516 lines
18 KiB
Python
516 lines
18 KiB
Python
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from sympy.core.numbers import igcd, mod_inverse
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from sympy.core.power import integer_nthroot
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from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power
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from sympy.ntheory import isprime
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from math import log, sqrt
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import random
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rgen = random.Random()
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class SievePolynomial:
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def __init__(self, modified_coeff=(), a=None, b=None):
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"""This class denotes the seive polynomial.
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If ``g(x) = (a*x + b)**2 - N``. `g(x)` can be expanded
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to ``a*x**2 + 2*a*b*x + b**2 - N``, so the coefficient
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is stored in the form `[a**2, 2*a*b, b**2 - N]`. This
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ensures faster `eval` method because we dont have to
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perform `a**2, 2*a*b, b**2` every time we call the
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`eval` method. As multiplication is more expensive
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than addition, by using modified_coefficient we get
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a faster seiving process.
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Parameters
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==========
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modified_coeff : modified_coefficient of sieve polynomial
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a : parameter of the sieve polynomial
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b : parameter of the sieve polynomial
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"""
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self.modified_coeff = modified_coeff
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self.a = a
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self.b = b
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def eval(self, x):
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"""
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Compute the value of the sieve polynomial at point x.
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Parameters
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==========
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x : Integer parameter for sieve polynomial
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"""
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ans = 0
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for coeff in self.modified_coeff:
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ans *= x
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ans += coeff
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return ans
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class FactorBaseElem:
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"""This class stores an element of the `factor_base`.
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"""
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def __init__(self, prime, tmem_p, log_p):
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"""
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Initialization of factor_base_elem.
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Parameters
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==========
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prime : prime number of the factor_base
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tmem_p : Integer square root of x**2 = n mod prime
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log_p : Compute Natural Logarithm of the prime
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"""
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self.prime = prime
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self.tmem_p = tmem_p
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self.log_p = log_p
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self.soln1 = None
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self.soln2 = None
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self.a_inv = None
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self.b_ainv = None
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def _generate_factor_base(prime_bound, n):
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"""Generate `factor_base` for Quadratic Sieve. The `factor_base`
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consists of all the points whose ``legendre_symbol(n, p) == 1``
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and ``p < num_primes``. Along with the prime `factor_base` also stores
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natural logarithm of prime and the residue n modulo p.
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It also returns the of primes numbers in the `factor_base` which are
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close to 1000 and 5000.
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Parameters
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==========
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prime_bound : upper prime bound of the factor_base
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n : integer to be factored
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"""
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from sympy.ntheory.generate import sieve
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factor_base = []
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idx_1000, idx_5000 = None, None
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for prime in sieve.primerange(1, prime_bound):
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if pow(n, (prime - 1) // 2, prime) == 1:
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if prime > 1000 and idx_1000 is None:
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idx_1000 = len(factor_base) - 1
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if prime > 5000 and idx_5000 is None:
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idx_5000 = len(factor_base) - 1
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residue = _sqrt_mod_prime_power(n, prime, 1)[0]
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log_p = round(log(prime)*2**10)
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factor_base.append(FactorBaseElem(prime, residue, log_p))
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return idx_1000, idx_5000, factor_base
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def _initialize_first_polynomial(N, M, factor_base, idx_1000, idx_5000, seed=None):
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"""This step is the initialization of the 1st sieve polynomial.
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Here `a` is selected as a product of several primes of the factor_base
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such that `a` is about to ``sqrt(2*N) / M``. Other initial values of
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factor_base elem are also initialized which includes a_inv, b_ainv, soln1,
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soln2 which are used when the sieve polynomial is changed. The b_ainv
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is required for fast polynomial change as we do not have to calculate
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`2*b*mod_inverse(a, prime)` every time.
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We also ensure that the `factor_base` primes which make `a` are between
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1000 and 5000.
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Parameters
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==========
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N : Number to be factored
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M : sieve interval
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factor_base : factor_base primes
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idx_1000 : index of prime number in the factor_base near 1000
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idx_5000 : index of prime number in the factor_base near to 5000
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seed : Generate pseudoprime numbers
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"""
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if seed is not None:
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rgen.seed(seed)
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approx_val = sqrt(2*N) / M
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# `a` is a parameter of the sieve polynomial and `q` is the prime factors of `a`
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# randomly search for a combination of primes whose multiplication is close to approx_val
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# This multiplication of primes will be `a` and the primes will be `q`
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# `best_a` denotes that `a` is close to approx_val in the random search of combination
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best_a, best_q, best_ratio = None, None, None
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start = 0 if idx_1000 is None else idx_1000
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end = len(factor_base) - 1 if idx_5000 is None else idx_5000
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for _ in range(50):
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a = 1
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q = []
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while(a < approx_val):
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rand_p = 0
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while(rand_p == 0 or rand_p in q):
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rand_p = rgen.randint(start, end)
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p = factor_base[rand_p].prime
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a *= p
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q.append(rand_p)
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ratio = a / approx_val
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if best_ratio is None or abs(ratio - 1) < abs(best_ratio - 1):
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best_q = q
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best_a = a
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best_ratio = ratio
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a = best_a
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q = best_q
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B = []
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for idx, val in enumerate(q):
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q_l = factor_base[val].prime
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gamma = factor_base[val].tmem_p * mod_inverse(a // q_l, q_l) % q_l
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if gamma > q_l / 2:
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gamma = q_l - gamma
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B.append(a//q_l*gamma)
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b = sum(B)
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g = SievePolynomial([a*a, 2*a*b, b*b - N], a, b)
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for fb in factor_base:
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if a % fb.prime == 0:
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continue
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fb.a_inv = mod_inverse(a, fb.prime)
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fb.b_ainv = [2*b_elem*fb.a_inv % fb.prime for b_elem in B]
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fb.soln1 = (fb.a_inv*(fb.tmem_p - b)) % fb.prime
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fb.soln2 = (fb.a_inv*(-fb.tmem_p - b)) % fb.prime
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return g, B
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def _initialize_ith_poly(N, factor_base, i, g, B):
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"""Initialization stage of ith poly. After we finish sieving 1`st polynomial
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here we quickly change to the next polynomial from which we will again
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start sieving. Suppose we generated ith sieve polynomial and now we
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want to generate (i + 1)th polynomial, where ``1 <= i <= 2**(j - 1) - 1``
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where `j` is the number of prime factors of the coefficient `a`
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then this function can be used to go to the next polynomial. If
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``i = 2**(j - 1) - 1`` then go to _initialize_first_polynomial stage.
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Parameters
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==========
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N : number to be factored
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factor_base : factor_base primes
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i : integer denoting ith polynomial
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g : (i - 1)th polynomial
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B : array that stores a//q_l*gamma
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"""
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from sympy.functions.elementary.integers import ceiling
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v = 1
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j = i
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while(j % 2 == 0):
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v += 1
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j //= 2
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if ceiling(i / (2**v)) % 2 == 1:
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neg_pow = -1
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else:
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neg_pow = 1
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b = g.b + 2*neg_pow*B[v - 1]
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a = g.a
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g = SievePolynomial([a*a, 2*a*b, b*b - N], a, b)
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for fb in factor_base:
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if a % fb.prime == 0:
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continue
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fb.soln1 = (fb.soln1 - neg_pow*fb.b_ainv[v - 1]) % fb.prime
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fb.soln2 = (fb.soln2 - neg_pow*fb.b_ainv[v - 1]) % fb.prime
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return g
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def _gen_sieve_array(M, factor_base):
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"""Sieve Stage of the Quadratic Sieve. For every prime in the factor_base
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that does not divide the coefficient `a` we add log_p over the sieve_array
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such that ``-M <= soln1 + i*p <= M`` and ``-M <= soln2 + i*p <= M`` where `i`
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is an integer. When p = 2 then log_p is only added using
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``-M <= soln1 + i*p <= M``.
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Parameters
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==========
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M : sieve interval
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factor_base : factor_base primes
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"""
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sieve_array = [0]*(2*M + 1)
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for factor in factor_base:
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if factor.soln1 is None: #The prime does not divides a
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continue
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for idx in range((M + factor.soln1) % factor.prime, 2*M, factor.prime):
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sieve_array[idx] += factor.log_p
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if factor.prime == 2:
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continue
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#if prime is 2 then sieve only with soln_1_p
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for idx in range((M + factor.soln2) % factor.prime, 2*M, factor.prime):
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sieve_array[idx] += factor.log_p
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return sieve_array
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def _check_smoothness(num, factor_base):
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"""Here we check that if `num` is a smooth number or not. If `a` is a smooth
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number then it returns a vector of prime exponents modulo 2. For example
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if a = 2 * 5**2 * 7**3 and the factor base contains {2, 3, 5, 7} then
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`a` is a smooth number and this function returns ([1, 0, 0, 1], True). If
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`a` is a partial relation which means that `a` a has one prime factor
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greater than the `factor_base` then it returns `(a, False)` which denotes `a`
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is a partial relation.
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Parameters
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==========
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a : integer whose smootheness is to be checked
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factor_base : factor_base primes
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"""
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vec = []
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if num < 0:
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vec.append(1)
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num *= -1
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else:
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vec.append(0)
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#-1 is not included in factor_base add -1 in vector
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for factor in factor_base:
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if num % factor.prime != 0:
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vec.append(0)
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continue
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factor_exp = 0
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while num % factor.prime == 0:
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factor_exp += 1
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num //= factor.prime
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vec.append(factor_exp % 2)
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if num == 1:
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return vec, True
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if isprime(num):
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return num, False
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return None, None
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def _trial_division_stage(N, M, factor_base, sieve_array, sieve_poly, partial_relations, ERROR_TERM):
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"""Trial division stage. Here we trial divide the values generetated
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by sieve_poly in the sieve interval and if it is a smooth number then
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it is stored in `smooth_relations`. Moreover, if we find two partial relations
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with same large prime then they are combined to form a smooth relation.
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First we iterate over sieve array and look for values which are greater
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than accumulated_val, as these values have a high chance of being smooth
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number. Then using these values we find smooth relations.
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In general, let ``t**2 = u*p modN`` and ``r**2 = v*p modN`` be two partial relations
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with the same large prime p. Then they can be combined ``(t*r/p)**2 = u*v modN``
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to form a smooth relation.
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Parameters
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==========
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N : Number to be factored
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M : sieve interval
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factor_base : factor_base primes
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sieve_array : stores log_p values
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sieve_poly : polynomial from which we find smooth relations
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partial_relations : stores partial relations with one large prime
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ERROR_TERM : error term for accumulated_val
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"""
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sqrt_n = sqrt(float(N))
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accumulated_val = log(M * sqrt_n)*2**10 - ERROR_TERM
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smooth_relations = []
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proper_factor = set()
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partial_relation_upper_bound = 128*factor_base[-1].prime
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for idx, val in enumerate(sieve_array):
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if val < accumulated_val:
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continue
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x = idx - M
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v = sieve_poly.eval(x)
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vec, is_smooth = _check_smoothness(v, factor_base)
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if is_smooth is None:#Neither smooth nor partial
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continue
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u = sieve_poly.a*x + sieve_poly.b
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# Update the partial relation
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# If 2 partial relation with same large prime is found then generate smooth relation
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if is_smooth is False:#partial relation found
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large_prime = vec
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#Consider the large_primes under 128*F
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if large_prime > partial_relation_upper_bound:
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continue
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if large_prime not in partial_relations:
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partial_relations[large_prime] = (u, v)
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continue
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else:
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u_prev, v_prev = partial_relations[large_prime]
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partial_relations.pop(large_prime)
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try:
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large_prime_inv = mod_inverse(large_prime, N)
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except ValueError:#if large_prine divides N
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proper_factor.add(large_prime)
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continue
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u = u*u_prev*large_prime_inv
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v = v*v_prev // (large_prime*large_prime)
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vec, is_smooth = _check_smoothness(v, factor_base)
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#assert u*u % N == v % N
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smooth_relations.append((u, v, vec))
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return smooth_relations, proper_factor
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#LINEAR ALGEBRA STAGE
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def _build_matrix(smooth_relations):
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"""Build a 2D matrix from smooth relations.
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Parameters
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==========
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smooth_relations : Stores smooth relations
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"""
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matrix = []
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for s_relation in smooth_relations:
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matrix.append(s_relation[2])
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return matrix
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def _gauss_mod_2(A):
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"""Fast gaussian reduction for modulo 2 matrix.
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Parameters
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==========
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A : Matrix
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Examples
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========
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>>> from sympy.ntheory.qs import _gauss_mod_2
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>>> _gauss_mod_2([[0, 1, 1], [1, 0, 1], [0, 1, 0], [1, 1, 1]])
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([[[1, 0, 1], 3]],
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[True, True, True, False],
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[[0, 1, 0], [1, 0, 0], [0, 0, 1], [1, 0, 1]])
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Reference
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==========
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.. [1] A fast algorithm for gaussian elimination over GF(2) and
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its implementation on the GAPP. Cetin K.Koc, Sarath N.Arachchige"""
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import copy
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matrix = copy.deepcopy(A)
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row = len(matrix)
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col = len(matrix[0])
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mark = [False]*row
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for c in range(col):
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for r in range(row):
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if matrix[r][c] == 1:
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break
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mark[r] = True
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for c1 in range(col):
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if c1 == c:
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continue
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if matrix[r][c1] == 1:
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for r2 in range(row):
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matrix[r2][c1] = (matrix[r2][c1] + matrix[r2][c]) % 2
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dependent_row = []
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for idx, val in enumerate(mark):
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if val == False:
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dependent_row.append([matrix[idx], idx])
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return dependent_row, mark, matrix
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def _find_factor(dependent_rows, mark, gauss_matrix, index, smooth_relations, N):
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"""Finds proper factor of N. Here, transform the dependent rows as a
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combination of independent rows of the gauss_matrix to form the desired
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relation of the form ``X**2 = Y**2 modN``. After obtaining the desired relation
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we obtain a proper factor of N by `gcd(X - Y, N)`.
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Parameters
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==========
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dependent_rows : denoted dependent rows in the reduced matrix form
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mark : boolean array to denoted dependent and independent rows
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gauss_matrix : Reduced form of the smooth relations matrix
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index : denoted the index of the dependent_rows
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smooth_relations : Smooth relations vectors matrix
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N : Number to be factored
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"""
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idx_in_smooth = dependent_rows[index][1]
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independent_u = [smooth_relations[idx_in_smooth][0]]
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independent_v = [smooth_relations[idx_in_smooth][1]]
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dept_row = dependent_rows[index][0]
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for idx, val in enumerate(dept_row):
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if val == 1:
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for row in range(len(gauss_matrix)):
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if gauss_matrix[row][idx] == 1 and mark[row] == True:
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independent_u.append(smooth_relations[row][0])
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independent_v.append(smooth_relations[row][1])
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break
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u = 1
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v = 1
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for i in independent_u:
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u *= i
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for i in independent_v:
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v *= i
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#assert u**2 % N == v % N
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v = integer_nthroot(v, 2)[0]
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return igcd(u - v, N)
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def qs(N, prime_bound, M, ERROR_TERM=25, seed=1234):
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"""Performs factorization using Self-Initializing Quadratic Sieve.
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In SIQS, let N be a number to be factored, and this N should not be a
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perfect power. If we find two integers such that ``X**2 = Y**2 modN`` and
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``X != +-Y modN``, then `gcd(X + Y, N)` will reveal a proper factor of N.
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In order to find these integers X and Y we try to find relations of form
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t**2 = u modN where u is a product of small primes. If we have enough of
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these relations then we can form ``(t1*t2...ti)**2 = u1*u2...ui modN`` such that
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the right hand side is a square, thus we found a relation of ``X**2 = Y**2 modN``.
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Here, several optimizations are done like using multiple polynomials for
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sieving, fast changing between polynomials and using partial relations.
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The use of partial relations can speeds up the factoring by 2 times.
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Parameters
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==========
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N : Number to be Factored
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prime_bound : upper bound for primes in the factor base
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M : Sieve Interval
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ERROR_TERM : Error term for checking smoothness
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threshold : Extra smooth relations for factorization
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seed : generate pseudo prime numbers
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Examples
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========
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>>> from sympy.ntheory import qs
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>>> qs(25645121643901801, 2000, 10000)
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{5394769, 4753701529}
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>>> qs(9804659461513846513, 2000, 10000)
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{4641991, 2112166839943}
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References
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==========
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.. [1] https://pdfs.semanticscholar.org/5c52/8a975c1405bd35c65993abf5a4edb667c1db.pdf
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.. [2] https://www.rieselprime.de/ziki/Self-initializing_quadratic_sieve
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"""
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ERROR_TERM*=2**10
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rgen.seed(seed)
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idx_1000, idx_5000, factor_base = _generate_factor_base(prime_bound, N)
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smooth_relations = []
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ith_poly = 0
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partial_relations = {}
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proper_factor = set()
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threshold = 5*len(factor_base) // 100
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while True:
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if ith_poly == 0:
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ith_sieve_poly, B_array = _initialize_first_polynomial(N, M, factor_base, idx_1000, idx_5000)
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else:
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ith_sieve_poly = _initialize_ith_poly(N, factor_base, ith_poly, ith_sieve_poly, B_array)
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ith_poly += 1
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if ith_poly >= 2**(len(B_array) - 1): # time to start with a new sieve polynomial
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ith_poly = 0
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sieve_array = _gen_sieve_array(M, factor_base)
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s_rel, p_f = _trial_division_stage(N, M, factor_base, sieve_array, ith_sieve_poly, partial_relations, ERROR_TERM)
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smooth_relations += s_rel
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proper_factor |= p_f
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if len(smooth_relations) >= len(factor_base) + threshold:
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break
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matrix = _build_matrix(smooth_relations)
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dependent_row, mark, gauss_matrix = _gauss_mod_2(matrix)
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N_copy = N
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for index in range(len(dependent_row)):
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factor = _find_factor(dependent_row, mark, gauss_matrix, index, smooth_relations, N)
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if factor > 1 and factor < N:
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proper_factor.add(factor)
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while(N_copy % factor == 0):
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N_copy //= factor
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if isprime(N_copy):
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proper_factor.add(N_copy)
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break
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if(N_copy == 1):
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break
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return proper_factor
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