wgGddlmZddlmZmZmZddlmZddl m Z ddl m Z mZGddZ Gdd Zd Zdd Zd ZdZdZdZdZdZdZddZy ))_randint)gcdinvertsqrt)_sqrt_mod_prime_power)isprime)logrceZdZddZdZy)SievePolynomialNc.||_||_||_y)aThis class denotes the seive polynomial. If ``g(x) = (a*x + b)**2 - N``. `g(x)` can be expanded to ``a*x**2 + 2*a*b*x + b**2 - N``, so the coefficient is stored in the form `[a**2, 2*a*b, b**2 - N]`. This ensures faster `eval` method because we dont have to perform `a**2, 2*a*b, b**2` every time we call the `eval` method. As multiplication is more expensive than addition, by using modified_coefficient we get a faster seiving process. Parameters ========== modified_coeff : modified_coefficient of sieve polynomial a : parameter of the sieve polynomial b : parameter of the sieve polynomial N)modified_coeffab)selfr rrs U/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/sympy/ntheory/qs.py__init__zSievePolynomial.__init__ s$-c@d}|jD] }||z}||z }|S)z Compute the value of the sieve polynomial at point x. Parameters ========== x : Integer parameter for sieve polynomial r)r )rxanscoeffs revalzSievePolynomial.evals7(( E 1HC 5LC  r)NN)__name__ __module__ __qualname__rrrrrr r s , rr ceZdZdZdZy)FactorBaseElemz7This class stores an element of the `factor_base`. cf||_||_||_d|_d|_d|_d|_y)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_psoln1soln2a_invb_ainv)rr r!r"s rrzFactorBaseElem.__init__2s5       rN)rrr__doc__rrrrrr/s rrc\ddlm}g}d\}}|jd|D]}t||dz dz|dk(s|dkDr|t |dz }|dkDr|t |dz }t ||dd}t t|dz}|jt||||||fS) aGenerate `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 r)sieveNNii) sympy.ntheory.generater) primerangepowlenrroundr appendr) prime_boundnr) factor_baseidx_1000idx_5000r residuer"s r_generate_factor_baser:Fs-K#Hh!!![1F q519"E *a /t| 0{+a/t| 0{+a/+Aua8;G#e*U*+E   ~eWeD EF X{ **rNcDt|}td|z|z }d\}} } |dn|} |t|dz n|} tdD]} d}g}||krJd}|dk(s||vr|| | }|dk(r||vr||j}||z}|j |||krJ||z }| t |dz t | dz ks{|} |}|} |}| }g}|D]W}||j}||jt||z|z|z}||dz kDr||z }|j ||z|zYt|}t||zd|z|z||z|z g||}|D]}||jzdk(rt||j|_ |Dcgc]!}d|z|jz|jz#c}|_ |j|j|z z|jz|_ |j|j |z z|jz|_||fScc}w)agThis step is the initialization of the 1st sieve polynomial. Here `a` is selected as a product of several primes of the factor_base such that `a` is about to ``sqrt(2*N) / M``. Other initial values of factor_base elem are also initialized which includes a_inv, b_ainv, soln1, soln2 which are used when the sieve polynomial is changed. The b_ainv is required for fast polynomial change as we do not have to calculate `2*b*invert(a, prime)` every time. We also ensure that the `factor_base` primes which make `a` are between 1000 and 5000. 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 seed : Generate pseudoprime numbers r,)NNNrr+2)rrr1ranger r3absr!rsumr r%r&r#r$)NMr6r7r8seedrandint approx_valbest_abest_q best_ratiostartend_rqrand_ppratioBvalq_lgammargfbb_elems r_initialize_first_polynomialrVcsg*tnGacQJ "2FFJ!AxE"*"2#k Q C 2Y  *nFA+1 ,A+1F#))A FA HHV  *nJ  UQY#j1n2E!EFFJ AA A#$$C ''&c3*??#E 37?%KE C  AA1ac!eQqS1W-q!4A: rxx<1  !RXX&@ABfQvXbhh&1B HHbii!m,8HHryyj1n-9 : a4KCs &Hcddlm}d}|}|dzdk(r|dz }|dz}|dzdk(r||d|zz dzdk(rd}nd}|jd|z||dz zz} |j} t | | zd| z| z| | z|z g| | }|D]}} | | j zdk(r| j || j|dz zz | j z| _| j|| j|dz zz | j z| _|S)aInitialization stage of ith poly. After we finish sieving 1`st polynomial here we quickly change to the next polynomial from which we will again start sieving. Suppose we generated ith sieve polynomial and now we want to generate (i + 1)th polynomial, where ``1 <= i <= 2**(j - 1) - 1`` where `j` is the number of prime factors of the coefficient `a` then this function can be used to go to the next polynomial. If ``i = 2**(j - 1) - 1`` then go to _initialize_first_polynomial stage. Parameters ========== N : number to be factored factor_base : factor_base primes i : integer denoting ith polynomial g : (i - 1)th polynomial B : array that stores a//q_l*gamma r)ceilingr+r,) #sympy.functions.elementary.integersrXrrr r r#r&r$) r@r6irSrOrXvjneg_powrrrTs r_initialize_ith_polyr_s3$< A A a%1* Q a a%1*qAqDzQ!# ai!a%  A A1ac!eQqS1W-q!4AD rxx<1  HHwryyQ'777288CHHwryyQ'777288C D Hrcdgd|zdzz}|D]}|jt||jz|jzd|z|jD]}||xx|jz cc<|jdk(rpt||jz|jzd|z|jD]}||xx|jz cc<|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+)r#r=r r"r$)rAr6 sieve_arrayfactoridxs r_gen_sieve_arrayrds#qsQw-K - <<  !fll*fll:AaCN -C   ,  - <<1  !fll*fll:AaCN -C   ,  - - rcg}|dkr|jd|dz}n|jd|D]u}||jzdk7r|jd'd}||jzdk(r'|dz }||jz}||jzdk(r'|j|dzw|dk(r|dfSt|r|dfSy)abHere we check that if `num` is a smooth number or not. If `a` is a smooth number then it returns a vector of prime exponents modulo 2. For example if a = 2 * 5**2 * 7**3 and the factor base contains {2, 3, 5, 7} then `a` is a smooth number and this function returns ([1, 0, 0, 1], True). If `a` is a partial relation which means that `a` a has one prime factor greater than the `factor_base` then it returns `(a, False)` which denotes `a` is a partial relation. Parameters ========== a : integer whose smootheness is to be checked factor_base : factor_base primes rr+rYr,TFr*)r3r r)numr6vecrb factor_exps r_check_smoothnessris C Qw 1  r  1 #   " JJqM  FLL A% !OJ FLL CFLL A% :>"# axDys|Ez rcft|}t||zdz|z }g} t} d|djz} t |D]\} } | |kr | |z }|j |}t ||\}}|4|j|z|jz}|dur\|}|| kDr\||vr||f||<h||\}}|j| t||}||z|z}||z||zz}t ||\}}| j|||f| | fS#t$r| j|YwxYw)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-rYF)isqrtr setr enumeraterrirrpoprZeroDivisionErroraddr3)r@rAr6ra sieve_polypartial_relations ERROR_TERMsqrt_naccumulated_valsmooth_relations proper_factorpartial_relation_upper_boundrcrPrr\rg is_smoothu large_primeu_prevv_prevlarge_prime_invs r_trial_division_stagers.1XF!f*oe+j8OEM#&{2'<'<#< k*-S   !G OOA *1k:Y    LLNZ\\ )  K99"3323Q!+.!2;!?!%%k2&,[!&}||k(r |||dk(st|D]} || ||| |zdz|| |<@rg} t |D]!\} } | dk(s | j || | g#| ||fS)aFast gaussian reduction for modulo 2 matrix. Parameters ========== A : Matrix Examples ======== >>> from sympy.ntheory.qs import _gauss_mod_2 >>> _gauss_mod_2([[0, 1, 1], [1, 0, 1], [0, 1, 0], [1, 1, 1]]) ([[[1, 0, 1], 3]], [True, True, True, False], [[0, 1, 0], [1, 0, 0], [0, 0, 1], [1, 0, 1]]) Reference ========== .. [1] A fast algorithm for gaussian elimination over GF(2) and its implementation on the GAPP. Cetin K.Koc, Sarath N.ArachchigerNFr+Tr,)copydeepcopyr1r=rnr3) Arrrowcolmarkcrc1r2 dependent_rowrcrPs r _gauss_mod_2r`s2, ]]1 F f+C fQi.C 73;D 3Z Js Aay|q  Q* JBQway}!*JB&,Rjnvbz!}&D%IF2JrNJ  J JMdO5S %<  &+s!3 45 $ &&rc||d}||dg}||dg}||d} t| D]h\} } | dk(s tt|D]F} || | dk(s|| dk(s|j|| d|j|| dhjd} d}|D]}| |z} |D]}||z} t |}t | |z |S)aFinds proper factor of N. Here, transform the dependent rows as a combination of independent rows of the gauss_matrix to form the desired relation of the form ``X**2 = Y**2 modN``. After obtaining the desired relation we obtain a proper factor of N by `gcd(X - Y, N)`. Parameters ========== dependent_rows : denoted dependent rows in the reduced matrix form mark : boolean array to denoted dependent and independent rows gauss_matrix : Reduced form of the smooth relations matrix index : denoted the index of the dependent_rows smooth_relations : Smooth relations vectors matrix N : Number to be factored r+rT)rnr=r1r3rlr)dependent_rowsr gauss_matrixindexrwr@ idx_in_smooth independent_u independent_vdept_rowrcrPrr{r\r[s r _find_factorrs' #5)!,M%m4Q78M%m4Q78Me$Q'Hh'S !8S./ $S)Q.493D!(()9#)>q)AB!(()9#)>q)AB   A A  Q  Q aA q1ua=rc |dz}t||\}}}g}d} i} t} dt|zdz} | dk(rt|||||\} }nt |||  } | dz } | dt|dz zk\rd} t ||}t ||||| | |\}}||z }| |z} t|t|| zk\rnt|}t|\}}}|}tt|D]l}t||||||}|dkDs||ks| j|||zdk(r||z}||zdk(rt|r| j|| S|dk(sk| S| 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 threshold : Extra smooth relations for factorization seed : generate pseudo prime numbers Examples ======== >>> from sympy.ntheory import qs >>> qs(25645121643901801, 2000, 10000) {5394769, 4753701529} >>> qs(9804659461513846513, 2000, 10000) {4641991, 2112166839943} References ========== .. [1] https://pdfs.semanticscholar.org/5c52/8a975c1405bd35c65993abf5a4edb667c1db.pdf .. [2] https://www.rieselprime.de/ziki/Self-initializing_quadratic_sieve r-rdr+r,) r:rmr1rVr_rdrrrr=rrqr)r@r4rArtrBr7r8r6rwith_polyrsrx thresholdith_sieve_polyB_arrayras_relp_frrrrN_copyrrbs rqsrsNJ&;K&K#Hh HEM#k""c)I  q=&B1aV^`h&i #NG1![(N\cdNA  q3w[lnxy sE!  C $4y$@ @  + ,F(4V(<%M4 Fs=)* mT<HXZ[\ A:&1*   f %6/Q&6!6/Q&v!!&) {   r)N)i)sympy.core.randomrsympy.external.gmpyrrrrlsympy.ntheory.residue_ntheoryr sympy.ntheoryrmathr r rr:rVr_rdrirrrrrrrrrsf&::?!$$N.+:CL% P6#L<+@ *'Z%PJr