 1 PDT} - {1635670800 -28800 0 PST} - {1648980000 -25200 1 PDT} - {1667120400 -28800 0 PST} - {1680429600 -25200 1 PDT} - {1698570000 -28800 0 PST} - {1712484000 -25200 1 PDT} - {1730019600 -28800 0 PST} - {1743933600 -25200 1 PDT} - {1761469200 -28800 0 PST} - {1775383200 -25200 1 PDT} - {1792918800 -28800 0 PST} - {1806832800 -25200 1 PDT} - {1824973200 -28800 0 PST} - {1838282400 -25200 1 PDT} - {1856422800 -28800 0 PST} - {1869732000 -25200 1 PDT} - {1887872400 -28800 0 PST} - {1901786400 -25200 1 PDT} - {1919322000 -28800 0 PST} - {1933236000 -25200 1 PDT} - {1950771600 -28800 0 PST} - {1964685600 -25200 1 PDT} - {1982826000 -28800 0 PST} - {1996135200 -25200 1 PDT} - {2014275600 -28800 0 PST} - {2027584800 -25200 1 PDT} - {2045725200 -28800 0 PST} - {2059034400 -25200 1 PDT} - {2077174800 -28800 0 PST} - {2091088800 -25200 1 PDT} - {2108624400 -28800 0 PST} - {2122538400 -25200 1 PDT} - {2140074000 -28800 0 PST} - {2153988000 -25200 1 PDT} - {2172128400 -28800 0 PST} - {2185437600 -25200 1 PDT} - {2203578000 -28800 0 PST} - {2216887200 -25200 1 PDT} - {2235027600 -28800 0 PST} - {2248941600 -25200 1 PDT} - {2266477200 -28800 0 PST} - {2280391200 -25200 1 PDT} - {2297926800 -28800 0 PST} - {2311840800 -25200 1 PDT} - {2329376400 -28800 0 PST} - {2343290400 -25200 1 PDT} - {2361430800 -28800 0 PST} - {2374740000 -25200 1 PDT} - {2392880400 -28800 0 PST} - {2406189600 -25200 1 PDT} - {2424330000 -28800 0 PST} - {2438244000 -25200 1 PDT} - {2455779600 -28800 0 PST} - {2469693600 -25200 1 PDT} - {2487229200 -28800 0 PST} - {2501143200 -25200 1 PDT} - {2519283600 -28800 0 PST} - {2532592800 -25200 1 PDT} - {2550733200 -28800 0 PST} - {2564042400 -25200 1 PDT} - {2582182800 -28800 0 PST} - {2596096800 -25200 1 PDT} - {2613632400 -28800 0 PST} - {2627546400 -25200 1 PDT} - {2645082000 -28800 0 PST} - {2658996000 -25200 1 PDT} - {2676531600 -28800 0 PST} - {2690445600 -25200 1 PDT} - {2708586000 -28800 0 PST} - {2721895200 -25200 1 PDT} - {2740035600 -28800 0 PST} - {2753344800 -25200 1 PDT} - {2771485200 -28800 0 PST} - {2785399200 -25200 1 PDT} - {2802934800 -28800 0 PST} - {2816848800 -25200 1 PDT} - {2834384400 -28800 0 PST} - {2848298400 -25200 1 PDT} - {2866438800 -28800 0 PST} - {2879748000 -25200 1 PDT} - {2897888400 -28800 0 PST} - {2911197600 -25200 1 PDT} - {2929338000 -28800 0 PST} - {2942647200 -25200 1 PDT} - {2960787600 -28800 0 PST} - {2974701600 -25200 1 PDT} - {2992237200 -28800 0 PST} - {3006151200 -25200 1 PDT} - {3023686800 -28800 0 PST} - {3037600800 -25200 1 PDT} - {3055741200 -28800 0 PST} - {3069050400 -25200 1 PDT} - {3087190800 -28800 0 PST} - {3100500000 -25200 1 PDT} - {3118640400 -28800 0 PST} - {3132554400 -25200 1 PDT} - {3150090000 -28800 0 PST} - {3164004000 -25200 1 PDT} - {3181539600 -28800 0 PST} - {3195453600 -25200 1 PDT} - {3212989200 -28800 0 PST} - {3226903200 -25200 1 PDT} - {3245043600 -28800 0 PST} - {3258352800 -25200 1 PDT} - {3276493200 -28800 0 PST} - {3289802400 -25200 1 PDT} - {3307942800 -28800 0 PST} - {3321856800 -25200 1 PDT} - {3339392400 -28800 0 PST} - {3353306400 -25200 1 PDT} - {3370842000 -28800 0 PST} - {3384756000 -25200 1 PDT} - {3402896400 -28800 0 PST} - {3416205600 -25200 1 PDT} - {3434346000 -28800 0 PST} - {3447655200 -25200 1 PDT} - {3465795600 -28800 0 PST} - {3479709600 -25200 1 PDT} - {3497245200 -28800 0 PST} - {3511159200 -25200 1 PDT} - {3528694800 -28800 0 PST} - {3542608800 -25200 1 PDT} - {3560144400 -28800 0 PST} - {3574058400 -25200 1 PDT} - {3592198800 -28800 0 PST} - {3605508000 -25200 1 PDT} - {3623648400 -28800 0 PST} - {3636957600 -25200 1 PDT} - {3655098000 -28800 0 PST} - {3669012000 -25200 1 PDT} - {3686547600 -28800 0 PST} - {3700461600 -25200 1 PDT} - {3717997200 -28800 0 PST} - {3731911200 -25200 1 PDT} - {3750051600 -28800 0 PST} - {3763360800 -25200 1 PDT} - {3781501200 -28800 0 PST} - {3794810400 -25200 1 PDT} - {3812950800 -28800 0 PST} - {3826260000 -25200 1 PDT} - {3844400400 -28800 0 PST} - {3858314400 -25200 1 PDT} - {3875850000 -28800 0 PST} - {3889764000 -25200 1 PDT} - {3907299600 -28800 0 PST} - {3921213600 -25200 1 PDT} - {3939354000 -28800 0 PST} - {3952663200 -25200 1 PDT} - {3970803600 -28800 0 PST} - {3984112800 -25200 1 PDT} - {4002253200 -28800 0 PST} - {4016167200 -25200 1 PDT} - {4033702800 -28800 0 PST} - {4047616800 -25200 1 PDT} - {4065152400 -28800 0 PST} - {4079066400 -25200 1 PDT} - {4096602000 -28800 0 PST} +if {![info exists TZData(America/Tijuana)]} { + LoadTimeZoneFile America/Tijuana } +set TZData(:America/Santa_Isabel) $TZData(:America/Tijuana) diff --git a/library/tzdata/Asia/Barnaul b/library/tzdata/Asia/Barnaul new file mode 100644 index 0000000..8072e34 --- /dev/null +++ b/library/tzdata/Asia/Barnaul @@ -0,0 +1,71 @@ +# created by tools/tclZIC.tcl - do not edit + +set TZData(:Asia/Barnaul) { + {-9223372036854775808 20100 0 LMT} + {-1579844100 21600 0 +06} + {-1247551200 25200 0 +08} + {354906000 28800 1 +08} + {370713600 25200 0 +07} + {386442000 28800 1 +08} + {402249600 25200 0 +07} + {417978000 28800 1 +08} + {433785600 25200 0 +07} + {449600400 28800 1 +08} + {465332400 25200 0 +07} + {481057200 28800 1 +08} + {496782000 25200 0 +07} + {512506800 28800 1 +08} + {528231600 25200 0 +07} + {543956400 28800 1 +08} + {559681200 25200 0 +07} + {575406000 28800 1 +08} + {591130800 25200 0 +07} + {606855600 28800 1 +08} + {622580400 25200 0 +07} + {638305200 28800 1 +08} + {654634800 25200 0 +07} + {670359600 28800 1 +08} + {686084400 25200 0 +07} + {701798400 28800 1 +08} + {717519600 25200 0 +07} + {733258800 28800 1 +08} + {748983600 25200 0 +07} + {764708400 28800 1 +08} + {780433200 25200 0 +07} + {796158000 28800 1 +08} + {801594000 25200 0 +07} + {811886400 21600 0 +06} + {828216000 25200 1 +07} + {846360000 21600 0 +06} + {859665600 25200 1 +07} + {877809600 21600 0 +06} + {891115200 25200 1 +07} + {909259200 21600 0 +06} + {922564800 25200 1 +07} + {941313600 21600 0 +06} + {954014400 25200 1 +07} + {972763200 21600 0 +06} + {985464000 25200 1 +07} + {1004212800 21600 0 +06} + {1017518400 25200 1 +07} + {1035662400 21600 0 +06} + {1048968000 25200 1 +07} + {1067112000 21600 0 +06} + {1080417600 25200 1 +07} + {1099166400 21600 0 +06} + {1111867200 25200 1 +07} + {1130616000 21600 0 +06} + {1143316800 25200 1 +07} + {1162065600 21600 0 +06} + {1174766400 25200 1 +07} + {1193515200 21600 0 +06} + {1206820800 25200 1 +07} + {1224964800 21600 0 +06} + {1238270400 25200 1 +07} + {1256414400 21600 0 +06} + {1269720000 25200 1 +07} + {1288468800 21600 0 +06} + {1301169600 25200 0 +07} + {1414263600 21600 0 +06} + {1459022400 25200 0 +07} +} diff --git a/library/tzdata/Asia/Chita b/library/tzdata/Asia/Chita index eabce7f..6aef523 100644 --- a/library/tzdata/Asia/Chita +++ b/library/tzdata/Asia/Chita @@ -68,4 +68,5 @@ set TZData(:Asia/Chita) { {1288458000 32400 0 YAKT} {1301158800 36000 0 YAKT} {1414252800 28800 0 IRKT} + {1459015200 32400 0 YAKT} } diff --git a/library/tzdata/Asia/Gaza b/library/tzdata/Asia/Gaza index 805b6b7..6adfe6d 100644 --- a/library/tzdata/Asia/Gaza +++ b/library/tzdata/Asia/Gaza @@ -107,172 +107,172 @@ set TZData(:Asia/Gaza) { {1414098000 7200 0 EET} {1427493600 10800 1 EEST} {1445547600 7200 0 EET} - {1458943200 10800 1 EEST} + {1458946800 10800 1 EEST} {1476997200 7200 0 EET} - {1490997600 10800 1 EEST} + {1490396400 10800 1 EEST} {1509051600 7200 0 EET} - {1522447200 10800 1 EEST} + {1522450800 10800 1 EEST} {1540501200 7200 0 EET} - {1553896800 10800 1 EEST} + {1553900400 10800 1 EEST} {1571950800 7200 0 EET} - {1585346400 10800 1 EEST} + {1585350000 10800 1 EEST} {1603400400 7200 0 EET} - {1616796000 10800 1 EEST} + {1616799600 10800 1 EEST} {1634850000 7200 0 EET} - {1648245600 10800 1 EEST} + {1648249200 10800 1 EEST} {1666299600 7200 0 EET} - {1680300000 10800 1 EEST} + {1679698800 10800 1 EEST} {1698354000 7200 0 EET} - {1711749600 10800 1 EEST} + {1711753200 10800 1 EEST} {1729803600 7200 0 EET} - {1743199200 10800 1 EEST} + {1743202800 10800 1 EEST} {1761253200 7200 0 EET} - {1774648800 10800 1 EEST} + {1774652400 10800 1 EEST} {1792702800 7200 0 EET} - {1806098400 10800 1 EEST} + {1806102000 10800 1 EEST} {1824152400 7200 0 EET} - {1838152800 10800 1 EEST} + {1837551600 10800 1 EEST} {1856206800 7200 0 EET} - {1869602400 10800 1 EEST} + {1869606000 10800 1 EEST} {1887656400 7200 0 EET} - {1901052000 10800 1 EEST} + {1901055600 10800 1 EEST} {1919106000 7200 0 EET} - {1932501600 10800 1 EEST} + {1932505200 10800 1 EEST} {1950555600 7200 0 EET} - {1963951200 10800 1 EEST} + {1963954800 10800 1 EEST} {1982005200 7200 0 EET} - {1995400800 10800 1 EEST} + {1995404400 10800 1 EEST} {2013454800 7200 0 EET} - {2027455200 10800 1 EEST} + {2026854000 10800 1 EEST} {2045509200 7200 0 EET} - {2058904800 10800 1 EEST} + {2058908400 10800 1 EEST} {2076958800 7200 0 EET} - {2090354400 10800 1 EEST} + {2090358000 10800 1 EEST} {2108408400 7200 0 EET} - {2121804000 10800 1 EEST} + {2121807600 10800 1 EEST} {2139858000 7200 0 EET} - {2153253600 10800 1 EEST} + {2153257200 10800 1 EEST} {2171307600 7200 0 EET} - {2184703200 10800 1 EEST} + {2184706800 10800 1 EEST} {2202757200 7200 0 EET} - {2216757600 10800 1 EEST} + {2216761200 10800 1 EEST} {2234811600 7200 0 EET} - {2248207200 10800 1 EEST} + {2248210800 10800 1 EEST} {2266261200 7200 0 EET} - {2279656800 10800 1 EEST} + {2279660400 10800 1 EEST} {2297710800 7200 0 EET} - {2311106400 10800 1 EEST} + {2311110000 10800 1 EEST} {2329160400 7200 0 EET} - {2342556000 10800 1 EEST} + {2342559600 10800 1 EEST} {2360610000 7200 0 EET} - {2374610400 10800 1 EEST} + {2374009200 10800 1 EEST} {2392664400 7200 0 EET} - {2406060000 10800 1 EEST} + {2406063600 10800 1 EEST} {2424114000 7200 0 EET} - {2437509600 10800 1 EEST} + {2437513200 10800 1 EEST} {2455563600 7200 0 EET} - {2468959200 10800 1 EEST} + {2468962800 10800 1 EEST} {2487013200 7200 0 EET} - {2500408800 10800 1 EEST} + {2500412400 10800 1 EEST} {2518462800 7200 0 EET} - {2531858400 10800 1 EEST} + {2531862000 10800 1 EEST} {2549912400 7200 0 EET} - {2563912800 10800 1 EEST} + {2563311600 10800 1 EEST} {2581966800 7200 0 EET} - {2595362400 10800 1 EEST} + {2595366000 10800 1 EEST} {2613416400 7200 0 EET} - {2626812000 10800 1 EEST} + {2626815600 10800 1 EEST} {2644866000 7200 0 EET} - {2658261600 10800 1 EEST} + {2658265200 10800 1 EEST} {2676315600 7200 0 EET} - {2689711200 10800 1 EEST} + {2689714800 10800 1 EEST} {2707765200 7200 0 EET} - {2721765600 10800 1 EEST} + {2721164400 10800 1 EEST} {2739819600 7200 0 EET} - {2753215200 10800 1 EEST} + {2753218800 10800 1 EEST} {2771269200 7200 0 EET} - {2784664800 10800 1 EEST} + {2784668400 10800 1 EEST} {2802718800 7200 0 EET} - {2816114400 10800 1 EEST} + {2816118000 10800 1 EEST} {2834168400 7200 0 EET} - {2847564000 10800 1 EEST} + {2847567600 10800 1 EEST} {2865618000 7200 0 EET} - {2879013600 10800 1 EEST} + {2879017200 10800 1 EEST} {2897067600 7200 0 EET} - {2911068000 10800 1 EEST} + {2910466800 10800 1 EEST} {2929122000 7200 0 EET} - {2942517600 10800 1 EEST} + {2942521200 10800 1 EEST} {2960571600 7200 0 EET} - {2973967200 10800 1 EEST} + {2973970800 10800 1 EEST} {2992021200 7200 0 EET} - {3005416800 10800 1 EEST} + {3005420400 10800 1 EEST} {3023470800 7200 0 EET} - {3036866400 10800 1 EEST} + {3036870000 10800 1 EEST} {3054920400 7200 0 EET} - {3068316000 10800 1 EEST} + {3068319600 10800 1 EEST} {3086370000 7200 0 EET} - {3100370400 10800 1 EEST} + {3100374000 10800 1 EEST} {3118424400 7200 0 EET} - {3131820000 10800 1 EEST} + {3131823600 10800 1 EEST} {3149874000 7200 0 EET} - {3163269600 10800 1 EEST} + {3163273200 10800 1 EEST} {3181323600 7200 0 EET} - {3194719200 10800 1 EEST} + {3194722800 10800 1 EEST} {3212773200 7200 0 EET} - {3226168800 10800 1 EEST} + {3226172400 10800 1 EEST} {3244222800 7200 0 EET} - {3258223200 10800 1 EEST} + {3257622000 10800 1 EEST} {3276277200 7200 0 EET} - {3289672800 10800 1 EEST} + {3289676400 10800 1 EEST} {3307726800 7200 0 EET} - {3321122400 10800 1 EEST} + {3321126000 10800 1 EEST} {3339176400 7200 0 EET} - {3352572000 10800 1 EEST} + {3352575600 10800 1 EEST} {3370626000 7200 0 EET} - {3384021600 10800 1 EEST} + {3384025200 10800 1 EEST} {3402075600 7200 0 EET} - {3415471200 10800 1 EEST} + {3415474800 10800 1 EEST} {3433525200 7200 0 EET} - {3447525600 10800 1 EEST} + {3446924400 10800 1 EEST} {3465579600 7200 0 EET} - {3478975200 10800 1 EEST} + {3478978800 10800 1 EEST} {3497029200 7200 0 EET} - {3510424800 10800 1 EEST} + {3510428400 10800 1 EEST} {3528478800 7200 0 EET} - {3541874400 10800 1 EEST} + {3541878000 10800 1 EEST} {3559928400 7200 0 EET} - {3573324000 10800 1 EEST} + {3573327600 10800 1 EEST} {3591378000 7200 0 EET} - {3605378400 10800 1 EEST} + {3604777200 10800 1 EEST} {3623432400 7200 0 EET} - {3636828000 10800 1 EEST} + {3636831600 10800 1 EEST} {3654882000 7200 0 EET} - {3668277600 10800 1 EEST} + {3668281200 10800 1 EEST} {3686331600 7200 0 EET} - {3699727200 10800 1 EEST} + {3699730800 10800 1 EEST} {3717781200 7200 0 EET} - {3731176800 10800 1 EEST} + {3731180400 10800 1 EEST} {3749230800 7200 0 EET} - {3762626400 10800 1 EEST} + {3762630000 10800 1 EEST} {3780680400 7200 0 EET} - {3794680800 10800 1 EEST} + {3794079600 10800 1 EEST} {3812734800 7200 0 EET} - {3826130400 10800 1 EEST} + {3826134000 10800 1 EEST} {3844184400 7200 0 EET} - {3857580000 10800 1 EEST} + {3857583600 10800 1 EEST} {3875634000 7200 0 EET} - {3889029600 10800 1 EEST} + {3889033200 10800 1 EEST} {3907083600 7200 0 EET} - {3920479200 10800 1 EEST} + {3920482800 10800 1 EEST} {3938533200 7200 0 EET} - {3951928800 10800 1 EEST} + {3951932400 10800 1 EEST} {3969982800 7200 0 EET} - {3983983200 10800 1 EEST} + {3983986800 10800 1 EEST} {4002037200 7200 0 EET} - {4015432800 10800 1 EEST} + {4015436400 10800 1 EEST} {4033486800 7200 0 EET} - {4046882400 10800 1 EEST} + {4046886000 10800 1 EEST} {4064936400 7200 0 EET} - {4078332000 10800 1 EEST} + {4078335600 10800 1 EEST} {4096386000 7200 0 EET} } diff --git a/library/tzdata/Asia/Hebron b/library/tzdata/Asia/Hebron index 9049d93..2db45f2 100644 --- a/library/tzdata/Asia/Hebron +++ b/library/tzdata/Asia/Hebron @@ -106,172 +106,172 @@ set TZData(:Asia/Hebron) { {1414098000 7200 0 EET} {1427493600 10800 1 EEST} {1445547600 7200 0 EET} - {1458943200 10800 1 EEST} + {1458946800 10800 1 EEST} {1476997200 7200 0 EET} - {1490997600 10800 1 EEST} + {1490396400 10800 1 EEST} {1509051600 7200 0 EET} - {1522447200 10800 1 EEST} + {1522450800 10800 1 EEST} {1540501200 7200 0 EET} - {1553896800 10800 1 EEST} + {1553900400 10800 1 EEST} {1571950800 7200 0 EET} - {1585346400 10800 1 EEST} + {1585350000 10800 1 EEST} {1603400400 7200 0 EET} - {1616796000 10800 1 EEST} + {1616799600 10800 1 EEST} {1634850000 7200 0 EET} - {1648245600 10800 1 EEST} + {1648249200 10800 1 EEST} {1666299600 7200 0 EET} - {1680300000 10800 1 EEST} + {1679698800 10800 1 EEST} {1698354000 7200 0 EET} - {1711749600 10800 1 EEST} + {1711753200 10800 1 EEST} {1729803600 7200 0 EET} - {1743199200 10800 1 EEST} + {1743202800 10800 1 EEST} {1761253200 7200 0 EET} - {1774648800 10800 1 EEST} + {1774652400 10800 1 EEST} {1792702800 7200 0 EET} - {1806098400 10800 1 EEST} + {1806102000 10800 1 EEST} {1824152400 7200 0 EET} - {1838152800 10800 1 EEST} + {1837551600 10800 1 EEST} {1856206800 7200 0 EET} - {1869602400 10800 1 EEST} + {1869606000 10800 1 EEST} {1887656400 7200 0 EET} - {1901052000 10800 1 EEST} + {1901055600 10800 1 EEST} {1919106000 7200 0 EET} - {1932501600 10800 1 EEST} + {1932505200 10800 1 EEST} {1950555600 7200 0 EET} - {1963951200 10800 1 EEST} + {1963954800 10800 1 EEST} {1982005200 7200 0 EET} - {1995400800 10800 1 EEST} + {1995404400 10800 1 EEST} {2013454800 7200 0 EET} - {2027455200 10800 1 EEST} + {2026854000 10800 1 EEST} {2045509200 7200 0 EET} - {2058904800 10800 1 EEST} + {2058908400 10800 1 EEST} {2076958800 7200 0 EET} - {2090354400 10800 1 EEST} + {2090358000 10800 1 EEST} {2108408400 7200 0 EET} - {2121804000 10800 1 EEST} + {2121807600 10800 1 EEST} {2139858000 7200 0 EET} - {2153253600 10800 1 EEST} + {2153257200 10800 1 EEST} {2171307600 7200 0 EET} - {2184703200 10800 1 EEST} + {2184706800 10800 1 EEST} {2202757200 7200 0 EET} - {2216757600 10800 1 EEST} + {2216761200 10800 1 EEST} {2234811600 7200 0 EET} - {2248207200 10800 1 EEST} + {2248210800 10800 1 EEST} {2266261200 7200 0 EET} - {2279656800 10800 1 EEST} + {2279660400 10800 1 EEST} {2297710800 7200 0 EET} - {2311106400 10800 1 EEST} + {2311110000 10800 1 EEST} {2329160400 7200 0 EET} - {2342556000 10800 1 EEST} + {2342559600 10800 1 EEST} {2360610000 7200 0 EET} - {2374610400 10800 1 EEST} + {2374009200 10800 1 EEST} {2392664400 7200 0 EET} - {2406060000 10800 1 EEST} + {2406063600 10800 1 EEST} {2424114000 7200 0 EET} - {2437509600 10800 1 EEST} + {2437513200 10800 1 EEST} {2455563600 7200 0 EET} - {2468959200 10800 1 EEST} + {2468962800 10800 1 EEST} {2487013200 7200 0 EET} - {2500408800 10800 1 EEST} + {2500412400 10800 1 EEST} {2518462800 7200 0 EET} - {2531858400 10800 1 EEST} + {2531862000 10800 1 EEST} {2549912400 7200 0 EET} - {2563912800 10800 1 EEST} + {2563311600 10800 1 EEST} {2581966800 7200 0 EET} - {2595362400 10800 1 EEST} + {2595366000 10800 1 EEST} {2613416400 7200 0 EET} - {2626812000 10800 1 EEST} + {2626815600 10800 1 EEST} {2644866000 7200 0 EET} - {2658261600 10800 1 EEST} + {2658265200 10800 1 EEST} {2676315600 7200 0 EET} - {2689711200 10800 1 EEST} + {2689714800 10800 1 EEST} {2707765200 7200 0 EET} - {2721765600 10800 1 EEST} + {2721164400 10800 1 EEST} {2739819600 7200 0 EET} - {2753215200 10800 1 EEST} + {2753218800 10800 1 EEST} {2771269200 7200 0 EET} - {2784664800 10800 1 EEST} + {2784668400 10800 1 EEST} {2802718800 7200 0 EET} - {2816114400 10800 1 EEST} + {2816118000 10800 1 EEST} {2834168400 7200 0 EET} - {2847564000 10800 1 EEST} + {2847567600 10800 1 EEST} {2865618000 7200 0 EET} - {2879013600 10800 1 EEST} + {2879017200 10800 1 EEST} {2897067600 7200 0 EET} - {2911068000 10800 1 EEST} + {2910466800 10800 1 EEST} {2929122000 7200 0 EET} - {2942517600 10800 1 EEST} + {2942521200 10800 1 EEST} {2960571600 7200 0 EET} - {2973967200 10800 1 EEST} + {2973970800 10800 1 EEST} {2992021200 7200 0 EET} - {3005416800 10800 1 EEST} + {3005420400 10800 1 EEST} {3023470800 7200 0 EET} - {3036866400 10800 1 EEST} + {3036870000 10800 1 EEST} {3054920400 7200 0 EET} - {3068316000 10800 1 EEST} + {3068319600 10800 1 EEST} {3086370000 7200 0 EET} - {3100370400 10800 1 EEST} + {3100374000 10800 1 EEST} {3118424400 7200 0 EET} - {3131820000 10800 1 EEST} + {3131823600 10800 1 EEST} {3149874000 7200 0 EET} - {3163269600 10800 1 EEST} + {3163273200 10800 1 EEST} {3181323600 7200 0 EET} - {3194719200 10800 1 EEST} + {3194722800 10800 1 EEST} {3212773200 7200 0 EET} - {3226168800 10800 1 EEST} + {3226172400 10800 1 EEST} {3244222800 7200 0 EET} - {3258223200 10800 1 EEST} + {3257622000 10800 1 EEST} {3276277200 7200 0 EET} - {3289672800 10800 1 EEST} + {3289676400 10800 1 EEST} {3307726800 7200 0 EET} - {3321122400 10800 1 EEST} + {3321126000 10800 1 EEST} {3339176400 7200 0 EET} - {3352572000 10800 1 EEST} + {3352575600 10800 1 EEST} {3370626000 7200 0 EET} - {3384021600 10800 1 EEST} + {3384025200 10800 1 EEST} {3402075600 7200 0 EET} - {3415471200 10800 1 EEST} + {3415474800 10800 1 EEST} {3433525200 7200 0 EET} - {3447525600 10800 1 EEST} + {3446924400 10800 1 EEST} {3465579600 7200 0 EET} - {3478975200 10800 1 EEST} + {3478978800 10800 1 EEST} {3497029200 7200 0 EET} - {3510424800 10800 1 EEST} + {3510428400 10800 1 EEST} {3528478800 7200 0 EET} - {3541874400 10800 1 EEST} + {3541878000 10800 1 EEST} {3559928400 7200 0 EET} - {3573324000 10800 1 EEST} + {3573327600 10800 1 EEST} {3591378000 7200 0 EET} - {3605378400 10800 1 EEST} + {3604777200 10800 1 EEST} {3623432400 7200 0 EET} - {3636828000 10800 1 EEST} + {3636831600 10800 1 EEST} {3654882000 7200 0 EET} - {3668277600 10800 1 EEST} + {3668281200 10800 1 EEST} {3686331600 7200 0 EET} - {3699727200 10800 1 EEST} + {3699730800 10800 1 EEST} {3717781200 7200 0 EET} - {3731176800 10800 1 EEST} + {3731180400 10800 1 EEST} {3749230800 7200 0 EET} - {3762626400 10800 1 EEST} + {3762630000 10800 1 EEST} {3780680400 7200 0 EET} - {3794680800 10800 1 EEST} + {3794079600 10800 1 EEST} {3812734800 7200 0 EET} - {3826130400 10800 1 EEST} + {3826134000 10800 1 EEST} {3844184400 7200 0 EET} - {3857580000 10800 1 EEST} + {3857583600 10800 1 EEST} {3875634000 7200 0 EET} - {3889029600 10800 1 EEST} + {3889033200 10800 1 EEST} {3907083600 7200 0 EET} - {3920479200 10800 1 EEST} + {3920482800 10800 1 EEST} {3938533200 7200 0 EET} - {3951928800 10800 1 EEST} + {3951932400 10800 1 EEST} {3969982800 7200 0 EET} - {3983983200 10800 1 EEST} + {3983986800 10800 1 EEST} {4002037200 7200 0 EET} - {4015432800 10800 1 EEST} + {4015436400 10800 1 EEST} {4033486800 7200 0 EET} - {4046882400 10800 1 EEST} + {4046886000 10800 1 EEST} {4064936400 7200 0 EET} - {4078332000 10800 1 EEST} + {4078335600 10800 1 EEST} {4096386000 7200 0 EET} } diff --git a/library/tzdata/Asia/Karachi b/library/tzdata/Asia/Karachi index 3faa31e..669c11a 100644 --- a/library/tzdata/Asia/Karachi +++ b/library/tzdata/Asia/Karachi @@ -7,8 +7,8 @@ set TZData(:Asia/Karachi) { {-764145000 19800 0 IST} {-576135000 18000 0 KART} {38775600 18000 0 PKT} - {1018119660 21600 1 PKST} - {1033840860 18000 0 PKT} + {1018119600 21600 1 PKST} + {1033840800 18000 0 PKT} {1212260400 21600 1 PKST} {1225476000 18000 0 PKT} {1239735600 21600 1 PKST} diff --git a/library/tzdata/Asia/Sakhalin b/library/tzdata/Asia/Sakhalin index eed20ba..9247046 100644 --- a/library/tzdata/Asia/Sakhalin +++ b/library/tzdata/Asia/Sakhalin @@ -70,4 +70,5 @@ set TZData(:Asia/Sakhalin) { {1288454400 36000 0 SAKT} {1301155200 39600 0 SAKT} {1414249200 36000 0 SAKT} + {1459008000 39600 0 SAKT} } diff --git a/library/tzdata/Asia/Tehran b/library/tzdata/Asia/Tehran index 7dca0ae..5fd840d 100644 --- a/library/tzdata/Asia/Tehran +++ b/library/tzdata/Asia/Tehran @@ -102,4 +102,128 @@ set TZData(:Asia/Tehran) { {2105551800 12600 0 IRST} {2121193800 16200 1 IRDT} {2137087800 12600 0 IRST} + {2152729800 16200 1 IRDT} + {2168623800 12600 0 IRST} + {2184265800 16200 1 IRDT} + {2200159800 12600 0 IRST} + {2215888200 16200 1 IRDT} + {2231782200 12600 0 IRST} + {2247424200 16200 1 IRDT} + {2263318200 12600 0 IRST} + {2278960200 16200 1 IRDT} + {2294854200 12600 0 IRST} + {2310496200 16200 1 IRDT} + {2326390200 12600 0 IRST} + {2342118600 16200 1 IRDT} + {2358012600 12600 0 IRST} + {2373654600 16200 1 IRDT} + {2389548600 12600 0 IRST} + {2405190600 16200 1 IRDT} + {2421084600 12600 0 IRST} + {2436726600 16200 1 IRDT} + {2452620600 12600 0 IRST} + {2468349000 16200 1 IRDT} + {2484243000 12600 0 IRST} + {2499885000 16200 1 IRDT} + {2515779000 12600 0 IRST} + {2531421000 16200 1 IRDT} + {2547315000 12600 0 IRST} + {2562957000 16200 1 IRDT} + {2578851000 12600 0 IRST} + {2594579400 16200 1 IRDT} + {2610473400 12600 0 IRST} + {2626115400 16200 1 IRDT} + {2642009400 12600 0 IRST} + {2657651400 16200 1 IRDT} + {2673545400 12600 0 IRST} + {2689187400 16200 1 IRDT} + {2705081400 12600 0 IRST} + {2720809800 16200 1 IRDT} + {2736703800 12600 0 IRST} + {2752345800 16200 1 IRDT} + {2768239800 12600 0 IRST} + {2783881800 16200 1 IRDT} + {2799775800 12600 0 IRST} + {2815417800 16200 1 IRDT} + {2831311800 12600 0 IRST} + {2847040200 16200 1 IRDT} + {2862934200 12600 0 IRST} + {2878576200 16200 1 IRDT} + {2894470200 12600 0 IRST} + {2910112200 16200 1 IRDT} + {2926006200 12600 0 IRST} + {2941648200 16200 1 IRDT} + {2957542200 12600 0 IRST} + {2973270600 16200 1 IRDT} + {2989164600 12600 0 IRST} + {3004806600 16200 1 IRDT} + {3020700600 12600 0 IRST} + {3036342600 16200 1 IRDT} + {3052236600 12600 0 IRST} + {3067878600 16200 1 IRDT} + {3083772600 12600 0 IRST} + {3099501000 16200 1 IRDT} + {3115395000 12600 0 IRST} + {3131037000 16200 1 IRDT} + {3146931000 12600 0 IRST} + {3162573000 16200 1 IRDT} + {3178467000 12600 0 IRST} + {3194109000 16200 1 IRDT} + {3210003000 12600 0 IRST} + {3225731400 16200 1 IRDT} + {3241625400 12600 0 IRST} + {3257267400 16200 1 IRDT} + {3273161400 12600 0 IRST} + {3288803400 16200 1 IRDT} + {3304697400 12600 0 IRST} + {3320339400 16200 1 IRDT} + {3336233400 12600 0 IRST} + {3351961800 16200 1 IRDT} + {3367855800 12600 0 IRST} + {3383497800 16200 1 IRDT} + {3399391800 12600 0 IRST} + {3415033800 16200 1 IRDT} + {3430927800 12600 0 IRST} + {3446569800 16200 1 IRDT} + {3462463800 12600 0 IRST} + {3478192200 16200 1 IRDT} + {3494086200 12600 0 IRST} + {3509728200 16200 1 IRDT} + {3525622200 12600 0 IRST} + {3541264200 16200 1 IRDT} + {3557158200 12600 0 IRST} + {3572800200 16200 1 IRDT} + {3588694200 12600 0 IRST} + {3604422600 16200 1 IRDT} + {3620316600 12600 0 IRST} + {3635958600 16200 1 IRDT} + {3651852600 12600 0 IRST} + {3667494600 16200 1 IRDT} + {3683388600 12600 0 IRST} + {3699030600 16200 1 IRDT} + {3714924600 12600 0 IRST} + {3730653000 16200 1 IRDT} + {3746547000 12600 0 IRST} + {3762189000 16200 1 IRDT} + {3778083000 12600 0 IRST} + {3793725000 16200 1 IRDT} + {3809619000 12600 0 IRST} + {3825261000 16200 1 IRDT} + {3841155000 12600 0 IRST} + {3856883400 16200 1 IRDT} + {3872777400 12600 0 IRST} + {3888419400 16200 1 IRDT} + {3904313400 12600 0 IRST} + {3919955400 16200 1 IRDT} + {3935849400 12600 0 IRST} + {3951491400 16200 1 IRDT} + {3967385400 12600 0 IRST} + {3983113800 16200 1 IRDT} + {3999007800 12600 0 IRST} + {4014649800 16200 1 IRDT} + {4030543800 12600 0 IRST} + {4046185800 16200 1 IRDT} + {4062079800 12600 0 IRST} + {4077721800 16200 1 IRDT} + {4093615800 12600 0 IRST} } diff --git a/library/tzdata/Europe/Astrakhan b/library/tzdata/Europe/Astrakhan new file mode 100644 index 0000000..e5e9c26 --- /dev/null +++ b/library/tzdata/Europe/Astrakhan @@ -0,0 +1,70 @@ +# created by tools/tclZIC.tcl - do not edit + +set TZData(:Europe/Astrakhan) { + {-9223372036854775808 11532 0 LMT} + {-1441249932 10800 0 +03} + {-1247540400 14400 0 +05} + {354916800 18000 1 +05} + {370724400 14400 0 +04} + {386452800 18000 1 +05} + {402260400 14400 0 +04} + {417988800 18000 1 +05} + {433796400 14400 0 +04} + {449611200 18000 1 +05} + {465343200 14400 0 +04} + {481068000 18000 1 +05} + {496792800 14400 0 +04} + {512517600 18000 1 +05} + {528242400 14400 0 +04} + {543967200 18000 1 +05} + {559692000 14400 0 +04} + {575416800 18000 1 +05} + {591141600 14400 0 +04} + {606866400 10800 0 +04} + {606870000 14400 1 +04} + {622594800 10800 0 +03} + {638319600 14400 1 +04} + {654649200 10800 0 +03} + {670374000 14400 0 +04} + {701820000 14400 0 +04} + {717534000 10800 0 +03} + {733273200 14400 1 +04} + {748998000 10800 0 +03} + {764722800 14400 1 +04} + {780447600 10800 0 +03} + {796172400 14400 1 +04} + {811897200 10800 0 +03} + {828226800 14400 1 +04} + {846370800 10800 0 +03} + {859676400 14400 1 +04} + {877820400 10800 0 +03} + {891126000 14400 1 +04} + {909270000 10800 0 +03} + {922575600 14400 1 +04} + {941324400 10800 0 +03} + {954025200 14400 1 +04} + {972774000 10800 0 +03} + {985474800 14400 1 +04} + {1004223600 10800 0 +03} + {1017529200 14400 1 +04} + {1035673200 10800 0 +03} + {1048978800 14400 1 +04} + {1067122800 10800 0 +03} + {1080428400 14400 1 +04} + {1099177200 10800 0 +03} + {1111878000 14400 1 +04} + {1130626800 10800 0 +03} + {1143327600 14400 1 +04} + {1162076400 10800 0 +03} + {1174777200 14400 1 +04} + {1193526000 10800 0 +03} + {1206831600 14400 1 +04} + {1224975600 10800 0 +03} + {1238281200 14400 1 +04} + {1256425200 10800 0 +03} + {1269730800 14400 1 +04} + {1288479600 10800 0 +03} + {1301180400 14400 0 +04} + {1414274400 10800 0 +03} + {1459033200 14400 0 +04} +} diff --git a/library/tzdata/Europe/Chisinau b/library/tzdata/Europe/Chisinau index 5c240e7..db4c6db 100644 --- a/library/tzdata/Europe/Chisinau +++ b/library/tzdata/Europe/Chisinau @@ -46,9 +46,9 @@ set TZData(:Europe/Chisinau) { {591145200 10800 0 MSK} {606870000 14400 1 MSD} {622594800 10800 0 MSK} - {631141200 10800 0 MSK} - {641941200 7200 0 EET} - {662680800 7200 0 EEMMTT} + {638319600 14400 1 MSD} + {641948400 10800 0 EEST} + {654652800 7200 0 EET} {670377600 10800 1 EEST} {686102400 7200 0 EET} {694216800 7200 0 EET} diff --git a/library/tzdata/Europe/Samara b/library/tzdata/Europe/Samara index ee9d989..47615de 100644 --- a/library/tzdata/Europe/Samara +++ b/library/tzdata/Europe/Samara @@ -28,7 +28,7 @@ set TZData(:Europe/Samara) { {654649200 10800 0 MSK} {670374000 7200 0 EEMMTT} {670377600 10800 1 EEST} - {686102400 10800 0 KUYT} + {686102400 10800 0 SAMT} {687916800 14400 0 SAMT} {701809200 18000 1 SAMST} {717530400 14400 0 SAMT} diff --git a/library/tzdata/Europe/Ulyanovsk b/library/tzdata/Europe/Ulyanovsk new file mode 100644 index 0000000..b975622 --- /dev/null +++ b/library/tzdata/Europe/Ulyanovsk @@ -0,0 +1,73 @@ +# created by tools/tclZIC.tcl - do not edit + +set TZData(:Europe/Ulyanovsk) { + {-9223372036854775808 11616 0 LMT} + {-1593825216 10800 0 +03} + {-1247540400 14400 0 +05} + {354916800 18000 1 +05} + {370724400 14400 0 +04} + {386452800 18000 1 +05} + {402260400 14400 0 +04} + {417988800 18000 1 +05} + {433796400 14400 0 +04} + {449611200 18000 1 +05} + {465343200 14400 0 +04} + {481068000 18000 1 +05} + {496792800 14400 0 +04} + {512517600 18000 1 +05} + {528242400 14400 0 +04} + {543967200 18000 1 +05} + {559692000 14400 0 +04} + {575416800 18000 1 +05} + {591141600 14400 0 +04} + {606866400 10800 0 +04} + {606870000 14400 1 +04} + {622594800 10800 0 +03} + {638319600 14400 1 +04} + {654649200 10800 0 +03} + {670374000 7200 0 +03} + {670377600 10800 1 +03} + {686102400 7200 0 +02} + {695779200 10800 0 +04} + {701812800 14400 1 +04} + {717534000 10800 0 +03} + {733273200 14400 1 +04} + {748998000 10800 0 +03} + {764722800 14400 1 +04} + {780447600 10800 0 +03} + {796172400 14400 1 +04} + {811897200 10800 0 +03} + {828226800 14400 1 +04} + {846370800 10800 0 +03} + {859676400 14400 1 +04} + {877820400 10800 0 +03} + {891126000 14400 1 +04} + {909270000 10800 0 +03} + {922575600 14400 1 +04} + {941324400 10800 0 +03} + {954025200 14400 1 +04} + {972774000 10800 0 +03} + {985474800 14400 1 +04} + {1004223600 10800 0 +03} + {1017529200 14400 1 +04} + {1035673200 10800 0 +03} + {1048978800 14400 1 +04} + {1067122800 10800 0 +03} + {1080428400 14400 1 +04} + {1099177200 10800 0 +03} + {1111878000 14400 1 +04} + {1130626800 10800 0 +03} + {1143327600 14400 1 +04} + {1162076400 10800 0 +03} + {1174777200 14400 1 +04} + {1193526000 10800 0 +03} + {1206831600 14400 1 +04} + {1224975600 10800 0 +03} + {1238281200 14400 1 +04} + {1256425200 10800 0 +03} + {1269730800 14400 1 +04} + {1288479600 10800 0 +03} + {1301180400 14400 0 +04} + {1414274400 10800 0 +03} + {1459033200 14400 0 +04} +} -- cgit v0.12 From 91a20bc018f04bcf8ce13412ad16cfcbe9e53489 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 16 Mar 2016 09:56:55 +0000 Subject: Remove undocumented TCL_HASH_KEY_STORE_HASH. Setting this to "0" is not supported (was it ever ????), and not tested for long time. --- generic/tcl.h | 18 ------------------ generic/tclHash.c | 37 ------------------------------------- 2 files changed, 55 deletions(-) diff --git a/generic/tcl.h b/generic/tcl.h index 58ba7e5..3cd90a9 100644 --- a/generic/tcl.h +++ b/generic/tcl.h @@ -1165,18 +1165,6 @@ typedef Tcl_HashEntry * (Tcl_AllocHashEntryProc) (Tcl_HashTable *tablePtr, typedef void (Tcl_FreeHashEntryProc) (Tcl_HashEntry *hPtr); /* - * This flag controls whether the hash table stores the hash of a key, or - * recalculates it. There should be no reason for turning this flag off as it - * is completely binary and source compatible unless you directly access the - * bucketPtr member of the Tcl_HashTableEntry structure. This member has been - * removed and the space used to store the hash value. - */ - -#ifndef TCL_HASH_KEY_STORE_HASH -# define TCL_HASH_KEY_STORE_HASH 1 -#endif - -/* * Structure definition for an entry in a hash table. No-one outside Tcl * should access any of these fields directly; use the macros defined below. */ @@ -1185,15 +1173,9 @@ struct Tcl_HashEntry { Tcl_HashEntry *nextPtr; /* Pointer to next entry in this hash bucket, * or NULL for end of chain. */ Tcl_HashTable *tablePtr; /* Pointer to table containing entry. */ -#if TCL_HASH_KEY_STORE_HASH void *hash; /* Hash value, stored as pointer to ensure * that the offsets of the fields in this * structure are not changed. */ -#else - Tcl_HashEntry **bucketPtr; /* Pointer to bucket that points to first - * entry in this entry's chain: used for - * deleting the entry. */ -#endif ClientData clientData; /* Application stores something here with * Tcl_SetHashValue. */ union { /* Key has one of these forms: */ diff --git a/generic/tclHash.c b/generic/tclHash.c index 1991aea..3ea9dd9 100644 --- a/generic/tclHash.c +++ b/generic/tclHash.c @@ -321,11 +321,9 @@ CreateHashEntry( for (hPtr = tablePtr->buckets[index]; hPtr != NULL; hPtr = hPtr->nextPtr) { -#if TCL_HASH_KEY_STORE_HASH if (hash != PTR2UINT(hPtr->hash)) { continue; } -#endif if (((void *) key == hPtr) || compareKeysProc((void *) key, hPtr)) { if (newPtr) { *newPtr = 0; @@ -336,11 +334,9 @@ CreateHashEntry( } else { for (hPtr = tablePtr->buckets[index]; hPtr != NULL; hPtr = hPtr->nextPtr) { -#if TCL_HASH_KEY_STORE_HASH if (hash != PTR2UINT(hPtr->hash)) { continue; } -#endif if (key == hPtr->key.oneWordValue) { if (newPtr) { *newPtr = 0; @@ -368,15 +364,9 @@ CreateHashEntry( } hPtr->tablePtr = tablePtr; -#if TCL_HASH_KEY_STORE_HASH hPtr->hash = UINT2PTR(hash); hPtr->nextPtr = tablePtr->buckets[index]; tablePtr->buckets[index] = hPtr; -#else - hPtr->bucketPtr = &tablePtr->buckets[index]; - hPtr->nextPtr = *hPtr->bucketPtr; - *hPtr->bucketPtr = hPtr; -#endif tablePtr->numEntries++; /* @@ -416,9 +406,7 @@ Tcl_DeleteHashEntry( const Tcl_HashKeyType *typePtr; Tcl_HashTable *tablePtr; Tcl_HashEntry **bucketPtr; -#if TCL_HASH_KEY_STORE_HASH int index; -#endif tablePtr = entryPtr->tablePtr; @@ -433,7 +421,6 @@ Tcl_DeleteHashEntry( typePtr = &tclArrayHashKeyType; } -#if TCL_HASH_KEY_STORE_HASH if (typePtr->hashKeyProc == NULL || typePtr->flags & TCL_HASH_KEY_RANDOMIZE_HASH) { index = RANDOM_INDEX(tablePtr, PTR2INT(entryPtr->hash)); @@ -442,9 +429,6 @@ Tcl_DeleteHashEntry( } bucketPtr = &tablePtr->buckets[index]; -#else - bucketPtr = entryPtr->bucketPtr; -#endif if (*bucketPtr == entryPtr) { *bucketPtr = entryPtr->nextPtr; @@ -1062,7 +1046,6 @@ RebuildTable( for (oldChainPtr = oldBuckets; oldSize > 0; oldSize--, oldChainPtr++) { for (hPtr = *oldChainPtr; hPtr != NULL; hPtr = *oldChainPtr) { *oldChainPtr = hPtr->nextPtr; -#if TCL_HASH_KEY_STORE_HASH if (typePtr->hashKeyProc == NULL || typePtr->flags & TCL_HASH_KEY_RANDOMIZE_HASH) { index = RANDOM_INDEX(tablePtr, PTR2INT(hPtr->hash)); @@ -1071,26 +1054,6 @@ RebuildTable( } hPtr->nextPtr = tablePtr->buckets[index]; tablePtr->buckets[index] = hPtr; -#else - void *key = Tcl_GetHashKey(tablePtr, hPtr); - - if (typePtr->hashKeyProc) { - unsigned int hash; - - hash = typePtr->hashKeyProc(tablePtr, key); - if (typePtr->flags & TCL_HASH_KEY_RANDOMIZE_HASH) { - index = RANDOM_INDEX(tablePtr, hash); - } else { - index = hash & tablePtr->mask; - } - } else { - index = RANDOM_INDEX(tablePtr, key); - } - - hPtr->bucketPtr = &tablePtr->buckets[index]; - hPtr->nextPtr = *hPtr->bucketPtr; - *hPtr->bucketPtr = hPtr; -#endif } } -- cgit v0.12 From 671e2228ac8024e945034191da52584381ce61ac Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 16 Mar 2016 09:58:08 +0000 Subject: (cherry-pick from trunk): Update tzdata to 2016b from ietf.org --- library/tzdata/America/Port-au-Prince | 168 ---------------------------------- library/tzdata/Asia/Barnaul | 71 ++++++++++++++ library/tzdata/Asia/Gaza | 168 +++++++++++++++++----------------- library/tzdata/Asia/Hebron | 168 +++++++++++++++++----------------- library/tzdata/Asia/Sakhalin | 1 + library/tzdata/Europe/Astrakhan | 70 ++++++++++++++ library/tzdata/Europe/Chisinau | 6 +- library/tzdata/Europe/Samara | 2 +- library/tzdata/Europe/Ulyanovsk | 73 +++++++++++++++ 9 files changed, 387 insertions(+), 340 deletions(-) create mode 100644 library/tzdata/Asia/Barnaul create mode 100644 library/tzdata/Europe/Astrakhan create mode 100644 library/tzdata/Europe/Ulyanovsk diff --git a/library/tzdata/America/Port-au-Prince b/library/tzdata/America/Port-au-Prince index f1d7fc4..b8b60d6 100644 --- a/library/tzdata/America/Port-au-Prince +++ b/library/tzdata/America/Port-au-Prince @@ -46,172 +46,4 @@ set TZData(:America/Port-au-Prince) { {1414908000 -18000 0 EST} {1425798000 -14400 1 EDT} {1446357600 -18000 0 EST} - {1457852400 -14400 1 EDT} - {1478412000 -18000 0 EST} - {1489302000 -14400 1 EDT} - {1509861600 -18000 0 EST} - {1520751600 -14400 1 EDT} - {1541311200 -18000 0 EST} - {1552201200 -14400 1 EDT} - {1572760800 -18000 0 EST} - {1583650800 -14400 1 EDT} - {1604210400 -18000 0 EST} - {1615705200 -14400 1 EDT} - {1636264800 -18000 0 EST} - {1647154800 -14400 1 EDT} - {1667714400 -18000 0 EST} - {1678604400 -14400 1 EDT} - {1699164000 -18000 0 EST} - {1710054000 -14400 1 EDT} - {1730613600 -18000 0 EST} - {1741503600 -14400 1 EDT} - {1762063200 -18000 0 EST} - {1772953200 -14400 1 EDT} - {1793512800 -18000 0 EST} - {1805007600 -14400 1 EDT} - {1825567200 -18000 0 EST} - {1836457200 -14400 1 EDT} - {1857016800 -18000 0 EST} - {1867906800 -14400 1 EDT} - {1888466400 -18000 0 EST} - {1899356400 -14400 1 EDT} - {1919916000 -18000 0 EST} - {1930806000 -14400 1 EDT} - {1951365600 -18000 0 EST} - {1962860400 -14400 1 EDT} - {1983420000 -18000 0 EST} - {1994310000 -14400 1 EDT} - {2014869600 -18000 0 EST} - {2025759600 -14400 1 EDT} - {2046319200 -18000 0 EST} - {2057209200 -14400 1 EDT} - {2077768800 -18000 0 EST} - {2088658800 -14400 1 EDT} - {2109218400 -18000 0 EST} - {2120108400 -14400 1 EDT} - {2140668000 -18000 0 EST} - {2152162800 -14400 1 EDT} - {2172722400 -18000 0 EST} - {2183612400 -14400 1 EDT} - {2204172000 -18000 0 EST} - {2215062000 -14400 1 EDT} - {2235621600 -18000 0 EST} - {2246511600 -14400 1 EDT} - {2267071200 -18000 0 EST} - {2277961200 -14400 1 EDT} - {2298520800 -18000 0 EST} - {2309410800 -14400 1 EDT} - {2329970400 -18000 0 EST} - {2341465200 -14400 1 EDT} - {2362024800 -18000 0 EST} - {2372914800 -14400 1 EDT} - {2393474400 -18000 0 EST} - {2404364400 -14400 1 EDT} - {2424924000 -18000 0 EST} - {2435814000 -14400 1 EDT} - {2456373600 -18000 0 EST} - {2467263600 -14400 1 EDT} - {2487823200 -18000 0 EST} - {2499318000 -14400 1 EDT} - {2519877600 -18000 0 EST} - {2530767600 -14400 1 EDT} - {2551327200 -18000 0 EST} - {2562217200 -14400 1 EDT} - {2582776800 -18000 0 EST} - {2593666800 -14400 1 EDT} - {2614226400 -18000 0 EST} - {2625116400 -14400 1 EDT} - {2645676000 -18000 0 EST} - {2656566000 -14400 1 EDT} - {2677125600 -18000 0 EST} - {2688620400 -14400 1 EDT} - {2709180000 -18000 0 EST} - {2720070000 -14400 1 EDT} - {2740629600 -18000 0 EST} - {2751519600 -14400 1 EDT} - {2772079200 -18000 0 EST} - {2782969200 -14400 1 EDT} - {2803528800 -18000 0 EST} - {2814418800 -14400 1 EDT} - {2834978400 -18000 0 EST} - {2846473200 -14400 1 EDT} - {2867032800 -18000 0 EST} - {2877922800 -14400 1 EDT} - {2898482400 -18000 0 EST} - {2909372400 -14400 1 EDT} - {2929932000 -18000 0 EST} - {2940822000 -14400 1 EDT} - {2961381600 -18000 0 EST} - {2972271600 -14400 1 EDT} - {2992831200 -18000 0 EST} - {3003721200 -14400 1 EDT} - {3024280800 -18000 0 EST} - {3035775600 -14400 1 EDT} - {3056335200 -18000 0 EST} - {3067225200 -14400 1 EDT} - {3087784800 -18000 0 EST} - {3098674800 -14400 1 EDT} - {3119234400 -18000 0 EST} - {3130124400 -14400 1 EDT} - {3150684000 -18000 0 EST} - {3161574000 -14400 1 EDT} - {3182133600 -18000 0 EST} - {3193023600 -14400 1 EDT} - {3213583200 -18000 0 EST} - {3225078000 -14400 1 EDT} - {3245637600 -18000 0 EST} - {3256527600 -14400 1 EDT} - {3277087200 -18000 0 EST} - {3287977200 -14400 1 EDT} - {3308536800 -18000 0 EST} - {3319426800 -14400 1 EDT} - {3339986400 -18000 0 EST} - {3350876400 -14400 1 EDT} - {3371436000 -18000 0 EST} - {3382930800 -14400 1 EDT} - {3403490400 -18000 0 EST} - {3414380400 -14400 1 EDT} - {3434940000 -18000 0 EST} - {3445830000 -14400 1 EDT} - {3466389600 -18000 0 EST} - {3477279600 -14400 1 EDT} - {3497839200 -18000 0 EST} - {3508729200 -14400 1 EDT} - {3529288800 -18000 0 EST} - {3540178800 -14400 1 EDT} - {3560738400 -18000 0 EST} - {3572233200 -14400 1 EDT} - {3592792800 -18000 0 EST} - {3603682800 -14400 1 EDT} - {3624242400 -18000 0 EST} - {3635132400 -14400 1 EDT} - {3655692000 -18000 0 EST} - {3666582000 -14400 1 EDT} - {3687141600 -18000 0 EST} - {3698031600 -14400 1 EDT} - {3718591200 -18000 0 EST} - {3730086000 -14400 1 EDT} - {3750645600 -18000 0 EST} - {3761535600 -14400 1 EDT} - {3782095200 -18000 0 EST} - {3792985200 -14400 1 EDT} - {3813544800 -18000 0 EST} - {3824434800 -14400 1 EDT} - {3844994400 -18000 0 EST} - {3855884400 -14400 1 EDT} - {3876444000 -18000 0 EST} - {3887334000 -14400 1 EDT} - {3907893600 -18000 0 EST} - {3919388400 -14400 1 EDT} - {3939948000 -18000 0 EST} - {3950838000 -14400 1 EDT} - {3971397600 -18000 0 EST} - {3982287600 -14400 1 EDT} - {4002847200 -18000 0 EST} - {4013737200 -14400 1 EDT} - {4034296800 -18000 0 EST} - {4045186800 -14400 1 EDT} - {4065746400 -18000 0 EST} - {4076636400 -14400 1 EDT} - {4097196000 -18000 0 EST} } diff --git a/library/tzdata/Asia/Barnaul b/library/tzdata/Asia/Barnaul new file mode 100644 index 0000000..8072e34 --- /dev/null +++ b/library/tzdata/Asia/Barnaul @@ -0,0 +1,71 @@ +# created by tools/tclZIC.tcl - do not edit + +set TZData(:Asia/Barnaul) { + {-9223372036854775808 20100 0 LMT} + {-1579844100 21600 0 +06} + {-1247551200 25200 0 +08} + {354906000 28800 1 +08} + {370713600 25200 0 +07} + {386442000 28800 1 +08} + {402249600 25200 0 +07} + {417978000 28800 1 +08} + {433785600 25200 0 +07} + {449600400 28800 1 +08} + {465332400 25200 0 +07} + {481057200 28800 1 +08} + {496782000 25200 0 +07} + {512506800 28800 1 +08} + {528231600 25200 0 +07} + {543956400 28800 1 +08} + {559681200 25200 0 +07} + {575406000 28800 1 +08} + {591130800 25200 0 +07} + {606855600 28800 1 +08} + {622580400 25200 0 +07} + {638305200 28800 1 +08} + {654634800 25200 0 +07} + {670359600 28800 1 +08} + {686084400 25200 0 +07} + {701798400 28800 1 +08} + {717519600 25200 0 +07} + {733258800 28800 1 +08} + {748983600 25200 0 +07} + {764708400 28800 1 +08} + {780433200 25200 0 +07} + {796158000 28800 1 +08} + {801594000 25200 0 +07} + {811886400 21600 0 +06} + {828216000 25200 1 +07} + {846360000 21600 0 +06} + {859665600 25200 1 +07} + {877809600 21600 0 +06} + {891115200 25200 1 +07} + {909259200 21600 0 +06} + {922564800 25200 1 +07} + {941313600 21600 0 +06} + {954014400 25200 1 +07} + {972763200 21600 0 +06} + {985464000 25200 1 +07} + {1004212800 21600 0 +06} + {1017518400 25200 1 +07} + {1035662400 21600 0 +06} + {1048968000 25200 1 +07} + {1067112000 21600 0 +06} + {1080417600 25200 1 +07} + {1099166400 21600 0 +06} + {1111867200 25200 1 +07} + {1130616000 21600 0 +06} + {1143316800 25200 1 +07} + {1162065600 21600 0 +06} + {1174766400 25200 1 +07} + {1193515200 21600 0 +06} + {1206820800 25200 1 +07} + {1224964800 21600 0 +06} + {1238270400 25200 1 +07} + {1256414400 21600 0 +06} + {1269720000 25200 1 +07} + {1288468800 21600 0 +06} + {1301169600 25200 0 +07} + {1414263600 21600 0 +06} + {1459022400 25200 0 +07} +} diff --git a/library/tzdata/Asia/Gaza b/library/tzdata/Asia/Gaza index 805b6b7..6adfe6d 100644 --- a/library/tzdata/Asia/Gaza +++ b/library/tzdata/Asia/Gaza @@ -107,172 +107,172 @@ set TZData(:Asia/Gaza) { {1414098000 7200 0 EET} {1427493600 10800 1 EEST} {1445547600 7200 0 EET} - {1458943200 10800 1 EEST} + {1458946800 10800 1 EEST} {1476997200 7200 0 EET} - {1490997600 10800 1 EEST} + {1490396400 10800 1 EEST} {1509051600 7200 0 EET} - {1522447200 10800 1 EEST} + {1522450800 10800 1 EEST} {1540501200 7200 0 EET} - {1553896800 10800 1 EEST} + {1553900400 10800 1 EEST} {1571950800 7200 0 EET} - {1585346400 10800 1 EEST} + {1585350000 10800 1 EEST} {1603400400 7200 0 EET} - {1616796000 10800 1 EEST} + {1616799600 10800 1 EEST} {1634850000 7200 0 EET} - {1648245600 10800 1 EEST} + {1648249200 10800 1 EEST} {1666299600 7200 0 EET} - {1680300000 10800 1 EEST} + {1679698800 10800 1 EEST} {1698354000 7200 0 EET} - {1711749600 10800 1 EEST} + {1711753200 10800 1 EEST} {1729803600 7200 0 EET} - {1743199200 10800 1 EEST} + {1743202800 10800 1 EEST} {1761253200 7200 0 EET} - {1774648800 10800 1 EEST} + {1774652400 10800 1 EEST} {1792702800 7200 0 EET} - {1806098400 10800 1 EEST} + {1806102000 10800 1 EEST} {1824152400 7200 0 EET} - {1838152800 10800 1 EEST} + {1837551600 10800 1 EEST} {1856206800 7200 0 EET} - {1869602400 10800 1 EEST} + {1869606000 10800 1 EEST} {1887656400 7200 0 EET} - {1901052000 10800 1 EEST} + {1901055600 10800 1 EEST} {1919106000 7200 0 EET} - {1932501600 10800 1 EEST} + {1932505200 10800 1 EEST} {1950555600 7200 0 EET} - {1963951200 10800 1 EEST} + {1963954800 10800 1 EEST} {1982005200 7200 0 EET} - {1995400800 10800 1 EEST} + {1995404400 10800 1 EEST} {2013454800 7200 0 EET} - {2027455200 10800 1 EEST} + {2026854000 10800 1 EEST} {2045509200 7200 0 EET} - {2058904800 10800 1 EEST} + {2058908400 10800 1 EEST} {2076958800 7200 0 EET} - {2090354400 10800 1 EEST} + {2090358000 10800 1 EEST} {2108408400 7200 0 EET} - {2121804000 10800 1 EEST} + {2121807600 10800 1 EEST} {2139858000 7200 0 EET} - {2153253600 10800 1 EEST} + {2153257200 10800 1 EEST} {2171307600 7200 0 EET} - {2184703200 10800 1 EEST} + {2184706800 10800 1 EEST} {2202757200 7200 0 EET} - {2216757600 10800 1 EEST} + {2216761200 10800 1 EEST} {2234811600 7200 0 EET} - {2248207200 10800 1 EEST} + {2248210800 10800 1 EEST} {2266261200 7200 0 EET} - {2279656800 10800 1 EEST} + {2279660400 10800 1 EEST} {2297710800 7200 0 EET} - {2311106400 10800 1 EEST} + {2311110000 10800 1 EEST} {2329160400 7200 0 EET} - {2342556000 10800 1 EEST} + {2342559600 10800 1 EEST} {2360610000 7200 0 EET} - {2374610400 10800 1 EEST} + {2374009200 10800 1 EEST} {2392664400 7200 0 EET} - {2406060000 10800 1 EEST} + {2406063600 10800 1 EEST} {2424114000 7200 0 EET} - {2437509600 10800 1 EEST} + {2437513200 10800 1 EEST} {2455563600 7200 0 EET} - {2468959200 10800 1 EEST} + {2468962800 10800 1 EEST} {2487013200 7200 0 EET} - {2500408800 10800 1 EEST} + {2500412400 10800 1 EEST} {2518462800 7200 0 EET} - {2531858400 10800 1 EEST} + {2531862000 10800 1 EEST} {2549912400 7200 0 EET} - {2563912800 10800 1 EEST} + {2563311600 10800 1 EEST} {2581966800 7200 0 EET} - {2595362400 10800 1 EEST} + {2595366000 10800 1 EEST} {2613416400 7200 0 EET} - {2626812000 10800 1 EEST} + {2626815600 10800 1 EEST} {2644866000 7200 0 EET} - {2658261600 10800 1 EEST} + {2658265200 10800 1 EEST} {2676315600 7200 0 EET} - {2689711200 10800 1 EEST} + {2689714800 10800 1 EEST} {2707765200 7200 0 EET} - {2721765600 10800 1 EEST} + {2721164400 10800 1 EEST} {2739819600 7200 0 EET} - {2753215200 10800 1 EEST} + {2753218800 10800 1 EEST} {2771269200 7200 0 EET} - {2784664800 10800 1 EEST} + {2784668400 10800 1 EEST} {2802718800 7200 0 EET} - {2816114400 10800 1 EEST} + {2816118000 10800 1 EEST} {2834168400 7200 0 EET} - {2847564000 10800 1 EEST} + {2847567600 10800 1 EEST} {2865618000 7200 0 EET} - {2879013600 10800 1 EEST} + {2879017200 10800 1 EEST} {2897067600 7200 0 EET} - {2911068000 10800 1 EEST} + {2910466800 10800 1 EEST} {2929122000 7200 0 EET} - {2942517600 10800 1 EEST} + {2942521200 10800 1 EEST} {2960571600 7200 0 EET} - {2973967200 10800 1 EEST} + {2973970800 10800 1 EEST} {2992021200 7200 0 EET} - {3005416800 10800 1 EEST} + {3005420400 10800 1 EEST} {3023470800 7200 0 EET} - {3036866400 10800 1 EEST} + {3036870000 10800 1 EEST} {3054920400 7200 0 EET} - {3068316000 10800 1 EEST} + {3068319600 10800 1 EEST} {3086370000 7200 0 EET} - {3100370400 10800 1 EEST} + {3100374000 10800 1 EEST} {3118424400 7200 0 EET} - {3131820000 10800 1 EEST} + {3131823600 10800 1 EEST} {3149874000 7200 0 EET} - {3163269600 10800 1 EEST} + {3163273200 10800 1 EEST} {3181323600 7200 0 EET} - {3194719200 10800 1 EEST} + {3194722800 10800 1 EEST} {3212773200 7200 0 EET} - {3226168800 10800 1 EEST} + {3226172400 10800 1 EEST} {3244222800 7200 0 EET} - {3258223200 10800 1 EEST} + {3257622000 10800 1 EEST} {3276277200 7200 0 EET} - {3289672800 10800 1 EEST} + {3289676400 10800 1 EEST} {3307726800 7200 0 EET} - {3321122400 10800 1 EEST} + {3321126000 10800 1 EEST} {3339176400 7200 0 EET} - {3352572000 10800 1 EEST} + {3352575600 10800 1 EEST} {3370626000 7200 0 EET} - {3384021600 10800 1 EEST} + {3384025200 10800 1 EEST} {3402075600 7200 0 EET} - {3415471200 10800 1 EEST} + {3415474800 10800 1 EEST} {3433525200 7200 0 EET} - {3447525600 10800 1 EEST} + {3446924400 10800 1 EEST} {3465579600 7200 0 EET} - {3478975200 10800 1 EEST} + {3478978800 10800 1 EEST} {3497029200 7200 0 EET} - {3510424800 10800 1 EEST} + {3510428400 10800 1 EEST} {3528478800 7200 0 EET} - {3541874400 10800 1 EEST} + {3541878000 10800 1 EEST} {3559928400 7200 0 EET} - {3573324000 10800 1 EEST} + {3573327600 10800 1 EEST} {3591378000 7200 0 EET} - {3605378400 10800 1 EEST} + {3604777200 10800 1 EEST} {3623432400 7200 0 EET} - {3636828000 10800 1 EEST} + {3636831600 10800 1 EEST} {3654882000 7200 0 EET} - {3668277600 10800 1 EEST} + {3668281200 10800 1 EEST} {3686331600 7200 0 EET} - {3699727200 10800 1 EEST} + {3699730800 10800 1 EEST} {3717781200 7200 0 EET} - {3731176800 10800 1 EEST} + {3731180400 10800 1 EEST} {3749230800 7200 0 EET} - {3762626400 10800 1 EEST} + {3762630000 10800 1 EEST} {3780680400 7200 0 EET} - {3794680800 10800 1 EEST} + {3794079600 10800 1 EEST} {3812734800 7200 0 EET} - {3826130400 10800 1 EEST} + {3826134000 10800 1 EEST} {3844184400 7200 0 EET} - {3857580000 10800 1 EEST} + {3857583600 10800 1 EEST} {3875634000 7200 0 EET} - {3889029600 10800 1 EEST} + {3889033200 10800 1 EEST} {3907083600 7200 0 EET} - {3920479200 10800 1 EEST} + {3920482800 10800 1 EEST} {3938533200 7200 0 EET} - {3951928800 10800 1 EEST} + {3951932400 10800 1 EEST} {3969982800 7200 0 EET} - {3983983200 10800 1 EEST} + {3983986800 10800 1 EEST} {4002037200 7200 0 EET} - {4015432800 10800 1 EEST} + {4015436400 10800 1 EEST} {4033486800 7200 0 EET} - {4046882400 10800 1 EEST} + {4046886000 10800 1 EEST} {4064936400 7200 0 EET} - {4078332000 10800 1 EEST} + {4078335600 10800 1 EEST} {4096386000 7200 0 EET} } diff --git a/library/tzdata/Asia/Hebron b/library/tzdata/Asia/Hebron index 9049d93..2db45f2 100644 --- a/library/tzdata/Asia/Hebron +++ b/library/tzdata/Asia/Hebron @@ -106,172 +106,172 @@ set TZData(:Asia/Hebron) { {1414098000 7200 0 EET} {1427493600 10800 1 EEST} {1445547600 7200 0 EET} - {1458943200 10800 1 EEST} + {1458946800 10800 1 EEST} {1476997200 7200 0 EET} - {1490997600 10800 1 EEST} + {1490396400 10800 1 EEST} {1509051600 7200 0 EET} - {1522447200 10800 1 EEST} + {1522450800 10800 1 EEST} {1540501200 7200 0 EET} - {1553896800 10800 1 EEST} + {1553900400 10800 1 EEST} {1571950800 7200 0 EET} - {1585346400 10800 1 EEST} + {1585350000 10800 1 EEST} {1603400400 7200 0 EET} - {1616796000 10800 1 EEST} + {1616799600 10800 1 EEST} {1634850000 7200 0 EET} - {1648245600 10800 1 EEST} + {1648249200 10800 1 EEST} {1666299600 7200 0 EET} - {1680300000 10800 1 EEST} + {1679698800 10800 1 EEST} {1698354000 7200 0 EET} - {1711749600 10800 1 EEST} + {1711753200 10800 1 EEST} {1729803600 7200 0 EET} - {1743199200 10800 1 EEST} + {1743202800 10800 1 EEST} {1761253200 7200 0 EET} - {1774648800 10800 1 EEST} + {1774652400 10800 1 EEST} {1792702800 7200 0 EET} - {1806098400 10800 1 EEST} + {1806102000 10800 1 EEST} {1824152400 7200 0 EET} - {1838152800 10800 1 EEST} + {1837551600 10800 1 EEST} {1856206800 7200 0 EET} - {1869602400 10800 1 EEST} + {1869606000 10800 1 EEST} {1887656400 7200 0 EET} - {1901052000 10800 1 EEST} + {1901055600 10800 1 EEST} {1919106000 7200 0 EET} - {1932501600 10800 1 EEST} + {1932505200 10800 1 EEST} {1950555600 7200 0 EET} - {1963951200 10800 1 EEST} + {1963954800 10800 1 EEST} {1982005200 7200 0 EET} - {1995400800 10800 1 EEST} + {1995404400 10800 1 EEST} {2013454800 7200 0 EET} - {2027455200 10800 1 EEST} + {2026854000 10800 1 EEST} {2045509200 7200 0 EET} - {2058904800 10800 1 EEST} + {2058908400 10800 1 EEST} {2076958800 7200 0 EET} - {2090354400 10800 1 EEST} + {2090358000 10800 1 EEST} {2108408400 7200 0 EET} - {2121804000 10800 1 EEST} + {2121807600 10800 1 EEST} {2139858000 7200 0 EET} - {2153253600 10800 1 EEST} + {2153257200 10800 1 EEST} {2171307600 7200 0 EET} - {2184703200 10800 1 EEST} + {2184706800 10800 1 EEST} {2202757200 7200 0 EET} - {2216757600 10800 1 EEST} + {2216761200 10800 1 EEST} {2234811600 7200 0 EET} - {2248207200 10800 1 EEST} + {2248210800 10800 1 EEST} {2266261200 7200 0 EET} - {2279656800 10800 1 EEST} + {2279660400 10800 1 EEST} {2297710800 7200 0 EET} - {2311106400 10800 1 EEST} + {2311110000 10800 1 EEST} {2329160400 7200 0 EET} - {2342556000 10800 1 EEST} + {2342559600 10800 1 EEST} {2360610000 7200 0 EET} - {2374610400 10800 1 EEST} + {2374009200 10800 1 EEST} {2392664400 7200 0 EET} - {2406060000 10800 1 EEST} + {2406063600 10800 1 EEST} {2424114000 7200 0 EET} - {2437509600 10800 1 EEST} + {2437513200 10800 1 EEST} {2455563600 7200 0 EET} - {2468959200 10800 1 EEST} + {2468962800 10800 1 EEST} {2487013200 7200 0 EET} - {2500408800 10800 1 EEST} + {2500412400 10800 1 EEST} {2518462800 7200 0 EET} - {2531858400 10800 1 EEST} + {2531862000 10800 1 EEST} {2549912400 7200 0 EET} - {2563912800 10800 1 EEST} + {2563311600 10800 1 EEST} {2581966800 7200 0 EET} - {2595362400 10800 1 EEST} + {2595366000 10800 1 EEST} {2613416400 7200 0 EET} - {2626812000 10800 1 EEST} + {2626815600 10800 1 EEST} {2644866000 7200 0 EET} - {2658261600 10800 1 EEST} + {2658265200 10800 1 EEST} {2676315600 7200 0 EET} - {2689711200 10800 1 EEST} + {2689714800 10800 1 EEST} {2707765200 7200 0 EET} - {2721765600 10800 1 EEST} + {2721164400 10800 1 EEST} {2739819600 7200 0 EET} - {2753215200 10800 1 EEST} + {2753218800 10800 1 EEST} {2771269200 7200 0 EET} - {2784664800 10800 1 EEST} + {2784668400 10800 1 EEST} {2802718800 7200 0 EET} - {2816114400 10800 1 EEST} + {2816118000 10800 1 EEST} {2834168400 7200 0 EET} - {2847564000 10800 1 EEST} + {2847567600 10800 1 EEST} {2865618000 7200 0 EET} - {2879013600 10800 1 EEST} + {2879017200 10800 1 EEST} {2897067600 7200 0 EET} - {2911068000 10800 1 EEST} + {2910466800 10800 1 EEST} {2929122000 7200 0 EET} - {2942517600 10800 1 EEST} + {2942521200 10800 1 EEST} {2960571600 7200 0 EET} - {2973967200 10800 1 EEST} + {2973970800 10800 1 EEST} {2992021200 7200 0 EET} - {3005416800 10800 1 EEST} + {3005420400 10800 1 EEST} {3023470800 7200 0 EET} - {3036866400 10800 1 EEST} + {3036870000 10800 1 EEST} {3054920400 7200 0 EET} - {3068316000 10800 1 EEST} + {3068319600 10800 1 EEST} {3086370000 7200 0 EET} - {3100370400 10800 1 EEST} + {3100374000 10800 1 EEST} {3118424400 7200 0 EET} - {3131820000 10800 1 EEST} + {3131823600 10800 1 EEST} {3149874000 7200 0 EET} - {3163269600 10800 1 EEST} + {3163273200 10800 1 EEST} {3181323600 7200 0 EET} - {3194719200 10800 1 EEST} + {3194722800 10800 1 EEST} {3212773200 7200 0 EET} - {3226168800 10800 1 EEST} + {3226172400 10800 1 EEST} {3244222800 7200 0 EET} - {3258223200 10800 1 EEST} + {3257622000 10800 1 EEST} {3276277200 7200 0 EET} - {3289672800 10800 1 EEST} + {3289676400 10800 1 EEST} {3307726800 7200 0 EET} - {3321122400 10800 1 EEST} + {3321126000 10800 1 EEST} {3339176400 7200 0 EET} - {3352572000 10800 1 EEST} + {3352575600 10800 1 EEST} {3370626000 7200 0 EET} - {3384021600 10800 1 EEST} + {3384025200 10800 1 EEST} {3402075600 7200 0 EET} - {3415471200 10800 1 EEST} + {3415474800 10800 1 EEST} {3433525200 7200 0 EET} - {3447525600 10800 1 EEST} + {3446924400 10800 1 EEST} {3465579600 7200 0 EET} - {3478975200 10800 1 EEST} + {3478978800 10800 1 EEST} {3497029200 7200 0 EET} - {3510424800 10800 1 EEST} + {3510428400 10800 1 EEST} {3528478800 7200 0 EET} - {3541874400 10800 1 EEST} + {3541878000 10800 1 EEST} {3559928400 7200 0 EET} - {3573324000 10800 1 EEST} + {3573327600 10800 1 EEST} {3591378000 7200 0 EET} - {3605378400 10800 1 EEST} + {3604777200 10800 1 EEST} {3623432400 7200 0 EET} - {3636828000 10800 1 EEST} + {3636831600 10800 1 EEST} {3654882000 7200 0 EET} - {3668277600 10800 1 EEST} + {3668281200 10800 1 EEST} {3686331600 7200 0 EET} - {3699727200 10800 1 EEST} + {3699730800 10800 1 EEST} {3717781200 7200 0 EET} - {3731176800 10800 1 EEST} + {3731180400 10800 1 EEST} {3749230800 7200 0 EET} - {3762626400 10800 1 EEST} + {3762630000 10800 1 EEST} {3780680400 7200 0 EET} - {3794680800 10800 1 EEST} + {3794079600 10800 1 EEST} {3812734800 7200 0 EET} - {3826130400 10800 1 EEST} + {3826134000 10800 1 EEST} {3844184400 7200 0 EET} - {3857580000 10800 1 EEST} + {3857583600 10800 1 EEST} {3875634000 7200 0 EET} - {3889029600 10800 1 EEST} + {3889033200 10800 1 EEST} {3907083600 7200 0 EET} - {3920479200 10800 1 EEST} + {3920482800 10800 1 EEST} {3938533200 7200 0 EET} - {3951928800 10800 1 EEST} + {3951932400 10800 1 EEST} {3969982800 7200 0 EET} - {3983983200 10800 1 EEST} + {3983986800 10800 1 EEST} {4002037200 7200 0 EET} - {4015432800 10800 1 EEST} + {4015436400 10800 1 EEST} {4033486800 7200 0 EET} - {4046882400 10800 1 EEST} + {4046886000 10800 1 EEST} {4064936400 7200 0 EET} - {4078332000 10800 1 EEST} + {4078335600 10800 1 EEST} {4096386000 7200 0 EET} } diff --git a/library/tzdata/Asia/Sakhalin b/library/tzdata/Asia/Sakhalin index eed20ba..9247046 100644 --- a/library/tzdata/Asia/Sakhalin +++ b/library/tzdata/Asia/Sakhalin @@ -70,4 +70,5 @@ set TZData(:Asia/Sakhalin) { {1288454400 36000 0 SAKT} {1301155200 39600 0 SAKT} {1414249200 36000 0 SAKT} + {1459008000 39600 0 SAKT} } diff --git a/library/tzdata/Europe/Astrakhan b/library/tzdata/Europe/Astrakhan new file mode 100644 index 0000000..e5e9c26 --- /dev/null +++ b/library/tzdata/Europe/Astrakhan @@ -0,0 +1,70 @@ +# created by tools/tclZIC.tcl - do not edit + +set TZData(:Europe/Astrakhan) { + {-9223372036854775808 11532 0 LMT} + {-1441249932 10800 0 +03} + {-1247540400 14400 0 +05} + {354916800 18000 1 +05} + {370724400 14400 0 +04} + {386452800 18000 1 +05} + {402260400 14400 0 +04} + {417988800 18000 1 +05} + {433796400 14400 0 +04} + {449611200 18000 1 +05} + {465343200 14400 0 +04} + {481068000 18000 1 +05} + {496792800 14400 0 +04} + {512517600 18000 1 +05} + {528242400 14400 0 +04} + {543967200 18000 1 +05} + {559692000 14400 0 +04} + {575416800 18000 1 +05} + {591141600 14400 0 +04} + {606866400 10800 0 +04} + {606870000 14400 1 +04} + {622594800 10800 0 +03} + {638319600 14400 1 +04} + {654649200 10800 0 +03} + {670374000 14400 0 +04} + {701820000 14400 0 +04} + {717534000 10800 0 +03} + {733273200 14400 1 +04} + {748998000 10800 0 +03} + {764722800 14400 1 +04} + {780447600 10800 0 +03} + {796172400 14400 1 +04} + {811897200 10800 0 +03} + {828226800 14400 1 +04} + {846370800 10800 0 +03} + {859676400 14400 1 +04} + {877820400 10800 0 +03} + {891126000 14400 1 +04} + {909270000 10800 0 +03} + {922575600 14400 1 +04} + {941324400 10800 0 +03} + {954025200 14400 1 +04} + {972774000 10800 0 +03} + {985474800 14400 1 +04} + {1004223600 10800 0 +03} + {1017529200 14400 1 +04} + {1035673200 10800 0 +03} + {1048978800 14400 1 +04} + {1067122800 10800 0 +03} + {1080428400 14400 1 +04} + {1099177200 10800 0 +03} + {1111878000 14400 1 +04} + {1130626800 10800 0 +03} + {1143327600 14400 1 +04} + {1162076400 10800 0 +03} + {1174777200 14400 1 +04} + {1193526000 10800 0 +03} + {1206831600 14400 1 +04} + {1224975600 10800 0 +03} + {1238281200 14400 1 +04} + {1256425200 10800 0 +03} + {1269730800 14400 1 +04} + {1288479600 10800 0 +03} + {1301180400 14400 0 +04} + {1414274400 10800 0 +03} + {1459033200 14400 0 +04} +} diff --git a/library/tzdata/Europe/Chisinau b/library/tzdata/Europe/Chisinau index 5c240e7..db4c6db 100644 --- a/library/tzdata/Europe/Chisinau +++ b/library/tzdata/Europe/Chisinau @@ -46,9 +46,9 @@ set TZData(:Europe/Chisinau) { {591145200 10800 0 MSK} {606870000 14400 1 MSD} {622594800 10800 0 MSK} - {631141200 10800 0 MSK} - {641941200 7200 0 EET} - {662680800 7200 0 EEMMTT} + {638319600 14400 1 MSD} + {641948400 10800 0 EEST} + {654652800 7200 0 EET} {670377600 10800 1 EEST} {686102400 7200 0 EET} {694216800 7200 0 EET} diff --git a/library/tzdata/Europe/Samara b/library/tzdata/Europe/Samara index ee9d989..47615de 100644 --- a/library/tzdata/Europe/Samara +++ b/library/tzdata/Europe/Samara @@ -28,7 +28,7 @@ set TZData(:Europe/Samara) { {654649200 10800 0 MSK} {670374000 7200 0 EEMMTT} {670377600 10800 1 EEST} - {686102400 10800 0 KUYT} + {686102400 10800 0 SAMT} {687916800 14400 0 SAMT} {701809200 18000 1 SAMST} {717530400 14400 0 SAMT} diff --git a/library/tzdata/Europe/Ulyanovsk b/library/tzdata/Europe/Ulyanovsk new file mode 100644 index 0000000..b975622 --- /dev/null +++ b/library/tzdata/Europe/Ulyanovsk @@ -0,0 +1,73 @@ +# created by tools/tclZIC.tcl - do not edit + +set TZData(:Europe/Ulyanovsk) { + {-9223372036854775808 11616 0 LMT} + {-1593825216 10800 0 +03} + {-1247540400 14400 0 +05} + {354916800 18000 1 +05} + {370724400 14400 0 +04} + {386452800 18000 1 +05} + {402260400 14400 0 +04} + {417988800 18000 1 +05} + {433796400 14400 0 +04} + {449611200 18000 1 +05} + {465343200 14400 0 +04} + {481068000 18000 1 +05} + {496792800 14400 0 +04} + {512517600 18000 1 +05} + {528242400 14400 0 +04} + {543967200 18000 1 +05} + {559692000 14400 0 +04} + {575416800 18000 1 +05} + {591141600 14400 0 +04} + {606866400 10800 0 +04} + {606870000 14400 1 +04} + {622594800 10800 0 +03} + {638319600 14400 1 +04} + {654649200 10800 0 +03} + {670374000 7200 0 +03} + {670377600 10800 1 +03} + {686102400 7200 0 +02} + {695779200 10800 0 +04} + {701812800 14400 1 +04} + {717534000 10800 0 +03} + {733273200 14400 1 +04} + {748998000 10800 0 +03} + {764722800 14400 1 +04} + {780447600 10800 0 +03} + {796172400 14400 1 +04} + {811897200 10800 0 +03} + {828226800 14400 1 +04} + {846370800 10800 0 +03} + {859676400 14400 1 +04} + {877820400 10800 0 +03} + {891126000 14400 1 +04} + {909270000 10800 0 +03} + {922575600 14400 1 +04} + {941324400 10800 0 +03} + {954025200 14400 1 +04} + {972774000 10800 0 +03} + {985474800 14400 1 +04} + {1004223600 10800 0 +03} + {1017529200 14400 1 +04} + {1035673200 10800 0 +03} + {1048978800 14400 1 +04} + {1067122800 10800 0 +03} + {1080428400 14400 1 +04} + {1099177200 10800 0 +03} + {1111878000 14400 1 +04} + {1130626800 10800 0 +03} + {1143327600 14400 1 +04} + {1162076400 10800 0 +03} + {1174777200 14400 1 +04} + {1193526000 10800 0 +03} + {1206831600 14400 1 +04} + {1224975600 10800 0 +03} + {1238281200 14400 1 +04} + {1256425200 10800 0 +03} + {1269730800 14400 1 +04} + {1288479600 10800 0 +03} + {1301180400 14400 0 +04} + {1414274400 10800 0 +03} + {1459033200 14400 0 +04} +} -- cgit v0.12 From bfa7cb3c5d224a86a6136bc063018638eb3b9a0b Mon Sep 17 00:00:00 2001 From: dgp Date: Thu, 17 Mar 2016 13:14:22 +0000 Subject: [ae38befcfb] Rewrite TclGetInnermostExceptionRange() for fewer iterations. --- generic/tclCompile.c | 21 ++++++++++----------- 1 file changed, 10 insertions(+), 11 deletions(-) diff --git a/generic/tclCompile.c b/generic/tclCompile.c index 4c259ab..c0b5dcc 100644 --- a/generic/tclCompile.c +++ b/generic/tclCompile.c @@ -3360,26 +3360,25 @@ TclGetInnermostExceptionRange( int returnCode, ExceptionAux **auxPtrPtr) { - int exnIdx = -1, i; + int i = envPtr->exceptArrayNext; + ExceptionRange *rangePtr = envPtr->exceptArrayPtr + i; - for (i=0 ; iexceptArrayNext ; i++) { - ExceptionRange *rangePtr = &envPtr->exceptArrayPtr[i]; + while (i > 0) { + rangePtr--; i--; if (CurrentOffset(envPtr) >= rangePtr->codeOffset && (rangePtr->numCodeBytes == -1 || CurrentOffset(envPtr) < rangePtr->codeOffset+rangePtr->numCodeBytes) && (returnCode != TCL_CONTINUE || envPtr->exceptAuxArrayPtr[i].supportsContinue)) { - exnIdx = i; + + if (auxPtrPtr) { + *auxPtrPtr = envPtr->exceptAuxArrayPtr + i; + } + return rangePtr; } } - if (exnIdx == -1) { - return NULL; - } - if (auxPtrPtr) { - *auxPtrPtr = &envPtr->exceptAuxArrayPtr[exnIdx]; - } - return &envPtr->exceptArrayPtr[exnIdx]; + return NULL; } /* -- cgit v0.12 From 01d1cdb5e8789884ae310c5e38fb976559157460 Mon Sep 17 00:00:00 2001 From: dkf Date: Sun, 20 Mar 2016 20:40:07 +0000 Subject: [1af8de570511] Fix crash in [string replace] caused by cut-n-paste. --- generic/tclExecute.c | 35 +++++++++++++---------------------- tests/stringComp.test | 10 ++++++++++ 2 files changed, 23 insertions(+), 22 deletions(-) diff --git a/generic/tclExecute.c b/generic/tclExecute.c index dacc9e2..c43cc40 100644 --- a/generic/tclExecute.c +++ b/generic/tclExecute.c @@ -5737,6 +5737,17 @@ TEBCresume( if (length3 - 1 == toIdx - fromIdx) { unsigned char *bytes1, *bytes2; + /* + * Flush the info in the string internal rep that refers to the + * about-to-be-invalidated UTF-8 rep. This sets the 'allocated' + * field of the String structure to 0 to indicate that a new + * buffer needs to be allocated. This assumes that the value is + * already of tclStringTypePtr type, which should be true provided + * we call it after Tcl_GetUnicodeFromObj. + */ +#define MarkStringInternalRepForFlush(objPtr) \ + (((int *) ((objPtr)->internalRep.twoPtrValue.ptr1))[1] = 0) + if (Tcl_IsShared(valuePtr)) { objResultPtr = Tcl_DuplicateObj(valuePtr); if (TclIsPureByteArray(objResultPtr) @@ -5749,17 +5760,7 @@ TEBCresume( ustring2 = Tcl_GetUnicodeFromObj(value3Ptr, NULL); memcpy(ustring1 + fromIdx, ustring2, length3 * sizeof(Tcl_UniChar)); - - /* - * Magic! Flush the info in the string internal rep that - * refers to the about-to-be-invalidated UTF-8 rep. This - * sets the 'allocated' field of the String structure to 0 - * to indicate that a new buffer needs to be allocated. - * This is safe; we know we've got a tclStringTypePtr set - * at this point (post Tcl_GetUnicodeFromObj). - */ - - ((int *) objResultPtr->internalRep.twoPtrValue.ptr1)[1] = 0; + MarkStringInternalRepForFlush(objResultPtr); } Tcl_InvalidateStringRep(objResultPtr); TclDecrRefCount(value3Ptr); @@ -5776,17 +5777,7 @@ TEBCresume( ustring2 = Tcl_GetUnicodeFromObj(value3Ptr, NULL); memcpy(ustring1 + fromIdx, ustring2, length3 * sizeof(Tcl_UniChar)); - - /* - * Magic! Flush the info in the string internal rep that - * refers to the about-to-be-invalidated UTF-8 rep. This - * sets the 'allocated' field of the String structure to 0 - * to indicate that a new buffer needs to be allocated. - * This is safe; we know we've got a tclStringTypePtr set - * at this point (post Tcl_GetUnicodeFromObj). - */ - - ((int *) objResultPtr->internalRep.twoPtrValue.ptr1)[1] = 0; + MarkStringInternalRepForFlush(valuePtr); } Tcl_InvalidateStringRep(valuePtr); TclDecrRefCount(value3Ptr); diff --git a/tests/stringComp.test b/tests/stringComp.test index a66525e..140a270 100644 --- a/tests/stringComp.test +++ b/tests/stringComp.test @@ -728,6 +728,16 @@ test stringComp-14.3 {Bug 0dca3bfa8f} { expr {$arg ne $argCopy} }} abcde } 1 +test stringComp-14.4 {Bug 1af8de570511} { + apply {{x y} { + # Generate an unshared string value + set val "" + for { set i 0 } { $i < $x } { incr i } { + set val [format "0%s" $val] + } + string replace $val[unset val] 1 1 $y + }} 4 x +} 0x00 ## string tolower ## not yet bc -- cgit v0.12 From d291d84481a5f1d56df04bfffcd53fae1a5c8a00 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Mon, 21 Mar 2016 09:05:34 +0000 Subject: Proposed fix for [d3071887dbc7aeac]: Fix SEGV in Tcl_FinalizeNotifier() --- unix/tclUnixNotfy.c | 2 ++ 1 file changed, 2 insertions(+) diff --git a/unix/tclUnixNotfy.c b/unix/tclUnixNotfy.c index 48ba0cc..3946c7d 100644 --- a/unix/tclUnixNotfy.c +++ b/unix/tclUnixNotfy.c @@ -433,9 +433,11 @@ Tcl_FinalizeNotifier( "unable to write q to triggerPipe"); } close(triggerPipe); + pthread_mutex_lock(¬ifierMutex); while(triggerPipe != -1) { pthread_cond_wait(¬ifierCV, ¬ifierMutex); } + pthread_mutex_lock(¬ifierMutex); if (notifierThreadRunning) { int result = pthread_join((pthread_t) notifierThread, NULL); -- cgit v0.12 From c031c59ec3e4290d29f9c819b2640e90a176ad13 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Mon, 21 Mar 2016 09:16:28 +0000 Subject: .... oops .... --- unix/tclUnixNotfy.c | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/unix/tclUnixNotfy.c b/unix/tclUnixNotfy.c index 3946c7d..1457890 100644 --- a/unix/tclUnixNotfy.c +++ b/unix/tclUnixNotfy.c @@ -437,7 +437,7 @@ Tcl_FinalizeNotifier( while(triggerPipe != -1) { pthread_cond_wait(¬ifierCV, ¬ifierMutex); } - pthread_mutex_lock(¬ifierMutex); + pthread_mutex_unlock(¬ifierMutex); if (notifierThreadRunning) { int result = pthread_join((pthread_t) notifierThread, NULL); -- cgit v0.12 From 72376991f5c43b445f266ab25f73c1c41d6393f0 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Mon, 21 Mar 2016 12:03:17 +0000 Subject: (cherry-pick): Fix [d3071887dbc7aeac]: Fix SEGV in Tcl_FinalizeNotifier(). Thanks to hirofumi for both the bug-report and the fix --- unix/tclUnixNotfy.c | 2 ++ 1 file changed, 2 insertions(+) diff --git a/unix/tclUnixNotfy.c b/unix/tclUnixNotfy.c index 48ba0cc..1457890 100644 --- a/unix/tclUnixNotfy.c +++ b/unix/tclUnixNotfy.c @@ -433,9 +433,11 @@ Tcl_FinalizeNotifier( "unable to write q to triggerPipe"); } close(triggerPipe); + pthread_mutex_lock(¬ifierMutex); while(triggerPipe != -1) { pthread_cond_wait(¬ifierCV, ¬ifierMutex); } + pthread_mutex_unlock(¬ifierMutex); if (notifierThreadRunning) { int result = pthread_join((pthread_t) notifierThread, NULL); -- cgit v0.12 From 06831f3ce25bb5b9778c4de1ad35dd50ed6b7b32 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Mon, 21 Mar 2016 12:08:34 +0000 Subject: (cherry-pick): Fix [d3071887dbc7aeac]: Fix SEGV in Tcl_FinalizeNotifier(). Thanks to hirofumi for both the bug-report and the fix --- unix/tclUnixNotfy.c | 2 ++ 1 file changed, 2 insertions(+) diff --git a/unix/tclUnixNotfy.c b/unix/tclUnixNotfy.c index a57a89a..e7ea7a1 100644 --- a/unix/tclUnixNotfy.c +++ b/unix/tclUnixNotfy.c @@ -432,9 +432,11 @@ Tcl_FinalizeNotifier( "unable to write q to triggerPipe"); } close(triggerPipe); + pthread_mutex_lock(¬ifierMutex); while(triggerPipe != -1) { pthread_cond_wait(¬ifierCV, ¬ifierMutex); } + pthread_mutex_unlock(¬ifierMutex); if (notifierThreadRunning) { int result = pthread_join((pthread_t) notifierThread, NULL); -- cgit v0.12 From a4933e22d0b56bf07cf35cb90eb1f6fd6c9e48cb Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Mon, 21 Mar 2016 14:22:52 +0000 Subject: (experiment) Use TclpMasterLock() in stead of a separate notifierInitMutex. One less mutex to be worried about. --- generic/tclInt.h | 1 + unix/tclUnixNotfy.c | 21 ++++++++++----------- unix/tclUnixThrd.c | 25 +++++++++++++++++++++++++ win/tclWinNotify.c | 26 ++++++++++++++++++-------- 4 files changed, 54 insertions(+), 19 deletions(-) diff --git a/generic/tclInt.h b/generic/tclInt.h index 42c13dd..da792aa 100644 --- a/generic/tclInt.h +++ b/generic/tclInt.h @@ -3059,6 +3059,7 @@ MODULE_SCOPE void TclpInitUnlock(void); MODULE_SCOPE Tcl_Obj * TclpObjListVolumes(void); MODULE_SCOPE void TclpMasterLock(void); MODULE_SCOPE void TclpMasterUnlock(void); +MODULE_SCOPE void TclpMasterReset(void); MODULE_SCOPE int TclpMatchFiles(Tcl_Interp *interp, char *separators, Tcl_DString *dirPtr, char *pattern, char *tail); MODULE_SCOPE int TclpObjNormalizePath(Tcl_Interp *interp, diff --git a/unix/tclUnixNotfy.c b/unix/tclUnixNotfy.c index 1457890..ca6a7ef 100644 --- a/unix/tclUnixNotfy.c +++ b/unix/tclUnixNotfy.c @@ -152,7 +152,6 @@ static int triggerPipe = -1; * The notifierMutex locks access to all of the global notifier state. */ -pthread_mutex_t notifierInitMutex = PTHREAD_MUTEX_INITIALIZER; pthread_mutex_t notifierMutex = PTHREAD_MUTEX_INITIALIZER; /* * The following static indicates if the notifier thread is running. @@ -281,7 +280,7 @@ static void StartNotifierThread(const char *proc) { if (!notifierThreadRunning) { - pthread_mutex_lock(¬ifierInitMutex); + TclpMasterLock(); if (!notifierThreadRunning) { if (TclpThreadCreate(¬ifierThread, NotifierThreadProc, NULL, TCL_THREAD_STACK_DEFAULT, TCL_THREAD_JOINABLE) != TCL_OK) { @@ -300,7 +299,7 @@ StartNotifierThread(const char *proc) notifierThreadRunning = 1; } - pthread_mutex_unlock(¬ifierInitMutex); + TclpMasterUnlock(); } } #endif /* TCL_THREADS */ @@ -362,7 +361,7 @@ Tcl_InitNotifier(void) tsdPtr->waitCVinitialized = 1; } - pthread_mutex_lock(¬ifierInitMutex); + TclpMasterLock(); #if defined(HAVE_PTHREAD_ATFORK) /* * Install pthread_atfork handlers to clean up the notifier in the @@ -381,7 +380,7 @@ Tcl_InitNotifier(void) notifierCount++; - pthread_mutex_unlock(¬ifierInitMutex); + TclpMasterUnlock(); #endif /* TCL_THREADS */ return tsdPtr; @@ -417,7 +416,7 @@ Tcl_FinalizeNotifier( #ifdef TCL_THREADS ThreadSpecificData *tsdPtr = TCL_TSD_INIT(&dataKey); - pthread_mutex_lock(¬ifierInitMutex); + TclpMasterLock(); notifierCount--; /* @@ -462,7 +461,7 @@ Tcl_FinalizeNotifier( #endif /* __CYGWIN__ */ tsdPtr->waitCVinitialized = 0; - pthread_mutex_unlock(¬ifierInitMutex); + TclpMasterUnlock(); #endif /* TCL_THREADS */ } } @@ -1368,7 +1367,7 @@ static void AtForkPrepare(void) { #if RESET_ATFORK_MUTEX == 0 - pthread_mutex_lock(¬ifierInitMutex); + TclpMasterLock(); #endif } @@ -1392,7 +1391,7 @@ static void AtForkParent(void) { #if RESET_ATFORK_MUTEX == 0 - pthread_mutex_unlock(¬ifierInitMutex); + TclpMasterUnlock(); #endif } @@ -1419,9 +1418,9 @@ AtForkChild(void) pthread_cond_destroy(¬ifierCV); } #if RESET_ATFORK_MUTEX == 0 - pthread_mutex_unlock(¬ifierInitMutex); + TclpMasterUnlock(); #else - pthread_mutex_init(¬ifierInitMutex, NULL); + TclpMasterReset(); pthread_mutex_init(¬ifierMutex, NULL); #endif pthread_cond_init(¬ifierCV, NULL); diff --git a/unix/tclUnixThrd.c b/unix/tclUnixThrd.c index 554a2dc..4130993 100644 --- a/unix/tclUnixThrd.c +++ b/unix/tclUnixThrd.c @@ -357,6 +357,31 @@ TclpMasterUnlock(void) /* *---------------------------------------------------------------------- * + * TclpMasterReset + * + * This procedure is used to reset a lock that serializes creation and + * finalization of synchronization objects. + * + * Results: + * None. + * + * Side effects: + * Reset the master mutex. + * + *---------------------------------------------------------------------- + */ + +void +TclpMasterReset(void) +{ +#ifdef TCL_THREADS + pthread_mutex_init(&masterLock, NULL); +#endif +} + +/* + *---------------------------------------------------------------------- + * * Tcl_GetAllocMutex * * This procedure returns a pointer to a statically initialized mutex for diff --git a/win/tclWinNotify.c b/win/tclWinNotify.c index 4543b02..985a769 100644 --- a/win/tclWinNotify.c +++ b/win/tclWinNotify.c @@ -51,7 +51,8 @@ static Tcl_ThreadDataKey dataKey; static int notifierCount = 0; static const TCHAR classname[] = TEXT("TclNotifier"); -TCL_DECLARE_MUTEX(notifierMutex) +static int initialized = 0; +static CRITICAL_SECTION notifierMutex; /* * Static routines defined in this file. @@ -85,12 +86,19 @@ Tcl_InitNotifier(void) ThreadSpecificData *tsdPtr = TCL_TSD_INIT(&dataKey); WNDCLASS class; + TclpMasterLock(); + if (!initialized) { + initialized = 1; + InitializeCriticalSection(¬ifierMutex); + } + TclpMasterUnlock(); + /* * Register Notifier window class if this is the first thread to use * this module. */ - Tcl_MutexLock(¬ifierMutex); + EnterCriticalSection(¬ifierMutex); if (notifierCount == 0) { class.style = 0; class.cbClsExtra = 0; @@ -108,7 +116,7 @@ Tcl_InitNotifier(void) } } notifierCount++; - Tcl_MutexUnlock(¬ifierMutex); + LeaveCriticalSection(¬ifierMutex); tsdPtr->pending = 0; tsdPtr->timerActive = 0; @@ -183,12 +191,14 @@ Tcl_FinalizeNotifier( * notifier window class. */ - Tcl_MutexLock(¬ifierMutex); - notifierCount--; - if (notifierCount == 0) { - UnregisterClass(classname, TclWinGetTclInstance()); + EnterCriticalSection(¬ifierMutex); + if (notifierCount) { + notifierCount--; + if (notifierCount == 0) { + UnregisterClass(classname, TclWinGetTclInstance()); + } } - Tcl_MutexUnlock(¬ifierMutex); + LeaveCriticalSection(¬ifierMutex); } } -- cgit v0.12 From c829651d80705243c9ea3665699f13ad901debbe Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Tue, 22 Mar 2016 09:05:10 +0000 Subject: =?UTF-8?q?Fix=20signed-unsigned-compare=20warning=20(reported=20b?= =?UTF-8?q?y=20Fran=C3=A7ois=20Vogel=20on=20Windows,=20but=20gcc=20can=20t?= =?UTF-8?q?rigger=20it=20as=20well)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- generic/tclCompExpr.c | 2 +- generic/tclEnsemble.c | 2 +- unix/configure | 2 +- unix/tcl.m4 | 2 +- win/configure | 2 +- win/tcl.m4 | 2 +- 6 files changed, 6 insertions(+), 6 deletions(-) diff --git a/generic/tclCompExpr.c b/generic/tclCompExpr.c index 50edbec..2ffcc68 100644 --- a/generic/tclCompExpr.c +++ b/generic/tclCompExpr.c @@ -666,7 +666,7 @@ ParseExpr( OpNode *newPtr = NULL; do { - if (size <= UINT_MAX/sizeof(OpNode)) { + if (size <= (int)(UINT_MAX/sizeof(OpNode))) { newPtr = attemptckrealloc(nodes, size * sizeof(OpNode)); } } while ((newPtr == NULL) diff --git a/generic/tclEnsemble.c b/generic/tclEnsemble.c index 986a553..6b6a156 100644 --- a/generic/tclEnsemble.c +++ b/generic/tclEnsemble.c @@ -3081,7 +3081,7 @@ TclAttemptCompileProc( int result, i; Tcl_Token *saveTokenPtr = parsePtr->tokenPtr; int savedStackDepth = envPtr->currStackDepth; - unsigned savedCodeNext = envPtr->codeNext - envPtr->codeStart; + int savedCodeNext = envPtr->codeNext - envPtr->codeStart; int savedAuxDataArrayNext = envPtr->auxDataArrayNext; int savedExceptArrayNext = envPtr->exceptArrayNext; #ifdef TCL_COMPILE_DEBUG diff --git a/unix/configure b/unix/configure index 3f9aa13..e999455 100755 --- a/unix/configure +++ b/unix/configure @@ -4985,7 +4985,7 @@ fi if test "$GCC" = yes; then : CFLAGS_OPTIMIZE=-O2 - CFLAGS_WARNING="-Wall" + CFLAGS_WARNING="-Wall -Wsign-compare -Wdeclaration-after-statement" else diff --git a/unix/tcl.m4 b/unix/tcl.m4 index 57d8ff0..f5aa84e 100644 --- a/unix/tcl.m4 +++ b/unix/tcl.m4 @@ -1096,7 +1096,7 @@ AC_DEFUN([SC_CONFIG_CFLAGS], [ CFLAGS_DEBUG=-g AS_IF([test "$GCC" = yes], [ CFLAGS_OPTIMIZE=-O2 - CFLAGS_WARNING="-Wall" + CFLAGS_WARNING="-Wall -Wsign-compare -Wdeclaration-after-statement" ], [ CFLAGS_OPTIMIZE=-O CFLAGS_WARNING="" diff --git a/win/configure b/win/configure index a1f57d1..73d6d9f 100755 --- a/win/configure +++ b/win/configure @@ -4165,7 +4165,7 @@ $as_echo "using shared flags" >&6; } CFLAGS_DEBUG=-g CFLAGS_OPTIMIZE="-O2 -fomit-frame-pointer" - CFLAGS_WARNING="-Wall -Wdeclaration-after-statement" + CFLAGS_WARNING="-Wall -Wsign-compare -Wdeclaration-after-statement" LDFLAGS_DEBUG= LDFLAGS_OPTIMIZE= diff --git a/win/tcl.m4 b/win/tcl.m4 index a75a936..7eff8e8 100644 --- a/win/tcl.m4 +++ b/win/tcl.m4 @@ -727,7 +727,7 @@ AC_DEFUN([SC_CONFIG_CFLAGS], [ CFLAGS_DEBUG=-g CFLAGS_OPTIMIZE="-O2 -fomit-frame-pointer" - CFLAGS_WARNING="-Wall -Wdeclaration-after-statement" + CFLAGS_WARNING="-Wall -Wsign-compare -Wdeclaration-after-statement" LDFLAGS_DEBUG= LDFLAGS_OPTIMIZE= -- cgit v0.12 From fd553241ab33882835ecf6065d61b775c9e2f1b1 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Tue, 22 Mar 2016 09:07:44 +0000 Subject: =?UTF-8?q?(cherry-pick):=20Fix=20signed-unsigned-compare=20warnin?= =?UTF-8?q?g=20(reported=20by=20Fran=C3=A7ois=20Vogel=20on=20Windows,=20bu?= =?UTF-8?q?t=20gcc=20can=20trigger=20it=20as=20well)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- generic/tclCompExpr.c | 2 +- generic/tclEnsemble.c | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/generic/tclCompExpr.c b/generic/tclCompExpr.c index 50edbec..2ffcc68 100644 --- a/generic/tclCompExpr.c +++ b/generic/tclCompExpr.c @@ -666,7 +666,7 @@ ParseExpr( OpNode *newPtr = NULL; do { - if (size <= UINT_MAX/sizeof(OpNode)) { + if (size <= (int)(UINT_MAX/sizeof(OpNode))) { newPtr = attemptckrealloc(nodes, size * sizeof(OpNode)); } } while ((newPtr == NULL) diff --git a/generic/tclEnsemble.c b/generic/tclEnsemble.c index 986a553..6b6a156 100644 --- a/generic/tclEnsemble.c +++ b/generic/tclEnsemble.c @@ -3081,7 +3081,7 @@ TclAttemptCompileProc( int result, i; Tcl_Token *saveTokenPtr = parsePtr->tokenPtr; int savedStackDepth = envPtr->currStackDepth; - unsigned savedCodeNext = envPtr->codeNext - envPtr->codeStart; + int savedCodeNext = envPtr->codeNext - envPtr->codeStart; int savedAuxDataArrayNext = envPtr->auxDataArrayNext; int savedExceptArrayNext = envPtr->exceptArrayNext; #ifdef TCL_COMPILE_DEBUG -- cgit v0.12 From f16557b5a03c04972b0b35eef114e92d2195e529 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Tue, 22 Mar 2016 13:15:21 +0000 Subject: Alternative solution for [f1253530cd] --- library/word.tcl | 16 ++++++++++++---- 1 file changed, 12 insertions(+), 4 deletions(-) diff --git a/library/word.tcl b/library/word.tcl index b8f34a5..3e4bc3a 100644 --- a/library/word.tcl +++ b/library/word.tcl @@ -15,12 +15,20 @@ if {$::tcl_platform(platform) eq "windows"} { # Windows style - any but a unicode space char - set ::tcl_wordchars {\S} - set ::tcl_nonwordchars {\s} + if {![info exists ::tcl_wordchars]} { + set ::tcl_wordchars {\S} + } + if {![info exists ::tcl_nonwordchars]} { + set ::tcl_nonwordchars {\s} + } } else { # Motif style - any unicode word char (number, letter, or underscore) - set ::tcl_wordchars {\w} - set ::tcl_nonwordchars {\W} + if {![info exists ::tcl_wordchars]} { + set ::tcl_wordchars {\w} + } + if {![info exists ::tcl_nonwordchars]} { + set ::tcl_nonwordchars {\W} + } } # Arrange for caches of the real matcher REs to be kept, which enables the REs -- cgit v0.12 From 08da639e50f42d9cb3447401a59db6669748f9a9 Mon Sep 17 00:00:00 2001 From: dgp Date: Tue, 22 Mar 2016 16:38:47 +0000 Subject: Don't compare signed and unsigned. Get types right. --- generic/tclCompile.h | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/generic/tclCompile.h b/generic/tclCompile.h index b5bfab1..5b747fe 100644 --- a/generic/tclCompile.h +++ b/generic/tclCompile.h @@ -135,7 +135,7 @@ typedef struct ExceptionAux { int numBreakTargets; /* The number of [break]s that want to be * targeted to the place where this loop * exception will be bound to. */ - int *breakTargets; /* The offsets of the INST_JUMP4 instructions + unsigned int *breakTargets; /* The offsets of the INST_JUMP4 instructions * issued by the [break]s that we must * update. Note that resizing a jump (via * TclFixupForwardJump) can cause the contents @@ -145,7 +145,7 @@ typedef struct ExceptionAux { int numContinueTargets; /* The number of [continue]s that want to be * targeted to the place where this loop * exception will be bound to. */ - int *continueTargets; /* The offsets of the INST_JUMP4 instructions + unsigned int *continueTargets; /* The offsets of the INST_JUMP4 instructions * issued by the [continue]s that we must * update. Note that resizing a jump (via * TclFixupForwardJump) can cause the contents -- cgit v0.12 From f0d0319ba6c340e680ec612d980061897bf07b4f Mon Sep 17 00:00:00 2001 From: dgp Date: Tue, 22 Mar 2016 17:08:50 +0000 Subject: More places where unsigned quantities should be decared such. --- generic/tclCompCmdsSZ.c | 2 +- generic/tclCompile.h | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/generic/tclCompCmdsSZ.c b/generic/tclCompCmdsSZ.c index ef9340e..101edbd 100644 --- a/generic/tclCompCmdsSZ.c +++ b/generic/tclCompCmdsSZ.c @@ -2028,7 +2028,7 @@ IssueSwitchChainedTests( int foundDefault; /* Flag to indicate whether a "default" clause * is present. */ JumpFixup *fixupArray; /* Array of forward-jump fixup records. */ - int *fixupTargetArray; /* Array of places for fixups to point at. */ + unsigned int *fixupTargetArray; /* Array of places for fixups to point at. */ int fixupCount; /* Number of places to fix up. */ int contFixIndex; /* Where the first of the jumps due to a group * of continuation bodies starts, or -1 if diff --git a/generic/tclCompile.h b/generic/tclCompile.h index 5b747fe..d5bc86b 100644 --- a/generic/tclCompile.h +++ b/generic/tclCompile.h @@ -928,7 +928,7 @@ typedef enum { typedef struct JumpFixup { TclJumpType jumpType; /* Indicates the kind of jump. */ - int codeOffset; /* Offset of the first byte of the one-byte + unsigned int codeOffset; /* Offset of the first byte of the one-byte * forward jump's code. */ int cmdIndex; /* Index of the first command after the one * for which the jump was emitted. Used to -- cgit v0.12 From d281007655b47cbc0a3449aaa0c605bce151fec4 Mon Sep 17 00:00:00 2001 From: dkf Date: Tue, 22 Mar 2016 20:30:08 +0000 Subject: Factor out string internal rep definition so fix for [1af8de570511] is less awful. --- generic/tclExecute.c | 8 ++--- generic/tclStringObj.c | 55 +--------------------------- generic/tclStringRep.h | 97 ++++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 102 insertions(+), 58 deletions(-) create mode 100644 generic/tclStringRep.h diff --git a/generic/tclExecute.c b/generic/tclExecute.c index c43cc40..d4077f5 100644 --- a/generic/tclExecute.c +++ b/generic/tclExecute.c @@ -19,6 +19,7 @@ #include "tclCompile.h" #include "tclOOInt.h" #include "tommath.h" +#include "tclStringRep.h" #include #include @@ -5739,14 +5740,13 @@ TEBCresume( /* * Flush the info in the string internal rep that refers to the - * about-to-be-invalidated UTF-8 rep. This sets the 'allocated' - * field of the String structure to 0 to indicate that a new - * buffer needs to be allocated. This assumes that the value is + * about-to-be-invalidated UTF-8 rep. This indicates that a new + * buffer needs to be allocated, and assumes that the value is * already of tclStringTypePtr type, which should be true provided * we call it after Tcl_GetUnicodeFromObj. */ #define MarkStringInternalRepForFlush(objPtr) \ - (((int *) ((objPtr)->internalRep.twoPtrValue.ptr1))[1] = 0) + (GET_STRING(objPtr)->allocated = 0) if (Tcl_IsShared(valuePtr)) { objResultPtr = Tcl_DuplicateObj(valuePtr); diff --git a/generic/tclStringObj.c b/generic/tclStringObj.c index 8d70d20..11a57e9 100644 --- a/generic/tclStringObj.c +++ b/generic/tclStringObj.c @@ -36,6 +36,7 @@ #include "tclInt.h" #include "tommath.h" +#include "tclStringRep.h" /* * Set COMPAT to 1 to restore the shimmering patterns to those of Tcl 8.5. @@ -89,60 +90,6 @@ const Tcl_ObjType tclStringType = { UpdateStringOfString, /* updateStringProc */ SetStringFromAny /* setFromAnyProc */ }; - -/* - * The following structure is the internal rep for a String object. It keeps - * track of how much memory has been used and how much has been allocated for - * the Unicode and UTF string to enable growing and shrinking of the UTF and - * Unicode reps of the String object with fewer mallocs. To optimize string - * length and indexing operations, this structure also stores the number of - * characters (same of UTF and Unicode!) once that value has been computed. - * - * Under normal configurations, what Tcl calls "Unicode" is actually UTF-16 - * restricted to the Basic Multilingual Plane (i.e. U+00000 to U+0FFFF). This - * can be officially modified by altering the definition of Tcl_UniChar in - * tcl.h, but do not do that unless you are sure what you're doing! - */ - -typedef struct String { - int numChars; /* The number of chars in the string. -1 means - * this value has not been calculated. >= 0 - * means that there is a valid Unicode rep, or - * that the number of UTF bytes == the number - * of chars. */ - int allocated; /* The amount of space actually allocated for - * the UTF string (minus 1 byte for the - * termination char). */ - int maxChars; /* Max number of chars that can fit in the - * space allocated for the unicode array. */ - int hasUnicode; /* Boolean determining whether the string has - * a Unicode representation. */ - Tcl_UniChar unicode[1]; /* The array of Unicode chars. The actual size - * of this field depends on the 'maxChars' - * field above. */ -} String; - -#define STRING_MAXCHARS \ - (int)(((size_t)UINT_MAX - sizeof(String))/sizeof(Tcl_UniChar)) -#define STRING_SIZE(numChars) \ - (sizeof(String) + ((numChars) * sizeof(Tcl_UniChar))) -#define stringCheckLimits(numChars) \ - if ((numChars) < 0 || (numChars) > STRING_MAXCHARS) { \ - Tcl_Panic("max length for a Tcl unicode value (%d chars) exceeded", \ - STRING_MAXCHARS); \ - } -#define stringAttemptAlloc(numChars) \ - (String *) attemptckalloc((unsigned) STRING_SIZE(numChars) ) -#define stringAlloc(numChars) \ - (String *) ckalloc((unsigned) STRING_SIZE(numChars) ) -#define stringRealloc(ptr, numChars) \ - (String *) ckrealloc((ptr), (unsigned) STRING_SIZE(numChars) ) -#define stringAttemptRealloc(ptr, numChars) \ - (String *) attemptckrealloc((ptr), (unsigned) STRING_SIZE(numChars) ) -#define GET_STRING(objPtr) \ - ((String *) (objPtr)->internalRep.twoPtrValue.ptr1) -#define SET_STRING(objPtr, stringPtr) \ - ((objPtr)->internalRep.twoPtrValue.ptr1 = (void *) (stringPtr)) /* * TCL STRING GROWTH ALGORITHM diff --git a/generic/tclStringRep.h b/generic/tclStringRep.h new file mode 100644 index 0000000..227e6bc --- /dev/null +++ b/generic/tclStringRep.h @@ -0,0 +1,97 @@ +/* + * tclStringRep.h -- + * + * This file contains the definition of the Unicode string internal + * representation and macros to access it. + * + * A Unicode string is an internationalized string. Conceptually, a + * Unicode string is an array of 16-bit quantities organized as a + * sequence of properly formed UTF-8 characters. There is a one-to-one + * map between Unicode and UTF characters. Because Unicode characters + * have a fixed width, operations such as indexing operate on Unicode + * data. The String object is optimized for the case where each UTF char + * in a string is only one byte. In this case, we store the value of + * numChars, but we don't store the Unicode data (unless Tcl_GetUnicode + * is explicitly called). + * + * The String object type stores one or both formats. The default + * behavior is to store UTF. Once Unicode is calculated by a function, it + * is stored in the internal rep for future access (without an additional + * O(n) cost). + * + * To allow many appends to be done to an object without constantly + * reallocating the space for the string or Unicode representation, we + * allocate double the space for the string or Unicode and use the + * internal representation to keep track of how much space is used vs. + * allocated. + * + * Copyright (c) 1995-1997 Sun Microsystems, Inc. + * Copyright (c) 1999 by Scriptics Corporation. + * + * See the file "license.terms" for information on usage and redistribution of + * this file, and for a DISCLAIMER OF ALL WARRANTIES. + */ + +/* + * The following structure is the internal rep for a String object. It keeps + * track of how much memory has been used and how much has been allocated for + * the Unicode and UTF string to enable growing and shrinking of the UTF and + * Unicode reps of the String object with fewer mallocs. To optimize string + * length and indexing operations, this structure also stores the number of + * characters (same of UTF and Unicode!) once that value has been computed. + * + * Under normal configurations, what Tcl calls "Unicode" is actually UTF-16 + * restricted to the Basic Multilingual Plane (i.e. U+00000 to U+0FFFF). This + * can be officially modified by altering the definition of Tcl_UniChar in + * tcl.h, but do not do that unless you are sure what you're doing! + */ + +typedef struct String { + int numChars; /* The number of chars in the string. -1 means + * this value has not been calculated. >= 0 + * means that there is a valid Unicode rep, or + * that the number of UTF bytes == the number + * of chars. */ + int allocated; /* The amount of space actually allocated for + * the UTF string (minus 1 byte for the + * termination char). */ + int maxChars; /* Max number of chars that can fit in the + * space allocated for the unicode array. */ + int hasUnicode; /* Boolean determining whether the string has + * a Unicode representation. */ + Tcl_UniChar unicode[1]; /* The array of Unicode chars. The actual size + * of this field depends on the 'maxChars' + * field above. */ +} String; + +#define STRING_MAXCHARS \ + (int)(((size_t)UINT_MAX - sizeof(String))/sizeof(Tcl_UniChar)) +#define STRING_SIZE(numChars) \ + (sizeof(String) + ((numChars) * sizeof(Tcl_UniChar))) +#define stringCheckLimits(numChars) \ + do { \ + if ((numChars) < 0 || (numChars) > STRING_MAXCHARS) { \ + Tcl_Panic("max length for a Tcl unicode value (%d chars) exceeded", \ + STRING_MAXCHARS); \ + } \ + } while (0) +#define stringAttemptAlloc(numChars) \ + (String *) attemptckalloc((unsigned) STRING_SIZE(numChars)) +#define stringAlloc(numChars) \ + (String *) ckalloc((unsigned) STRING_SIZE(numChars)) +#define stringRealloc(ptr, numChars) \ + (String *) ckrealloc((ptr), (unsigned) STRING_SIZE(numChars)) +#define stringAttemptRealloc(ptr, numChars) \ + (String *) attemptckrealloc((ptr), (unsigned) STRING_SIZE(numChars)) +#define GET_STRING(objPtr) \ + ((String *) (objPtr)->internalRep.twoPtrValue.ptr1) +#define SET_STRING(objPtr, stringPtr) \ + ((objPtr)->internalRep.twoPtrValue.ptr1 = (void *) (stringPtr)) + +/* + * Local Variables: + * mode: c + * c-basic-offset: 4 + * fill-column: 78 + * End: + */ -- cgit v0.12 From 9ac95b002e2b6c6b66c04d1964c22ccc25d4e883 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 23 Mar 2016 08:48:54 +0000 Subject: =?UTF-8?q?Fix=20[7d0db7c388f52de8]:=20Occasional=20build=20failur?= =?UTF-8?q?es=20with=20parallel=20make.=20Thanks=20to=20Jaroslav=20=C5=A0k?= =?UTF-8?q?arvada=20for=20bug=20report=20and=20fix.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- unix/Makefile.in | 128 +++++++++++++++++++++++++++---------------------------- 1 file changed, 64 insertions(+), 64 deletions(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index 570bce0..0cb9c44 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -1351,196 +1351,196 @@ tclThreadTest.o: $(GENERIC_DIR)/tclThreadTest.c tclTomMathInterface.o: $(GENERIC_DIR)/tclTomMathInterface.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(GENERIC_DIR)/tclTomMathInterface.c -bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) +bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bncore.c -bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) +bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_reverse.c -bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) +bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c -bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) +bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c -bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) +bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add.c -bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) +bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add_d.c -bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) +bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_and.c -bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) +bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clamp.c -bn_mp_clear.o: $(TOMMATH_DIR)/bn_mp_clear.c $(MATHHDRS) +bn_mp_clear.o: $(TOMMATH_DIR)/bn_mp_clear.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clear.c -bn_mp_clear_multi.o: $(TOMMATH_DIR)/bn_mp_clear_multi.c $(MATHHDRS) +bn_mp_clear_multi.o: $(TOMMATH_DIR)/bn_mp_clear_multi.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clear_multi.c -bn_mp_cmp.o: $(TOMMATH_DIR)/bn_mp_cmp.c $(MATHHDRS) +bn_mp_cmp.o: $(TOMMATH_DIR)/bn_mp_cmp.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp.c -bn_mp_cmp_d.o: $(TOMMATH_DIR)/bn_mp_cmp_d.c $(MATHHDRS) +bn_mp_cmp_d.o: $(TOMMATH_DIR)/bn_mp_cmp_d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp_d.c -bn_mp_cmp_mag.o: $(TOMMATH_DIR)/bn_mp_cmp_mag.c $(MATHHDRS) +bn_mp_cmp_mag.o: $(TOMMATH_DIR)/bn_mp_cmp_mag.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp_mag.c -bn_mp_cnt_lsb.o: $(TOMMATH_DIR)/bn_mp_cnt_lsb.c $(MATHHDRS) +bn_mp_cnt_lsb.o: $(TOMMATH_DIR)/bn_mp_cnt_lsb.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cnt_lsb.c -bn_mp_copy.o: $(TOMMATH_DIR)/bn_mp_copy.c $(MATHHDRS) +bn_mp_copy.o: $(TOMMATH_DIR)/bn_mp_copy.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_copy.c -bn_mp_count_bits.o: $(TOMMATH_DIR)/bn_mp_count_bits.c $(MATHHDRS) +bn_mp_count_bits.o: $(TOMMATH_DIR)/bn_mp_count_bits.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_count_bits.c -bn_mp_div.o: $(TOMMATH_DIR)/bn_mp_div.c $(MATHHDRS) +bn_mp_div.o: $(TOMMATH_DIR)/bn_mp_div.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div.c -bn_mp_div_d.o: $(TOMMATH_DIR)/bn_mp_div_d.c $(MATHHDRS) +bn_mp_div_d.o: $(TOMMATH_DIR)/bn_mp_div_d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_d.c -bn_mp_div_2.o: $(TOMMATH_DIR)/bn_mp_div_2.c $(MATHHDRS) +bn_mp_div_2.o: $(TOMMATH_DIR)/bn_mp_div_2.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_2.c -bn_mp_div_2d.o: $(TOMMATH_DIR)/bn_mp_div_2d.c $(MATHHDRS) +bn_mp_div_2d.o: $(TOMMATH_DIR)/bn_mp_div_2d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_2d.c -bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS) +bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_3.c -bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS) +bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_exch.c -bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS) +bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d.c -bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS) +bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_grow.c -bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS) +bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init.c -bn_mp_init_copy.o: $(TOMMATH_DIR)/bn_mp_init_copy.c $(MATHHDRS) +bn_mp_init_copy.o: $(TOMMATH_DIR)/bn_mp_init_copy.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_copy.c -bn_mp_init_multi.o: $(TOMMATH_DIR)/bn_mp_init_multi.c 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-bn_mp_karatsuba_sqr.o: $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c $(MATHHDRS) +bn_mp_karatsuba_sqr.o: $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c -bn_mp_lshd.o: $(TOMMATH_DIR)/bn_mp_lshd.c $(MATHHDRS) +bn_mp_lshd.o: $(TOMMATH_DIR)/bn_mp_lshd.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_lshd.c -bn_mp_mod.o: $(TOMMATH_DIR)/bn_mp_mod.c $(MATHHDRS) +bn_mp_mod.o: $(TOMMATH_DIR)/bn_mp_mod.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mod.c -bn_mp_mod_2d.o: $(TOMMATH_DIR)/bn_mp_mod_2d.c $(MATHHDRS) +bn_mp_mod_2d.o: $(TOMMATH_DIR)/bn_mp_mod_2d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mod_2d.c -bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) +bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul.c -bn_mp_mul_2.o: $(TOMMATH_DIR)/bn_mp_mul_2.c $(MATHHDRS) +bn_mp_mul_2.o: 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$(MATHHDRS) +bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sqr.c -bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) +bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sub.c tclUnixChan.o: $(UNIX_DIR)/tclUnixChan.c $(IOHDR) -- cgit v0.12 From 6217f91705eb0d6550f5376c362942971284a102 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 23 Mar 2016 08:59:25 +0000 Subject: =?UTF-8?q?(cherry-pick):=20Fix=20[7d0db7c388f52de8]:=20Occasional?= =?UTF-8?q?=20build=20failures=20with=20parallel=20make.=20Thanks=20to=20J?= =?UTF-8?q?aroslav=20=C5=A0karvada=20for=20bug=20report=20and=20fix.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- unix/Makefile.in | 128 +++++++++++++++++++++++++++---------------------------- 1 file changed, 64 insertions(+), 64 deletions(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index bc73118..cc438a4 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -1351,196 +1351,196 @@ tclThreadTest.o: $(GENERIC_DIR)/tclThreadTest.c tclTomMathInterface.o: $(GENERIC_DIR)/tclTomMathInterface.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(GENERIC_DIR)/tclTomMathInterface.c -bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) +bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bncore.c -bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) +bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_reverse.c -bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) +bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c -bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) +bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) 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+bn_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_mul_digs.c -bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) +bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sqr.c -bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) +bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sub.c tclUnixChan.o: $(UNIX_DIR)/tclUnixChan.c $(IOHDR) -- cgit v0.12 From a65c0e7103d10d4dc444c2cd443ecc4ecff5f6cb Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 23 Mar 2016 09:00:54 +0000 Subject: =?UTF-8?q?(cherry-pick):=20Fix=20[7d0db7c388f52de8]:=20Occasional?= =?UTF-8?q?=20build=20failures=20with=20parallel=20make.=20Thanks=20to=20J?= =?UTF-8?q?aroslav=20=C5=A0karvada=20for=20bug=20report=20and=20fix.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- unix/Makefile.in | 128 +++++++++++++++++++++++++++---------------------------- 1 file changed, 64 insertions(+), 64 deletions(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index 84d0391..6e2970f 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -1238,196 +1238,196 @@ tclThreadTest.o: $(GENERIC_DIR)/tclThreadTest.c tclTomMathInterface.o: $(GENERIC_DIR)/tclTomMathInterface.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(GENERIC_DIR)/tclTomMathInterface.c -bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) +bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bncore.c -bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) +bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_reverse.c -bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) +bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c -bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) +bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c -bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) +bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add.c -bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) +bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add_d.c -bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) +bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_and.c -bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) +bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clamp.c -bn_mp_clear.o: 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+bn_mp_mod_2d.o: $(TOMMATH_DIR)/bn_mp_mod_2d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mod_2d.c -bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) +bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul.c -bn_mp_mul_2.o: $(TOMMATH_DIR)/bn_mp_mul_2.c $(MATHHDRS) +bn_mp_mul_2.o: $(TOMMATH_DIR)/bn_mp_mul_2.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_2.c -bn_mp_mul_2d.o: $(TOMMATH_DIR)/bn_mp_mul_2d.c $(MATHHDRS) +bn_mp_mul_2d.o: $(TOMMATH_DIR)/bn_mp_mul_2d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_2d.c -bn_mp_mul_d.o: $(TOMMATH_DIR)/bn_mp_mul_d.c $(MATHHDRS) +bn_mp_mul_d.o: $(TOMMATH_DIR)/bn_mp_mul_d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_d.c -bn_mp_neg.o: $(TOMMATH_DIR)/bn_mp_neg.c $(MATHHDRS) +bn_mp_neg.o: $(TOMMATH_DIR)/bn_mp_neg.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_neg.c -bn_mp_or.o: $(TOMMATH_DIR)/bn_mp_or.c $(MATHHDRS) +bn_mp_or.o: $(TOMMATH_DIR)/bn_mp_or.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_or.c -bn_mp_radix_size.o: $(TOMMATH_DIR)/bn_mp_radix_size.c $(MATHHDRS) +bn_mp_radix_size.o: $(TOMMATH_DIR)/bn_mp_radix_size.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_radix_size.c -bn_mp_radix_smap.o: $(TOMMATH_DIR)/bn_mp_radix_smap.c $(MATHHDRS) +bn_mp_radix_smap.o: $(TOMMATH_DIR)/bn_mp_radix_smap.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_radix_smap.c -bn_mp_read_radix.o: $(TOMMATH_DIR)/bn_mp_read_radix.c $(MATHHDRS) +bn_mp_read_radix.o: $(TOMMATH_DIR)/bn_mp_read_radix.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_read_radix.c -bn_mp_rshd.o: $(TOMMATH_DIR)/bn_mp_rshd.c $(MATHHDRS) +bn_mp_rshd.o: $(TOMMATH_DIR)/bn_mp_rshd.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_rshd.c -bn_mp_set.o: $(TOMMATH_DIR)/bn_mp_set.c $(MATHHDRS) +bn_mp_set.o: $(TOMMATH_DIR)/bn_mp_set.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_set.c -bn_mp_set_int.o: $(TOMMATH_DIR)/bn_mp_set_int.c $(MATHHDRS) +bn_mp_set_int.o: $(TOMMATH_DIR)/bn_mp_set_int.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_set_int.c -bn_mp_shrink.o: $(TOMMATH_DIR)/bn_mp_shrink.c $(MATHHDRS) +bn_mp_shrink.o: $(TOMMATH_DIR)/bn_mp_shrink.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_shrink.c -bn_mp_sqr.o: $(TOMMATH_DIR)/bn_mp_sqr.c $(MATHHDRS) +bn_mp_sqr.o: $(TOMMATH_DIR)/bn_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sqr.c -bn_mp_sqrt.o: $(TOMMATH_DIR)/bn_mp_sqrt.c $(MATHHDRS) +bn_mp_sqrt.o: $(TOMMATH_DIR)/bn_mp_sqrt.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sqrt.c -bn_mp_sub.o: $(TOMMATH_DIR)/bn_mp_sub.c $(MATHHDRS) +bn_mp_sub.o: $(TOMMATH_DIR)/bn_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sub.c -bn_mp_sub_d.o: $(TOMMATH_DIR)/bn_mp_sub_d.c $(MATHHDRS) +bn_mp_sub_d.o: $(TOMMATH_DIR)/bn_mp_sub_d.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sub_d.c -bn_mp_to_unsigned_bin.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c $(MATHHDRS) +bn_mp_to_unsigned_bin.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c -bn_mp_to_unsigned_bin_n.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c $(MATHHDRS) +bn_mp_to_unsigned_bin_n.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c -bn_mp_toom_mul.o: $(TOMMATH_DIR)/bn_mp_toom_mul.c $(MATHHDRS) +bn_mp_toom_mul.o: $(TOMMATH_DIR)/bn_mp_toom_mul.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toom_mul.c -bn_mp_toom_sqr.o: $(TOMMATH_DIR)/bn_mp_toom_sqr.c $(MATHHDRS) +bn_mp_toom_sqr.o: $(TOMMATH_DIR)/bn_mp_toom_sqr.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toom_sqr.c -bn_mp_toradix_n.o: $(TOMMATH_DIR)/bn_mp_toradix_n.c $(MATHHDRS) +bn_mp_toradix_n.o: $(TOMMATH_DIR)/bn_mp_toradix_n.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toradix_n.c -bn_mp_unsigned_bin_size.o: $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c $(MATHHDRS) +bn_mp_unsigned_bin_size.o: $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c -bn_mp_xor.o: $(TOMMATH_DIR)/bn_mp_xor.c $(MATHHDRS) +bn_mp_xor.o: $(TOMMATH_DIR)/bn_mp_xor.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_xor.c -bn_mp_zero.o: $(TOMMATH_DIR)/bn_mp_zero.c $(MATHHDRS) +bn_mp_zero.o: $(TOMMATH_DIR)/bn_mp_zero.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_zero.c -bn_s_mp_add.o: $(TOMMATH_DIR)/bn_s_mp_add.c $(MATHHDRS) +bn_s_mp_add.o: $(TOMMATH_DIR)/bn_s_mp_add.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_add.c -bn_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_s_mp_mul_digs.c $(MATHHDRS) +bn_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_mul_digs.c -bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) +bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sqr.c -bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) +bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sub.c tclUnixChan.o: $(UNIX_DIR)/tclUnixChan.c $(IOHDR) -- cgit v0.12 From 62af51b8f98d93a42b8242eafa267316ad222973 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 23 Mar 2016 09:55:06 +0000 Subject: Fix [f1253530cdd83e66]: Different Windows / *nix default bindings for text widget. Still - most likely - not the complete story for trunk/tcl9, but whatever we do this is a better starting-point than how it was. --- library/word.tcl | 24 +++++++----------------- 1 file changed, 7 insertions(+), 17 deletions(-) diff --git a/library/word.tcl b/library/word.tcl index 3e4bc3a..0246530 100644 --- a/library/word.tcl +++ b/library/word.tcl @@ -11,24 +11,14 @@ # of this file, and for a DISCLAIMER OF ALL WARRANTIES. # The following variables are used to determine which characters are -# interpreted as white space. +# interpreted as word characters. See bug [f1253530cdd8]. Will +# probably be removed in Tcl 9. -if {$::tcl_platform(platform) eq "windows"} { - # Windows style - any but a unicode space char - if {![info exists ::tcl_wordchars]} { - set ::tcl_wordchars {\S} - } - if {![info exists ::tcl_nonwordchars]} { - set ::tcl_nonwordchars {\s} - } -} else { - # Motif style - any unicode word char (number, letter, or underscore) - if {![info exists ::tcl_wordchars]} { - set ::tcl_wordchars {\w} - } - if {![info exists ::tcl_nonwordchars]} { - set ::tcl_nonwordchars {\W} - } +if {![info exists ::tcl_wordchars]} { + set ::tcl_wordchars {\w} +} +if {![info exists ::tcl_nonwordchars]} { + set ::tcl_nonwordchars {\W} } # Arrange for caches of the real matcher REs to be kept, which enables the REs -- cgit v0.12 From ff955a7522d3517a71ee5427c360075be87f056d Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 23 Mar 2016 09:59:08 +0000 Subject: (cherry-pick): Fix [f1253530cdd83e66]: Different Windows / *nix default bindings for text widget See: [https://groups.google.com/forum/#!topic/comp.lang.tcl/ZZ_WwfQdmoE]. People like Eric Brunel, who want the most logical behavior for any Tcl release can simply do: set ::tcl_wordchars {\w} set ::tcl_nonwordchars {\W} --- library/word.tcl | 16 ++++++++++++---- 1 file changed, 12 insertions(+), 4 deletions(-) diff --git a/library/word.tcl b/library/word.tcl index b8f34a5..3e4bc3a 100644 --- a/library/word.tcl +++ b/library/word.tcl @@ -15,12 +15,20 @@ if {$::tcl_platform(platform) eq "windows"} { # Windows style - any but a unicode space char - set ::tcl_wordchars {\S} - set ::tcl_nonwordchars {\s} + if {![info exists ::tcl_wordchars]} { + set ::tcl_wordchars {\S} + } + if {![info exists ::tcl_nonwordchars]} { + set ::tcl_nonwordchars {\s} + } } else { # Motif style - any unicode word char (number, letter, or underscore) - set ::tcl_wordchars {\w} - set ::tcl_nonwordchars {\W} + if {![info exists ::tcl_wordchars]} { + set ::tcl_wordchars {\w} + } + if {![info exists ::tcl_nonwordchars]} { + set ::tcl_nonwordchars {\W} + } } # Arrange for caches of the real matcher REs to be kept, which enables the REs -- cgit v0.12 From 4936a3f9d6993a339e97da0e54cdf7694d58769c Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 23 Mar 2016 10:00:40 +0000 Subject: (cherry-pick): Fix [f1253530cdd83e66]: Different Windows / *nix default bindings for text widget See: [https://groups.google.com/forum/#!topic/comp.lang.tcl/ZZ_WwfQdmoE]. People like Eric Brunel, who want the most logical behavior for any Tcl release can simply do: set ::tcl_wordchars {\w} set ::tcl_nonwordchars {\W} --- library/word.tcl | 16 ++++++++++++---- 1 file changed, 12 insertions(+), 4 deletions(-) diff --git a/library/word.tcl b/library/word.tcl index b8f34a5..3e4bc3a 100644 --- a/library/word.tcl +++ b/library/word.tcl @@ -15,12 +15,20 @@ if {$::tcl_platform(platform) eq "windows"} { # Windows style - any but a unicode space char - set ::tcl_wordchars {\S} - set ::tcl_nonwordchars {\s} + if {![info exists ::tcl_wordchars]} { + set ::tcl_wordchars {\S} + } + if {![info exists ::tcl_nonwordchars]} { + set ::tcl_nonwordchars {\s} + } } else { # Motif style - any unicode word char (number, letter, or underscore) - set ::tcl_wordchars {\w} - set ::tcl_nonwordchars {\W} + if {![info exists ::tcl_wordchars]} { + set ::tcl_wordchars {\w} + } + if {![info exists ::tcl_nonwordchars]} { + set ::tcl_nonwordchars {\W} + } } # Arrange for caches of the real matcher REs to be kept, which enables the REs -- cgit v0.12 From 143c2731049b73d59148a7f19d47b78b8c24b8b2 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 23 Mar 2016 13:03:54 +0000 Subject: Fix DTRACE_HDR value if tracing is diabled. Follow-up to [7d0db7c388] --- unix/Makefile.in | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index 0cb9c44..897dd1f 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -584,7 +584,7 @@ MAC_OSX_SRCS = \ CYGWIN_SRCS = \ $(TOP_DIR)/win/tclWinError.c -DTRACE_HDR = tclDTrace.h +DTRACE_HDR = @DTRACE_HDR@ DTRACE_SRC = $(GENERIC_DIR)/tclDTrace.d -- cgit v0.12 From df30a3a3ac6e6eb01c81e221d015d20f0a9dffa1 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 23 Mar 2016 13:04:52 +0000 Subject: (cherry-pick): Fix DTRACE_HDR value if tracing is diabled. Follow-up to [7d0db7c388] --- unix/Makefile.in | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index cc438a4..7604bc7 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -584,7 +584,7 @@ MAC_OSX_SRCS = \ CYGWIN_SRCS = \ $(TOP_DIR)/win/tclWinError.c -DTRACE_HDR = tclDTrace.h +DTRACE_HDR = @DTRACE_HDR@ DTRACE_SRC = $(GENERIC_DIR)/tclDTrace.d -- cgit v0.12 From 0c78aaa138bc51f0f2a003a187161b60ec32a2ab Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Wed, 23 Mar 2016 13:06:09 +0000 Subject: (cherry-pick): Fix DTRACE_HDR value if tracing is diabled. Follow-up to [7d0db7c388] --- unix/Makefile.in | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index 6e2970f..722f85d 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -529,7 +529,7 @@ MAC_OSX_SRCS = \ $(MAC_OSX_DIR)/tclMacOSXFCmd.c \ $(MAC_OSX_DIR)/tclMacOSXNotify.c -DTRACE_HDR = tclDTrace.h +DTRACE_HDR = @DTRACE_HDR@ DTRACE_SRC = $(GENERIC_DIR)/tclDTrace.d -- cgit v0.12 From 77cc7b40aa68fe4c7ef545f76495ba18c4ace487 Mon Sep 17 00:00:00 2001 From: dgp Date: Thu, 24 Mar 2016 21:21:42 +0000 Subject: No longer a need to keep around 'revert to 8.5' code. --- generic/tclStringObj.c | 83 ++------------------------------------------------ 1 file changed, 3 insertions(+), 80 deletions(-) diff --git a/generic/tclStringObj.c b/generic/tclStringObj.c index 11a57e9..e718749 100644 --- a/generic/tclStringObj.c +++ b/generic/tclStringObj.c @@ -39,15 +39,6 @@ #include "tclStringRep.h" /* - * Set COMPAT to 1 to restore the shimmering patterns to those of Tcl 8.5. - * This is an escape hatch in case the changes have some unexpected unwelcome - * impact on performance. If things go well, this mechanism can go away when - * post-8.6 development begins. - */ - -#define COMPAT 0 - -/* * Prototypes for functions defined later in this file: */ @@ -445,18 +436,6 @@ Tcl_GetCharLength( if (numChars == -1) { TclNumUtfChars(numChars, objPtr->bytes, objPtr->length); stringPtr->numChars = numChars; - -#if COMPAT - if (numChars < objPtr->length) { - /* - * Since we've just computed the number of chars, and not all UTF - * chars are 1-byte long, go ahead and populate the unicode - * string. - */ - - FillUnicodeRep(objPtr); - } -#endif } return numChars; } @@ -1173,11 +1152,7 @@ Tcl_AppendUnicodeToObj( * objPtr's string rep. */ - if (stringPtr->hasUnicode -#if COMPAT - && stringPtr->numChars > 0 -#endif - ) { + if (stringPtr->hasUnicode) { AppendUnicodeToUnicodeRep(objPtr, unicode, length); } else { AppendUnicodeToUtfRep(objPtr, unicode, length); @@ -1281,11 +1256,7 @@ Tcl_AppendObjToObj( * appendObjPtr and append it. */ - if (stringPtr->hasUnicode -#if COMPAT - && stringPtr->numChars > 0 -#endif - ) { + if (stringPtr->hasUnicode) { /* * If appendObjPtr is not of the "String" type, don't convert it. */ @@ -1318,11 +1289,7 @@ Tcl_AppendObjToObj( AppendUtfToUtfRep(objPtr, bytes, length); - if (numChars >= 0 && appendNumChars >= 0 -#if COMPAT - && appendNumChars == length -#endif - ) { + if (numChars >= 0 && appendNumChars >= 0) { stringPtr->numChars = numChars + appendNumChars; } } @@ -1446,14 +1413,6 @@ AppendUnicodeToUtfRep( if (stringPtr->numChars != -1) { stringPtr->numChars += numChars; } - -#if COMPAT - /* - * Invalidate the unicode rep. - */ - - stringPtr->hasUnicode = 0; -#endif } /* @@ -2871,7 +2830,6 @@ DupStringInternalRep( String *srcStringPtr = GET_STRING(srcPtr); String *copyStringPtr = NULL; -#if COMPAT==0 if (srcStringPtr->numChars == -1) { /* * The String struct in the source value holds zero useful data. Don't @@ -2914,41 +2872,6 @@ DupStringInternalRep( */ copyStringPtr->allocated = copyPtr->bytes ? copyPtr->length : 0; -#else /* COMPAT!=0 */ - /* - * If the src obj is a string of 1-byte Utf chars, then copy the string - * rep of the source object and create an "empty" Unicode internal rep for - * the new object. Otherwise, copy Unicode internal rep, and invalidate - * the string rep of the new object. - */ - - if (srcStringPtr->hasUnicode && srcStringPtr->numChars > 0) { - /* - * Copy the full allocation for the Unicode buffer. - */ - - copyStringPtr = stringAlloc(srcStringPtr->maxChars); - copyStringPtr->maxChars = srcStringPtr->maxChars; - memcpy(copyStringPtr->unicode, srcStringPtr->unicode, - srcStringPtr->numChars * sizeof(Tcl_UniChar)); - copyStringPtr->unicode[srcStringPtr->numChars] = 0; - copyStringPtr->allocated = 0; - } else { - copyStringPtr = stringAlloc(0); - copyStringPtr->unicode[0] = 0; - copyStringPtr->maxChars = 0; - - /* - * Tricky point: the string value was copied by generic object - * management code, so it doesn't contain any extra bytes that might - * exist in the source object. - */ - - copyStringPtr->allocated = copyPtr->length; - } - copyStringPtr->numChars = srcStringPtr->numChars; - copyStringPtr->hasUnicode = srcStringPtr->hasUnicode; -#endif /* COMPAT==0 */ SET_STRING(copyPtr, copyStringPtr); copyPtr->typePtr = &tclStringType; -- cgit v0.12 From cd945198884b9249ee89c47775c2eb9f725a750d Mon Sep 17 00:00:00 2001 From: venkat Date: Sat, 26 Mar 2016 18:10:48 +0000 Subject: Update to tzdata 2016c from ietf.org --- library/tzdata/America/Santiago | 169 +++++++++++++++++++++++++++++++++++++- library/tzdata/Antarctica/Palmer | 169 +++++++++++++++++++++++++++++++++++++- library/tzdata/Asia/Baku | 168 ------------------------------------- library/tzdata/Asia/Barnaul | 6 +- library/tzdata/Europe/Kaliningrad | 10 +-- library/tzdata/Europe/Vilnius | 11 +-- library/tzdata/Europe/Volgograd | 6 +- library/tzdata/Pacific/Easter | 169 +++++++++++++++++++++++++++++++++++++- 8 files changed, 522 insertions(+), 186 deletions(-) diff --git a/library/tzdata/America/Santiago b/library/tzdata/America/Santiago index b6d9b38..3a5c0fd 100644 --- a/library/tzdata/America/Santiago +++ b/library/tzdata/America/Santiago @@ -118,5 +118,172 @@ set TZData(:America/Santiago) { {1378612800 -10800 1 CLST} {1398567600 -14400 0 CLT} {1410062400 -10800 1 CLST} - {1430017200 -10800 0 CLT} + {1463281200 -14400 0 CLT} + {1471147200 -10800 1 CLST} + {1494730800 -14400 0 CLT} + {1502596800 -10800 1 CLST} + {1526180400 -14400 0 CLT} + {1534046400 -10800 1 CLST} + {1557630000 -14400 0 CLT} + {1565496000 -10800 1 CLST} + {1589079600 -14400 0 CLT} + {1596945600 -10800 1 CLST} + {1620529200 -14400 0 CLT} + {1629000000 -10800 1 CLST} + {1652583600 -14400 0 CLT} + {1660449600 -10800 1 CLST} + {1684033200 -14400 0 CLT} + {1691899200 -10800 1 CLST} + {1715482800 -14400 0 CLT} + {1723348800 -10800 1 CLST} + {1746932400 -14400 0 CLT} + {1754798400 -10800 1 CLST} + {1778382000 -14400 0 CLT} + {1786248000 -10800 1 CLST} + {1809831600 -14400 0 CLT} + {1818302400 -10800 1 CLST} + {1841886000 -14400 0 CLT} + {1849752000 -10800 1 CLST} + {1873335600 -14400 0 CLT} + {1881201600 -10800 1 CLST} + {1904785200 -14400 0 CLT} + {1912651200 -10800 1 CLST} + {1936234800 -14400 0 CLT} + {1944100800 -10800 1 CLST} + {1967684400 -14400 0 CLT} + {1976155200 -10800 1 CLST} + {1999738800 -14400 0 CLT} + {2007604800 -10800 1 CLST} + {2031188400 -14400 0 CLT} + {2039054400 -10800 1 CLST} + {2062638000 -14400 0 CLT} + {2070504000 -10800 1 CLST} + {2094087600 -14400 0 CLT} + {2101953600 -10800 1 CLST} + {2125537200 -14400 0 CLT} + {2133403200 -10800 1 CLST} + {2156986800 -14400 0 CLT} + {2165457600 -10800 1 CLST} + {2189041200 -14400 0 CLT} + {2196907200 -10800 1 CLST} + {2220490800 -14400 0 CLT} + {2228356800 -10800 1 CLST} + {2251940400 -14400 0 CLT} + {2259806400 -10800 1 CLST} + {2283390000 -14400 0 CLT} + {2291256000 -10800 1 CLST} + {2314839600 -14400 0 CLT} + {2322705600 -10800 1 CLST} + {2346894000 -14400 0 CLT} + {2354760000 -10800 1 CLST} + {2378343600 -14400 0 CLT} + {2386209600 -10800 1 CLST} + {2409793200 -14400 0 CLT} + {2417659200 -10800 1 CLST} + {2441242800 -14400 0 CLT} + {2449108800 -10800 1 CLST} + {2472692400 -14400 0 CLT} + {2480558400 -10800 1 CLST} + {2504142000 -14400 0 CLT} + {2512612800 -10800 1 CLST} + {2536196400 -14400 0 CLT} + {2544062400 -10800 1 CLST} + {2567646000 -14400 0 CLT} + {2575512000 -10800 1 CLST} + {2599095600 -14400 0 CLT} + {2606961600 -10800 1 CLST} + {2630545200 -14400 0 CLT} + {2638411200 -10800 1 CLST} + {2661994800 -14400 0 CLT} + {2669860800 -10800 1 CLST} + {2693444400 -14400 0 CLT} + {2701915200 -10800 1 CLST} + {2725498800 -14400 0 CLT} + {2733364800 -10800 1 CLST} + {2756948400 -14400 0 CLT} + {2764814400 -10800 1 CLST} + {2788398000 -14400 0 CLT} + {2796264000 -10800 1 CLST} + {2819847600 -14400 0 CLT} + {2827713600 -10800 1 CLST} + {2851297200 -14400 0 CLT} + {2859768000 -10800 1 CLST} + {2883351600 -14400 0 CLT} + {2891217600 -10800 1 CLST} + {2914801200 -14400 0 CLT} + {2922667200 -10800 1 CLST} + {2946250800 -14400 0 CLT} + {2954116800 -10800 1 CLST} + {2977700400 -14400 0 CLT} + {2985566400 -10800 1 CLST} + {3009150000 -14400 0 CLT} + {3017016000 -10800 1 CLST} + {3040599600 -14400 0 CLT} + {3049070400 -10800 1 CLST} + {3072654000 -14400 0 CLT} + {3080520000 -10800 1 CLST} + {3104103600 -14400 0 CLT} + {3111969600 -10800 1 CLST} + {3135553200 -14400 0 CLT} + {3143419200 -10800 1 CLST} + {3167002800 -14400 0 CLT} + {3174868800 -10800 1 CLST} + {3198452400 -14400 0 CLT} + {3206318400 -10800 1 CLST} + {3230506800 -14400 0 CLT} + {3238372800 -10800 1 CLST} + {3261956400 -14400 0 CLT} + {3269822400 -10800 1 CLST} + {3293406000 -14400 0 CLT} + {3301272000 -10800 1 CLST} + {3324855600 -14400 0 CLT} + {3332721600 -10800 1 CLST} + {3356305200 -14400 0 CLT} + {3364171200 -10800 1 CLST} + {3387754800 -14400 0 CLT} + {3396225600 -10800 1 CLST} + {3419809200 -14400 0 CLT} + {3427675200 -10800 1 CLST} + {3451258800 -14400 0 CLT} + {3459124800 -10800 1 CLST} + {3482708400 -14400 0 CLT} + {3490574400 -10800 1 CLST} + {3514158000 -14400 0 CLT} + {3522024000 -10800 1 CLST} + {3545607600 -14400 0 CLT} + {3553473600 -10800 1 CLST} + {3577057200 -14400 0 CLT} + {3585528000 -10800 1 CLST} + {3609111600 -14400 0 CLT} + {3616977600 -10800 1 CLST} + {3640561200 -14400 0 CLT} + {3648427200 -10800 1 CLST} + {3672010800 -14400 0 CLT} + {3679876800 -10800 1 CLST} + {3703460400 -14400 0 CLT} + {3711326400 -10800 1 CLST} + {3734910000 -14400 0 CLT} + {3743380800 -10800 1 CLST} + {3766964400 -14400 0 CLT} + {3774830400 -10800 1 CLST} + {3798414000 -14400 0 CLT} + {3806280000 -10800 1 CLST} + {3829863600 -14400 0 CLT} + {3837729600 -10800 1 CLST} + {3861313200 -14400 0 CLT} + {3869179200 -10800 1 CLST} + {3892762800 -14400 0 CLT} + {3900628800 -10800 1 CLST} + {3924212400 -14400 0 CLT} + {3932683200 -10800 1 CLST} + {3956266800 -14400 0 CLT} + {3964132800 -10800 1 CLST} + {3987716400 -14400 0 CLT} + {3995582400 -10800 1 CLST} + {4019166000 -14400 0 CLT} + {4027032000 -10800 1 CLST} + {4050615600 -14400 0 CLT} + {4058481600 -10800 1 CLST} + {4082065200 -14400 0 CLT} + {4089931200 -10800 1 CLST} } diff --git a/library/tzdata/Antarctica/Palmer b/library/tzdata/Antarctica/Palmer index 2c43861..5767985 100644 --- a/library/tzdata/Antarctica/Palmer +++ b/library/tzdata/Antarctica/Palmer @@ -81,5 +81,172 @@ set TZData(:Antarctica/Palmer) { {1378612800 -10800 1 CLST} {1398567600 -14400 0 CLT} {1410062400 -10800 1 CLST} - {1430017200 -10800 0 CLT} + {1463281200 -14400 0 CLT} + {1471147200 -10800 1 CLST} + {1494730800 -14400 0 CLT} + {1502596800 -10800 1 CLST} + {1526180400 -14400 0 CLT} + {1534046400 -10800 1 CLST} + {1557630000 -14400 0 CLT} + {1565496000 -10800 1 CLST} + {1589079600 -14400 0 CLT} + {1596945600 -10800 1 CLST} + {1620529200 -14400 0 CLT} + {1629000000 -10800 1 CLST} + {1652583600 -14400 0 CLT} + {1660449600 -10800 1 CLST} + {1684033200 -14400 0 CLT} + {1691899200 -10800 1 CLST} + {1715482800 -14400 0 CLT} + {1723348800 -10800 1 CLST} + {1746932400 -14400 0 CLT} + {1754798400 -10800 1 CLST} + {1778382000 -14400 0 CLT} + {1786248000 -10800 1 CLST} + {1809831600 -14400 0 CLT} + {1818302400 -10800 1 CLST} + {1841886000 -14400 0 CLT} + {1849752000 -10800 1 CLST} + {1873335600 -14400 0 CLT} + {1881201600 -10800 1 CLST} + {1904785200 -14400 0 CLT} + {1912651200 -10800 1 CLST} + {1936234800 -14400 0 CLT} + {1944100800 -10800 1 CLST} + {1967684400 -14400 0 CLT} + {1976155200 -10800 1 CLST} + {1999738800 -14400 0 CLT} + {2007604800 -10800 1 CLST} + {2031188400 -14400 0 CLT} + {2039054400 -10800 1 CLST} + {2062638000 -14400 0 CLT} + {2070504000 -10800 1 CLST} + {2094087600 -14400 0 CLT} + {2101953600 -10800 1 CLST} + {2125537200 -14400 0 CLT} + {2133403200 -10800 1 CLST} + {2156986800 -14400 0 CLT} + {2165457600 -10800 1 CLST} + {2189041200 -14400 0 CLT} + {2196907200 -10800 1 CLST} + {2220490800 -14400 0 CLT} + {2228356800 -10800 1 CLST} + {2251940400 -14400 0 CLT} + {2259806400 -10800 1 CLST} + {2283390000 -14400 0 CLT} + {2291256000 -10800 1 CLST} + {2314839600 -14400 0 CLT} + {2322705600 -10800 1 CLST} + {2346894000 -14400 0 CLT} + {2354760000 -10800 1 CLST} + {2378343600 -14400 0 CLT} + {2386209600 -10800 1 CLST} + {2409793200 -14400 0 CLT} + {2417659200 -10800 1 CLST} + {2441242800 -14400 0 CLT} + {2449108800 -10800 1 CLST} + {2472692400 -14400 0 CLT} + {2480558400 -10800 1 CLST} + {2504142000 -14400 0 CLT} + {2512612800 -10800 1 CLST} + {2536196400 -14400 0 CLT} + {2544062400 -10800 1 CLST} + {2567646000 -14400 0 CLT} + {2575512000 -10800 1 CLST} + {2599095600 -14400 0 CLT} + {2606961600 -10800 1 CLST} + {2630545200 -14400 0 CLT} + {2638411200 -10800 1 CLST} + {2661994800 -14400 0 CLT} + {2669860800 -10800 1 CLST} + {2693444400 -14400 0 CLT} + {2701915200 -10800 1 CLST} + {2725498800 -14400 0 CLT} + {2733364800 -10800 1 CLST} + {2756948400 -14400 0 CLT} + {2764814400 -10800 1 CLST} + {2788398000 -14400 0 CLT} + {2796264000 -10800 1 CLST} + {2819847600 -14400 0 CLT} + {2827713600 -10800 1 CLST} + {2851297200 -14400 0 CLT} + {2859768000 -10800 1 CLST} + {2883351600 -14400 0 CLT} + {2891217600 -10800 1 CLST} + {2914801200 -14400 0 CLT} + {2922667200 -10800 1 CLST} + {2946250800 -14400 0 CLT} + {2954116800 -10800 1 CLST} + {2977700400 -14400 0 CLT} + {2985566400 -10800 1 CLST} + {3009150000 -14400 0 CLT} + {3017016000 -10800 1 CLST} + {3040599600 -14400 0 CLT} + {3049070400 -10800 1 CLST} + {3072654000 -14400 0 CLT} + {3080520000 -10800 1 CLST} + {3104103600 -14400 0 CLT} + {3111969600 -10800 1 CLST} + {3135553200 -14400 0 CLT} + {3143419200 -10800 1 CLST} + {3167002800 -14400 0 CLT} + {3174868800 -10800 1 CLST} + {3198452400 -14400 0 CLT} + {3206318400 -10800 1 CLST} + {3230506800 -14400 0 CLT} + {3238372800 -10800 1 CLST} + {3261956400 -14400 0 CLT} + {3269822400 -10800 1 CLST} + {3293406000 -14400 0 CLT} + {3301272000 -10800 1 CLST} + {3324855600 -14400 0 CLT} + {3332721600 -10800 1 CLST} + {3356305200 -14400 0 CLT} + {3364171200 -10800 1 CLST} + {3387754800 -14400 0 CLT} + {3396225600 -10800 1 CLST} + {3419809200 -14400 0 CLT} + {3427675200 -10800 1 CLST} + {3451258800 -14400 0 CLT} + {3459124800 -10800 1 CLST} + {3482708400 -14400 0 CLT} + {3490574400 -10800 1 CLST} + {3514158000 -14400 0 CLT} + {3522024000 -10800 1 CLST} + {3545607600 -14400 0 CLT} + {3553473600 -10800 1 CLST} + {3577057200 -14400 0 CLT} + {3585528000 -10800 1 CLST} + {3609111600 -14400 0 CLT} + {3616977600 -10800 1 CLST} + {3640561200 -14400 0 CLT} + {3648427200 -10800 1 CLST} + {3672010800 -14400 0 CLT} + {3679876800 -10800 1 CLST} + {3703460400 -14400 0 CLT} + {3711326400 -10800 1 CLST} + {3734910000 -14400 0 CLT} + {3743380800 -10800 1 CLST} + {3766964400 -14400 0 CLT} + {3774830400 -10800 1 CLST} + {3798414000 -14400 0 CLT} + {3806280000 -10800 1 CLST} + {3829863600 -14400 0 CLT} + {3837729600 -10800 1 CLST} + {3861313200 -14400 0 CLT} + {3869179200 -10800 1 CLST} + {3892762800 -14400 0 CLT} + {3900628800 -10800 1 CLST} + {3924212400 -14400 0 CLT} + {3932683200 -10800 1 CLST} + {3956266800 -14400 0 CLT} + {3964132800 -10800 1 CLST} + {3987716400 -14400 0 CLT} + {3995582400 -10800 1 CLST} + {4019166000 -14400 0 CLT} + {4027032000 -10800 1 CLST} + {4050615600 -14400 0 CLT} + {4058481600 -10800 1 CLST} + {4082065200 -14400 0 CLT} + {4089931200 -10800 1 CLST} } diff --git a/library/tzdata/Asia/Baku b/library/tzdata/Asia/Baku index e50071b..c26a2f5 100644 --- a/library/tzdata/Asia/Baku +++ b/library/tzdata/Asia/Baku @@ -71,172 +71,4 @@ set TZData(:Asia/Baku) { {1414281600 14400 0 AZT} {1427587200 18000 1 AZST} {1445731200 14400 0 AZT} - {1459036800 18000 1 AZST} - {1477785600 14400 0 AZT} - {1490486400 18000 1 AZST} - {1509235200 14400 0 AZT} - {1521936000 18000 1 AZST} - {1540684800 14400 0 AZT} - {1553990400 18000 1 AZST} - {1572134400 14400 0 AZT} - {1585440000 18000 1 AZST} - {1603584000 14400 0 AZT} - {1616889600 18000 1 AZST} - {1635638400 14400 0 AZT} - {1648339200 18000 1 AZST} - {1667088000 14400 0 AZT} - {1679788800 18000 1 AZST} - {1698537600 14400 0 AZT} - {1711843200 18000 1 AZST} - {1729987200 14400 0 AZT} - {1743292800 18000 1 AZST} - {1761436800 14400 0 AZT} - {1774742400 18000 1 AZST} - {1792886400 14400 0 AZT} - {1806192000 18000 1 AZST} - {1824940800 14400 0 AZT} - {1837641600 18000 1 AZST} - {1856390400 14400 0 AZT} - {1869091200 18000 1 AZST} - {1887840000 14400 0 AZT} - {1901145600 18000 1 AZST} - {1919289600 14400 0 AZT} - {1932595200 18000 1 AZST} - {1950739200 14400 0 AZT} - {1964044800 18000 1 AZST} - {1982793600 14400 0 AZT} - {1995494400 18000 1 AZST} - {2014243200 14400 0 AZT} - {2026944000 18000 1 AZST} - {2045692800 14400 0 AZT} - {2058393600 18000 1 AZST} - {2077142400 14400 0 AZT} - {2090448000 18000 1 AZST} - {2108592000 14400 0 AZT} - {2121897600 18000 1 AZST} - {2140041600 14400 0 AZT} - {2153347200 18000 1 AZST} - {2172096000 14400 0 AZT} - {2184796800 18000 1 AZST} - {2203545600 14400 0 AZT} - {2216246400 18000 1 AZST} - {2234995200 14400 0 AZT} - {2248300800 18000 1 AZST} - {2266444800 14400 0 AZT} - {2279750400 18000 1 AZST} - {2297894400 14400 0 AZT} - {2311200000 18000 1 AZST} - {2329344000 14400 0 AZT} - {2342649600 18000 1 AZST} - {2361398400 14400 0 AZT} - {2374099200 18000 1 AZST} - {2392848000 14400 0 AZT} - {2405548800 18000 1 AZST} - {2424297600 14400 0 AZT} - {2437603200 18000 1 AZST} - {2455747200 14400 0 AZT} - {2469052800 18000 1 AZST} - {2487196800 14400 0 AZT} - {2500502400 18000 1 AZST} - {2519251200 14400 0 AZT} - {2531952000 18000 1 AZST} - {2550700800 14400 0 AZT} - {2563401600 18000 1 AZST} - {2582150400 14400 0 AZT} - {2595456000 18000 1 AZST} - {2613600000 14400 0 AZT} - {2626905600 18000 1 AZST} - {2645049600 14400 0 AZT} - {2658355200 18000 1 AZST} - {2676499200 14400 0 AZT} - {2689804800 18000 1 AZST} - {2708553600 14400 0 AZT} - {2721254400 18000 1 AZST} - {2740003200 14400 0 AZT} - {2752704000 18000 1 AZST} - {2771452800 14400 0 AZT} - {2784758400 18000 1 AZST} - {2802902400 14400 0 AZT} - {2816208000 18000 1 AZST} - {2834352000 14400 0 AZT} - {2847657600 18000 1 AZST} - {2866406400 14400 0 AZT} - {2879107200 18000 1 AZST} - {2897856000 14400 0 AZT} - {2910556800 18000 1 AZST} - {2929305600 14400 0 AZT} - {2942006400 18000 1 AZST} - {2960755200 14400 0 AZT} - {2974060800 18000 1 AZST} - {2992204800 14400 0 AZT} - {3005510400 18000 1 AZST} - {3023654400 14400 0 AZT} - {3036960000 18000 1 AZST} - {3055708800 14400 0 AZT} - {3068409600 18000 1 AZST} - {3087158400 14400 0 AZT} - {3099859200 18000 1 AZST} - {3118608000 14400 0 AZT} - {3131913600 18000 1 AZST} - {3150057600 14400 0 AZT} - {3163363200 18000 1 AZST} - {3181507200 14400 0 AZT} - {3194812800 18000 1 AZST} - {3212956800 14400 0 AZT} - {3226262400 18000 1 AZST} - {3245011200 14400 0 AZT} - {3257712000 18000 1 AZST} - {3276460800 14400 0 AZT} - {3289161600 18000 1 AZST} - {3307910400 14400 0 AZT} - {3321216000 18000 1 AZST} - {3339360000 14400 0 AZT} - {3352665600 18000 1 AZST} - {3370809600 14400 0 AZT} - {3384115200 18000 1 AZST} - {3402864000 14400 0 AZT} - {3415564800 18000 1 AZST} - {3434313600 14400 0 AZT} - {3447014400 18000 1 AZST} - {3465763200 14400 0 AZT} - {3479068800 18000 1 AZST} - {3497212800 14400 0 AZT} - {3510518400 18000 1 AZST} - {3528662400 14400 0 AZT} - {3541968000 18000 1 AZST} - {3560112000 14400 0 AZT} - {3573417600 18000 1 AZST} - {3592166400 14400 0 AZT} - {3604867200 18000 1 AZST} - {3623616000 14400 0 AZT} - {3636316800 18000 1 AZST} - {3655065600 14400 0 AZT} - {3668371200 18000 1 AZST} - {3686515200 14400 0 AZT} - {3699820800 18000 1 AZST} - {3717964800 14400 0 AZT} - {3731270400 18000 1 AZST} - {3750019200 14400 0 AZT} - {3762720000 18000 1 AZST} - {3781468800 14400 0 AZT} - {3794169600 18000 1 AZST} - {3812918400 14400 0 AZT} - {3825619200 18000 1 AZST} - {3844368000 14400 0 AZT} - {3857673600 18000 1 AZST} - {3875817600 14400 0 AZT} - {3889123200 18000 1 AZST} - {3907267200 14400 0 AZT} - {3920572800 18000 1 AZST} - {3939321600 14400 0 AZT} - {3952022400 18000 1 AZST} - {3970771200 14400 0 AZT} - {3983472000 18000 1 AZST} - {4002220800 14400 0 AZT} - {4015526400 18000 1 AZST} - {4033670400 14400 0 AZT} - {4046976000 18000 1 AZST} - {4065120000 14400 0 AZT} - {4078425600 18000 1 AZST} - {4096569600 14400 0 AZT} } diff --git a/library/tzdata/Asia/Barnaul b/library/tzdata/Asia/Barnaul index 8072e34..f6a45da 100644 --- a/library/tzdata/Asia/Barnaul +++ b/library/tzdata/Asia/Barnaul @@ -24,8 +24,10 @@ set TZData(:Asia/Barnaul) { {622580400 25200 0 +07} {638305200 28800 1 +08} {654634800 25200 0 +07} - {670359600 28800 1 +08} - {686084400 25200 0 +07} + {670359600 21600 0 +07} + {670363200 25200 1 +07} + {686088000 21600 0 +06} + {695764800 25200 0 +08} {701798400 28800 1 +08} {717519600 25200 0 +07} {733258800 28800 1 +08} diff --git a/library/tzdata/Europe/Kaliningrad b/library/tzdata/Europe/Kaliningrad index d03f7d0..32d7aaa 100644 --- a/library/tzdata/Europe/Kaliningrad +++ b/library/tzdata/Europe/Kaliningrad @@ -35,11 +35,11 @@ set TZData(:Europe/Kaliningrad) { {559695600 10800 0 MSK} {575420400 14400 1 MSD} {591145200 10800 0 MSK} - {606870000 14400 1 MSD} - {622594800 10800 0 MSK} - {638319600 14400 1 MSD} - {654649200 10800 0 MSK} - {670374000 7200 0 EEMMTT} + {606870000 7200 0 EEMMTT} + {606873600 10800 1 EEST} + {622598400 7200 0 EET} + {638323200 10800 1 EEST} + {654652800 7200 0 EET} {670377600 10800 1 EEST} {686102400 7200 0 EET} {701816400 10800 1 EEST} diff --git a/library/tzdata/Europe/Vilnius b/library/tzdata/Europe/Vilnius index 62d5d87..5e73150 100644 --- a/library/tzdata/Europe/Vilnius +++ b/library/tzdata/Europe/Vilnius @@ -30,11 +30,12 @@ set TZData(:Europe/Vilnius) { {559695600 10800 0 MSK} {575420400 14400 1 MSD} {591145200 10800 0 MSK} - {606870000 14400 1 MSD} - {622594800 10800 0 MSK} - {638319600 14400 1 MSD} - {654649200 10800 0 MSK} - {670374000 10800 1 EEST} + {606870000 7200 0 EEMMTT} + {606873600 10800 1 EEST} + {622598400 7200 0 EET} + {638323200 10800 1 EEST} + {654652800 7200 0 EET} + {670377600 10800 1 EEST} {686102400 7200 0 EET} {701827200 10800 1 EEST} {717552000 7200 0 EET} diff --git a/library/tzdata/Europe/Volgograd b/library/tzdata/Europe/Volgograd index d71fb0b..ef33d4b 100644 --- a/library/tzdata/Europe/Volgograd +++ b/library/tzdata/Europe/Volgograd @@ -20,9 +20,9 @@ set TZData(:Europe/Volgograd) { {528242400 14400 0 VOLT} {543967200 18000 1 VOLST} {559692000 14400 0 VOLT} - {575416800 18000 1 VOLST} - {591141600 14400 0 VOLT} - {606866400 10800 0 VOLMMTT} + {575416800 10800 0 VOLMMTT} + {575420400 14400 1 VOLST} + {591145200 10800 0 VOLT} {606870000 14400 1 VOLST} {622594800 10800 0 VOLT} {638319600 14400 1 VOLST} diff --git a/library/tzdata/Pacific/Easter b/library/tzdata/Pacific/Easter index 4b45ba2..ef0f2d5 100644 --- a/library/tzdata/Pacific/Easter +++ b/library/tzdata/Pacific/Easter @@ -97,5 +97,172 @@ set TZData(:Pacific/Easter) { {1378612800 -18000 1 EASST} {1398567600 -21600 0 EAST} {1410062400 -18000 1 EASST} - {1430017200 -18000 0 EAST} + {1463281200 -21600 0 EAST} + {1471147200 -18000 1 EASST} + {1494730800 -21600 0 EAST} + {1502596800 -18000 1 EASST} + {1526180400 -21600 0 EAST} + {1534046400 -18000 1 EASST} + {1557630000 -21600 0 EAST} + {1565496000 -18000 1 EASST} + {1589079600 -21600 0 EAST} + {1596945600 -18000 1 EASST} + {1620529200 -21600 0 EAST} + {1629000000 -18000 1 EASST} + {1652583600 -21600 0 EAST} + {1660449600 -18000 1 EASST} + {1684033200 -21600 0 EAST} + {1691899200 -18000 1 EASST} + {1715482800 -21600 0 EAST} + {1723348800 -18000 1 EASST} + {1746932400 -21600 0 EAST} + {1754798400 -18000 1 EASST} + {1778382000 -21600 0 EAST} + {1786248000 -18000 1 EASST} + {1809831600 -21600 0 EAST} + {1818302400 -18000 1 EASST} + {1841886000 -21600 0 EAST} + {1849752000 -18000 1 EASST} + {1873335600 -21600 0 EAST} + {1881201600 -18000 1 EASST} + {1904785200 -21600 0 EAST} + {1912651200 -18000 1 EASST} + {1936234800 -21600 0 EAST} + {1944100800 -18000 1 EASST} + {1967684400 -21600 0 EAST} + {1976155200 -18000 1 EASST} + {1999738800 -21600 0 EAST} + {2007604800 -18000 1 EASST} + {2031188400 -21600 0 EAST} + {2039054400 -18000 1 EASST} + {2062638000 -21600 0 EAST} + {2070504000 -18000 1 EASST} + {2094087600 -21600 0 EAST} + {2101953600 -18000 1 EASST} + {2125537200 -21600 0 EAST} + {2133403200 -18000 1 EASST} + {2156986800 -21600 0 EAST} + {2165457600 -18000 1 EASST} + {2189041200 -21600 0 EAST} + {2196907200 -18000 1 EASST} + {2220490800 -21600 0 EAST} + {2228356800 -18000 1 EASST} + {2251940400 -21600 0 EAST} + {2259806400 -18000 1 EASST} + {2283390000 -21600 0 EAST} + {2291256000 -18000 1 EASST} + {2314839600 -21600 0 EAST} + {2322705600 -18000 1 EASST} + {2346894000 -21600 0 EAST} + {2354760000 -18000 1 EASST} + {2378343600 -21600 0 EAST} + {2386209600 -18000 1 EASST} + {2409793200 -21600 0 EAST} + {2417659200 -18000 1 EASST} + {2441242800 -21600 0 EAST} + {2449108800 -18000 1 EASST} + {2472692400 -21600 0 EAST} + {2480558400 -18000 1 EASST} + {2504142000 -21600 0 EAST} + {2512612800 -18000 1 EASST} + {2536196400 -21600 0 EAST} + {2544062400 -18000 1 EASST} + {2567646000 -21600 0 EAST} + {2575512000 -18000 1 EASST} + {2599095600 -21600 0 EAST} + {2606961600 -18000 1 EASST} + {2630545200 -21600 0 EAST} + {2638411200 -18000 1 EASST} + {2661994800 -21600 0 EAST} + {2669860800 -18000 1 EASST} + {2693444400 -21600 0 EAST} + {2701915200 -18000 1 EASST} + {2725498800 -21600 0 EAST} + {2733364800 -18000 1 EASST} + {2756948400 -21600 0 EAST} + {2764814400 -18000 1 EASST} + {2788398000 -21600 0 EAST} + {2796264000 -18000 1 EASST} + {2819847600 -21600 0 EAST} + {2827713600 -18000 1 EASST} + {2851297200 -21600 0 EAST} + {2859768000 -18000 1 EASST} + {2883351600 -21600 0 EAST} + {2891217600 -18000 1 EASST} + {2914801200 -21600 0 EAST} + {2922667200 -18000 1 EASST} + {2946250800 -21600 0 EAST} + {2954116800 -18000 1 EASST} + {2977700400 -21600 0 EAST} + {2985566400 -18000 1 EASST} + {3009150000 -21600 0 EAST} + {3017016000 -18000 1 EASST} + {3040599600 -21600 0 EAST} + {3049070400 -18000 1 EASST} + {3072654000 -21600 0 EAST} + {3080520000 -18000 1 EASST} + {3104103600 -21600 0 EAST} + {3111969600 -18000 1 EASST} + {3135553200 -21600 0 EAST} + {3143419200 -18000 1 EASST} + {3167002800 -21600 0 EAST} + {3174868800 -18000 1 EASST} + {3198452400 -21600 0 EAST} + {3206318400 -18000 1 EASST} + {3230506800 -21600 0 EAST} + {3238372800 -18000 1 EASST} + {3261956400 -21600 0 EAST} + {3269822400 -18000 1 EASST} + {3293406000 -21600 0 EAST} + {3301272000 -18000 1 EASST} + {3324855600 -21600 0 EAST} + {3332721600 -18000 1 EASST} + {3356305200 -21600 0 EAST} + {3364171200 -18000 1 EASST} + {3387754800 -21600 0 EAST} + {3396225600 -18000 1 EASST} + {3419809200 -21600 0 EAST} + {3427675200 -18000 1 EASST} + {3451258800 -21600 0 EAST} + {3459124800 -18000 1 EASST} + {3482708400 -21600 0 EAST} + {3490574400 -18000 1 EASST} + {3514158000 -21600 0 EAST} + {3522024000 -18000 1 EASST} + {3545607600 -21600 0 EAST} + {3553473600 -18000 1 EASST} + {3577057200 -21600 0 EAST} + {3585528000 -18000 1 EASST} + {3609111600 -21600 0 EAST} + {3616977600 -18000 1 EASST} + {3640561200 -21600 0 EAST} + {3648427200 -18000 1 EASST} + {3672010800 -21600 0 EAST} + {3679876800 -18000 1 EASST} + {3703460400 -21600 0 EAST} + {3711326400 -18000 1 EASST} + {3734910000 -21600 0 EAST} + {3743380800 -18000 1 EASST} + {3766964400 -21600 0 EAST} + {3774830400 -18000 1 EASST} + {3798414000 -21600 0 EAST} + {3806280000 -18000 1 EASST} + {3829863600 -21600 0 EAST} + {3837729600 -18000 1 EASST} + {3861313200 -21600 0 EAST} + {3869179200 -18000 1 EASST} + {3892762800 -21600 0 EAST} + {3900628800 -18000 1 EASST} + {3924212400 -21600 0 EAST} + {3932683200 -18000 1 EASST} + {3956266800 -21600 0 EAST} + {3964132800 -18000 1 EASST} + {3987716400 -21600 0 EAST} + {3995582400 -18000 1 EASST} + {4019166000 -21600 0 EAST} + {4027032000 -18000 1 EASST} + {4050615600 -21600 0 EAST} + {4058481600 -18000 1 EASST} + {4082065200 -21600 0 EAST} + {4089931200 -18000 1 EASST} } -- cgit v0.12 From b8cacee84f43fc8f72c924e0c0fcc7c2f53f5a04 Mon Sep 17 00:00:00 2001 From: dkf Date: Sun, 27 Mar 2016 15:43:07 +0000 Subject: Some OSX symbols builds create libtcl8.6.dylib.dSYM which is a directory. --- unix/Makefile.in | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index 7604bc7..233584e 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -657,7 +657,7 @@ Makefile: $(UNIX_DIR)/Makefile.in $(DLTEST_DIR)/Makefile.in # $(SHELL) config.status clean: clean-packages - rm -f *.a *.o libtcl* core errs *~ \#* TAGS *.E a.out \ + rm -rf *.a *.o libtcl* core errs *~ \#* TAGS *.E a.out \ errors ${TCL_EXE} ${TCLTEST_EXE} lib.exp Tcl @DTRACE_HDR@ cd dltest ; $(MAKE) clean -- cgit v0.12 From 365eefedf170cdc365c5d948509ffb6d12b5ce23 Mon Sep 17 00:00:00 2001 From: dkf Date: Sun, 27 Mar 2016 16:37:11 +0000 Subject: [47ac84309b] Fix problems with [lreplace] compilation. --- generic/tclCompCmdsGR.c | 14 ++++++++++++-- tests/lreplace.test | 11 +++++++++++ 2 files changed, 23 insertions(+), 2 deletions(-) diff --git a/generic/tclCompCmdsGR.c b/generic/tclCompCmdsGR.c index 87ed745..9f430ea 100644 --- a/generic/tclCompCmdsGR.c +++ b/generic/tclCompCmdsGR.c @@ -1488,8 +1488,18 @@ TclCompileLreplaceCmd( return TCL_ERROR; } - if(idx2 != INDEX_END && idx2 >= 0 && idx2 < idx1) { - idx2 = idx1-1; + /* + * Compilation fails when one index is end-based but the other isn't. + * Fixing this will require more bytecodes, but this is a workaround for + * now. [Bug 47ac84309b] + */ + + if ((idx1 <= INDEX_END) != (idx2 <= INDEX_END)) { + return TCL_ERROR; + } + + if (idx2 != INDEX_END && idx2 >= 0 && idx2 < idx1) { + idx2 = idx1 - 1; } /* diff --git a/tests/lreplace.test b/tests/lreplace.test index e66a331..55a36a8 100644 --- a/tests/lreplace.test +++ b/tests/lreplace.test @@ -181,6 +181,17 @@ test lreplace-4.11 {lreplace end index first} { test lreplace-4.12 {lreplace end index first} { lreplace {0 1 2 3 4} end-2 2 a b c } {0 1 a b c 3 4} + +test lreplace-5.1 {compiled lreplace: Bug 47ac84309b} { + apply {x { + lreplace $x end 0 + }} {a b c} +} {a b c} +test lreplace-5.2 {compiled lreplace: Bug 47ac84309b} { + apply {x { + lreplace $x end 0 A + }} {a b c} +} {a b A c} # cleanup catch {unset foo} -- cgit v0.12 From 43308bf3c516fbe56b41a2d8e4ea948c63603b3e Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Sun, 27 Mar 2016 21:28:15 +0000 Subject: Better fix for [7d0db7c388f52de81]: In stead of adding dependencies to multiple Makefile lines, combine them --- unix/Makefile.in | 132 +++++++++++++++++++++++++++---------------------------- 1 file changed, 66 insertions(+), 66 deletions(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index 16cfad7..d9f8bbc 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -584,7 +584,7 @@ MAC_OSX_SRCS = \ CYGWIN_SRCS = \ $(TOP_DIR)/win/tclWinError.c -DTRACE_HDR = @DTRACE_HDR@ +DTRACE_HDR = tclDTrace.h DTRACE_SRC = $(GENERIC_DIR)/tclDTrace.d @@ -1351,196 +1351,196 @@ tclThreadTest.o: $(GENERIC_DIR)/tclThreadTest.c tclTomMathInterface.o: $(GENERIC_DIR)/tclTomMathInterface.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(GENERIC_DIR)/tclTomMathInterface.c -bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) $(DTRACE_HDR) +bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bncore.c -bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) $(DTRACE_HDR) +bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_reverse.c -bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) +bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c -bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c -bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add.c -bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add_d.c -bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_and.c -bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clamp.c -bn_mp_clear.o: $(TOMMATH_DIR)/bn_mp_clear.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_clear.o: $(TOMMATH_DIR)/bn_mp_clear.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clear.c -bn_mp_clear_multi.o: $(TOMMATH_DIR)/bn_mp_clear_multi.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_clear_multi.o: $(TOMMATH_DIR)/bn_mp_clear_multi.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clear_multi.c -bn_mp_cmp.o: $(TOMMATH_DIR)/bn_mp_cmp.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cmp.o: $(TOMMATH_DIR)/bn_mp_cmp.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp.c -bn_mp_cmp_d.o: $(TOMMATH_DIR)/bn_mp_cmp_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cmp_d.o: $(TOMMATH_DIR)/bn_mp_cmp_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp_d.c -bn_mp_cmp_mag.o: $(TOMMATH_DIR)/bn_mp_cmp_mag.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cmp_mag.o: $(TOMMATH_DIR)/bn_mp_cmp_mag.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp_mag.c -bn_mp_cnt_lsb.o: $(TOMMATH_DIR)/bn_mp_cnt_lsb.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cnt_lsb.o: $(TOMMATH_DIR)/bn_mp_cnt_lsb.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cnt_lsb.c -bn_mp_copy.o: $(TOMMATH_DIR)/bn_mp_copy.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_copy.o: $(TOMMATH_DIR)/bn_mp_copy.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_copy.c -bn_mp_count_bits.o: $(TOMMATH_DIR)/bn_mp_count_bits.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_count_bits.o: $(TOMMATH_DIR)/bn_mp_count_bits.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_count_bits.c -bn_mp_div.o: $(TOMMATH_DIR)/bn_mp_div.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div.o: $(TOMMATH_DIR)/bn_mp_div.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div.c -bn_mp_div_d.o: $(TOMMATH_DIR)/bn_mp_div_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_d.o: $(TOMMATH_DIR)/bn_mp_div_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_d.c -bn_mp_div_2.o: $(TOMMATH_DIR)/bn_mp_div_2.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_2.o: $(TOMMATH_DIR)/bn_mp_div_2.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_2.c -bn_mp_div_2d.o: $(TOMMATH_DIR)/bn_mp_div_2d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_2d.o: $(TOMMATH_DIR)/bn_mp_div_2d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_2d.c -bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_3.c -bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_exch.c -bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d.c -bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_grow.c -bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init.c -bn_mp_init_copy.o: $(TOMMATH_DIR)/bn_mp_init_copy.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_copy.o: $(TOMMATH_DIR)/bn_mp_init_copy.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_copy.c -bn_mp_init_multi.o: $(TOMMATH_DIR)/bn_mp_init_multi.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_multi.o: $(TOMMATH_DIR)/bn_mp_init_multi.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_multi.c -bn_mp_init_set.o: $(TOMMATH_DIR)/bn_mp_init_set.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_set.o: $(TOMMATH_DIR)/bn_mp_init_set.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_set.c -bn_mp_init_set_int.o: $(TOMMATH_DIR)/bn_mp_init_set_int.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_set_int.o: $(TOMMATH_DIR)/bn_mp_init_set_int.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_set_int.c -bn_mp_init_size.o:$(TOMMATH_DIR)/bn_mp_init_size.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_size.o:$(TOMMATH_DIR)/bn_mp_init_size.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_size.c -bn_mp_karatsuba_mul.o: $(TOMMATH_DIR)/bn_mp_karatsuba_mul.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_karatsuba_mul.o: $(TOMMATH_DIR)/bn_mp_karatsuba_mul.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_karatsuba_mul.c -bn_mp_karatsuba_sqr.o: $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_karatsuba_sqr.o: $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c -bn_mp_lshd.o: $(TOMMATH_DIR)/bn_mp_lshd.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_lshd.o: $(TOMMATH_DIR)/bn_mp_lshd.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_lshd.c -bn_mp_mod.o: $(TOMMATH_DIR)/bn_mp_mod.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mod.o: $(TOMMATH_DIR)/bn_mp_mod.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mod.c -bn_mp_mod_2d.o: $(TOMMATH_DIR)/bn_mp_mod_2d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mod_2d.o: $(TOMMATH_DIR)/bn_mp_mod_2d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mod_2d.c -bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul.c -bn_mp_mul_2.o: $(TOMMATH_DIR)/bn_mp_mul_2.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul_2.o: $(TOMMATH_DIR)/bn_mp_mul_2.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_2.c -bn_mp_mul_2d.o: $(TOMMATH_DIR)/bn_mp_mul_2d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul_2d.o: $(TOMMATH_DIR)/bn_mp_mul_2d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_2d.c -bn_mp_mul_d.o: $(TOMMATH_DIR)/bn_mp_mul_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul_d.o: $(TOMMATH_DIR)/bn_mp_mul_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_d.c -bn_mp_neg.o: $(TOMMATH_DIR)/bn_mp_neg.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_neg.o: $(TOMMATH_DIR)/bn_mp_neg.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_neg.c -bn_mp_or.o: $(TOMMATH_DIR)/bn_mp_or.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_or.o: $(TOMMATH_DIR)/bn_mp_or.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_or.c -bn_mp_radix_size.o: $(TOMMATH_DIR)/bn_mp_radix_size.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_radix_size.o: $(TOMMATH_DIR)/bn_mp_radix_size.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_radix_size.c -bn_mp_radix_smap.o: $(TOMMATH_DIR)/bn_mp_radix_smap.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_radix_smap.o: $(TOMMATH_DIR)/bn_mp_radix_smap.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_radix_smap.c -bn_mp_read_radix.o: $(TOMMATH_DIR)/bn_mp_read_radix.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_read_radix.o: $(TOMMATH_DIR)/bn_mp_read_radix.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_read_radix.c -bn_mp_rshd.o: $(TOMMATH_DIR)/bn_mp_rshd.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_rshd.o: $(TOMMATH_DIR)/bn_mp_rshd.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_rshd.c -bn_mp_set.o: $(TOMMATH_DIR)/bn_mp_set.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_set.o: $(TOMMATH_DIR)/bn_mp_set.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_set.c -bn_mp_set_int.o: $(TOMMATH_DIR)/bn_mp_set_int.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_set_int.o: $(TOMMATH_DIR)/bn_mp_set_int.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_set_int.c -bn_mp_shrink.o: $(TOMMATH_DIR)/bn_mp_shrink.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_shrink.o: $(TOMMATH_DIR)/bn_mp_shrink.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_shrink.c -bn_mp_sqr.o: $(TOMMATH_DIR)/bn_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sqr.o: $(TOMMATH_DIR)/bn_mp_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sqr.c -bn_mp_sqrt.o: $(TOMMATH_DIR)/bn_mp_sqrt.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sqrt.o: $(TOMMATH_DIR)/bn_mp_sqrt.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sqrt.c -bn_mp_sub.o: $(TOMMATH_DIR)/bn_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sub.o: $(TOMMATH_DIR)/bn_mp_sub.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sub.c -bn_mp_sub_d.o: $(TOMMATH_DIR)/bn_mp_sub_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sub_d.o: $(TOMMATH_DIR)/bn_mp_sub_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sub_d.c -bn_mp_to_unsigned_bin.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_to_unsigned_bin.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c -bn_mp_to_unsigned_bin_n.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_to_unsigned_bin_n.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c -bn_mp_toom_mul.o: $(TOMMATH_DIR)/bn_mp_toom_mul.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_toom_mul.o: $(TOMMATH_DIR)/bn_mp_toom_mul.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toom_mul.c -bn_mp_toom_sqr.o: $(TOMMATH_DIR)/bn_mp_toom_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_toom_sqr.o: $(TOMMATH_DIR)/bn_mp_toom_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toom_sqr.c -bn_mp_toradix_n.o: $(TOMMATH_DIR)/bn_mp_toradix_n.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_toradix_n.o: $(TOMMATH_DIR)/bn_mp_toradix_n.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toradix_n.c -bn_mp_unsigned_bin_size.o: $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_unsigned_bin_size.o: $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c -bn_mp_xor.o: $(TOMMATH_DIR)/bn_mp_xor.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_xor.o: $(TOMMATH_DIR)/bn_mp_xor.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_xor.c -bn_mp_zero.o: $(TOMMATH_DIR)/bn_mp_zero.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_zero.o: $(TOMMATH_DIR)/bn_mp_zero.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_zero.c -bn_s_mp_add.o: $(TOMMATH_DIR)/bn_s_mp_add.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_add.o: $(TOMMATH_DIR)/bn_s_mp_add.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_add.c -bn_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_s_mp_mul_digs.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_mul_digs.c -bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sqr.c -bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sub.c tclUnixChan.o: $(UNIX_DIR)/tclUnixChan.c $(IOHDR) @@ -1596,7 +1596,7 @@ tclWinError.o: $(TOP_DIR)/win/tclWinError.c # DTrace support -$(TCL_OBJS) $(STUB_LIB_OBJS) $(TCLSH_OBJS) $(TCLTEST_OBJS) $(XTTEST_OBJS): @DTRACE_HDR@ +$(TCL_OBJS) $(STUB_LIB_OBJS) $(TCLSH_OBJS) $(TCLTEST_OBJS) $(XTTEST_OBJS) $(TOMMATH_OBJS): @DTRACE_HDR@ $(DTRACE_HDR): $(DTRACE_SRC) $(DTRACE) -h $(DTRACE_SWITCHES) -o $@ -s $(DTRACE_SRC) -- cgit v0.12 From d5ff54059dda1b7a925361e016193915d3e1cccd Mon Sep 17 00:00:00 2001 From: dgp Date: Mon, 28 Mar 2016 19:03:14 +0000 Subject: There's a "parsedVarName" Tcl_ObjType that remembers how a variable name breaks down into the name of an array and the name of an element. It has been storing them in an intrep as a Tcl_Obj holding the array name and an allocated string holding the element name. This branch revises the intrep strategy to use Tcl_Obj's to hold both parts. This reduces copying and seems to simplify the code. Also "nulled out" the UpdateStringProc for the type which can never be called. I think this is a better answer, but I'd like any other informed opinions. --- generic/tclVar.c | 45 ++++++++++++++++++++++++++++++++++----------- 1 file changed, 34 insertions(+), 11 deletions(-) diff --git a/generic/tclVar.c b/generic/tclVar.c index 5574f30..3c1b01f 100644 --- a/generic/tclVar.c +++ b/generic/tclVar.c @@ -206,7 +206,9 @@ static Tcl_UpdateStringProc PanicOnUpdateVarName; static Tcl_FreeInternalRepProc FreeParsedVarName; static Tcl_DupInternalRepProc DupParsedVarName; +#if 0 static Tcl_UpdateStringProc UpdateParsedVarName; +#endif static Tcl_UpdateStringProc PanicOnUpdateVarName; static Tcl_SetFromAnyProc PanicOnSetVarName; @@ -237,7 +239,13 @@ static const Tcl_ObjType localVarNameType = { static const Tcl_ObjType tclParsedVarNameType = { "parsedVarName", - FreeParsedVarName, DupParsedVarName, UpdateParsedVarName, PanicOnSetVarName + FreeParsedVarName, DupParsedVarName, +#if 0 +UpdateParsedVarName, +#else +PanicOnUpdateVarName, +#endif + PanicOnSetVarName }; /* @@ -536,7 +544,9 @@ TclObjLookupVarEx( const char *errMsg = NULL; CallFrame *varFramePtr = iPtr->varFramePtr; const char *part2 = part2Ptr? TclGetString(part2Ptr):NULL; +#if 0 char *newPart2 = NULL; +#endif *arrayPtrPtr = NULL; if (typePtr == &localVarNameType) { @@ -583,9 +593,13 @@ TclObjLookupVarEx( } return NULL; } +#if 0 part2 = newPart2 = part1Ptr->internalRep.twoPtrValue.ptr2; if (newPart2) { part2Ptr = Tcl_NewStringObj(newPart2, -1); +#else + if ((part2Ptr = part1Ptr->internalRep.twoPtrValue.ptr2)) { +#endif if (createPart2) { Tcl_IncrRefCount(part2Ptr); } @@ -629,11 +643,15 @@ TclObjLookupVarEx( len2 = len1 - i - 2; len1 = i; +#if 0 newPart2 = ckalloc(len2 + 1); memcpy(newPart2, part2, (unsigned) len2); *(newPart2+len2) = '\0'; part2 = newPart2; part2Ptr = Tcl_NewStringObj(newPart2, -1); +#else + part2Ptr = Tcl_NewStringObj(part2, len2); +#endif if (createPart2) { Tcl_IncrRefCount(part2Ptr); } @@ -658,7 +676,12 @@ TclObjLookupVarEx( Tcl_IncrRefCount(part1Ptr); objPtr->internalRep.twoPtrValue.ptr1 = part1Ptr; +#if 0 objPtr->internalRep.twoPtrValue.ptr2 = (void *) part2; +#else + Tcl_IncrRefCount(part2Ptr); + objPtr->internalRep.twoPtrValue.ptr2 = part2Ptr; +#endif typePtr = part1Ptr->typePtr; part1 = TclGetString(part1Ptr); @@ -683,9 +706,11 @@ TclObjLookupVarEx( Tcl_SetErrorCode(interp, "TCL", "LOOKUP", "VARNAME", TclGetString(part1Ptr), NULL); } +#if 0 if (newPart2) { Tcl_DecrRefCount(part2Ptr); } +#endif return NULL; } @@ -734,9 +759,11 @@ TclObjLookupVarEx( *arrayPtrPtr = varPtr; varPtr = TclLookupArrayElement(interp, part1Ptr, part2Ptr, flags, msg, createPart1, createPart2, varPtr, -1); +#if 0 if (newPart2) { Tcl_DecrRefCount(part2Ptr); } +#endif } return varPtr; } @@ -5592,11 +5619,11 @@ FreeParsedVarName( Tcl_Obj *objPtr) { register Tcl_Obj *arrayPtr = objPtr->internalRep.twoPtrValue.ptr1; - register char *elem = objPtr->internalRep.twoPtrValue.ptr2; + register Tcl_Obj *elem = objPtr->internalRep.twoPtrValue.ptr2; if (arrayPtr != NULL) { TclDecrRefCount(arrayPtr); - ckfree(elem); + TclDecrRefCount(elem); } objPtr->typePtr = NULL; } @@ -5607,17 +5634,11 @@ DupParsedVarName( Tcl_Obj *dupPtr) { register Tcl_Obj *arrayPtr = srcPtr->internalRep.twoPtrValue.ptr1; - register char *elem = srcPtr->internalRep.twoPtrValue.ptr2; - char *elemCopy; - unsigned elemLen; + register Tcl_Obj *elem = srcPtr->internalRep.twoPtrValue.ptr2; if (arrayPtr != NULL) { Tcl_IncrRefCount(arrayPtr); - elemLen = strlen(elem); - elemCopy = ckalloc(elemLen + 1); - memcpy(elemCopy, elem, elemLen); - *(elemCopy + elemLen) = '\0'; - elem = elemCopy; + Tcl_IncrRefCount(elem); } dupPtr->internalRep.twoPtrValue.ptr1 = arrayPtr; @@ -5625,6 +5646,7 @@ DupParsedVarName( dupPtr->typePtr = &tclParsedVarNameType; } +#if 0 static void UpdateParsedVarName( Tcl_Obj *objPtr) @@ -5659,6 +5681,7 @@ UpdateParsedVarName( *p++ = ')'; *p = '\0'; } +#endif /* *---------------------------------------------------------------------- -- cgit v0.12 From 20baf86d1e03655bb6d7fae562091e95fe52db15 Mon Sep 17 00:00:00 2001 From: dgp Date: Mon, 28 Mar 2016 21:24:22 +0000 Subject: The DupEncodingIntRep() routine is broken. It fails to set the typePtr field of the duplicated Tcl_Obj to indicate that the intrep is that of the &encodingType. The impact must be that refcounting of Tcl_Encodings are out of balance, at least in the (unusual?) case where "encoding" values need duplication. I hesitate to actually commit this fix until I have a test demonstrating the need for it, and a chance to see the impact on actual encoding operations. The lifetime management of encodings has a history of being tricky and raising controversy. More work required. --- generic/tclEncoding.c | 1 + 1 file changed, 1 insertion(+) diff --git a/generic/tclEncoding.c b/generic/tclEncoding.c index 4edebcf..32055a3 100644 --- a/generic/tclEncoding.c +++ b/generic/tclEncoding.c @@ -355,6 +355,7 @@ DupEncodingIntRep( Tcl_Obj *dupPtr) { dupPtr->internalRep.twoPtrValue.ptr1 = Tcl_GetEncoding(NULL, srcPtr->bytes); + dupPtr->typePtr = &encodingType; } /* -- cgit v0.12 From e586c259bcbae989eca56b696377488dfe656b20 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Tue, 29 Mar 2016 08:34:38 +0000 Subject: (cherry-pick): Better fix for [7d0db7c388f52de81]: In stead of adding dependencies to multiple Makefile lines, combine them --- unix/Makefile.in | 132 +++++++++++++++++++++++++++---------------------------- 1 file changed, 66 insertions(+), 66 deletions(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index 233584e..b336074 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -584,7 +584,7 @@ MAC_OSX_SRCS = \ CYGWIN_SRCS = \ $(TOP_DIR)/win/tclWinError.c -DTRACE_HDR = @DTRACE_HDR@ +DTRACE_HDR = tclDTrace.h DTRACE_SRC = $(GENERIC_DIR)/tclDTrace.d @@ -1351,196 +1351,196 @@ tclThreadTest.o: $(GENERIC_DIR)/tclThreadTest.c tclTomMathInterface.o: $(GENERIC_DIR)/tclTomMathInterface.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(GENERIC_DIR)/tclTomMathInterface.c -bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) $(DTRACE_HDR) +bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bncore.c -bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) $(DTRACE_HDR) +bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_reverse.c -bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) +bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c -bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c -bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add.c -bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add_d.c -bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_and.c -bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clamp.c -bn_mp_clear.o: $(TOMMATH_DIR)/bn_mp_clear.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_clear.o: $(TOMMATH_DIR)/bn_mp_clear.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clear.c -bn_mp_clear_multi.o: $(TOMMATH_DIR)/bn_mp_clear_multi.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_clear_multi.o: $(TOMMATH_DIR)/bn_mp_clear_multi.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clear_multi.c -bn_mp_cmp.o: $(TOMMATH_DIR)/bn_mp_cmp.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cmp.o: $(TOMMATH_DIR)/bn_mp_cmp.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp.c -bn_mp_cmp_d.o: $(TOMMATH_DIR)/bn_mp_cmp_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cmp_d.o: $(TOMMATH_DIR)/bn_mp_cmp_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp_d.c -bn_mp_cmp_mag.o: $(TOMMATH_DIR)/bn_mp_cmp_mag.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cmp_mag.o: $(TOMMATH_DIR)/bn_mp_cmp_mag.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp_mag.c -bn_mp_cnt_lsb.o: $(TOMMATH_DIR)/bn_mp_cnt_lsb.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cnt_lsb.o: $(TOMMATH_DIR)/bn_mp_cnt_lsb.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cnt_lsb.c -bn_mp_copy.o: $(TOMMATH_DIR)/bn_mp_copy.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_copy.o: $(TOMMATH_DIR)/bn_mp_copy.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_copy.c -bn_mp_count_bits.o: $(TOMMATH_DIR)/bn_mp_count_bits.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_count_bits.o: $(TOMMATH_DIR)/bn_mp_count_bits.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_count_bits.c -bn_mp_div.o: $(TOMMATH_DIR)/bn_mp_div.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div.o: $(TOMMATH_DIR)/bn_mp_div.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div.c -bn_mp_div_d.o: $(TOMMATH_DIR)/bn_mp_div_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_d.o: $(TOMMATH_DIR)/bn_mp_div_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_d.c -bn_mp_div_2.o: $(TOMMATH_DIR)/bn_mp_div_2.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_2.o: $(TOMMATH_DIR)/bn_mp_div_2.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_2.c -bn_mp_div_2d.o: $(TOMMATH_DIR)/bn_mp_div_2d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_2d.o: $(TOMMATH_DIR)/bn_mp_div_2d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_2d.c -bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_3.c -bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_exch.c -bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d.c -bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_grow.c -bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init.c -bn_mp_init_copy.o: $(TOMMATH_DIR)/bn_mp_init_copy.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_copy.o: $(TOMMATH_DIR)/bn_mp_init_copy.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_copy.c -bn_mp_init_multi.o: $(TOMMATH_DIR)/bn_mp_init_multi.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_multi.o: $(TOMMATH_DIR)/bn_mp_init_multi.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_multi.c -bn_mp_init_set.o: $(TOMMATH_DIR)/bn_mp_init_set.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_set.o: $(TOMMATH_DIR)/bn_mp_init_set.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_set.c -bn_mp_init_set_int.o: $(TOMMATH_DIR)/bn_mp_init_set_int.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_set_int.o: $(TOMMATH_DIR)/bn_mp_init_set_int.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_set_int.c -bn_mp_init_size.o:$(TOMMATH_DIR)/bn_mp_init_size.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_size.o:$(TOMMATH_DIR)/bn_mp_init_size.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_size.c -bn_mp_karatsuba_mul.o: $(TOMMATH_DIR)/bn_mp_karatsuba_mul.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_karatsuba_mul.o: $(TOMMATH_DIR)/bn_mp_karatsuba_mul.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_karatsuba_mul.c -bn_mp_karatsuba_sqr.o: $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_karatsuba_sqr.o: $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c -bn_mp_lshd.o: $(TOMMATH_DIR)/bn_mp_lshd.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_lshd.o: $(TOMMATH_DIR)/bn_mp_lshd.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_lshd.c -bn_mp_mod.o: $(TOMMATH_DIR)/bn_mp_mod.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mod.o: $(TOMMATH_DIR)/bn_mp_mod.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mod.c -bn_mp_mod_2d.o: $(TOMMATH_DIR)/bn_mp_mod_2d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mod_2d.o: $(TOMMATH_DIR)/bn_mp_mod_2d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mod_2d.c -bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul.c -bn_mp_mul_2.o: $(TOMMATH_DIR)/bn_mp_mul_2.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul_2.o: $(TOMMATH_DIR)/bn_mp_mul_2.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_2.c -bn_mp_mul_2d.o: $(TOMMATH_DIR)/bn_mp_mul_2d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul_2d.o: $(TOMMATH_DIR)/bn_mp_mul_2d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_2d.c -bn_mp_mul_d.o: $(TOMMATH_DIR)/bn_mp_mul_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul_d.o: $(TOMMATH_DIR)/bn_mp_mul_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_d.c -bn_mp_neg.o: $(TOMMATH_DIR)/bn_mp_neg.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_neg.o: $(TOMMATH_DIR)/bn_mp_neg.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_neg.c -bn_mp_or.o: $(TOMMATH_DIR)/bn_mp_or.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_or.o: $(TOMMATH_DIR)/bn_mp_or.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_or.c -bn_mp_radix_size.o: $(TOMMATH_DIR)/bn_mp_radix_size.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_radix_size.o: $(TOMMATH_DIR)/bn_mp_radix_size.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_radix_size.c -bn_mp_radix_smap.o: $(TOMMATH_DIR)/bn_mp_radix_smap.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_radix_smap.o: $(TOMMATH_DIR)/bn_mp_radix_smap.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_radix_smap.c -bn_mp_read_radix.o: $(TOMMATH_DIR)/bn_mp_read_radix.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_read_radix.o: $(TOMMATH_DIR)/bn_mp_read_radix.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_read_radix.c -bn_mp_rshd.o: $(TOMMATH_DIR)/bn_mp_rshd.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_rshd.o: $(TOMMATH_DIR)/bn_mp_rshd.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_rshd.c -bn_mp_set.o: $(TOMMATH_DIR)/bn_mp_set.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_set.o: $(TOMMATH_DIR)/bn_mp_set.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_set.c -bn_mp_set_int.o: $(TOMMATH_DIR)/bn_mp_set_int.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_set_int.o: $(TOMMATH_DIR)/bn_mp_set_int.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_set_int.c -bn_mp_shrink.o: $(TOMMATH_DIR)/bn_mp_shrink.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_shrink.o: $(TOMMATH_DIR)/bn_mp_shrink.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_shrink.c -bn_mp_sqr.o: $(TOMMATH_DIR)/bn_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sqr.o: $(TOMMATH_DIR)/bn_mp_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sqr.c -bn_mp_sqrt.o: $(TOMMATH_DIR)/bn_mp_sqrt.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sqrt.o: $(TOMMATH_DIR)/bn_mp_sqrt.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sqrt.c -bn_mp_sub.o: $(TOMMATH_DIR)/bn_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sub.o: $(TOMMATH_DIR)/bn_mp_sub.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sub.c -bn_mp_sub_d.o: $(TOMMATH_DIR)/bn_mp_sub_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sub_d.o: $(TOMMATH_DIR)/bn_mp_sub_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sub_d.c -bn_mp_to_unsigned_bin.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_to_unsigned_bin.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c -bn_mp_to_unsigned_bin_n.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_to_unsigned_bin_n.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c -bn_mp_toom_mul.o: $(TOMMATH_DIR)/bn_mp_toom_mul.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_toom_mul.o: $(TOMMATH_DIR)/bn_mp_toom_mul.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toom_mul.c -bn_mp_toom_sqr.o: $(TOMMATH_DIR)/bn_mp_toom_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_toom_sqr.o: $(TOMMATH_DIR)/bn_mp_toom_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toom_sqr.c -bn_mp_toradix_n.o: $(TOMMATH_DIR)/bn_mp_toradix_n.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_toradix_n.o: $(TOMMATH_DIR)/bn_mp_toradix_n.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toradix_n.c -bn_mp_unsigned_bin_size.o: $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_unsigned_bin_size.o: $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c -bn_mp_xor.o: $(TOMMATH_DIR)/bn_mp_xor.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_xor.o: $(TOMMATH_DIR)/bn_mp_xor.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_xor.c -bn_mp_zero.o: $(TOMMATH_DIR)/bn_mp_zero.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_zero.o: $(TOMMATH_DIR)/bn_mp_zero.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_zero.c -bn_s_mp_add.o: $(TOMMATH_DIR)/bn_s_mp_add.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_add.o: $(TOMMATH_DIR)/bn_s_mp_add.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_add.c -bn_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_s_mp_mul_digs.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_mul_digs.c -bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sqr.c -bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sub.c tclUnixChan.o: $(UNIX_DIR)/tclUnixChan.c $(IOHDR) @@ -1596,7 +1596,7 @@ tclWinError.o: $(TOP_DIR)/win/tclWinError.c # DTrace support -$(TCL_OBJS) $(STUB_LIB_OBJS) $(TCLSH_OBJS) $(TCLTEST_OBJS) $(XTTEST_OBJS): @DTRACE_HDR@ +$(TCL_OBJS) $(STUB_LIB_OBJS) $(TCLSH_OBJS) $(TCLTEST_OBJS) $(XTTEST_OBJS) $(TOMMATH_OBJS): @DTRACE_HDR@ $(DTRACE_HDR): $(DTRACE_SRC) $(DTRACE) -h $(DTRACE_SWITCHES) -o $@ -s $(DTRACE_SRC) -- cgit v0.12 From 75cbe0aa43accb4642e041fee2c68a893130a53a Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Tue, 29 Mar 2016 08:36:15 +0000 Subject: (cherry-pick): Better fix for [7d0db7c388f52de81]: In stead of adding dependencies to multiple Makefile lines, combine them --- unix/Makefile.in | 132 +++++++++++++++++++++++++++---------------------------- 1 file changed, 66 insertions(+), 66 deletions(-) diff --git a/unix/Makefile.in b/unix/Makefile.in index 722f85d..fc5b03f 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -529,7 +529,7 @@ MAC_OSX_SRCS = \ $(MAC_OSX_DIR)/tclMacOSXFCmd.c \ $(MAC_OSX_DIR)/tclMacOSXNotify.c -DTRACE_HDR = @DTRACE_HDR@ +DTRACE_HDR = tclDTrace.h DTRACE_SRC = $(GENERIC_DIR)/tclDTrace.d @@ -1238,196 +1238,196 @@ tclThreadTest.o: $(GENERIC_DIR)/tclThreadTest.c tclTomMathInterface.o: $(GENERIC_DIR)/tclTomMathInterface.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(GENERIC_DIR)/tclTomMathInterface.c -bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) $(DTRACE_HDR) +bncore.o: $(TOMMATH_DIR)/bncore.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bncore.c -bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) $(DTRACE_HDR) +bn_reverse.o: $(TOMMATH_DIR)/bn_reverse.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_reverse.c -bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) +bn_fast_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_mul_digs.c -bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_fast_s_mp_sqr.o: $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_fast_s_mp_sqr.c -bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_add.o: $(TOMMATH_DIR)/bn_mp_add.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add.c -bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_add_d.o: $(TOMMATH_DIR)/bn_mp_add_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_add_d.c -bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_and.o: $(TOMMATH_DIR)/bn_mp_and.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_and.c -bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_clamp.o: $(TOMMATH_DIR)/bn_mp_clamp.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clamp.c -bn_mp_clear.o: $(TOMMATH_DIR)/bn_mp_clear.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_clear.o: $(TOMMATH_DIR)/bn_mp_clear.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clear.c -bn_mp_clear_multi.o: $(TOMMATH_DIR)/bn_mp_clear_multi.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_clear_multi.o: $(TOMMATH_DIR)/bn_mp_clear_multi.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_clear_multi.c -bn_mp_cmp.o: $(TOMMATH_DIR)/bn_mp_cmp.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cmp.o: $(TOMMATH_DIR)/bn_mp_cmp.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp.c -bn_mp_cmp_d.o: $(TOMMATH_DIR)/bn_mp_cmp_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cmp_d.o: $(TOMMATH_DIR)/bn_mp_cmp_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp_d.c -bn_mp_cmp_mag.o: $(TOMMATH_DIR)/bn_mp_cmp_mag.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cmp_mag.o: $(TOMMATH_DIR)/bn_mp_cmp_mag.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cmp_mag.c -bn_mp_cnt_lsb.o: $(TOMMATH_DIR)/bn_mp_cnt_lsb.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_cnt_lsb.o: $(TOMMATH_DIR)/bn_mp_cnt_lsb.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_cnt_lsb.c -bn_mp_copy.o: $(TOMMATH_DIR)/bn_mp_copy.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_copy.o: $(TOMMATH_DIR)/bn_mp_copy.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_copy.c -bn_mp_count_bits.o: $(TOMMATH_DIR)/bn_mp_count_bits.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_count_bits.o: $(TOMMATH_DIR)/bn_mp_count_bits.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_count_bits.c -bn_mp_div.o: $(TOMMATH_DIR)/bn_mp_div.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div.o: $(TOMMATH_DIR)/bn_mp_div.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div.c -bn_mp_div_d.o: $(TOMMATH_DIR)/bn_mp_div_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_d.o: $(TOMMATH_DIR)/bn_mp_div_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_d.c -bn_mp_div_2.o: $(TOMMATH_DIR)/bn_mp_div_2.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_2.o: $(TOMMATH_DIR)/bn_mp_div_2.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_2.c -bn_mp_div_2d.o: $(TOMMATH_DIR)/bn_mp_div_2d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_2d.o: $(TOMMATH_DIR)/bn_mp_div_2d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_2d.c -bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_div_3.o: $(TOMMATH_DIR)/bn_mp_div_3.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_div_3.c -bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_exch.o: $(TOMMATH_DIR)/bn_mp_exch.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_exch.c -bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_expt_d.o: $(TOMMATH_DIR)/bn_mp_expt_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_expt_d.c -bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_grow.o: $(TOMMATH_DIR)/bn_mp_grow.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_grow.c -bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init.o: $(TOMMATH_DIR)/bn_mp_init.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init.c -bn_mp_init_copy.o: $(TOMMATH_DIR)/bn_mp_init_copy.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_copy.o: $(TOMMATH_DIR)/bn_mp_init_copy.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_copy.c -bn_mp_init_multi.o: $(TOMMATH_DIR)/bn_mp_init_multi.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_multi.o: $(TOMMATH_DIR)/bn_mp_init_multi.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_multi.c -bn_mp_init_set.o: $(TOMMATH_DIR)/bn_mp_init_set.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_set.o: $(TOMMATH_DIR)/bn_mp_init_set.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_set.c -bn_mp_init_set_int.o: $(TOMMATH_DIR)/bn_mp_init_set_int.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_set_int.o: $(TOMMATH_DIR)/bn_mp_init_set_int.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_set_int.c -bn_mp_init_size.o:$(TOMMATH_DIR)/bn_mp_init_size.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_init_size.o:$(TOMMATH_DIR)/bn_mp_init_size.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_init_size.c -bn_mp_karatsuba_mul.o: $(TOMMATH_DIR)/bn_mp_karatsuba_mul.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_karatsuba_mul.o: $(TOMMATH_DIR)/bn_mp_karatsuba_mul.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_karatsuba_mul.c -bn_mp_karatsuba_sqr.o: $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_karatsuba_sqr.o: $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_karatsuba_sqr.c -bn_mp_lshd.o: $(TOMMATH_DIR)/bn_mp_lshd.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_lshd.o: $(TOMMATH_DIR)/bn_mp_lshd.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_lshd.c -bn_mp_mod.o: $(TOMMATH_DIR)/bn_mp_mod.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mod.o: $(TOMMATH_DIR)/bn_mp_mod.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mod.c -bn_mp_mod_2d.o: $(TOMMATH_DIR)/bn_mp_mod_2d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mod_2d.o: $(TOMMATH_DIR)/bn_mp_mod_2d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mod_2d.c -bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul.o: $(TOMMATH_DIR)/bn_mp_mul.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul.c -bn_mp_mul_2.o: $(TOMMATH_DIR)/bn_mp_mul_2.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul_2.o: $(TOMMATH_DIR)/bn_mp_mul_2.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_2.c -bn_mp_mul_2d.o: $(TOMMATH_DIR)/bn_mp_mul_2d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul_2d.o: $(TOMMATH_DIR)/bn_mp_mul_2d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_2d.c -bn_mp_mul_d.o: $(TOMMATH_DIR)/bn_mp_mul_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_mul_d.o: $(TOMMATH_DIR)/bn_mp_mul_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_mul_d.c -bn_mp_neg.o: $(TOMMATH_DIR)/bn_mp_neg.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_neg.o: $(TOMMATH_DIR)/bn_mp_neg.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_neg.c -bn_mp_or.o: $(TOMMATH_DIR)/bn_mp_or.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_or.o: $(TOMMATH_DIR)/bn_mp_or.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_or.c -bn_mp_radix_size.o: $(TOMMATH_DIR)/bn_mp_radix_size.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_radix_size.o: $(TOMMATH_DIR)/bn_mp_radix_size.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_radix_size.c -bn_mp_radix_smap.o: $(TOMMATH_DIR)/bn_mp_radix_smap.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_radix_smap.o: $(TOMMATH_DIR)/bn_mp_radix_smap.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_radix_smap.c -bn_mp_read_radix.o: $(TOMMATH_DIR)/bn_mp_read_radix.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_read_radix.o: $(TOMMATH_DIR)/bn_mp_read_radix.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_read_radix.c -bn_mp_rshd.o: $(TOMMATH_DIR)/bn_mp_rshd.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_rshd.o: $(TOMMATH_DIR)/bn_mp_rshd.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_rshd.c -bn_mp_set.o: $(TOMMATH_DIR)/bn_mp_set.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_set.o: $(TOMMATH_DIR)/bn_mp_set.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_set.c -bn_mp_set_int.o: $(TOMMATH_DIR)/bn_mp_set_int.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_set_int.o: $(TOMMATH_DIR)/bn_mp_set_int.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_set_int.c -bn_mp_shrink.o: $(TOMMATH_DIR)/bn_mp_shrink.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_shrink.o: $(TOMMATH_DIR)/bn_mp_shrink.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_shrink.c -bn_mp_sqr.o: $(TOMMATH_DIR)/bn_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sqr.o: $(TOMMATH_DIR)/bn_mp_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sqr.c -bn_mp_sqrt.o: $(TOMMATH_DIR)/bn_mp_sqrt.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sqrt.o: $(TOMMATH_DIR)/bn_mp_sqrt.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sqrt.c -bn_mp_sub.o: $(TOMMATH_DIR)/bn_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sub.o: $(TOMMATH_DIR)/bn_mp_sub.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sub.c -bn_mp_sub_d.o: $(TOMMATH_DIR)/bn_mp_sub_d.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_sub_d.o: $(TOMMATH_DIR)/bn_mp_sub_d.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_sub_d.c -bn_mp_to_unsigned_bin.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_to_unsigned_bin.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_to_unsigned_bin.c -bn_mp_to_unsigned_bin_n.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_to_unsigned_bin_n.o: $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_to_unsigned_bin_n.c -bn_mp_toom_mul.o: $(TOMMATH_DIR)/bn_mp_toom_mul.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_toom_mul.o: $(TOMMATH_DIR)/bn_mp_toom_mul.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toom_mul.c -bn_mp_toom_sqr.o: $(TOMMATH_DIR)/bn_mp_toom_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_toom_sqr.o: $(TOMMATH_DIR)/bn_mp_toom_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toom_sqr.c -bn_mp_toradix_n.o: $(TOMMATH_DIR)/bn_mp_toradix_n.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_toradix_n.o: $(TOMMATH_DIR)/bn_mp_toradix_n.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_toradix_n.c -bn_mp_unsigned_bin_size.o: $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_unsigned_bin_size.o: $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_unsigned_bin_size.c -bn_mp_xor.o: $(TOMMATH_DIR)/bn_mp_xor.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_xor.o: $(TOMMATH_DIR)/bn_mp_xor.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_xor.c -bn_mp_zero.o: $(TOMMATH_DIR)/bn_mp_zero.c $(MATHHDRS) $(DTRACE_HDR) +bn_mp_zero.o: $(TOMMATH_DIR)/bn_mp_zero.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_mp_zero.c -bn_s_mp_add.o: $(TOMMATH_DIR)/bn_s_mp_add.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_add.o: $(TOMMATH_DIR)/bn_s_mp_add.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_add.c -bn_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_s_mp_mul_digs.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_mul_digs.o: $(TOMMATH_DIR)/bn_s_mp_mul_digs.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_mul_digs.c -bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_sqr.o: $(TOMMATH_DIR)/bn_s_mp_sqr.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sqr.c -bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) $(DTRACE_HDR) +bn_s_mp_sub.o: $(TOMMATH_DIR)/bn_s_mp_sub.c $(MATHHDRS) $(CC) -c $(CC_SWITCHES) $(TOMMATH_DIR)/bn_s_mp_sub.c tclUnixChan.o: $(UNIX_DIR)/tclUnixChan.c $(IOHDR) @@ -1484,7 +1484,7 @@ tclWinError.o: $(TOP_DIR)/win/tclWinError.c # DTrace support -$(TCL_OBJS) $(STUB_LIB_OBJS) $(TCLSH_OBJS) $(TCLTEST_OBJS) $(XTTEST_OBJS): @DTRACE_HDR@ +$(TCL_OBJS) $(STUB_LIB_OBJS) $(TCLSH_OBJS) $(TCLTEST_OBJS) $(XTTEST_OBJS) $(TOMMATH_OBJS): @DTRACE_HDR@ $(DTRACE_HDR): $(DTRACE_SRC) $(DTRACE) -h $(DTRACE_SWITCHES) -o $@ -s $(DTRACE_SRC) -- cgit v0.12 From 77ccaefbb1482145de9b9276979318e88b9e542e Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Tue, 29 Mar 2016 10:17:02 +0000 Subject: Eliminate AT_FORK_INIT_VALUE/RESET_ATFORK_MUTEX macro's, since other values than the default are not supported anyway. This results in the elimination of (empty functions anyway) AtForkPrepare/AtForkParent. Also improve consistancy in some variable names. No change of functionality. --- unix/tclUnixNotfy.c | 70 ++++++----------------------------------------------- win/tclWinNotify.c | 8 +++--- win/tclWinSock.c | 17 +++---------- 3 files changed, 15 insertions(+), 80 deletions(-) diff --git a/unix/tclUnixNotfy.c b/unix/tclUnixNotfy.c index 1457890..3422089 100644 --- a/unix/tclUnixNotfy.c +++ b/unix/tclUnixNotfy.c @@ -1,7 +1,5 @@ -#define AT_FORK_INIT_VALUE 0 -#define RESET_ATFORK_MUTEX 1 /* - * tclUnixNotify.c -- + * tclUnixNotfy.c -- * * This file contains the implementation of the select()-based * Unix-specific notifier, which is the lowest-level part of the Tcl @@ -197,9 +195,7 @@ static Tcl_ThreadId notifierThread; #ifdef TCL_THREADS static void NotifierThreadProc(ClientData clientData); #if defined(HAVE_PTHREAD_ATFORK) -static int atForkInit = AT_FORK_INIT_VALUE; -static void AtForkPrepare(void); -static void AtForkParent(void); +static int atForkInit = 0; static void AtForkChild(void); #endif /* HAVE_PTHREAD_ATFORK */ #endif /* TCL_THREADS */ @@ -256,7 +252,7 @@ extern unsigned char __stdcall TranslateMessage(const MSG *); * Threaded-cygwin specific constants and functions in this file: */ -static const WCHAR NotfyClassName[] = L"TclNotifier"; +static const WCHAR className[] = L"TclNotifier"; static DWORD __stdcall NotifierProc(void *hwnd, unsigned int message, void *wParam, void *lParam); #endif /* TCL_THREADS && __CYGWIN__ */ @@ -345,7 +341,7 @@ Tcl_InitNotifier(void) class.hInstance = TclWinGetTclInstance(); class.hbrBackground = NULL; class.lpszMenuName = NULL; - class.lpszClassName = NotfyClassName; + class.lpszClassName = className; class.lpfnWndProc = NotifierProc; class.hIcon = NULL; class.hCursor = NULL; @@ -370,7 +366,7 @@ Tcl_InitNotifier(void) */ if (!atForkInit) { - int result = pthread_atfork(AtForkPrepare, AtForkParent, AtForkChild); + int result = pthread_atfork(NULL, NULL, AtForkChild); if (result) { Tcl_Panic("Tcl_InitNotifier: pthread_atfork failed"); @@ -1351,54 +1347,6 @@ NotifierThreadProc( /* *---------------------------------------------------------------------- * - * AtForkPrepare -- - * - * Lock the notifier in preparation for a fork. - * - * Results: - * None. - * - * Side effects: - * None. - * - *---------------------------------------------------------------------- - */ - -static void -AtForkPrepare(void) -{ -#if RESET_ATFORK_MUTEX == 0 - pthread_mutex_lock(¬ifierInitMutex); -#endif -} - -/* - *---------------------------------------------------------------------- - * - * AtForkParent -- - * - * Unlock the notifier in the parent after a fork. - * - * Results: - * None. - * - * Side effects: - * None. - * - *---------------------------------------------------------------------- - */ - -static void -AtForkParent(void) -{ -#if RESET_ATFORK_MUTEX == 0 - pthread_mutex_unlock(¬ifierInitMutex); -#endif -} - -/* - *---------------------------------------------------------------------- - * * AtForkChild -- * * Unlock and reinstall the notifier in the child after a fork. @@ -1418,12 +1366,8 @@ AtForkChild(void) if (notifierThreadRunning == 1) { pthread_cond_destroy(¬ifierCV); } -#if RESET_ATFORK_MUTEX == 0 - pthread_mutex_unlock(¬ifierInitMutex); -#else pthread_mutex_init(¬ifierInitMutex, NULL); pthread_mutex_init(¬ifierMutex, NULL); -#endif pthread_cond_init(¬ifierCV, NULL); /* @@ -1455,8 +1399,8 @@ AtForkChild(void) */ #ifdef __CYGWIN__ DestroyWindow(tsdPtr->hwnd); - tsdPtr->hwnd = CreateWindowExW(NULL, NotfyClassName, - NotfyClassName, 0, 0, 0, 0, 0, NULL, NULL, + tsdPtr->hwnd = CreateWindowExW(NULL, className, + className, 0, 0, 0, 0, 0, NULL, NULL, TclWinGetTclInstance(), NULL); ResetEvent(tsdPtr->event); #else diff --git a/win/tclWinNotify.c b/win/tclWinNotify.c index 4543b02..ea4035b 100644 --- a/win/tclWinNotify.c +++ b/win/tclWinNotify.c @@ -50,7 +50,7 @@ static Tcl_ThreadDataKey dataKey; */ static int notifierCount = 0; -static const TCHAR classname[] = TEXT("TclNotifier"); +static const TCHAR className[] = TEXT("TclNotifier"); TCL_DECLARE_MUTEX(notifierMutex) /* @@ -98,7 +98,7 @@ Tcl_InitNotifier(void) class.hInstance = TclWinGetTclInstance(); class.hbrBackground = NULL; class.lpszMenuName = NULL; - class.lpszClassName = classname; + class.lpszClassName = className; class.lpfnWndProc = NotifierProc; class.hIcon = NULL; class.hCursor = NULL; @@ -186,7 +186,7 @@ Tcl_FinalizeNotifier( Tcl_MutexLock(¬ifierMutex); notifierCount--; if (notifierCount == 0) { - UnregisterClass(classname, TclWinGetTclInstance()); + UnregisterClass(className, TclWinGetTclInstance()); } Tcl_MutexUnlock(¬ifierMutex); } @@ -350,7 +350,7 @@ Tcl_ServiceModeHook( */ if (mode == TCL_SERVICE_ALL && !tsdPtr->hwnd) { - tsdPtr->hwnd = CreateWindow(classname, classname, + tsdPtr->hwnd = CreateWindow(className, className, WS_TILED, 0, 0, 0, 0, NULL, NULL, TclWinGetTclInstance(), NULL); diff --git a/win/tclWinSock.c b/win/tclWinSock.c index a022ed5..c71aa96 100644 --- a/win/tclWinSock.c +++ b/win/tclWinSock.c @@ -62,15 +62,6 @@ #undef TCL_FEATURE_KEEPALIVE_NAGLE /* - * Make sure to remove the redirection defines set in tclWinPort.h that is in - * use in other sections of the core, except for us. - */ - -#undef getservbyname -#undef getsockopt -#undef setsockopt - -/* * Helper macros to make parts of this file clearer. The macros do exactly * what they say on the tin. :-) They also only ever refer to their arguments * once, and so can be used without regard to side effects. @@ -90,7 +81,7 @@ */ static int initialized = 0; -static const TCHAR classname[] = TEXT("TclSocket"); +static const TCHAR className[] = TEXT("TclSocket"); TCL_DECLARE_MUTEX(socketMutex) /* @@ -2336,7 +2327,7 @@ InitSockets(void) windowClass.hInstance = TclWinGetTclInstance(); windowClass.hbrBackground = NULL; windowClass.lpszMenuName = NULL; - windowClass.lpszClassName = classname; + windowClass.lpszClassName = className; windowClass.lpfnWndProc = SocketProc; windowClass.hIcon = NULL; windowClass.hCursor = NULL; @@ -2466,7 +2457,7 @@ SocketExitHandler( */ TclpFinalizeSockets(); - UnregisterClass(classname, TclWinGetTclInstance()); + UnregisterClass(className, TclWinGetTclInstance()); initialized = 0; Tcl_MutexUnlock(&socketMutex); } @@ -2992,7 +2983,7 @@ SocketThread( * Create a dummy window receiving socket events. */ - tsdPtr->hwnd = CreateWindow(classname, classname, WS_TILED, 0, 0, 0, 0, + tsdPtr->hwnd = CreateWindow(className, className, WS_TILED, 0, 0, 0, 0, NULL, NULL, windowClass.hInstance, arg); /* -- cgit v0.12 From dedc1d86a28c562377215bc2d0bf10e75ed00f1d Mon Sep 17 00:00:00 2001 From: dkf Date: Wed, 30 Mar 2016 08:46:43 +0000 Subject: [47ac84309b] Import of aspect's branch from his personal repository on chiselapp. --- generic/tclCmdIL.c | 2 +- generic/tclCompCmdsGR.c | 47 ++++++++++++++++++++++++++++++++++++++++------- tests/lreplace.test | 43 ++++++++++++++++++++++++++++++++++++++++--- 3 files changed, 81 insertions(+), 11 deletions(-) diff --git a/generic/tclCmdIL.c b/generic/tclCmdIL.c index 739dca9..0d35397 100644 --- a/generic/tclCmdIL.c +++ b/generic/tclCmdIL.c @@ -2755,7 +2755,7 @@ Tcl_LreplaceObjCmd( * (to allow for replacing the last elem). */ - if ((first >= listLen) && (listLen > 0)) { + if ((first > listLen) && (listLen > 0)) { Tcl_SetObjResult(interp, Tcl_ObjPrintf( "list doesn't contain element %s", TclGetString(objv[2]))); Tcl_SetErrorCode(interp, "TCL", "OPERATION", "LREPLACE", "BADIDX", diff --git a/generic/tclCompCmdsGR.c b/generic/tclCompCmdsGR.c index 9f430ea..ffe39ba 100644 --- a/generic/tclCompCmdsGR.c +++ b/generic/tclCompCmdsGR.c @@ -1489,6 +1489,15 @@ TclCompileLreplaceCmd( } /* + * idx1, idx2 are now in canonical form: + * + * - integer: [0,len+1] + * - end index: INDEX_END + * - -ive offset: INDEX_END-[len-1,0] + * - +ive offset: INDEX_END+1 + */ + + /* * Compilation fails when one index is end-based but the other isn't. * Fixing this will require more bytecodes, but this is a workaround for * now. [Bug 47ac84309b] @@ -1521,6 +1530,9 @@ TclCompileLreplaceCmd( idx1 = 0; goto dropEnd; } else { + if (idx2 < idx1) { + idx2 = idx1 - 1; + } if (idx1 > 0) { tmpObj = Tcl_NewIntObj(idx1); Tcl_IncrRefCount(tmpObj); @@ -1548,9 +1560,7 @@ TclCompileLreplaceCmd( idx1 = 0; goto replaceTail; } else { - if (idx1 > 0 && idx2 > 0 && idx2 < idx1) { - idx2 = idx1 - 1; - } else if (idx1 < 0 && idx2 < 0 && idx2 < idx1) { + if (idx2 < idx1) { idx2 = idx1 - 1; } if (idx1 > 0) { @@ -1566,7 +1576,7 @@ TclCompileLreplaceCmd( * operate on. */ - dropAll: + dropAll: /* This just ensures the arg is a list. */ TclEmitOpcode( INST_LIST_LENGTH, envPtr); TclEmitOpcode( INST_POP, envPtr); PushStringLiteral(envPtr, ""); @@ -1579,12 +1589,21 @@ TclCompileLreplaceCmd( dropRange: if (tmpObj != NULL) { + /* + * Emit bytecode to check the list length. + */ + TclEmitOpcode( INST_DUP, envPtr); TclEmitOpcode( INST_LIST_LENGTH, envPtr); TclEmitPush(TclAddLiteralObj(envPtr, tmpObj, NULL), envPtr); - TclEmitOpcode( INST_GT, envPtr); + TclEmitOpcode( INST_GE, envPtr); offset = CurrentOffset(envPtr); TclEmitInstInt1( INST_JUMP_TRUE1, 0, envPtr); + + /* + * Emit an error if we've been given an empty list. + */ + TclEmitOpcode( INST_DUP, envPtr); TclEmitOpcode( INST_LIST_LENGTH, envPtr); offset2 = CurrentOffset(envPtr); @@ -1635,16 +1654,30 @@ TclCompileLreplaceCmd( replaceRange: if (tmpObj != NULL) { + /* + * Emit bytecode to check the list length. + */ + TclEmitOpcode( INST_DUP, envPtr); TclEmitOpcode( INST_LIST_LENGTH, envPtr); + + /* + * Check the list length vs idx1. + */ + TclEmitPush(TclAddLiteralObj(envPtr, tmpObj, NULL), envPtr); - TclEmitOpcode( INST_GT, envPtr); + TclEmitOpcode( INST_GE, envPtr); offset = CurrentOffset(envPtr); TclEmitInstInt1( INST_JUMP_TRUE1, 0, envPtr); + + /* + * Emit an error if we've been given an empty list. + */ + TclEmitOpcode( INST_DUP, envPtr); TclEmitOpcode( INST_LIST_LENGTH, envPtr); offset2 = CurrentOffset(envPtr); - TclEmitInstInt1( INST_JUMP_TRUE1, 0, envPtr); + TclEmitInstInt1( INST_JUMP_FALSE1, 0, envPtr); TclEmitPush(TclAddLiteralObj(envPtr, Tcl_ObjPrintf( "list doesn't contain element %d", idx1), NULL), envPtr); CompileReturnInternal(envPtr, INST_RETURN_IMM, TCL_ERROR, 0, diff --git a/tests/lreplace.test b/tests/lreplace.test index 55a36a8..d7f8226 100644 --- a/tests/lreplace.test +++ b/tests/lreplace.test @@ -98,7 +98,12 @@ test lreplace-1.26 {lreplace command} { [set foo [lreplace $foo end end]] \ [set foo [lreplace $foo end end]] } {a {} {}} - +test lreplace-1.27 {lreplace command} { + lreplace x 1 1 +} x +test lreplace-1.28 {lreplace command} { + lreplace x 1 1 y +} {x y} test lreplace-2.1 {lreplace errors} { list [catch lreplace msg] $msg @@ -119,8 +124,8 @@ test lreplace-2.6 {lreplace errors} { list [catch {lreplace x 3 2} msg] $msg } {1 {list doesn't contain element 3}} test lreplace-2.7 {lreplace errors} { - list [catch {lreplace x 1 1} msg] $msg -} {1 {list doesn't contain element 1}} + list [catch {lreplace x 2 2} msg] $msg +} {1 {list doesn't contain element 2}} test lreplace-3.1 {lreplace won't modify shared argument objects} { proc p {} { @@ -181,6 +186,12 @@ test lreplace-4.11 {lreplace end index first} { test lreplace-4.12 {lreplace end index first} { lreplace {0 1 2 3 4} end-2 2 a b c } {0 1 a b c 3 4} +test lreplace-4.13 {lreplace empty list} { + lreplace {} 1 1 1 +} 1 +test lreplace-4.14 {lreplace empty list} { + lreplace {} 2 2 2 +} 2 test lreplace-5.1 {compiled lreplace: Bug 47ac84309b} { apply {x { @@ -192,6 +203,32 @@ test lreplace-5.2 {compiled lreplace: Bug 47ac84309b} { lreplace $x end 0 A }} {a b c} } {a b A c} + +# Testing for compiled behaviour. Far too many variations to check with +# spelt-out tests. Note that this *just* checks whether the compiled version +# and the interpreted version are the same, not whether the interpreted +# version is correct. +apply {{} { + set lss {{} {a} {a b c} {a b c d}} + set ins {{} A {A B}} + set idxs {-2 -1 0 1 2 3 end-3 end-2 end-1 end end+1 end+2} + set lreplace lreplace + + foreach ls $lss { + foreach a $idxs { + foreach b $idxs { + foreach i $ins { + set expected [list [catch {$lreplace $ls $a $b {*}$i} m] $m] + set tester [list lreplace $ls $a $b {*}$i] + set script [list catch $tester m] + set script "list \[$script\] \$m" + test lreplace-6.[incr n] {lreplace battery} \ + [list apply [list {} $script]] $expected + } + } + } + } +}} # cleanup catch {unset foo} -- cgit v0.12 From d1d3c40c1a9a1ee762a59b4f429bd05315440914 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Fri, 1 Apr 2016 13:00:25 +0000 Subject: RFE [0ef5e653ff4caf5f896ec1182e0aac38ab9a0c46|0ef5e653]: Add nsf to coffbase.txt --- win/coffbase.txt | 1 + 1 file changed, 1 insertion(+) diff --git a/win/coffbase.txt b/win/coffbase.txt index 0ebe18a..3314f26 100644 --- a/win/coffbase.txt +++ b/win/coffbase.txt @@ -34,6 +34,7 @@ tclsdl 0x10B20000 0x00080000 vqtcl 0x10C00000 0x00010000 tdbc 0x10C40000 0x00010000 thread 0x10C80000 0x00020000 +nsf 0x10ca0000 0x00080000 ; ; insert new packages here ; -- cgit v0.12 From 514885892ef31d0ce71e8b1740b7ef838f459d99 Mon Sep 17 00:00:00 2001 From: dkf Date: Sun, 3 Apr 2016 11:04:23 +0000 Subject: Working on fixing problem with short reads from compressing streams that breaks PNG creation in Tk. --- tests/zlib.test | 18 ++++++++++++++++++ 1 file changed, 18 insertions(+) diff --git a/tests/zlib.test b/tests/zlib.test index 7a486ba..93c00f1 100644 --- a/tests/zlib.test +++ b/tests/zlib.test @@ -875,6 +875,24 @@ test zlib-11.3 {Bug 3595576 variant} -setup { } -cleanup { removeFile $file } -returnCodes error -result {can't set "noSuchNs::foo": parent namespace doesn't exist} + +test zlib-12.1 {Tk Bug 9eb55debc5} -constraints zlib -setup { + set stream [zlib stream compress] +} -body { + set opts {} + for {set y 0} {$y < 60} {incr y} { + for {set line {};set x 0} {$x < 100} {incr x} { + append line [binary format ccc $x $y 128] + } + if {$y == 59} { + set opts -finalize + } + $stream put {*}$opts $line + } + string length [$stream get] +} -cleanup { + $stream close +} -result 12026 ::tcltest::cleanupTests return -- cgit v0.12 From 082197d5c8f618d1cf78dffd199915806079b7de Mon Sep 17 00:00:00 2001 From: dkf Date: Mon, 4 Apr 2016 10:03:40 +0000 Subject: Was handling the flushing at the end of the stream wrongly. --- generic/tclZlib.c | 9 +++++---- tests/zlib.test | 8 ++++---- 2 files changed, 9 insertions(+), 8 deletions(-) diff --git a/generic/tclZlib.c b/generic/tclZlib.c index 50d9a30..691d57a 100644 --- a/generic/tclZlib.c +++ b/generic/tclZlib.c @@ -1194,11 +1194,12 @@ Tcl_ZlibStreamPut( zshPtr->stream.next_out = (Bytef *) dataTmp; e = deflate(&zshPtr->stream, flush); - while (e == Z_BUF_ERROR) { + while (e == Z_BUF_ERROR || (flush == Z_FINISH && e == Z_OK)) { /* - * Output buffer too small to hold the data being generated; so - * put a new buffer into place after saving the old generated - * data to the outData list. + * Output buffer too small to hold the data being generated or we + * are doing the end-of-stream flush (which can spit out masses of + * data). This means we need to put a new buffer into place after + * saving the old generated data to the outData list. */ obj = Tcl_NewByteArrayObj((unsigned char *) dataTmp, outSize); diff --git a/tests/zlib.test b/tests/zlib.test index 93c00f1..968469d 100644 --- a/tests/zlib.test +++ b/tests/zlib.test @@ -879,8 +879,7 @@ test zlib-11.3 {Bug 3595576 variant} -setup { test zlib-12.1 {Tk Bug 9eb55debc5} -constraints zlib -setup { set stream [zlib stream compress] } -body { - set opts {} - for {set y 0} {$y < 60} {incr y} { + for {set opts {};set y 0} {$y < 60} {incr y} { for {set line {};set x 0} {$x < 100} {incr x} { append line [binary format ccc $x $y 128] } @@ -889,10 +888,11 @@ test zlib-12.1 {Tk Bug 9eb55debc5} -constraints zlib -setup { } $stream put {*}$opts $line } - string length [$stream get] + set data [$stream get] + list [string length $data] [string length [zlib decompress $data]] } -cleanup { $stream close -} -result 12026 +} -result {12026 18000} ::tcltest::cleanupTests return -- cgit v0.12 From 5b2c3db492538b64ee77ef4873492a9a101f55bc Mon Sep 17 00:00:00 2001 From: dgp Date: Mon, 4 Apr 2016 15:43:49 +0000 Subject: Merge *both* commits to get the TclAsyncReady optimization. Without both parts, the test interp-34.3.1 hangs. --- generic/tclExecute.c | 27 +++++++++++++++++---------- 1 file changed, 17 insertions(+), 10 deletions(-) diff --git a/generic/tclExecute.c b/generic/tclExecute.c index d4077f5..823f619 100644 --- a/generic/tclExecute.c +++ b/generic/tclExecute.c @@ -35,14 +35,14 @@ #endif /* - * A mask (should be 2**n-1) that is used to work out when the bytecode engine - * should call Tcl_AsyncReady() to see whether there is a signal that needs - * handling. + * A counter that is used to work out when the bytecode engine should call + * Tcl_AsyncReady() to see whether there is a signal that needs handling, and + * other expensive periodic operations. */ -#ifndef ASYNC_CHECK_COUNT_MASK -# define ASYNC_CHECK_COUNT_MASK 63 -#endif /* !ASYNC_CHECK_COUNT_MASK */ +#ifndef ASYNC_CHECK_COUNT +# define ASYNC_CHECK_COUNT 64 +#endif /* !ASYNC_CHECK_COUNT */ /* * Boolean flag indicating whether the Tcl bytecode interpreter has been @@ -2115,8 +2115,14 @@ TEBCresume( * sporadically: no special need for speed. */ - int instructionCount = 0; /* Counter that is used to work out when to - * call Tcl_AsyncReady() */ + unsigned interruptCounter = 1; + /* Counter that is used to work out when to + * call Tcl_AsyncReady(). This must be 1 + * initially so that we call the async-check + * stanza early, otherwise there are command + * sequences that can make the interpreter + * busy-loop without an opportunity to + * recognise an interrupt. */ const char *curInstName; #ifdef TCL_COMPILE_DEBUG int traceInstructions; /* Whether we are doing instruction-level @@ -2314,10 +2320,11 @@ TEBCresume( /* * Check for asynchronous handlers [Bug 746722]; we do the check every - * ASYNC_CHECK_COUNT_MASK instruction, of the form (2**n-1). + * ASYNC_CHECK_COUNT instructions. */ - if ((instructionCount++ & ASYNC_CHECK_COUNT_MASK) == 0) { + if ((--interruptCounter) == 0) { + interruptCounter = ASYNC_CHECK_COUNT; DECACHE_STACK_INFO(); if (TclAsyncReady(iPtr)) { result = Tcl_AsyncInvoke(interp, result); -- cgit v0.12 From 7032562591b12990b1b45ad8815bb30513a4d8e1 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Tue, 5 Apr 2016 09:32:43 +0000 Subject: Rename UtfCount() to TclUtfCount() and use it in more places. Suggested by pspjuth here: [e99a79a32650e7e5] --- generic/tclInt.h | 1 + generic/tclStringObj.c | 4 ++-- generic/tclUtf.c | 22 ++++++++-------------- 3 files changed, 11 insertions(+), 16 deletions(-) diff --git a/generic/tclInt.h b/generic/tclInt.h index 34430c8..1dab0cb 100644 --- a/generic/tclInt.h +++ b/generic/tclInt.h @@ -3133,6 +3133,7 @@ MODULE_SCOPE int TclTrimLeft(const char *bytes, int numBytes, MODULE_SCOPE int TclTrimRight(const char *bytes, int numBytes, const char *trim, int numTrim); MODULE_SCOPE int TclUtfCasecmp(const char *cs, const char *ct); +MODULE_SCOPE int TclUtfCount(int ch); MODULE_SCOPE Tcl_Obj * TclpNativeToNormalized(ClientData clientData); MODULE_SCOPE Tcl_Obj * TclpFilesystemPathType(Tcl_Obj *pathPtr); MODULE_SCOPE int TclpDlopen(Tcl_Interp *interp, Tcl_Obj *pathPtr, diff --git a/generic/tclStringObj.c b/generic/tclStringObj.c index e718749..b480735 100644 --- a/generic/tclStringObj.c +++ b/generic/tclStringObj.c @@ -2967,7 +2967,7 @@ ExtendStringRepWithUnicode( */ int i, origLength, size = 0; - char *dst, buf[TCL_UTF_MAX]; + char *dst; String *stringPtr = GET_STRING(objPtr); if (numChars < 0) { @@ -2993,7 +2993,7 @@ ExtendStringRepWithUnicode( } for (i = 0; i < numChars && size >= 0; i++) { - size += Tcl_UniCharToUtf((int) unicode[i], buf); + size += TclUtfCount(unicode[i]); } if (size < 0) { Tcl_Panic("max size for a Tcl value (%d bytes) exceeded", INT_MAX); diff --git a/generic/tclUtf.c b/generic/tclUtf.c index b878149..6c4cb7f 100644 --- a/generic/tclUtf.c +++ b/generic/tclUtf.c @@ -84,17 +84,11 @@ static const unsigned char totalBytes[256] = { 1,1,1,1 #endif }; - -/* - * Functions used only in this module. - */ - -static int UtfCount(int ch); /* *--------------------------------------------------------------------------- * - * UtfCount -- + * TclUtfCount -- * * Find the number of bytes in the Utf character "ch". * @@ -107,8 +101,8 @@ static int UtfCount(int ch); *--------------------------------------------------------------------------- */ -INLINE static int -UtfCount( +int +TclUtfCount( int ch) /* The Tcl_UniChar whose size is returned. */ { if ((ch > 0) && (ch < UNICODE_SELF)) { @@ -143,7 +137,7 @@ UtfCount( *--------------------------------------------------------------------------- */ -INLINE int +int Tcl_UniCharToUtf( int ch, /* The Tcl_UniChar to be stored in the * buffer. */ @@ -829,7 +823,7 @@ Tcl_UtfToUpper( * char to dst if its size is <= the original char. */ - if (bytes < UtfCount(upChar)) { + if (bytes < TclUtfCount(upChar)) { memcpy(dst, src, (size_t) bytes); dst += bytes; } else { @@ -882,7 +876,7 @@ Tcl_UtfToLower( * char to dst if its size is <= the original char. */ - if (bytes < UtfCount(lowChar)) { + if (bytes < TclUtfCount(lowChar)) { memcpy(dst, src, (size_t) bytes); dst += bytes; } else { @@ -932,7 +926,7 @@ Tcl_UtfToTitle( bytes = TclUtfToUniChar(src, &ch); titleChar = Tcl_UniCharToTitle(ch); - if (bytes < UtfCount(titleChar)) { + if (bytes < TclUtfCount(titleChar)) { memcpy(dst, src, (size_t) bytes); dst += bytes; } else { @@ -944,7 +938,7 @@ Tcl_UtfToTitle( bytes = TclUtfToUniChar(src, &ch); lowChar = Tcl_UniCharToLower(ch); - if (bytes < UtfCount(lowChar)) { + if (bytes < TclUtfCount(lowChar)) { memcpy(dst, src, (size_t) bytes); dst += bytes; } else { -- cgit v0.12 From 365f647c5fa50a235d51f440e9ecc29ecb014607 Mon Sep 17 00:00:00 2001 From: dgp Date: Wed, 6 Apr 2016 18:24:05 +0000 Subject: [213b6a2b9d] Make level parsing honor EIAS. --- generic/tclProc.c | 118 +++++++++++++++++++++-------------------------------- tests/uplevel.test | 99 ++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 145 insertions(+), 72 deletions(-) diff --git a/generic/tclProc.c b/generic/tclProc.c index ac65bde..9770d26 100644 --- a/generic/tclProc.c +++ b/generic/tclProc.c @@ -808,94 +808,68 @@ TclObjGetFrame( register Interp *iPtr = (Interp *) interp; int curLevel, level, result; CallFrame *framePtr; - const char *name; + const char *name = NULL; /* * Parse object to figure out which level number to go to. */ - result = 1; + result = 0; curLevel = iPtr->varFramePtr->level; - if (objPtr == NULL) { - name = "1"; - goto haveLevel1; - } - name = TclGetString(objPtr); - if (objPtr->typePtr == &levelReferenceType) { - if (objPtr->internalRep.twoPtrValue.ptr1) { - level = curLevel - PTR2INT(objPtr->internalRep.twoPtrValue.ptr2); - } else { - level = PTR2INT(objPtr->internalRep.twoPtrValue.ptr2); - } - if (level < 0) { - goto levelError; - } - /* TODO: Consider skipping the typePtr checks */ - } else if (objPtr->typePtr == &tclIntType -#ifndef TCL_WIDE_INT_IS_LONG - || objPtr->typePtr == &tclWideIntType -#endif - ) { - if (TclGetIntFromObj(NULL, objPtr, &level) != TCL_OK || level < 0) { - goto levelError; - } - level = curLevel - level; - } else if (*name == '#') { - if (Tcl_GetInt(interp, name+1, &level) != TCL_OK || level < 0) { - goto levelError; - } - - /* - * Cache for future reference. - */ - - TclFreeIntRep(objPtr); - objPtr->typePtr = &levelReferenceType; - objPtr->internalRep.twoPtrValue.ptr1 = (void *) 0; - objPtr->internalRep.twoPtrValue.ptr2 = INT2PTR(level); - } else if (isdigit(UCHAR(*name))) { /* INTL: digit */ - if (Tcl_GetInt(interp, name, &level) != TCL_OK) { - return -1; - } - - /* - * Cache for future reference. - */ + /* + * Check for integer first, since that has potential to spare us + * a generation of a stringrep. + */ - TclFreeIntRep(objPtr); - objPtr->typePtr = &levelReferenceType; - objPtr->internalRep.twoPtrValue.ptr1 = (void *) 1; - objPtr->internalRep.twoPtrValue.ptr2 = INT2PTR(level); + if (objPtr == NULL) { + /* Do nothing */ + } else if (TCL_OK == Tcl_GetIntFromObj(NULL, objPtr, &level) + && (level >= 0)) { level = curLevel - level; + result = 1; + } else if (objPtr->typePtr == &levelReferenceType) { + level = PTR2INT(objPtr->internalRep.twoPtrValue.ptr1); + result = 1; } else { - /* - * Don't cache as the object *isn't* a level reference (might even be - * NULL...) - */ + name = TclGetString(objPtr); + if (name[0] == '#') { + if (TCL_OK == Tcl_GetInt(NULL, name+1, &level) && level >= 0) { + TclFreeIntRep(objPtr); + objPtr->typePtr = &levelReferenceType; + objPtr->internalRep.twoPtrValue.ptr1 = INT2PTR(level); + result = 1; + } else { + result = -1; + } + } else if (isdigit(UCHAR(name[0]))) { /* INTL: digit */ + /* + * If this were an integer, we'd have succeeded already. + * Docs say we have to treat this as a 'bad level' error. + */ + result = -1; + } + } - haveLevel1: + if (result == 0) { level = curLevel - 1; - result = 0; + name = "1"; } - - /* - * Figure out which frame to use, and return it to the caller. - */ - - for (framePtr = iPtr->varFramePtr; framePtr != NULL; - framePtr = framePtr->callerVarPtr) { - if (framePtr->level == level) { - break; + if (result != -1) { + if (level >= 0) { + for (framePtr = iPtr->varFramePtr; framePtr != NULL; + framePtr = framePtr->callerVarPtr) { + if (framePtr->level == level) { + *framePtrPtr = framePtr; + return result; + } + } + } + if (name == NULL) { + name = TclGetString(objPtr); } } - if (framePtr == NULL) { - goto levelError; - } - *framePtrPtr = framePtr; - return result; - levelError: Tcl_SetObjResult(interp, Tcl_ObjPrintf("bad level \"%s\"", name)); Tcl_SetErrorCode(interp, "TCL", "VALUE", "STACKLEVEL", NULL); return -1; diff --git a/tests/uplevel.test b/tests/uplevel.test index 0410469..9ecc0d5 100644 --- a/tests/uplevel.test +++ b/tests/uplevel.test @@ -101,6 +101,105 @@ test uplevel-4.4 {error: not enough args} -returnCodes error -body { uplevel 1 }} } -result {wrong # args: should be "uplevel ?level? command ?arg ...?"} +test uplevel-4.5 {level parsing} { + apply {{} {uplevel 0 {}}} +} {} +test uplevel-4.6 {level parsing} { + apply {{} {uplevel #0 {}}} +} {} +test uplevel-4.7 {level parsing} { + apply {{} {uplevel [expr 0] {}}} +} {} +test uplevel-4.8 {level parsing} { + apply {{} {uplevel #[expr 0] {}}} +} {} +test uplevel-4.9 {level parsing} { + apply {{} {uplevel -0 {}}} +} {} +test uplevel-4.10 {level parsing} { + apply {{} {uplevel #-0 {}}} +} {} +test uplevel-4.11 {level parsing} { + apply {{} {uplevel [expr -0] {}}} +} {} +test uplevel-4.12 {level parsing} { + apply {{} {uplevel #[expr -0] {}}} +} {} +test uplevel-4.13 {level parsing} { + apply {{} {uplevel 1 {}}} +} {} +test uplevel-4.14 {level parsing} { + apply {{} {uplevel #1 {}}} +} {} +test uplevel-4.15 {level parsing} { + apply {{} {uplevel [expr 1] {}}} +} {} +test uplevel-4.16 {level parsing} { + apply {{} {uplevel #[expr 1] {}}} +} {} +test uplevel-4.17 {level parsing} { + apply {{} {uplevel -0xffffffff {}}} +} {} +test uplevel-4.18 {level parsing} { + apply {{} {uplevel #-0xffffffff {}}} +} {} +test uplevel-4.19 {level parsing} { + apply {{} {uplevel [expr -0xffffffff] {}}} +} {} +test uplevel-4.20 {level parsing} { + apply {{} {uplevel #[expr -0xffffffff] {}}} +} {} +test uplevel-4.21 {level parsing} -body { + apply {{} {uplevel -1 {}}} +} -returnCodes error -result {invalid command name "-1"} +test uplevel-4.22 {level parsing} -body { + apply {{} {uplevel #-1 {}}} +} -returnCodes error -result {bad level "#-1"} +test uplevel-4.23 {level parsing} -body { + apply {{} {uplevel [expr -1] {}}} +} -returnCodes error -result {invalid command name "-1"} +test uplevel-4.24 {level parsing} -body { + apply {{} {uplevel #[expr -1] {}}} +} -returnCodes error -result {bad level "#-1"} +test uplevel-4.25 {level parsing} -body { + apply {{} {uplevel 0xffffffff {}}} +} -returnCodes error -result {bad level "0xffffffff"} +test uplevel-4.26 {level parsing} -body { + apply {{} {uplevel #0xffffffff {}}} +} -returnCodes error -result {bad level "#0xffffffff"} +test uplevel-4.27 {level parsing} -body { + apply {{} {uplevel [expr 0xffffffff] {}}} +} -returnCodes error -result {bad level "4294967295"} +test uplevel-4.28 {level parsing} -body { + apply {{} {uplevel #[expr 0xffffffff] {}}} +} -returnCodes error -result {bad level "#4294967295"} +test uplevel-4.29 {level parsing} -body { + apply {{} {uplevel 0.2 {}}} +} -returnCodes error -result {bad level "0.2"} +test uplevel-4.30 {level parsing} -body { + apply {{} {uplevel #0.2 {}}} +} -returnCodes error -result {bad level "#0.2"} +test uplevel-4.31 {level parsing} -body { + apply {{} {uplevel [expr 0.2] {}}} +} -returnCodes error -result {bad level "0.2"} +test uplevel-4.32 {level parsing} -body { + apply {{} {uplevel #[expr 0.2] {}}} +} -returnCodes error -result {bad level "#0.2"} +test uplevel-4.33 {level parsing} -body { + apply {{} {uplevel .2 {}}} +} -returnCodes error -result {invalid command name ".2"} +test uplevel-4.34 {level parsing} -body { + apply {{} {uplevel #.2 {}}} +} -returnCodes error -result {bad level "#.2"} +test uplevel-4.35 {level parsing} -body { + apply {{} {uplevel [expr .2] {}}} +} -returnCodes error -result {bad level "0.2"} +test uplevel-4.36 {level parsing} -body { + apply {{} {uplevel #[expr .2] {}}} +} -returnCodes error -result {bad level "#0.2"} + + + proc a2 {} { uplevel a3 -- cgit v0.12 From b6e7e4e4283f3809245d72b487eb5cc65cc6e95b Mon Sep 17 00:00:00 2001 From: dgp Date: Thu, 7 Apr 2016 14:54:39 +0000 Subject: Tidy up the last commit. --- generic/tclProc.c | 13 ++++++------- 1 file changed, 6 insertions(+), 7 deletions(-) diff --git a/generic/tclProc.c b/generic/tclProc.c index 9770d26..172b860 100644 --- a/generic/tclProc.c +++ b/generic/tclProc.c @@ -69,9 +69,8 @@ const Tcl_ObjType tclProcBodyType = { }; /* - * The [upvar]/[uplevel] level reference type. Uses the twoPtrValue field, - * encoding the type of level reference in ptr and the actual parsed out - * offset in ptr2. + * The [upvar]/[uplevel] level reference type. Uses the longValue field + * to remember the integer value of a parsed # format. * * Uses the default behaviour throughout, and never disposes of the string * rep; it's just a cache type. @@ -785,7 +784,7 @@ TclGetFrame( * Results: * The return value is -1 if an error occurred in finding the frame (in * this case an error message is left in the interp's result). 1 is - * returned if objPtr was either a number or a number preceded by "#" and + * returned if objPtr was either an int or an int preceded by "#" and * it specified a valid frame. 0 is returned if objPtr isn't one of the * two things above (in this case, the lookup acts as if objPtr were * "1"). The variable pointed to by framePtrPtr is filled in with the @@ -807,7 +806,6 @@ TclObjGetFrame( { register Interp *iPtr = (Interp *) interp; int curLevel, level, result; - CallFrame *framePtr; const char *name = NULL; /* @@ -829,7 +827,7 @@ TclObjGetFrame( level = curLevel - level; result = 1; } else if (objPtr->typePtr == &levelReferenceType) { - level = PTR2INT(objPtr->internalRep.twoPtrValue.ptr1); + level = (int) objPtr->internalRep.longValue; result = 1; } else { name = TclGetString(objPtr); @@ -837,7 +835,7 @@ TclObjGetFrame( if (TCL_OK == Tcl_GetInt(NULL, name+1, &level) && level >= 0) { TclFreeIntRep(objPtr); objPtr->typePtr = &levelReferenceType; - objPtr->internalRep.twoPtrValue.ptr1 = INT2PTR(level); + objPtr->internalRep.longValue = level; result = 1; } else { result = -1; @@ -857,6 +855,7 @@ TclObjGetFrame( } if (result != -1) { if (level >= 0) { + CallFrame *framePtr; for (framePtr = iPtr->varFramePtr; framePtr != NULL; framePtr = framePtr->callerVarPtr) { if (framePtr->level == level) { -- cgit v0.12 From 78b187fe71dba0f3710be978050e08fde81efd85 Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Fri, 8 Apr 2016 12:23:50 +0000 Subject: Fix [8663689908d3304a74fee525cd04aa4162e86391|8663689908d3]: regexp \\w missing characters --- generic/regc_lex.c | 19 ++++++++++++++++--- tests/utf.test | 2 +- 2 files changed, 17 insertions(+), 4 deletions(-) diff --git a/generic/regc_lex.c b/generic/regc_lex.c index 8d07c59..51cf8a4 100644 --- a/generic/regc_lex.c +++ b/generic/regc_lex.c @@ -256,20 +256,33 @@ static const chr brbacks[] = { /* \s within brackets */ CHR('s'), CHR('p'), CHR('a'), CHR('c'), CHR('e'), CHR(':'), CHR(']') }; + +#define PUNCT_CONN \ + CHR('_'), \ + 0x203f /* UNDERTIE */, \ + 0x2040 /* CHARACTER TIE */,\ + 0x2054 /* INVERTED UNDERTIE */,\ + 0xfe33 /* PRESENTATION FORM FOR VERTICAL LOW LINE */, \ + 0xfe34 /* PRESENTATION FORM FOR VERTICAL WAVY LOW LINE */, \ + 0xfe4d /* DASHED LOW LINE */, \ + 0xfe4e /* CENTRELINE LOW LINE */, \ + 0xfe4f /* WAVY LOW LINE */, \ + 0xff3f /* FULLWIDTH LOW LINE */ + static const chr backw[] = { /* \w */ CHR('['), CHR('['), CHR(':'), CHR('a'), CHR('l'), CHR('n'), CHR('u'), CHR('m'), - CHR(':'), CHR(']'), CHR('_'), CHR(']') + CHR(':'), CHR(']'), PUNCT_CONN, CHR(']') }; static const chr backW[] = { /* \W */ CHR('['), CHR('^'), CHR('['), CHR(':'), CHR('a'), CHR('l'), CHR('n'), CHR('u'), CHR('m'), - CHR(':'), CHR(']'), CHR('_'), CHR(']') + CHR(':'), CHR(']'), PUNCT_CONN, CHR(']') }; static const chr brbackw[] = { /* \w within brackets */ CHR('['), CHR(':'), CHR('a'), CHR('l'), CHR('n'), CHR('u'), CHR('m'), - CHR(':'), CHR(']'), CHR('_') + CHR(':'), CHR(']'), PUNCT_CONN }; /* diff --git a/tests/utf.test b/tests/utf.test index 30200c1..e8ee374 100644 --- a/tests/utf.test +++ b/tests/utf.test @@ -283,7 +283,7 @@ test utf-21.1 {TclUniCharIsAlnum} { } {1} test utf-21.2 {unicode alnum char in regc_locale.c} { # this returns 1 with Unicode 7 compliance - list [regexp {^[[:alnum:]]+$} \u1040\u021f\u0220] [regexp {^\w+$} \u1040\u021f\u0220] + list [regexp {^[[:alnum:]]+$} \u1040\u021f\u0220] [regexp {^\w+$} \u1040\u021f\u0220_\u203f\u2040\u2054\ufe33\ufe34\ufe4d\ufe4e\ufe4f\uff3f] } {1 1} test utf-21.3 {unicode print char in regc_locale.c} { # this returns 1 with Unicode 7 compliance -- cgit v0.12 From daaf750d95e22d23e47f846f89b0d7033b4eb0bd Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Fri, 8 Apr 2016 13:27:57 +0000 Subject: Fix [2538f373ffc78d6dba9c3d973c147a84fdd9bbd8|2538f373ffc78d6d]: crash in Tcl_OpenTcpServer() on Windows --- win/tclWinSock.c | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/win/tclWinSock.c b/win/tclWinSock.c index a022ed5..da2e60a 100644 --- a/win/tclWinSock.c +++ b/win/tclWinSock.c @@ -2061,7 +2061,6 @@ Tcl_OpenTcpServer( char channelName[SOCK_CHAN_LENGTH]; u_long flag = 1; /* Indicates nonblocking mode. */ const char *errorMsg = NULL; - ThreadSpecificData *tsdPtr = TclThreadDataKeyGet(&dataKey); if (TclpHasSockets(interp) != TCL_OK) { return NULL; @@ -2177,6 +2176,7 @@ error: } if (statePtr != NULL) { + ThreadSpecificData *tsdPtr = TclThreadDataKeyGet(&dataKey); statePtr->acceptProc = acceptProc; statePtr->acceptProcData = acceptProcData; -- cgit v0.12 From 63b9112f89a2f6938b767af98c7747fcbc167f2a Mon Sep 17 00:00:00 2001 From: "jan.nijtmans" Date: Sun, 10 Apr 2016 15:58:19 +0000 Subject: Fix [07d13d99b0a9]: Who broke TCL 8.6 and Tclblend ? --- generic/tclObj.c | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/generic/tclObj.c b/generic/tclObj.c index c641152..628c3a7 100644 --- a/generic/tclObj.c +++ b/generic/tclObj.c @@ -320,7 +320,7 @@ const Tcl_HashKeyType tclObjHashKeyType = { * does allow them to delete a command when references to it are gone, which * is fragile but useful given their somewhat-OO style. Because of this, this * structure MUST NOT be const so that the C compiler puts the data in - * writable memory. [Bug 2558422] + * writable memory. [Bug 2558422] [Bug 07d13d99b0a9] * TODO: Provide a better API for those extensions so that they can coexist... */ @@ -4176,7 +4176,8 @@ Tcl_GetCommandFromObj( * had is invalid one way or another. */ - if (SetCmdNameFromAny(interp, objPtr) != TCL_OK) { + /* See [] why we cannot call SetCmdNameFromAny() directly here. */ + if (tclCmdNameType.setFromAnyProc(interp, objPtr) != TCL_OK) { return NULL; } resPtr = objPtr->internalRep.twoPtrValue.ptr1; -- cgit v0.12 From 5f08f4b52618056de417f6d543d5bd596d9197eb Mon Sep 17 00:00:00 2001 From: dgp Date: Mon, 11 Apr 2016 17:29:47 +0000 Subject: [d1f55451c6] Remove unnecessary panic routines. --- generic/tclVar.c | 30 ++---------------------------- 1 file changed, 2 insertions(+), 28 deletions(-) diff --git a/generic/tclVar.c b/generic/tclVar.c index be035f7..b5b5b10 100644 --- a/generic/tclVar.c +++ b/generic/tclVar.c @@ -202,14 +202,10 @@ MODULE_SCOPE Var * TclLookupSimpleVar(Tcl_Interp *interp, static Tcl_DupInternalRepProc DupLocalVarName; static Tcl_FreeInternalRepProc FreeLocalVarName; -static Tcl_UpdateStringProc PanicOnUpdateVarName; static Tcl_FreeInternalRepProc FreeParsedVarName; static Tcl_DupInternalRepProc DupParsedVarName; -static Tcl_UpdateStringProc PanicOnUpdateVarName; -static Tcl_SetFromAnyProc PanicOnSetVarName; - /* * Types of Tcl_Objs used to cache variable lookups. * @@ -231,12 +227,12 @@ static Tcl_SetFromAnyProc PanicOnSetVarName; static const Tcl_ObjType localVarNameType = { "localVarName", - FreeLocalVarName, DupLocalVarName, PanicOnUpdateVarName, PanicOnSetVarName + FreeLocalVarName, DupLocalVarName, NULL, NULL }; static const Tcl_ObjType tclParsedVarNameType = { "parsedVarName", - FreeParsedVarName, DupParsedVarName, PanicOnUpdateVarName, PanicOnSetVarName + FreeParsedVarName, DupParsedVarName, NULL, NULL }; /* @@ -5505,28 +5501,6 @@ TclObjVarErrMsg( */ /* - * Panic functions that should never be called in normal operation. - */ - -static void -PanicOnUpdateVarName( - Tcl_Obj *objPtr) -{ - Tcl_Panic("%s of type %s should not be called", "updateStringProc", - objPtr->typePtr->name); -} - -static int -PanicOnSetVarName( - Tcl_Interp *interp, - Tcl_Obj *objPtr) -{ - Tcl_Panic("%s of type %s should not be called", "setFromAnyProc", - objPtr->typePtr->name); - return TCL_ERROR; -} - -/* * localVarName - * * INTERNALREP DEFINITION: -- cgit v0.12 From 9a625d23f10c30494d1e132eb1bb57bf7486d26b Mon Sep 17 00:00:00 2001 From: gahr Date: Wed, 20 Apr 2016 19:27:31 +0000 Subject: Implement msec and usec verbosity levels in tcltest::configure --- doc/tcltest.n | 8 ++++++-- library/tcltest/tcltest.tcl | 36 ++++++++++++++++++++++++++++++++---- 2 files changed, 38 insertions(+), 6 deletions(-) diff --git a/doc/tcltest.n b/doc/tcltest.n index cedc763..ac8b73b 100644 --- a/doc/tcltest.n +++ b/doc/tcltest.n @@ -872,8 +872,8 @@ harness are doing. . Sets the type of output verbosity desired to \fIlevel\fR, a list of zero or more of the elements \fBbody\fR, \fBpass\fR, -\fBskip\fR, \fBstart\fR, \fBerror\fR and \fBline\fR. Default value -is +\fBskip\fR, \fBstart\fR, \fBerror\fR, \fBline\fR, \fBmsec\fR and \fBusec\fR. +Default value is .QW "\fBbody error\fR" . Levels are defined as: .RS @@ -890,6 +890,10 @@ Print errorInfo and errorCode, if they exist, when a test return code does not match its expected return code .IP "line (\fBl\fR)" Print source file line information of failed tests +.IP "msec (\fBm\fR)" +Print each test's execution time in milliseconds +.IP "usec (\fBu\fR)" +Print each test's execution time in microseconds .PP The single letter abbreviations noted above are also recognized so that diff --git a/library/tcltest/tcltest.tcl b/library/tcltest/tcltest.tcl index 29ef778..30965b8 100644 --- a/library/tcltest/tcltest.tcl +++ b/library/tcltest/tcltest.tcl @@ -611,16 +611,27 @@ namespace eval tcltest { proc AcceptVerbose { level } { set level [AcceptList $level] + set levelMap { + l list + p pass + b body + s skip + t start + e error + l line + m msec + u usec + } + set levelRegexp "^([join [dict values $levelMap] |])\$" if {[llength $level] == 1} { - if {![regexp {^(pass|body|skip|start|error|line)$} $level]} { + if {![regexp $levelRegexp $level]} { # translate single characters abbreviations to expanded list - set level [string map {p pass b body s skip t start e error l line} \ - [split $level {}]] + set level [string map $levelMap [split $level {}]] } } set valid [list] foreach v $level { - if {[regexp {^(pass|body|skip|start|error|line)$} $v]} { + if {[regexp $levelRegexp $v]} { lappend valid $v } } @@ -1972,6 +1983,14 @@ proc tcltest::test {name description args} { # Only run the test body if the setup was successful if {!$setupFailure} { + # Register startup time + if {[IsVerbose msec]} { + set msStart [clock milliseconds] + } + if {[IsVerbose usec]} { + set usStart [clock microseconds] + } + # Verbose notification of $body start if {[IsVerbose start]} { puts [outputChannel] "---- $name start" @@ -2076,6 +2095,15 @@ proc tcltest::test {name description args} { } } + if {[IsVerbose msec]} { + set elapsed [expr {[clock milliseconds] - $msStart}] + puts [outputChannel] "++++ $name took $elapsed ms" + } + if {[IsVerbose usec]} { + set elapsed [expr {[clock microseconds] - $usStart}] + puts [outputChannel] "++++ $name took $elapsed μs" + } + # if we didn't experience any failures, then we passed variable numTests if {!($setupFailure || $cleanupFailure || $coreFailure -- cgit v0.12 From 46069c41c59226ee28c31faca3a66afc9582bc31 Mon Sep 17 00:00:00 2001 From: gahr Date: Wed, 20 Apr 2016 19:37:31 +0000 Subject: Bump tcltest version to 2.3.9 --- library/tcltest/pkgIndex.tcl | 2 +- library/tcltest/tcltest.tcl | 2 +- unix/Makefile.in | 4 ++-- win/Makefile.in | 4 ++-- 4 files changed, 6 insertions(+), 6 deletions(-) diff --git a/library/tcltest/pkgIndex.tcl b/library/tcltest/pkgIndex.tcl index 987725f..06097fd 100644 --- a/library/tcltest/pkgIndex.tcl +++ b/library/tcltest/pkgIndex.tcl @@ -9,4 +9,4 @@ # full path name of this file's directory. if {![package vsatisfies [package provide Tcl] 8.5]} {return} -package ifneeded tcltest 2.3.8 [list source [file join $dir tcltest.tcl]] +package ifneeded tcltest 2.3.9 [list source [file join $dir tcltest.tcl]] diff --git a/library/tcltest/tcltest.tcl b/library/tcltest/tcltest.tcl index 30965b8..3d8ae5f 100644 --- a/library/tcltest/tcltest.tcl +++ b/library/tcltest/tcltest.tcl @@ -22,7 +22,7 @@ namespace eval tcltest { # When the version number changes, be sure to update the pkgIndex.tcl file, # and the install directory in the Makefiles. When the minor version # changes (new feature) be sure to update the man page as well. - variable Version 2.3.8 + variable Version 2.3.9 # Compatibility support for dumb variables defined in tcltest 1 # Do not use these. Call [package provide Tcl] and [info patchlevel] diff --git a/unix/Makefile.in b/unix/Makefile.in index d9f8bbc..2c96973 100644 --- a/unix/Makefile.in +++ b/unix/Makefile.in @@ -849,8 +849,8 @@ install-libraries: libraries done; @echo "Installing package msgcat 1.6.0 as a Tcl Module"; @$(INSTALL_DATA) $(TOP_DIR)/library/msgcat/msgcat.tcl "$(SCRIPT_INSTALL_DIR)"/../tcl8/8.5/msgcat-1.6.0.tm; - @echo "Installing package tcltest 2.3.8 as a Tcl Module"; - @$(INSTALL_DATA) $(TOP_DIR)/library/tcltest/tcltest.tcl "$(SCRIPT_INSTALL_DIR)"/../tcl8/8.5/tcltest-2.3.8.tm; + @echo "Installing package tcltest 2.3.9 as a Tcl Module"; + @$(INSTALL_DATA) $(TOP_DIR)/library/tcltest/tcltest.tcl "$(SCRIPT_INSTALL_DIR)"/../tcl8/8.5/tcltest-2.3.9.tm; @echo "Installing package platform 1.0.14 as a Tcl Module"; @$(INSTALL_DATA) $(TOP_DIR)/library/platform/platform.tcl "$(SCRIPT_INSTALL_DIR)"/../tcl8/8.4/platform-1.0.14.tm; diff --git a/win/Makefile.in b/win/Makefile.in index 2d27a41..1368671 100644 --- a/win/Makefile.in +++ b/win/Makefile.in @@ -660,8 +660,8 @@ install-libraries: libraries install-tzdata install-msgs done; @echo "Installing package msgcat 1.6.0 as a Tcl Module"; @$(COPY) $(ROOT_DIR)/library/msgcat/msgcat.tcl $(SCRIPT_INSTALL_DIR)/../tcl8/8.5/msgcat-1.6.0.tm; - @echo "Installing package tcltest 2.3.8 as a Tcl Module"; - @$(COPY) $(ROOT_DIR)/library/tcltest/tcltest.tcl $(SCRIPT_INSTALL_DIR)/../tcl8/8.5/tcltest-2.3.8.tm; + @echo "Installing package tcltest 2.3.9 as a Tcl Module"; + @$(COPY) $(ROOT_DIR)/library/tcltest/tcltest.tcl $(SCRIPT_INSTALL_DIR)/../tcl8/8.5/tcltest-2.3.9.tm; @echo "Installing package platform 1.0.14 as a Tcl Module"; @$(COPY) $(ROOT_DIR)/library/platform/platform.tcl $(SCRIPT_INSTALL_DIR)/../tcl8/8.4/platform-1.0.14.tm; @echo "Installing package platform::shell 1.1.4 as a Tcl Module"; -- cgit v0.12 From b46c53fec03b48c96092ef01990c03a00891fb8c Mon Sep 17 00:00:00 2001 From: dgp Date: Fri, 22 Apr 2016 18:13:17 +0000 Subject: Refactor bytecode cleanup. --- generic/tclAssembly.c | 8 ++------ generic/tclCompile.c | 41 +++++++++++++++++++++++++++-------------- generic/tclCompile.h | 3 ++- generic/tclExecute.c | 17 +++++------------ 4 files changed, 36 insertions(+), 33 deletions(-) diff --git a/generic/tclAssembly.c b/generic/tclAssembly.c index 6d5676b..7db5d69 100644 --- a/generic/tclAssembly.c +++ b/generic/tclAssembly.c @@ -866,7 +866,7 @@ CompileAssembleObj( * Not valid, so free it and regenerate. */ - FreeAssembleCodeInternalRep(objPtr); + TclFreeIntRep(objPtr); } /* @@ -4313,11 +4313,7 @@ FreeAssembleCodeInternalRep( { ByteCode *codePtr = objPtr->internalRep.twoPtrValue.ptr1; - codePtr->refCount--; - if (codePtr->refCount <= 0) { - TclCleanupByteCode(codePtr); - } - objPtr->typePtr = NULL; + TclReleaseByteCode(codePtr); } /* diff --git a/generic/tclCompile.c b/generic/tclCompile.c index c0b5dcc..002b551 100644 --- a/generic/tclCompile.c +++ b/generic/tclCompile.c @@ -661,6 +661,7 @@ InstructionDesc const tclInstructionTable[] = { * Prototypes for procedures defined later in this file: */ +static void CleanupByteCode(ByteCode *codePtr); static ByteCode * CompileSubstObj(Tcl_Interp *interp, Tcl_Obj *objPtr, int flags); static void DupByteCodeInternalRep(Tcl_Obj *srcPtr, @@ -967,16 +968,13 @@ FreeByteCodeInternalRep( { register ByteCode *codePtr = objPtr->internalRep.twoPtrValue.ptr1; - objPtr->typePtr = NULL; - if (codePtr->refCount-- <= 1) { - TclCleanupByteCode(codePtr); - } + TclReleaseByteCode(codePtr); } /* *---------------------------------------------------------------------- * - * TclCleanupByteCode -- + * TclReleaseByteCode -- * * This procedure does all the real work of freeing up a bytecode * object's ByteCode structure. It's called only when the structure's @@ -993,7 +991,26 @@ FreeByteCodeInternalRep( */ void -TclCleanupByteCode( +TclPreserveByteCode( + register ByteCode *codePtr) +{ + codePtr->refCount++; +} + +void +TclReleaseByteCode( + register ByteCode *codePtr) +{ + if (--codePtr->refCount) { + return; + } + + /* Just dropped to refcount==0. Clean up. */ + CleanupByteCode(codePtr); +} + +static void +CleanupByteCode( register ByteCode *codePtr) /* Points to the ByteCode to free. */ { Tcl_Interp *interp = (Tcl_Interp *) *codePtr->interpHandle; @@ -1260,8 +1277,6 @@ Tcl_NRSubstObj( * * Results: * A (ByteCode *) is returned pointing to the resulting ByteCode. - * The caller must manage its refCount and arrange for a call to - * TclCleanupByteCode() when the last reference disappears. * * Side effects: * The Tcl_ObjType of objPtr is changed to the "substcode" type, and the @@ -1292,7 +1307,7 @@ CompileSubstObj( || (codePtr->nsEpoch != nsPtr->resolverEpoch) || (codePtr->localCachePtr != iPtr->varFramePtr->localCachePtr)) { - FreeSubstCodeInternalRep(objPtr); + TclFreeIntRep(objPtr); } } if (objPtr->typePtr != &substCodeType) { @@ -1353,10 +1368,7 @@ FreeSubstCodeInternalRep( { register ByteCode *codePtr = objPtr->internalRep.twoPtrValue.ptr1; - objPtr->typePtr = NULL; - if (codePtr->refCount-- <= 1) { - TclCleanupByteCode(codePtr); - } + TclReleaseByteCode(codePtr); } static void @@ -2752,7 +2764,8 @@ TclInitByteCodeObj( codePtr->compileEpoch = iPtr->compileEpoch; codePtr->nsPtr = namespacePtr; codePtr->nsEpoch = namespacePtr->resolverEpoch; - codePtr->refCount = 1; + codePtr->refCount = 0; + TclPreserveByteCode(codePtr); if (namespacePtr->compiledVarResProc || iPtr->resolverPtr) { codePtr->flags = TCL_BYTECODE_RESOLVE_VARS; } else { diff --git a/generic/tclCompile.h b/generic/tclCompile.h index d5bc86b..86a9db0 100644 --- a/generic/tclCompile.h +++ b/generic/tclCompile.h @@ -1067,7 +1067,6 @@ MODULE_SCOPE ByteCode * TclCompileObj(Tcl_Interp *interp, Tcl_Obj *objPtr, MODULE_SCOPE int TclAttemptCompileProc(Tcl_Interp *interp, Tcl_Parse *parsePtr, int depth, Command *cmdPtr, CompileEnv *envPtr); -MODULE_SCOPE void TclCleanupByteCode(ByteCode *codePtr); MODULE_SCOPE void TclCleanupStackForBreakContinue(CompileEnv *envPtr, ExceptionAux *auxPtr); MODULE_SCOPE void TclCompileCmdWord(Tcl_Interp *interp, @@ -1157,6 +1156,8 @@ MODULE_SCOPE void TclPushVarName(Tcl_Interp *interp, Tcl_Token *varTokenPtr, CompileEnv *envPtr, int flags, int *localIndexPtr, int *isScalarPtr); +MODULE_SCOPE void TclPreserveByteCode(ByteCode *codePtr); +MODULE_SCOPE void TclReleaseByteCode(ByteCode *codePtr); MODULE_SCOPE void TclReleaseLiteral(Tcl_Interp *interp, Tcl_Obj *objPtr); MODULE_SCOPE void TclInvalidateCmdLiteral(Tcl_Interp *interp, const char *name, Namespace *nsPtr); diff --git a/generic/tclExecute.c b/generic/tclExecute.c index 823f619..2a2b951 100644 --- a/generic/tclExecute.c +++ b/generic/tclExecute.c @@ -1499,11 +1499,9 @@ ExprObjCallback( * * Results: * A (ByteCode *) is returned pointing to the resulting ByteCode. - * The caller must manage its refCount and arrange for a call to - * TclCleanupByteCode() when the last reference disappears. * * Side effects: - * The Tcl_ObjType of objPtr is changed to the "bytecode" type, + * The Tcl_ObjType of objPtr is changed to the "exprcode" type, * and the ByteCode is kept in the internal rep (along with context * data for checking validity) for faster operations the next time * CompileExprObj is called on the same value. @@ -1536,7 +1534,7 @@ CompileExprObj( || (codePtr->nsPtr != namespacePtr) || (codePtr->nsEpoch != namespacePtr->resolverEpoch) || (codePtr->localCachePtr != iPtr->varFramePtr->localCachePtr)) { - FreeExprCodeInternalRep(objPtr); + TclFreeIntRep(objPtr); } } if (objPtr->typePtr != &exprCodeType) { @@ -1644,10 +1642,7 @@ FreeExprCodeInternalRep( { ByteCode *codePtr = objPtr->internalRep.twoPtrValue.ptr1; - objPtr->typePtr = NULL; - if (codePtr->refCount-- <= 1) { - TclCleanupByteCode(codePtr); - } + TclReleaseByteCode(codePtr); } /* @@ -2033,7 +2028,7 @@ TclNRExecuteByteCode( * sizeof(void *); int numWords = (size + sizeof(Tcl_Obj *) - 1) / sizeof(Tcl_Obj *); - codePtr->refCount++; + TclPreserveByteCode(codePtr); /* * Reserve the stack, setup the TEBCdataPtr (TD) and CallFrame @@ -8189,9 +8184,7 @@ TEBCresume( } iPtr->cmdFramePtr = bcFramePtr->nextPtr; - if (codePtr->refCount-- <= 1) { - TclCleanupByteCode(codePtr); - } + TclReleaseByteCode(codePtr); TclStackFree(interp, TD); /* free my stack */ return result; -- cgit v0.12 From 3f9f0e85ff34afdd6cfbc82c66b5e670d0deb4bb Mon Sep 17 00:00:00 2001 From: venkat Date: Fri, 22 Apr 2016 19:43:59 +0000 Subject: Update to tzdata2016d from IETF --- library/tzdata/America/Caracas | 1 + library/tzdata/Asia/Almaty | 103 ++++++++++++++++++------------------ library/tzdata/Asia/Anadyr | 4 +- library/tzdata/Asia/Aqtau | 107 +++++++++++++++++++------------------- library/tzdata/Asia/Aqtobe | 105 +++++++++++++++++++------------------ library/tzdata/Asia/Baku | 2 +- library/tzdata/Asia/Barnaul | 4 +- library/tzdata/Asia/Chita | 4 +- library/tzdata/Asia/Irkutsk | 4 +- library/tzdata/Asia/Kamchatka | 4 +- library/tzdata/Asia/Khandyga | 4 +- library/tzdata/Asia/Krasnoyarsk | 4 +- library/tzdata/Asia/Magadan | 5 +- library/tzdata/Asia/Novokuznetsk | 4 +- library/tzdata/Asia/Novosibirsk | 4 +- library/tzdata/Asia/Omsk | 4 +- library/tzdata/Asia/Oral | 106 ++++++++++++++++++------------------- library/tzdata/Asia/Qyzylorda | 105 ++++++++++++++++++------------------- library/tzdata/Asia/Sakhalin | 4 +- library/tzdata/Asia/Srednekolymsk | 4 +- library/tzdata/Asia/Tomsk | 73 ++++++++++++++++++++++++++ library/tzdata/Asia/Ust-Nera | 4 +- library/tzdata/Asia/Vladivostok | 4 +- library/tzdata/Asia/Yakutsk | 4 +- library/tzdata/Asia/Yekaterinburg | 4 +- library/tzdata/Asia/Yerevan | 6 +-- library/tzdata/Europe/Astrakhan | 5 +- library/tzdata/Europe/Kaliningrad | 4 +- library/tzdata/Europe/Kirov | 70 +++++++++++++++++++++++++ library/tzdata/Europe/Minsk | 3 +- library/tzdata/Europe/Moscow | 4 +- library/tzdata/Europe/Samara | 4 +- library/tzdata/Europe/Ulyanovsk | 4 +- library/tzdata/Europe/Volgograd | 5 +- 34 files changed, 463 insertions(+), 313 deletions(-) create mode 100644 library/tzdata/Asia/Tomsk create mode 100644 library/tzdata/Europe/Kirov diff --git a/library/tzdata/America/Caracas b/library/tzdata/America/Caracas index 2ba87ae..253c4ce 100644 --- a/library/tzdata/America/Caracas +++ b/library/tzdata/America/Caracas @@ -6,4 +6,5 @@ set TZData(:America/Caracas) { {-1826739140 -16200 0 VET} {-157750200 -14400 0 VET} {1197183600 -16200 0 VET} + {1462086000 -14400 0 VET} } diff --git a/library/tzdata/Asia/Almaty b/library/tzdata/Asia/Almaty index 68dee29..2b83197 100644 --- a/library/tzdata/Asia/Almaty +++ b/library/tzdata/Asia/Almaty @@ -2,55 +2,56 @@ set TZData(:Asia/Almaty) { {-9223372036854775808 18468 0 LMT} - {-1441170468 18000 0 ALMT} - {-1247547600 21600 0 ALMT} - {354909600 25200 1 ALMST} - {370717200 21600 0 ALMT} - {386445600 25200 1 ALMST} - {402253200 21600 0 ALMT} - {417981600 25200 1 ALMST} - {433789200 21600 0 ALMT} - {449604000 25200 1 ALMST} - {465336000 21600 0 ALMT} - {481060800 25200 1 ALMST} - {496785600 21600 0 ALMT} - {512510400 25200 1 ALMST} - {528235200 21600 0 ALMT} - {543960000 25200 1 ALMST} - {559684800 21600 0 ALMT} - {575409600 25200 1 ALMST} - {591134400 21600 0 ALMT} - {606859200 25200 1 ALMST} - {622584000 21600 0 ALMT} - {638308800 25200 1 ALMST} - {654638400 21600 0 ALMT} - {662666400 21600 0 ALMT} - {694202400 21600 0 ALMT} - {701802000 25200 1 ALMST} - {717523200 21600 0 ALMT} - {733262400 25200 1 ALMST} - {748987200 21600 0 ALMT} - {764712000 25200 1 ALMST} - {780436800 21600 0 ALMT} - {796161600 25200 1 ALMST} - {811886400 21600 0 ALMT} - {828216000 25200 1 ALMST} - {846360000 21600 0 ALMT} - {859665600 25200 1 ALMST} - {877809600 21600 0 ALMT} - {891115200 25200 1 ALMST} - {909259200 21600 0 ALMT} - {922564800 25200 1 ALMST} - {941313600 21600 0 ALMT} - {954014400 25200 1 ALMST} - {972763200 21600 0 ALMT} - {985464000 25200 1 ALMST} - {1004212800 21600 0 ALMT} - {1017518400 25200 1 ALMST} - {1035662400 21600 0 ALMT} - {1048968000 25200 1 ALMST} - {1067112000 21600 0 ALMT} - {1080417600 25200 1 ALMST} - {1099166400 21600 0 ALMT} - {1110823200 21600 0 ALMT} + {-1441170468 18000 0 +05} + {-1247547600 21600 0 +06} + {354909600 25200 1 +07} + {370717200 21600 0 +06} + {386445600 25200 1 +07} + {402253200 21600 0 +06} + {417981600 25200 1 +07} + {433789200 21600 0 +06} + {449604000 25200 1 +07} + {465336000 21600 0 +06} + {481060800 25200 1 +07} + {496785600 21600 0 +06} + {512510400 25200 1 +07} + {528235200 21600 0 +06} + {543960000 25200 1 +07} + {559684800 21600 0 +06} + {575409600 25200 1 +07} + {591134400 21600 0 +06} + {606859200 25200 1 +07} + {622584000 21600 0 +06} + {638308800 25200 1 +07} + {654638400 21600 0 +06} + {670363200 18000 0 +05} + {670366800 21600 1 +06} + {686091600 18000 0 +05} + {695768400 21600 0 +06} + {701812800 25200 1 +07} + {717537600 21600 0 +06} + {733262400 25200 1 +07} + {748987200 21600 0 +06} + {764712000 25200 1 +07} + {780436800 21600 0 +06} + {796161600 25200 1 +07} + {811886400 21600 0 +06} + {828216000 25200 1 +07} + {846360000 21600 0 +06} + {859665600 25200 1 +07} + {877809600 21600 0 +06} + {891115200 25200 1 +07} + {909259200 21600 0 +06} + {922564800 25200 1 +07} + {941313600 21600 0 +06} + {954014400 25200 1 +07} + {972763200 21600 0 +06} + {985464000 25200 1 +07} + {1004212800 21600 0 +06} + {1017518400 25200 1 +07} + {1035662400 21600 0 +06} + {1048968000 25200 1 +07} + {1067112000 21600 0 +06} + {1080417600 25200 1 +07} + {1099166400 21600 0 +06} } diff --git a/library/tzdata/Asia/Anadyr b/library/tzdata/Asia/Anadyr index 50ace50..2e8ffc7 100644 --- a/library/tzdata/Asia/Anadyr +++ b/library/tzdata/Asia/Anadyr @@ -29,8 +29,8 @@ set TZData(:Asia/Anadyr) { {670345200 43200 1 ANAST} {686070000 39600 0 ANAT} {695746800 43200 0 ANAMMTT} - {701780400 46800 1 ANAST} - {717501600 43200 0 ANAT} + {701791200 46800 1 ANAST} + {717516000 43200 0 ANAT} {733240800 46800 1 ANAST} {748965600 43200 0 ANAT} {764690400 46800 1 ANAST} diff --git a/library/tzdata/Asia/Aqtau b/library/tzdata/Asia/Aqtau index 11e89a2..90cc94d 100644 --- a/library/tzdata/Asia/Aqtau +++ b/library/tzdata/Asia/Aqtau @@ -2,57 +2,58 @@ set TZData(:Asia/Aqtau) { {-9223372036854775808 12064 0 LMT} - {-1441164064 14400 0 FORT} - {-1247544000 18000 0 FORT} - {-220942800 18000 0 SHET} - {370724400 21600 0 SHET} - {386445600 18000 0 SHET} - {386449200 21600 1 SHEST} - {402256800 18000 0 SHET} - {417985200 21600 1 SHEST} - {433792800 18000 0 SHET} - {449607600 21600 1 SHEST} - {465339600 18000 0 SHET} - {481064400 21600 1 SHEST} - {496789200 18000 0 SHET} - {512514000 21600 1 SHEST} - {528238800 18000 0 SHET} - {543963600 21600 1 SHEST} - {559688400 18000 0 SHET} - {575413200 21600 1 SHEST} - {591138000 18000 0 SHET} - {606862800 21600 1 SHEST} - {622587600 18000 0 SHET} - {638312400 21600 1 SHEST} - {654642000 18000 0 SHET} - {662670000 18000 0 SHET} - {692823600 18000 0 AQTT} - {701805600 21600 1 AQTST} - {717526800 18000 0 AQTT} - {733266000 21600 1 AQTST} - {748990800 18000 0 AQTT} - {764715600 21600 1 AQTST} - {780440400 18000 0 AQTT} - {796165200 14400 0 AQTT} - {796168800 18000 1 AQTST} - {811893600 14400 0 AQTT} - {828223200 18000 1 AQTST} - {846367200 14400 0 AQTT} - {859672800 18000 1 AQTST} - {877816800 14400 0 AQTT} - {891122400 18000 1 AQTST} - {909266400 14400 0 AQTT} - {922572000 18000 1 AQTST} - {941320800 14400 0 AQTT} - {954021600 18000 1 AQTST} - {972770400 14400 0 AQTT} - {985471200 18000 1 AQTST} - {1004220000 14400 0 AQTT} - {1017525600 18000 1 AQTST} - {1035669600 14400 0 AQTT} - {1048975200 18000 1 AQTST} - {1067119200 14400 0 AQTT} - {1080424800 18000 1 AQTST} - {1099173600 14400 0 AQTT} - {1110830400 18000 0 AQTT} + {-1441164064 14400 0 +04} + {-1247544000 18000 0 +05} + {-220942800 18000 0 +05} + {370724400 21600 0 +06} + {386445600 18000 0 +05} + {386449200 21600 1 +06} + {402256800 18000 0 +05} + {417985200 21600 1 +06} + {433792800 18000 0 +05} + {449607600 21600 1 +06} + {465339600 18000 0 +05} + {481064400 21600 1 +06} + {496789200 18000 0 +05} + {512514000 21600 1 +06} + {528238800 18000 0 +05} + {543963600 21600 1 +06} + {559688400 18000 0 +05} + {575413200 21600 1 +06} + {591138000 18000 0 +05} + {606862800 21600 1 +06} + {622587600 18000 0 +05} + {638312400 21600 1 +06} + {654642000 18000 0 +05} + {670366800 14400 0 +04} + {670370400 18000 1 +05} + {686095200 14400 0 +04} + {695772000 18000 0 +05} + {701816400 21600 1 +06} + {717541200 18000 0 +05} + {733266000 21600 1 +06} + {748990800 18000 0 +05} + {764715600 21600 1 +06} + {780440400 18000 0 +05} + {780444000 14400 0 +04} + {796168800 18000 1 +05} + {811893600 14400 0 +04} + {828223200 18000 1 +05} + {846367200 14400 0 +04} + {859672800 18000 1 +05} + {877816800 14400 0 +04} + {891122400 18000 1 +05} + {909266400 14400 0 +04} + {922572000 18000 1 +05} + {941320800 14400 0 +04} + {954021600 18000 1 +05} + {972770400 14400 0 +04} + {985471200 18000 1 +05} + {1004220000 14400 0 +04} + {1017525600 18000 1 +05} + {1035669600 14400 0 +04} + {1048975200 18000 1 +05} + {1067119200 14400 0 +04} + {1080424800 18000 1 +05} + {1099173600 18000 0 +05} } diff --git a/library/tzdata/Asia/Aqtobe b/library/tzdata/Asia/Aqtobe index c857491..55ef556 100644 --- a/library/tzdata/Asia/Aqtobe +++ b/library/tzdata/Asia/Aqtobe @@ -2,56 +2,57 @@ set TZData(:Asia/Aqtobe) { {-9223372036854775808 13720 0 LMT} - {-1441165720 14400 0 AKTT} - {-1247544000 18000 0 AKTT} - {354913200 21600 1 AKTST} - {370720800 21600 0 AKTT} - {386445600 18000 0 AKTT} - {386449200 21600 1 AKTST} - {402256800 18000 0 AKTT} - {417985200 21600 1 AKTST} - {433792800 18000 0 AKTT} - {449607600 21600 1 AKTST} - {465339600 18000 0 AKTT} - {481064400 21600 1 AKTST} - {496789200 18000 0 AKTT} - {512514000 21600 1 AKTST} - {528238800 18000 0 AKTT} - {543963600 21600 1 AKTST} - {559688400 18000 0 AKTT} - {575413200 21600 1 AKTST} - {591138000 18000 0 AKTT} - {606862800 21600 1 AKTST} - {622587600 18000 0 AKTT} - {638312400 21600 1 AKTST} - {654642000 18000 0 AKTT} - {662670000 18000 0 AKTT} - {692823600 18000 0 AQTT} - {701805600 21600 1 AQTST} - {717526800 18000 0 AQTT} - {733266000 21600 1 AQTST} - {748990800 18000 0 AQTT} - {764715600 21600 1 AQTST} - {780440400 18000 0 AQTT} - {796165200 21600 1 AQTST} - {811890000 18000 0 AQTT} - {828219600 21600 1 AQTST} - {846363600 18000 0 AQTT} - {859669200 21600 1 AQTST} - {877813200 18000 0 AQTT} - {891118800 21600 1 AQTST} - {909262800 18000 0 AQTT} - {922568400 21600 1 AQTST} - {941317200 18000 0 AQTT} - {954018000 21600 1 AQTST} - {972766800 18000 0 AQTT} - {985467600 21600 1 AQTST} - {1004216400 18000 0 AQTT} - {1017522000 21600 1 AQTST} - {1035666000 18000 0 AQTT} - {1048971600 21600 1 AQTST} - {1067115600 18000 0 AQTT} - {1080421200 21600 1 AQTST} - {1099170000 18000 0 AQTT} - {1110826800 18000 0 AQTT} + {-1441165720 14400 0 +04} + {-1247544000 18000 0 +05} + {354913200 21600 1 +06} + {370720800 21600 0 +06} + {386445600 18000 0 +05} + {386449200 21600 1 +06} + {402256800 18000 0 +05} + {417985200 21600 1 +06} + {433792800 18000 0 +05} + {449607600 21600 1 +06} + {465339600 18000 0 +05} + {481064400 21600 1 +06} + {496789200 18000 0 +05} + {512514000 21600 1 +06} + {528238800 18000 0 +05} + {543963600 21600 1 +06} + {559688400 18000 0 +05} + {575413200 21600 1 +06} + {591138000 18000 0 +05} + {606862800 21600 1 +06} + {622587600 18000 0 +05} + {638312400 21600 1 +06} + {654642000 18000 0 +05} + {670366800 14400 0 +04} + {670370400 18000 1 +05} + {686095200 14400 0 +04} + {695772000 18000 0 +05} + {701816400 21600 1 +06} + {717541200 18000 0 +05} + {733266000 21600 1 +06} + {748990800 18000 0 +05} + {764715600 21600 1 +06} + {780440400 18000 0 +05} + {796165200 21600 1 +06} + {811890000 18000 0 +05} + {828219600 21600 1 +06} + {846363600 18000 0 +05} + {859669200 21600 1 +06} + {877813200 18000 0 +05} + {891118800 21600 1 +06} + {909262800 18000 0 +05} + {922568400 21600 1 +06} + {941317200 18000 0 +05} + {954018000 21600 1 +06} + {972766800 18000 0 +05} + {985467600 21600 1 +06} + {1004216400 18000 0 +05} + {1017522000 21600 1 +06} + {1035666000 18000 0 +05} + {1048971600 21600 1 +06} + {1067115600 18000 0 +05} + {1080421200 21600 1 +06} + {1099170000 18000 0 +05} } diff --git a/library/tzdata/Asia/Baku b/library/tzdata/Asia/Baku index c26a2f5..bc0701a 100644 --- a/library/tzdata/Asia/Baku +++ b/library/tzdata/Asia/Baku @@ -27,7 +27,7 @@ set TZData(:Asia/Baku) { {670370400 14400 1 BAKST} {683496000 14400 0 AZST} {686098800 10800 0 AZT} - {701812800 14400 1 AZST} + {701823600 14400 1 AZST} {717537600 14400 0 AZT} {820440000 14400 0 AZT} {828234000 18000 1 AZST} diff --git a/library/tzdata/Asia/Barnaul b/library/tzdata/Asia/Barnaul index f6a45da..bf6abbf 100644 --- a/library/tzdata/Asia/Barnaul +++ b/library/tzdata/Asia/Barnaul @@ -28,8 +28,8 @@ set TZData(:Asia/Barnaul) { {670363200 25200 1 +07} {686088000 21600 0 +06} {695764800 25200 0 +08} - {701798400 28800 1 +08} - {717519600 25200 0 +07} + {701809200 28800 1 +08} + {717534000 25200 0 +07} {733258800 28800 1 +08} {748983600 25200 0 +07} {764708400 28800 1 +08} diff --git a/library/tzdata/Asia/Chita b/library/tzdata/Asia/Chita index 6aef523..2517beb 100644 --- a/library/tzdata/Asia/Chita +++ b/library/tzdata/Asia/Chita @@ -28,8 +28,8 @@ set TZData(:Asia/Chita) { {670356000 32400 1 YAKST} {686080800 28800 0 YAKT} {695757600 32400 0 YAKMMTT} - {701791200 36000 1 YAKST} - {717512400 32400 0 YAKT} + {701802000 36000 1 YAKST} + {717526800 32400 0 YAKT} {733251600 36000 1 YAKST} {748976400 32400 0 YAKT} {764701200 36000 1 YAKST} diff --git a/library/tzdata/Asia/Irkutsk b/library/tzdata/Asia/Irkutsk index 08e5798..ebe00c3 100644 --- a/library/tzdata/Asia/Irkutsk +++ b/library/tzdata/Asia/Irkutsk @@ -29,8 +29,8 @@ set TZData(:Asia/Irkutsk) { {670359600 28800 1 IRKST} {686084400 25200 0 IRKT} {695761200 28800 0 IRKMMTT} - {701794800 32400 1 IRKST} - {717516000 28800 0 IRKT} + {701805600 32400 1 IRKST} + {717530400 28800 0 IRKT} {733255200 32400 1 IRKST} {748980000 28800 0 IRKT} {764704800 32400 1 IRKST} diff --git a/library/tzdata/Asia/Kamchatka b/library/tzdata/Asia/Kamchatka index 82abcfa..2b77154 100644 --- a/library/tzdata/Asia/Kamchatka +++ b/library/tzdata/Asia/Kamchatka @@ -28,8 +28,8 @@ set TZData(:Asia/Kamchatka) { {670345200 43200 1 PETST} {686070000 39600 0 PETT} {695746800 43200 0 PETMMTT} - {701780400 46800 1 PETST} - {717501600 43200 0 PETT} + {701791200 46800 1 PETST} + {717516000 43200 0 PETT} {733240800 46800 1 PETST} {748965600 43200 0 PETT} {764690400 46800 1 PETST} diff --git a/library/tzdata/Asia/Khandyga b/library/tzdata/Asia/Khandyga index b2dc97a..b898e0d 100644 --- a/library/tzdata/Asia/Khandyga +++ b/library/tzdata/Asia/Khandyga @@ -28,8 +28,8 @@ set TZData(:Asia/Khandyga) { {670356000 32400 1 YAKST} {686080800 28800 0 YAKT} {695757600 32400 0 YAKMMTT} - {701791200 36000 1 YAKST} - {717512400 32400 0 YAKT} + {701802000 36000 1 YAKST} + {717526800 32400 0 YAKT} {733251600 36000 1 YAKST} {748976400 32400 0 YAKT} {764701200 36000 1 YAKST} diff --git a/library/tzdata/Asia/Krasnoyarsk b/library/tzdata/Asia


Period:
Authors:

Commits per author per week (path 'PC/VC6/python.dsp')

Addition and Subtraction

Multipliers

Exptmod

Modular Inverse