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# Copyright (c) 2014, Legrandin <helderijs@gmail.com>
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"""Functions to create and test prime numbers.
:undocumented: __package__
"""
from Crypto import Random
from Crypto.Math.Numbers import Integer
from Crypto.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,
randfunc=randfunc)
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 Crypto.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