451 lines
15 KiB
Python
451 lines
15 KiB
Python
from math import exp, log
|
|
from sympy.core.random import _randint
|
|
from sympy.external.gmpy import bit_scan1, gcd, invert, sqrt as isqrt
|
|
from sympy.ntheory.factor_ import _perfect_power
|
|
from sympy.ntheory.primetest import isprime
|
|
from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power
|
|
|
|
|
|
class SievePolynomial:
|
|
def __init__(self, a, b, N):
|
|
"""This class denotes the sieve polynomial.
|
|
Provide methods to compute `(a*x + b)**2 - N` and
|
|
`a*x + b` when given `x`.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
a : parameter of the sieve polynomial
|
|
b : parameter of the sieve polynomial
|
|
N : number to be factored
|
|
|
|
"""
|
|
self.a = a
|
|
self.b = b
|
|
self.a2 = a**2
|
|
self.ab = 2*a*b
|
|
self.b2 = b**2 - N
|
|
|
|
def eval_u(self, x):
|
|
return self.a*x + self.b
|
|
|
|
def eval_v(self, x):
|
|
return (self.a2*x + self.ab)*x + self.b2
|
|
|
|
|
|
class FactorBaseElem:
|
|
"""This class stores an element of the `factor_base`.
|
|
"""
|
|
def __init__(self, prime, tmem_p, log_p):
|
|
"""
|
|
Initialization of factor_base_elem.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
prime : prime number of the factor_base
|
|
tmem_p : Integer square root of x**2 = n mod prime
|
|
log_p : Compute Natural Logarithm of the prime
|
|
"""
|
|
self.prime = prime
|
|
self.tmem_p = tmem_p
|
|
self.log_p = log_p
|
|
# `soln1` and `soln2` are solutions to
|
|
# the equation `(a*x + b)**2 - N = 0 (mod p)`.
|
|
self.soln1 = None
|
|
self.soln2 = None
|
|
self.b_ainv = None
|
|
|
|
|
|
def _generate_factor_base(prime_bound, n):
|
|
"""Generate `factor_base` for Quadratic Sieve. The `factor_base`
|
|
consists of all the points whose ``legendre_symbol(n, p) == 1``
|
|
and ``p < num_primes``. Along with the prime `factor_base` also stores
|
|
natural logarithm of prime and the residue n modulo p.
|
|
It also returns the of primes numbers in the `factor_base` which are
|
|
close to 1000 and 5000.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
prime_bound : upper prime bound of the factor_base
|
|
n : integer to be factored
|
|
"""
|
|
from sympy.ntheory.generate import sieve
|
|
factor_base = []
|
|
idx_1000, idx_5000 = None, None
|
|
for prime in sieve.primerange(1, prime_bound):
|
|
if pow(n, (prime - 1) // 2, prime) == 1:
|
|
if prime > 1000 and idx_1000 is None:
|
|
idx_1000 = len(factor_base) - 1
|
|
if prime > 5000 and idx_5000 is None:
|
|
idx_5000 = len(factor_base) - 1
|
|
residue = _sqrt_mod_prime_power(n, prime, 1)[0]
|
|
log_p = round(log(prime)*2**10)
|
|
factor_base.append(FactorBaseElem(prime, residue, log_p))
|
|
return idx_1000, idx_5000, factor_base
|
|
|
|
|
|
def _generate_polynomial(N, M, factor_base, idx_1000, idx_5000, randint):
|
|
""" Generate sieve polynomials indefinitely.
|
|
Information such as `soln1` in the `factor_base` associated with
|
|
the polynomial is modified in place.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
N : Number to be factored
|
|
M : sieve interval
|
|
factor_base : factor_base primes
|
|
idx_1000 : index of prime number in the factor_base near 1000
|
|
idx_5000 : index of prime number in the factor_base near to 5000
|
|
randint : A callable that takes two integers (a, b) and returns a random integer
|
|
n such that a <= n <= b, similar to `random.randint`.
|
|
"""
|
|
approx_val = log(2*N)/2 - log(M)
|
|
start = idx_1000 or 0
|
|
end = idx_5000 or (len(factor_base) - 1)
|
|
while True:
|
|
# Choose `a` that is close to `sqrt(2*N) / M`
|
|
best_a, best_q, best_ratio = None, None, None
|
|
for _ in range(50):
|
|
a = 1
|
|
q = []
|
|
while log(a) < approx_val:
|
|
rand_p = 0
|
|
while(rand_p == 0 or rand_p in q):
|
|
rand_p = randint(start, end)
|
|
p = factor_base[rand_p].prime
|
|
a *= p
|
|
q.append(rand_p)
|
|
ratio = exp(log(a) - approx_val)
|
|
if best_ratio is None or abs(ratio - 1) < abs(best_ratio - 1):
|
|
best_q = q
|
|
best_a = a
|
|
best_ratio = ratio
|
|
|
|
# Set `b` using the Chinese remainder theorem
|
|
a = best_a
|
|
q = best_q
|
|
B = []
|
|
for val in q:
|
|
q_l = factor_base[val].prime
|
|
gamma = factor_base[val].tmem_p * invert(a // q_l, q_l) % q_l
|
|
if 2*gamma > q_l:
|
|
gamma = q_l - gamma
|
|
B.append(a//q_l*gamma)
|
|
b = sum(B)
|
|
g = SievePolynomial(a, b, N)
|
|
for fb in factor_base:
|
|
if a % fb.prime == 0:
|
|
fb.soln1 = None
|
|
continue
|
|
a_inv = invert(a, fb.prime)
|
|
fb.b_ainv = [2*b_elem*a_inv % fb.prime for b_elem in B]
|
|
fb.soln1 = (a_inv*(fb.tmem_p - b)) % fb.prime
|
|
fb.soln2 = (a_inv*(-fb.tmem_p - b)) % fb.prime
|
|
yield g
|
|
|
|
# Update `b` with Gray code
|
|
for i in range(1, 2**(len(B)-1)):
|
|
v = bit_scan1(i)
|
|
neg_pow = 2*((i >> (v + 1)) % 2) - 1
|
|
b = g.b + 2*neg_pow*B[v]
|
|
a = g.a
|
|
g = SievePolynomial(a, b, N)
|
|
for fb in factor_base:
|
|
if fb.soln1 is None:
|
|
continue
|
|
fb.soln1 = (fb.soln1 - neg_pow*fb.b_ainv[v]) % fb.prime
|
|
fb.soln2 = (fb.soln2 - neg_pow*fb.b_ainv[v]) % fb.prime
|
|
yield g
|
|
|
|
|
|
def _gen_sieve_array(M, factor_base):
|
|
"""Sieve Stage of the Quadratic Sieve. For every prime in the factor_base
|
|
that does not divide the coefficient `a` we add log_p over the sieve_array
|
|
such that ``-M <= soln1 + i*p <= M`` and ``-M <= soln2 + i*p <= M`` where `i`
|
|
is an integer. When p = 2 then log_p is only added using
|
|
``-M <= soln1 + i*p <= M``.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
M : sieve interval
|
|
factor_base : factor_base primes
|
|
"""
|
|
sieve_array = [0]*(2*M + 1)
|
|
for factor in factor_base:
|
|
if factor.soln1 is None: #The prime does not divides a
|
|
continue
|
|
for idx in range((M + factor.soln1) % factor.prime, 2*M, factor.prime):
|
|
sieve_array[idx] += factor.log_p
|
|
if factor.prime == 2:
|
|
continue
|
|
#if prime is 2 then sieve only with soln_1_p
|
|
for idx in range((M + factor.soln2) % factor.prime, 2*M, factor.prime):
|
|
sieve_array[idx] += factor.log_p
|
|
return sieve_array
|
|
|
|
|
|
def _check_smoothness(num, factor_base):
|
|
r""" Check if `num` is smooth with respect to the given `factor_base`
|
|
and compute its factorization vector.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
num : integer whose smootheness is to be checked
|
|
factor_base : factor_base primes
|
|
"""
|
|
if num < 0:
|
|
num *= -1
|
|
vec = 1
|
|
else:
|
|
vec = 0
|
|
for i, fb in enumerate(factor_base, 1):
|
|
if num % fb.prime:
|
|
continue
|
|
e = 1
|
|
num //= fb.prime
|
|
while num % fb.prime == 0:
|
|
e += 1
|
|
num //= fb.prime
|
|
if e % 2:
|
|
vec += 1 << i
|
|
return vec, num
|
|
|
|
|
|
def _trial_division_stage(N, M, factor_base, sieve_array, sieve_poly, partial_relations, ERROR_TERM):
|
|
"""Trial division stage. Here we trial divide the values generetated
|
|
by sieve_poly in the sieve interval and if it is a smooth number then
|
|
it is stored in `smooth_relations`. Moreover, if we find two partial relations
|
|
with same large prime then they are combined to form a smooth relation.
|
|
First we iterate over sieve array and look for values which are greater
|
|
than accumulated_val, as these values have a high chance of being smooth
|
|
number. Then using these values we find smooth relations.
|
|
In general, let ``t**2 = u*p modN`` and ``r**2 = v*p modN`` be two partial relations
|
|
with the same large prime p. Then they can be combined ``(t*r/p)**2 = u*v modN``
|
|
to form a smooth relation.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
N : Number to be factored
|
|
M : sieve interval
|
|
factor_base : factor_base primes
|
|
sieve_array : stores log_p values
|
|
sieve_poly : polynomial from which we find smooth relations
|
|
partial_relations : stores partial relations with one large prime
|
|
ERROR_TERM : error term for accumulated_val
|
|
"""
|
|
accumulated_val = (log(M) + log(N)/2 - ERROR_TERM) * 2**10
|
|
smooth_relations = []
|
|
proper_factor = set()
|
|
partial_relation_upper_bound = 128*factor_base[-1].prime
|
|
for x, val in enumerate(sieve_array, -M):
|
|
if val < accumulated_val:
|
|
continue
|
|
v = sieve_poly.eval_v(x)
|
|
vec, num = _check_smoothness(v, factor_base)
|
|
if num == 1:
|
|
smooth_relations.append((sieve_poly.eval_u(x), v, vec))
|
|
elif num < partial_relation_upper_bound and isprime(num):
|
|
if N % num == 0:
|
|
proper_factor.add(num)
|
|
continue
|
|
u = sieve_poly.eval_u(x)
|
|
if num in partial_relations:
|
|
u_prev, v_prev, vec_prev = partial_relations.pop(num)
|
|
u = u*u_prev*invert(num, N) % N
|
|
v = v*v_prev // num**2
|
|
vec ^= vec_prev
|
|
smooth_relations.append((u, v, vec))
|
|
else:
|
|
partial_relations[num] = (u, v, vec)
|
|
return smooth_relations, proper_factor
|
|
|
|
|
|
def _find_factor(N, smooth_relations, col):
|
|
""" Finds proper factor of N using fast gaussian reduction for modulo 2 matrix.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
N : Number to be factored
|
|
smooth_relations : Smooth relations vectors matrix
|
|
col : Number of columns in the matrix
|
|
|
|
Reference
|
|
==========
|
|
|
|
.. [1] A fast algorithm for gaussian elimination over GF(2) and
|
|
its implementation on the GAPP. Cetin K.Koc, Sarath N.Arachchige
|
|
"""
|
|
matrix = [s_relation[2] for s_relation in smooth_relations]
|
|
row = len(matrix)
|
|
mark = [False] * row
|
|
for pos in range(col):
|
|
m = 1 << pos
|
|
for i in range(row):
|
|
if p := matrix[i] & m:
|
|
add_col = p ^ matrix[i]
|
|
matrix[i] = m
|
|
mark[i] = True
|
|
for j in range(i + 1, row):
|
|
if matrix[j] & m:
|
|
matrix[j] ^= add_col
|
|
break
|
|
|
|
for m, mat, rel in zip(mark, matrix, smooth_relations):
|
|
if m:
|
|
continue
|
|
u, v = rel[0], rel[1]
|
|
for m1, mat1, rel1 in zip(mark, matrix, smooth_relations):
|
|
if m1 and mat & mat1:
|
|
u *= rel1[0]
|
|
v *= rel1[1]
|
|
# assert is_square(v)
|
|
v = isqrt(v)
|
|
if 1 < (g := gcd(u - v, N)) < N:
|
|
yield g
|
|
|
|
|
|
def qs(N, prime_bound, M, ERROR_TERM=25, seed=1234):
|
|
"""Performs factorization using Self-Initializing Quadratic Sieve.
|
|
In SIQS, let N be a number to be factored, and this N should not be a
|
|
perfect power. If we find two integers such that ``X**2 = Y**2 modN`` and
|
|
``X != +-Y modN``, then `gcd(X + Y, N)` will reveal a proper factor of N.
|
|
In order to find these integers X and Y we try to find relations of form
|
|
t**2 = u modN where u is a product of small primes. If we have enough of
|
|
these relations then we can form ``(t1*t2...ti)**2 = u1*u2...ui modN`` such that
|
|
the right hand side is a square, thus we found a relation of ``X**2 = Y**2 modN``.
|
|
|
|
Here, several optimizations are done like using multiple polynomials for
|
|
sieving, fast changing between polynomials and using partial relations.
|
|
The use of partial relations can speeds up the factoring by 2 times.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
N : Number to be Factored
|
|
prime_bound : upper bound for primes in the factor base
|
|
M : Sieve Interval
|
|
ERROR_TERM : Error term for checking smoothness
|
|
seed : seed of random number generator
|
|
|
|
Returns
|
|
=======
|
|
|
|
set(int) : A set of factors of N without considering multiplicity.
|
|
Returns ``{N}`` if factorization fails.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.ntheory import qs
|
|
>>> qs(25645121643901801, 2000, 10000)
|
|
{5394769, 4753701529}
|
|
>>> qs(9804659461513846513, 2000, 10000)
|
|
{4641991, 2112166839943}
|
|
|
|
See Also
|
|
========
|
|
|
|
qs_factor
|
|
|
|
References
|
|
==========
|
|
|
|
.. [1] https://pdfs.semanticscholar.org/5c52/8a975c1405bd35c65993abf5a4edb667c1db.pdf
|
|
.. [2] https://www.rieselprime.de/ziki/Self-initializing_quadratic_sieve
|
|
"""
|
|
return set(qs_factor(N, prime_bound, M, ERROR_TERM, seed))
|
|
|
|
|
|
def qs_factor(N, prime_bound, M, ERROR_TERM=25, seed=1234):
|
|
""" Performs factorization using Self-Initializing Quadratic Sieve.
|
|
|
|
Parameters
|
|
==========
|
|
|
|
N : Number to be Factored
|
|
prime_bound : upper bound for primes in the factor base
|
|
M : Sieve Interval
|
|
ERROR_TERM : Error term for checking smoothness
|
|
seed : seed of random number generator
|
|
|
|
Returns
|
|
=======
|
|
|
|
dict[int, int] : Factors of N.
|
|
Returns ``{N: 1}`` if factorization fails.
|
|
Note that the key is not always a prime number.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.ntheory import qs_factor
|
|
>>> qs_factor(1009 * 100003, 2000, 10000)
|
|
{1009: 1, 100003: 1}
|
|
|
|
See Also
|
|
========
|
|
|
|
qs
|
|
|
|
"""
|
|
if N < 2:
|
|
raise ValueError("N should be greater than 1")
|
|
factors = {}
|
|
smooth_relations = []
|
|
partial_relations = {}
|
|
# Eliminate the possibility of even numbers,
|
|
# prime numbers, and perfect powers.
|
|
if N % 2 == 0:
|
|
e = 1
|
|
N //= 2
|
|
while N % 2 == 0:
|
|
N //= 2
|
|
e += 1
|
|
factors[2] = e
|
|
if isprime(N):
|
|
factors[N] = 1
|
|
return factors
|
|
if result := _perfect_power(N, 3):
|
|
n, e = result
|
|
factors[n] = e
|
|
return factors
|
|
N_copy = N
|
|
randint = _randint(seed)
|
|
idx_1000, idx_5000, factor_base = _generate_factor_base(prime_bound, N)
|
|
threshold = len(factor_base) * 105//100
|
|
for g in _generate_polynomial(N, M, factor_base, idx_1000, idx_5000, randint):
|
|
sieve_array = _gen_sieve_array(M, factor_base)
|
|
s_rel, p_f = _trial_division_stage(N, M, factor_base, sieve_array, g, partial_relations, ERROR_TERM)
|
|
smooth_relations += s_rel
|
|
for p in p_f:
|
|
if N_copy % p:
|
|
continue
|
|
e = 1
|
|
N_copy //= p
|
|
while N_copy % p == 0:
|
|
N_copy //= p
|
|
e += 1
|
|
factors[p] = e
|
|
if threshold <= len(smooth_relations):
|
|
break
|
|
|
|
for factor in _find_factor(N, smooth_relations, len(factor_base) + 1):
|
|
if N_copy % factor == 0:
|
|
e = 1
|
|
N_copy //= factor
|
|
while N_copy % factor == 0:
|
|
N_copy //= factor
|
|
e += 1
|
|
factors[factor] = e
|
|
if N_copy == 1 or isprime(N_copy):
|
|
break
|
|
if N_copy != 1:
|
|
factors[N_copy] = 1
|
|
return factors
|