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337 lines
10 KiB
Python
337 lines
10 KiB
Python
# ===================================================================
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#
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# Copyright (c) 2014, Legrandin <helderijs@gmail.com>
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# All rights reserved.
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#
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# Redistribution and use in source and binary forms, with or without
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# modification, are permitted provided that the following conditions
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# are met:
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#
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# 1. Redistributions of source code must retain the above copyright
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# notice, this list of conditions and the following disclaimer.
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# 2. Redistributions in binary form must reproduce the above copyright
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# notice, this list of conditions and the following disclaimer in
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# the documentation and/or other materials provided with the
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# distribution.
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#
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# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
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# FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
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# COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
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# INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
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# BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
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# LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
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# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
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# ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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# POSSIBILITY OF SUCH DAMAGE.
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# ===================================================================
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from Crypto.Math.Numbers import Integer
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from Crypto import Random
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COMPOSITE = 0
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PROBABLY_PRIME = 1
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def miller_rabin_test(candidate, iterations, randfunc=None):
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"""Perform a Miller-Rabin primality test on an integer.
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The test is specified in Section C.3.1 of `FIPS PUB 186-4`__.
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:Parameters:
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:candidate: integer
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The number to test for primality.
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:iterations: integer
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The maximum number of iterations to perform before
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declaring a candidate a probable prime.
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:randfunc: callable
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An RNG function where bases are taken from.
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:Returns:
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``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``.
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.. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
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"""
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if not isinstance(candidate, Integer):
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candidate = Integer(candidate)
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if candidate.is_even():
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return COMPOSITE
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one = Integer(1)
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minus_one = Integer(candidate - 1)
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if randfunc is None:
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randfunc = Random.new().read
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# Step 1 and 2
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m = Integer(minus_one)
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a = 0
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while m.is_even():
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m >>= 1
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a += 1
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# Skip step 3
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# Step 4
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for i in xrange(iterations):
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# Step 4.1-2
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base = 1
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while base in (one, minus_one):
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base = Integer.random_range(2, candidate - 2)
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# Step 4.3-4.4
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z = pow(base, m, candidate)
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if z in (one, minus_one):
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continue
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# Step 4.5
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for j in xrange(1, a):
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z = pow(z, 2, candidate)
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if z == minus_one:
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break
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if z == one:
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return COMPOSITE
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else:
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return COMPOSITE
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# Step 5
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return PROBABLY_PRIME
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def lucas_test(candidate):
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"""Perform a Lucas primality test on an integer.
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The test is specified in Section C.3.3 of `FIPS PUB 186-4`__.
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:Parameters:
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:candidate: integer
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The number to test for primality.
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:Returns:
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``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``.
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.. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
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"""
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if not isinstance(candidate, Integer):
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candidate = Integer(candidate)
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# Step 1
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if candidate.is_even() or candidate.is_perfect_square():
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return COMPOSITE
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# Step 2
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def alternate():
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sgn = 1
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value = 5
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for x in xrange(10):
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yield sgn * value
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sgn, value = -sgn, value + 2
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for D in alternate():
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js = Integer.jacobi_symbol(D, candidate)
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if js == 0:
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return COMPOSITE
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if js == -1:
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break
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else:
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return COMPOSITE
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# Found D. P=1 and Q=(1-D)/4 (note that Q is guaranteed to be an integer)
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# Step 3
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# This is \delta(n) = n - jacobi(D/n)
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K = candidate + 1
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# Step 4
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r = K.size_in_bits() - 1
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# Step 5
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# U_1=1 and V_1=P
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U_i = Integer(1)
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V_i = Integer(1)
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U_temp = Integer(0)
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V_temp = Integer(0)
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# Step 6
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for i in xrange(r - 1, -1, -1):
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# Square
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# U_temp = U_i * V_i % candidate
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U_temp.set(U_i)
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U_temp *= V_i
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U_temp %= candidate
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# V_temp = (((V_i ** 2 + (U_i ** 2 * D)) * K) >> 1) % candidate
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V_temp.set(U_i)
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V_temp *= U_i
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V_temp *= D
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V_temp.multiply_accumulate(V_i, V_i)
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if V_temp.is_odd():
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V_temp += candidate
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V_temp >>= 1
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V_temp %= candidate
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# Multiply
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if K.get_bit(i):
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# U_i = (((U_temp + V_temp) * K) >> 1) % candidate
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U_i.set(U_temp)
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U_i += V_temp
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if U_i.is_odd():
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U_i += candidate
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U_i >>= 1
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U_i %= candidate
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# V_i = (((V_temp + U_temp * D) * K) >> 1) % candidate
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V_i.set(V_temp)
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V_i.multiply_accumulate(U_temp, D)
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if V_i.is_odd():
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V_i += candidate
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V_i >>= 1
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V_i %= candidate
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else:
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U_i.set(U_temp)
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V_i.set(V_temp)
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# Step 7
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if U_i == 0:
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return PROBABLY_PRIME
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return COMPOSITE
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from Crypto.Util.number import sieve_base as _sieve_base
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## The optimal number of small primes to use for the sieve
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## is probably dependent on the platform and the candidate size
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_sieve_base = _sieve_base[:100]
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def test_probable_prime(candidate, randfunc=None):
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"""Test if a number is prime.
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The probability of a false positive (a composite number that
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this routine declares as prime) is negligeable, but not zero.
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:Parameters:
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candidate : integer
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The number to test for primality.
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randfunc: callable
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The routine to draw random bytes from to select Miller-Rabin bases.
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:Returns:
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``PROBABLE_PRIME`` if the number if prime with very high probability.
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``COMPOSITE`` if the number is a composite.
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For efficiency reasons, ``COMPOSITE`` is also returned for small primes.
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"""
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if randfunc is None:
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randfunc = Random.new().read
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if not isinstance(candidate, Integer):
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candidate = Integer(candidate)
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# First, check trial division by the smallest primes
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try:
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map(candidate.fail_if_divisible_by, _sieve_base)
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except ValueError:
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return False
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# These are the number of Miller-Rabin iterations s.t. p(k, t) < 1E-30,
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# with p(k, t) being the probability that a randomly chosen k-bit number
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# is composite but still survives t MR iterations.
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mr_ranges = ((220, 30), (280, 20), (390, 15), (512, 10),
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(620, 7), (740, 6), (890, 5), (1200, 4),
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(1700, 3), (3700, 2))
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bit_size = candidate.size_in_bits()
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try:
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mr_iterations = list(filter(lambda x: bit_size < x[0],
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mr_ranges))[0][1]
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except IndexError:
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mr_iterations = 1
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if miller_rabin_test(candidate, mr_iterations,
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randfunc=randfunc) == COMPOSITE:
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return COMPOSITE
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if lucas_test(candidate) == COMPOSITE:
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return COMPOSITE
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return PROBABLY_PRIME
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def generate_probable_prime(**kwargs):
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"""Generate a random probable prime.
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The prime will not have any specific properties
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(E.g. it will not be a _strong prime_).
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Random numbers are evaluated for primality until one
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passes all tests, consisting of a certain number of
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Miller-Rabin tests with random bases followed by
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a single Lucas test.
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The number of Miller-Rabin iterations is chosen such that
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the probability that the output number is a non-prime is
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less than 1E-30 (roughly 2**{-100}).
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This approach is compliant to `FIPS PUB 186-4`__.
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:Keywords:
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:exact_bits:
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The desired size in bits of the probable prime.
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It must be at least 160.
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:randfunc: callable
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An RNG function where candidate primes are taken from.
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:Return:
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A probable prime in the range 2**exact_bits > p > 2**(exact_bits-1).
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.. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
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"""
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exact_bits = kwargs.pop("exact_bits", None)
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randfunc = kwargs.pop("randfunc", None)
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if kwargs:
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print "Unknown parameters:", kwargs.keys()
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if exact_bits is None:
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raise ValueError("Missing exact_bits parameter")
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if exact_bits < 160:
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raise ValueError("Prime number is not big enough.")
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if randfunc is None:
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randfunc = Random.new().read
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result = COMPOSITE
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while result == COMPOSITE:
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candidate = Integer.random(exact_bits=exact_bits,
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randfunc=randfunc) | 1
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result = test_probable_prime(candidate, randfunc)
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return candidate
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def generate_probable_safe_prime(**kwargs):
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"""Generate a random, probable safe prime.
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Note this operation is much slower than generating a simple prime.
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:Keywords:
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:exact_bits:
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The desired size in bits of the probable safe prime.
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:randfunc: callable
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An RNG function where candidate primes are taken from.
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:Return:
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A probable safe prime in the range
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2**exact_bits > p > 2**(exact_bits-1).
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"""
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exact_bits = kwargs.pop("exact_bits", None)
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randfunc = kwargs.pop("randfunc", None)
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if kwargs:
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print "Unknown parameters:", kwargs.keys()
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if randfunc is None:
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randfunc = Random.new().read
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result = COMPOSITE
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while result == COMPOSITE:
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q = generate_probable_prime(exact_bits=exact_bits - 1, randfunc=randfunc)
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candidate = q * 2 + 1
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if candidate.size_in_bits() != exact_bits:
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continue
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result = test_probable_prime(candidate, randfunc=randfunc)
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return candidate
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