diff options
Diffstat (limited to 'frozen_deps/Cryptodome/Math/Primality.py')
-rw-r--r-- | frozen_deps/Cryptodome/Math/Primality.py | 368 |
1 files changed, 368 insertions, 0 deletions
diff --git a/frozen_deps/Cryptodome/Math/Primality.py b/frozen_deps/Cryptodome/Math/Primality.py new file mode 100644 index 0000000..08ea3ff --- /dev/null +++ b/frozen_deps/Cryptodome/Math/Primality.py @@ -0,0 +1,368 @@ +# =================================================================== +# +# Copyright (c) 2014, Legrandin <[email protected]> +# All rights reserved. +# +# Redistribution and use in source and binary forms, with or without +# modification, are permitted provided that the following conditions +# are met: +# +# 1. Redistributions of source code must retain the above copyright +# notice, this list of conditions and the following disclaimer. +# 2. Redistributions in binary form must reproduce the above copyright +# notice, this list of conditions and the following disclaimer in +# the documentation and/or other materials provided with the +# distribution. +# +# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS +# FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE +# COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, +# INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, +# BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; +# LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER +# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT +# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN +# ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE +# POSSIBILITY OF SUCH DAMAGE. +# =================================================================== + +"""Functions to create and test prime numbers. + +:undocumented: __package__ +""" + +from Cryptodome import Random +from Cryptodome.Math.Numbers import Integer + +from Cryptodome.Util.py3compat import iter_range + +COMPOSITE = 0 +PROBABLY_PRIME = 1 + + +def miller_rabin_test(candidate, iterations, randfunc=None): + """Perform a Miller-Rabin primality test on an integer. + + The test is specified in Section C.3.1 of `FIPS PUB 186-4`__. + + :Parameters: + candidate : integer + The number to test for primality. + iterations : integer + The maximum number of iterations to perform before + declaring a candidate a probable prime. + randfunc : callable + An RNG function where bases are taken from. + + :Returns: + ``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``. + + .. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf + """ + + if not isinstance(candidate, Integer): + candidate = Integer(candidate) + + if candidate in (1, 2, 3, 5): + return PROBABLY_PRIME + + if candidate.is_even(): + return COMPOSITE + + one = Integer(1) + minus_one = Integer(candidate - 1) + + if randfunc is None: + randfunc = Random.new().read + + # Step 1 and 2 + m = Integer(minus_one) + a = 0 + while m.is_even(): + m >>= 1 + a += 1 + + # Skip step 3 + + # Step 4 + for i in iter_range(iterations): + + # Step 4.1-2 + base = 1 + while base in (one, minus_one): + base = Integer.random_range(min_inclusive=2, + max_inclusive=candidate - 2) + assert(2 <= base <= candidate - 2) + + # Step 4.3-4.4 + z = pow(base, m, candidate) + if z in (one, minus_one): + continue + + # Step 4.5 + for j in iter_range(1, a): + z = pow(z, 2, candidate) + if z == minus_one: + break + if z == one: + return COMPOSITE + else: + return COMPOSITE + + # Step 5 + return PROBABLY_PRIME + + +def lucas_test(candidate): + """Perform a Lucas primality test on an integer. + + The test is specified in Section C.3.3 of `FIPS PUB 186-4`__. + + :Parameters: + candidate : integer + The number to test for primality. + + :Returns: + ``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``. + + .. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf + """ + + if not isinstance(candidate, Integer): + candidate = Integer(candidate) + + # Step 1 + if candidate in (1, 2, 3, 5): + return PROBABLY_PRIME + if candidate.is_even() or candidate.is_perfect_square(): + return COMPOSITE + + # Step 2 + def alternate(): + value = 5 + while True: + yield value + if value > 0: + value += 2 + else: + value -= 2 + value = -value + + for D in alternate(): + if candidate in (D, -D): + continue + js = Integer.jacobi_symbol(D, candidate) + if js == 0: + return COMPOSITE + if js == -1: + break + # Found D. P=1 and Q=(1-D)/4 (note that Q is guaranteed to be an integer) + + # Step 3 + # This is \delta(n) = n - jacobi(D/n) + K = candidate + 1 + # Step 4 + r = K.size_in_bits() - 1 + # Step 5 + # U_1=1 and V_1=P + U_i = Integer(1) + V_i = Integer(1) + U_temp = Integer(0) + V_temp = Integer(0) + # Step 6 + for i in iter_range(r - 1, -1, -1): + # Square + # U_temp = U_i * V_i % candidate + U_temp.set(U_i) + U_temp *= V_i + U_temp %= candidate + # V_temp = (((V_i ** 2 + (U_i ** 2 * D)) * K) >> 1) % candidate + V_temp.set(U_i) + V_temp *= U_i + V_temp *= D + V_temp.multiply_accumulate(V_i, V_i) + if V_temp.is_odd(): + V_temp += candidate + V_temp >>= 1 + V_temp %= candidate + # Multiply + if K.get_bit(i): + # U_i = (((U_temp + V_temp) * K) >> 1) % candidate + U_i.set(U_temp) + U_i += V_temp + if U_i.is_odd(): + U_i += candidate + U_i >>= 1 + U_i %= candidate + # V_i = (((V_temp + U_temp * D) * K) >> 1) % candidate + V_i.set(V_temp) + V_i.multiply_accumulate(U_temp, D) + if V_i.is_odd(): + V_i += candidate + V_i >>= 1 + V_i %= candidate + else: + U_i.set(U_temp) + V_i.set(V_temp) + # Step 7 + if U_i == 0: + return PROBABLY_PRIME + return COMPOSITE + + +from Cryptodome.Util.number import sieve_base as _sieve_base_large +## The optimal number of small primes to use for the sieve +## is probably dependent on the platform and the candidate size +_sieve_base = set(_sieve_base_large[:100]) + + +def test_probable_prime(candidate, randfunc=None): + """Test if a number is prime. + + A number is qualified as prime if it passes a certain + number of Miller-Rabin tests (dependent on the size + of the number, but such that probability of a false + positive is less than 10^-30) and a single Lucas test. + + For instance, a 1024-bit candidate will need to pass + 4 Miller-Rabin tests. + + :Parameters: + candidate : integer + The number to test for primality. + randfunc : callable + The routine to draw random bytes from to select Miller-Rabin bases. + :Returns: + ``PROBABLE_PRIME`` if the number if prime with very high probability. + ``COMPOSITE`` if the number is a composite. + For efficiency reasons, ``COMPOSITE`` is also returned for small primes. + """ + + if randfunc is None: + randfunc = Random.new().read + + if not isinstance(candidate, Integer): + candidate = Integer(candidate) + + # First, check trial division by the smallest primes + if int(candidate) in _sieve_base: + return PROBABLY_PRIME + try: + map(candidate.fail_if_divisible_by, _sieve_base) + except ValueError: + return COMPOSITE + + # These are the number of Miller-Rabin iterations s.t. p(k, t) < 1E-30, + # with p(k, t) being the probability that a randomly chosen k-bit number + # is composite but still survives t MR iterations. + mr_ranges = ((220, 30), (280, 20), (390, 15), (512, 10), + (620, 7), (740, 6), (890, 5), (1200, 4), + (1700, 3), (3700, 2)) + + bit_size = candidate.size_in_bits() + try: + mr_iterations = list(filter(lambda x: bit_size < x[0], + mr_ranges))[0][1] + except IndexError: + mr_iterations = 1 + + if miller_rabin_test(candidate, mr_iterations, + randfunc=randfunc) == COMPOSITE: + return COMPOSITE + if lucas_test(candidate) == COMPOSITE: + return COMPOSITE + return PROBABLY_PRIME + + +def generate_probable_prime(**kwargs): + """Generate a random probable prime. + + The prime will not have any specific properties + (e.g. it will not be a *strong* prime). + + Random numbers are evaluated for primality until one + passes all tests, consisting of a certain number of + Miller-Rabin tests with random bases followed by + a single Lucas test. + + The number of Miller-Rabin iterations is chosen such that + the probability that the output number is a non-prime is + less than 1E-30 (roughly 2^{-100}). + + This approach is compliant to `FIPS PUB 186-4`__. + + :Keywords: + exact_bits : integer + The desired size in bits of the probable prime. + It must be at least 160. + randfunc : callable + An RNG function where candidate primes are taken from. + prime_filter : callable + A function that takes an Integer as parameter and returns + True if the number can be passed to further primality tests, + False if it should be immediately discarded. + + :Return: + A probable prime in the range 2^exact_bits > p > 2^(exact_bits-1). + + .. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf + """ + + exact_bits = kwargs.pop("exact_bits", None) + randfunc = kwargs.pop("randfunc", None) + prime_filter = kwargs.pop("prime_filter", lambda x: True) + if kwargs: + raise ValueError("Unknown parameters: " + kwargs.keys()) + + if exact_bits is None: + raise ValueError("Missing exact_bits parameter") + if exact_bits < 160: + raise ValueError("Prime number is not big enough.") + + if randfunc is None: + randfunc = Random.new().read + + result = COMPOSITE + while result == COMPOSITE: + candidate = Integer.random(exact_bits=exact_bits, + randfunc=randfunc) | 1 + if not prime_filter(candidate): + continue + result = test_probable_prime(candidate, randfunc) + return candidate + + +def generate_probable_safe_prime(**kwargs): + """Generate a random, probable safe prime. + + Note this operation is much slower than generating a simple prime. + + :Keywords: + exact_bits : integer + The desired size in bits of the probable safe prime. + randfunc : callable + An RNG function where candidate primes are taken from. + + :Return: + A probable safe prime in the range + 2^exact_bits > p > 2^(exact_bits-1). + """ + + exact_bits = kwargs.pop("exact_bits", None) + randfunc = kwargs.pop("randfunc", None) + if kwargs: + raise ValueError("Unknown parameters: " + kwargs.keys()) + + if randfunc is None: + randfunc = Random.new().read + + result = COMPOSITE + while result == COMPOSITE: + q = generate_probable_prime(exact_bits=exact_bits - 1, randfunc=randfunc) + candidate = q * 2 + 1 + if candidate.size_in_bits() != exact_bits: + continue + result = test_probable_prime(candidate, randfunc=randfunc) + return candidate |