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-rw-r--r--Doc/library/functions.rst21
-rw-r--r--Doc/whatsnew/3.8.rst6
-rw-r--r--Lib/test/test_builtin.py3
-rw-r--r--Lib/test/test_pow.py26
-rw-r--r--Misc/NEWS.d/next/Core and Builtins/2019-05-12-18-46-50.bpo-36027.Q4YatQ.rst3
-rw-r--r--Objects/longobject.c130
6 files changed, 173 insertions, 16 deletions
diff --git a/Doc/library/functions.rst b/Doc/library/functions.rst
index d5c9f18..415a65b 100644
--- a/Doc/library/functions.rst
+++ b/Doc/library/functions.rst
@@ -1277,9 +1277,24 @@ are always available. They are listed here in alphabetical order.
operands, the result has the same type as the operands (after coercion)
unless the second argument is negative; in that case, all arguments are
converted to float and a float result is delivered. For example, ``10**2``
- returns ``100``, but ``10**-2`` returns ``0.01``. If the second argument is
- negative, the third argument must be omitted. If *z* is present, *x* and *y*
- must be of integer types, and *y* must be non-negative.
+ returns ``100``, but ``10**-2`` returns ``0.01``.
+
+ For :class:`int` operands *x* and *y*, if *z* is present, *z* must also be
+ of integer type and *z* must be nonzero. If *z* is present and *y* is
+ negative, *x* must be relatively prime to *z*. In that case, ``pow(inv_x,
+ -y, z)`` is returned, where *inv_x* is an inverse to *x* modulo *z*.
+
+ Here's an example of computing an inverse for ``38`` modulo ``97``::
+
+ >>> pow(38, -1, 97)
+ 23
+ >>> 23 * 38 % 97 == 1
+ True
+
+ .. versionchanged:: 3.8
+ For :class:`int` operands, the three-argument form of ``pow`` now allows
+ the second argument to be negative, permitting computation of modular
+ inverses.
.. function:: print(*objects, sep=' ', end='\\n', file=sys.stdout, flush=False)
diff --git a/Doc/whatsnew/3.8.rst b/Doc/whatsnew/3.8.rst
index 591b454..74d0079 100644
--- a/Doc/whatsnew/3.8.rst
+++ b/Doc/whatsnew/3.8.rst
@@ -304,6 +304,12 @@ Other Language Changes
* Added new ``replace()`` method to the code type (:class:`types.CodeType`).
(Contributed by Victor Stinner in :issue:`37032`.)
+* For integers, the three-argument form of the :func:`pow` function now permits
+ the exponent to be negative in the case where the base is relatively prime to
+ the modulus. It then computes a modular inverse to the base when the exponent
+ is ``-1``, and a suitable power of that inverse for other negative exponents.
+ (Contributed by Mark Dickinson in :issue:`36027`.)
+
New Modules
===========
diff --git a/Lib/test/test_builtin.py b/Lib/test/test_builtin.py
index e32fb75..b536cec 100644
--- a/Lib/test/test_builtin.py
+++ b/Lib/test/test_builtin.py
@@ -1195,7 +1195,8 @@ class BuiltinTest(unittest.TestCase):
self.assertAlmostEqual(pow(-1, 0.5), 1j)
self.assertAlmostEqual(pow(-1, 1/3), 0.5 + 0.8660254037844386j)
- self.assertRaises(ValueError, pow, -1, -2, 3)
+ # See test_pow for additional tests for three-argument pow.
+ self.assertEqual(pow(-1, -2, 3), 1)
self.assertRaises(ValueError, pow, 1, 2, 0)
self.assertRaises(TypeError, pow)
diff --git a/Lib/test/test_pow.py b/Lib/test/test_pow.py
index cac1ae5..660ff80 100644
--- a/Lib/test/test_pow.py
+++ b/Lib/test/test_pow.py
@@ -1,3 +1,4 @@
+import math
import unittest
class PowTest(unittest.TestCase):
@@ -119,5 +120,30 @@ class PowTest(unittest.TestCase):
eq(pow(a, -fiveto), expected)
eq(expected, 1.0) # else we didn't push fiveto to evenness
+ def test_negative_exponent(self):
+ for a in range(-50, 50):
+ for m in range(-50, 50):
+ with self.subTest(a=a, m=m):
+ if m != 0 and math.gcd(a, m) == 1:
+ # Exponent -1 should give an inverse, with the
+ # same sign as m.
+ inv = pow(a, -1, m)
+ self.assertEqual(inv, inv % m)
+ self.assertEqual((inv * a - 1) % m, 0)
+
+ # Larger exponents
+ self.assertEqual(pow(a, -2, m), pow(inv, 2, m))
+ self.assertEqual(pow(a, -3, m), pow(inv, 3, m))
+ self.assertEqual(pow(a, -1001, m), pow(inv, 1001, m))
+
+ else:
+ with self.assertRaises(ValueError):
+ pow(a, -1, m)
+ with self.assertRaises(ValueError):
+ pow(a, -2, m)
+ with self.assertRaises(ValueError):
+ pow(a, -1001, m)
+
+
if __name__ == "__main__":
unittest.main()
diff --git a/Misc/NEWS.d/next/Core and Builtins/2019-05-12-18-46-50.bpo-36027.Q4YatQ.rst b/Misc/NEWS.d/next/Core and Builtins/2019-05-12-18-46-50.bpo-36027.Q4YatQ.rst
new file mode 100644
index 0000000..866309c
--- /dev/null
+++ b/Misc/NEWS.d/next/Core and Builtins/2019-05-12-18-46-50.bpo-36027.Q4YatQ.rst
@@ -0,0 +1,3 @@
+Allow computation of modular inverses via three-argument ``pow``: the second
+argument is now permitted to be negative in the case where the first and
+third arguments are relatively prime.
diff --git a/Objects/longobject.c b/Objects/longobject.c
index 5d2b595..49f1420 100644
--- a/Objects/longobject.c
+++ b/Objects/longobject.c
@@ -4174,6 +4174,98 @@ long_divmod(PyObject *a, PyObject *b)
return z;
}
+
+/* Compute an inverse to a modulo n, or raise ValueError if a is not
+ invertible modulo n. Assumes n is positive. The inverse returned
+ is whatever falls out of the extended Euclidean algorithm: it may
+ be either positive or negative, but will be smaller than n in
+ absolute value.
+
+ Pure Python equivalent for long_invmod:
+
+ def invmod(a, n):
+ b, c = 1, 0
+ while n:
+ q, r = divmod(a, n)
+ a, b, c, n = n, c, b - q*c, r
+
+ # at this point a is the gcd of the original inputs
+ if a == 1:
+ return b
+ raise ValueError("Not invertible")
+*/
+
+static PyLongObject *
+long_invmod(PyLongObject *a, PyLongObject *n)
+{
+ PyLongObject *b, *c;
+
+ /* Should only ever be called for positive n */
+ assert(Py_SIZE(n) > 0);
+
+ b = (PyLongObject *)PyLong_FromLong(1L);
+ if (b == NULL) {
+ return NULL;
+ }
+ c = (PyLongObject *)PyLong_FromLong(0L);
+ if (c == NULL) {
+ Py_DECREF(b);
+ return NULL;
+ }
+ Py_INCREF(a);
+ Py_INCREF(n);
+
+ /* references now owned: a, b, c, n */
+ while (Py_SIZE(n) != 0) {
+ PyLongObject *q, *r, *s, *t;
+
+ if (l_divmod(a, n, &q, &r) == -1) {
+ goto Error;
+ }
+ Py_DECREF(a);
+ a = n;
+ n = r;
+ t = (PyLongObject *)long_mul(q, c);
+ Py_DECREF(q);
+ if (t == NULL) {
+ goto Error;
+ }
+ s = (PyLongObject *)long_sub(b, t);
+ Py_DECREF(t);
+ if (s == NULL) {
+ goto Error;
+ }
+ Py_DECREF(b);
+ b = c;
+ c = s;
+ }
+ /* references now owned: a, b, c, n */
+
+ Py_DECREF(c);
+ Py_DECREF(n);
+ if (long_compare(a, _PyLong_One)) {
+ /* a != 1; we don't have an inverse. */
+ Py_DECREF(a);
+ Py_DECREF(b);
+ PyErr_SetString(PyExc_ValueError,
+ "base is not invertible for the given modulus");
+ return NULL;
+ }
+ else {
+ /* a == 1; b gives an inverse modulo n */
+ Py_DECREF(a);
+ return b;
+ }
+
+ Error:
+ Py_DECREF(a);
+ Py_DECREF(b);
+ Py_DECREF(c);
+ Py_DECREF(n);
+ return NULL;
+}
+
+
/* pow(v, w, x) */
static PyObject *
long_pow(PyObject *v, PyObject *w, PyObject *x)
@@ -4207,20 +4299,14 @@ long_pow(PyObject *v, PyObject *w, PyObject *x)
Py_RETURN_NOTIMPLEMENTED;
}
- if (Py_SIZE(b) < 0) { /* if exponent is negative */
- if (c) {
- PyErr_SetString(PyExc_ValueError, "pow() 2nd argument "
- "cannot be negative when 3rd argument specified");
- goto Error;
- }
- else {
- /* else return a float. This works because we know
+ if (Py_SIZE(b) < 0 && c == NULL) {
+ /* if exponent is negative and there's no modulus:
+ return a float. This works because we know
that this calls float_pow() which converts its
arguments to double. */
- Py_DECREF(a);
- Py_DECREF(b);
- return PyFloat_Type.tp_as_number->nb_power(v, w, x);
- }
+ Py_DECREF(a);
+ Py_DECREF(b);
+ return PyFloat_Type.tp_as_number->nb_power(v, w, x);
}
if (c) {
@@ -4255,6 +4341,26 @@ long_pow(PyObject *v, PyObject *w, PyObject *x)
goto Done;
}
+ /* if exponent is negative, negate the exponent and
+ replace the base with a modular inverse */
+ if (Py_SIZE(b) < 0) {
+ temp = (PyLongObject *)_PyLong_Copy(b);
+ if (temp == NULL)
+ goto Error;
+ Py_DECREF(b);
+ b = temp;
+ temp = NULL;
+ _PyLong_Negate(&b);
+ if (b == NULL)
+ goto Error;
+
+ temp = long_invmod(a, c);
+ if (temp == NULL)
+ goto Error;
+ Py_DECREF(a);
+ a = temp;
+ }
+
/* Reduce base by modulus in some cases:
1. If base < 0. Forcing the base non-negative makes things easier.
2. If base is obviously larger than the modulus. The "small
generated by cgit v0.12 at 2024-11-22 02:15:16 (GMT)