o ThZ:@sddlmZmZddlmZddlmZmZmZm Z ddl m Z ddl mZddlmZGdddZGd d d Zd d Zd dZddZddZddZddZdddZdddZdS))explog)_randint) bit_scan1gcdinvertsqrt)_perfect_power)isprime)_sqrt_mod_prime_powerc@s$eZdZddZddZddZdS)SievePolynomialcCs6||_||_|d|_d|||_|d||_dS)a4This 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 N)aba2abb2)selfrrNrQ/home/air/segue/gemini/back/venv/lib/python3.10/site-packages/sympy/ntheory/qs.py__init__ s   zSievePolynomial.__init__cCs|j||jSN)rrrxrrreval_uszSievePolynomial.eval_ucCs|j||j||jSr)rrrrrrreval_v szSievePolynomial.eval_vN)__name__ __module__ __qualname__rrrrrrrr s r c@seZdZdZddZdS)FactorBaseElemz7This class stores an element of the `factor_base`. cCs(||_||_||_d|_d|_d|_dS)z 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 N)primetmem_plog_psoln1soln2b_ainv)rr!r"r#rrrr's   zFactorBaseElem.__init__N)rrr__doc__rrrrrr $s r c Csddlm}g}d\}}|d|D]C}t||dd|dkrU|dkr.|dur.t|d}|dkr<|dursz(_generate_polynomial..)rr.ranger!r0rabsr"rsumr r$r&r%rrr)rMr3r4r5randint approx_valstartendbest_abest_q best_ratio_rqrand_ppratioBvalq_lgammargivneg_powrr<r_generate_polynomialYsr               rXcCsdgd|d}|D]D}|jdurq t||j|jd||jD] }|||j7<q"|jdkr4q t||j|jd||jD] }|||j7<qCq |S)aSieve 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 rr r)N)r$r@r!r#r%)rCr3 sieve_arrayfactoridxrrr_gen_sieve_arrays  " "r\cCs|dkr |d9}d}nd}t|dD]2\}}||jrqd}||j}||jdkr:|d7}||j}||jdks*|drD|d|>7}q||fS)z 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 rr)r ) enumerater!)numr3vecrUr>errr_check_smoothnesss"     rbcCs,t|t|d|d}g}t} d|dj} t|| D]q\} } | |kr)q || } t| |\}}|dkrE||| | |fq || krt|r||dkrY| |q || }||vr| |\}}}||t |||}| ||d} ||N}||| |fq || |f||<q || fS)a)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 r r*r]r)r) rsetr!r^rrbr0rr addpopr)rrCr3rY sieve_polypartial_relations ERROR_TERMaccumulated_valsmooth_relations proper_factorpartial_relation_upper_boundrrQrVr`r_uu_prevv_prevvec_prevrrr_trial_division_stages2    rrccsJdd|D}t|}dg|}t|D]=}d|>}t|D]2}|||@} rQ| ||A} |||<d||<t|d|D]} || |@rN|| | N<q>nqqt|||D]I\}} } |raqY| d| d}}t|||D]\}}}|r| |@r||d9}||d9}qpt|}dt|||}kr|krnqY|VqYdS)a 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 cSsg|]}|dqSr9r)r: s_relationrrrr?sz _find_factor..Fr)TrN)r.r@zipisqrtr)rrkcolmatrixrowmarkposmrUrNadd_coljmatrelrnrVm1mat1rel1rTrrr _find_factor s@         &rcCstt|||||S)aPerforms 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 )rd qs_factor)rr1rCriseedrrrqs:s1rc Cs|dkrtdi}g}i}|ddkr2d}|d}|ddkr.|d}|d7}|ddks ||d<t|r>> from sympy.ntheory import qs_factor >>> qs_factor(1009 * 100003, 2000, 10000) {1009: 1, 100003: 1} See Also ======== qs r zN should be greater than 1rr)id) ValueErrorr r rr7r.rXr\rrr)rr1rCrirfactorsrkrhraresultr2N_copyrDr4r5r3 thresholdrTrYs_relp_frNrZrrrrnsp           rN)rr)mathrrsympy.core.randomrsympy.external.gmpyrrrrrusympy.ntheory.factor_r sympy.ntheory.primetestr sympy.ntheory.residue_ntheoryr r r r7rXr\rbrrrrrrrrrs     K2 -4