U 9%e͞@sddlmZddlmZddlmZmZmZddlm Z ddl m Z ddl m Z ddlmZddlmZmZmZd d lmZd d lmZmZmZmZmZdd lmZdd lmZm Z ddl!m"Z"m#Z#ddZ$ddZ%ddZ&ddZ'ddZ(dXddZ)ddZ*e+fddZ,d d!Z-d"d#Z.d$d%Z/d&d'Z0d(d)Z1d*d+Z2d,d-Z3d.d/Z4d0d1Z5d2d3Z6dYd4d5Z7d6d7d8d9Z8d:d;Z9dd?d?eZ;dZdAdBZd]dHdIZ?d^dJdKZ@dLdMZAdNdOZBdPdQZCdRdSZDdTdUZEdVdWZFd@S)_) annotations)Function)igcdigcdex mod_inverse)isqrt)S)Poly)ZZ)gf_crt1gf_crt2linear_congruence)isprime) factorinttrailingtotient multiplicity perfect_power)as_int)_randintrandint)cycleproductc Cs&ddlm}t|t|}}|dkr.td||}|dkrBdSt||dkrXtd|t}t|D]N\}}|dkr|||d7<t|dD]\}}|||7<qqld}|D]\}}|||9}q|D]:\} } t| D](} t ||| |dkr|| }qqqq|S)aReturns the order of ``a`` modulo ``n``. The order of ``a`` modulo ``n`` is the smallest integer ``k`` such that ``a**k`` leaves a remainder of 1 with ``n``. Parameters ========== a : integer n : integer, n > 1. a and n should be relatively prime Examples ======== >>> from sympy.ntheory import n_order >>> n_order(3, 7) 6 >>> n_order(4, 7) 3 r) defaultdictrz%n should be an integer greater than 1*The two numbers should be relatively prime) collectionsrr ValueErrorrintritemsrangepow) anrfactorspxZkxpykyorderpe_r,\/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/ntheory/residue_ntheory.pyn_orders0   r.c#s\fddtdD}d}|krX|D]}t||dkr.qNq.|V|d7}q"dS)aJ Generates the primitive roots for a prime ``p`` Examples ======== >>> from sympy.ntheory.residue_ntheory import _primitive_root_prime_iter >>> list(_primitive_root_prime_iter(19)) [2, 3, 10, 13, 14, 15] References ========== .. [1] W. Stein "Elementary Number Theory" (2011), page 44 csg|]}d|qS)rr,.0ir)r,r- Ysz._primitive_root_prime_iter..rN)rkeysr!)r)vr"pwr,r2r-_primitive_root_prime_iterGsr8cCsNt|}|dkrtd|dkr$dSt|}t|dkr>> from sympy.ntheory.residue_ntheory import primitive_root >>> primitive_root(19) 2 References ========== .. [1] W. Stein "Elementary Number Theory" (2011), page 44 .. [2] P. Hackman "Elementary Number Theory" (2009), Chapter C rzp is required to be positiver4Nr) rrrlenris_primitive_rootlistprimitive_rootr rnextr8)r)fp1e1r1r#gr,r,r-r>esD      r>csttdkr"tdtdkr@tddkrTdkSt}|dkrhdS|?}t|r|dtt|d}nVt|}|sdS|\}}t|sdS||d|dtt|d}| |t fdd|DS)ac Returns True if ``a`` is a primitive root of ``p``. ``a`` is said to be the primitive root of ``p`` if gcd(a, p) == 1 and totient(p) is the smallest positive number s.t. a**totient(p) cong 1 mod(p) Parameters ========== a : integer p : integer, p > 1. a and p should be relatively prime Examples ======== >>> from sympy.ntheory import is_primitive_root, n_order, totient >>> is_primitive_root(3, 10) True >>> is_primitive_root(9, 10) False >>> n_order(3, 10) == totient(10) True >>> n_order(9, 10) == totient(10) False rz%p should be an integer greater than 1rr9Fc3s"|]}t|dkVqdSrNr!)r0primer"Z group_orderr)r,r- sz$is_primitive_root..) rrrrrsetrr5raddall)r"r)tqr$mr*r,rGr-r<s2  r<c Cst|d}||?}td|d}t||}|dkrq8qt|||}t|||}d}t|D]L} |t||||} t| d|d| |} | ||dkr\|d| 7}q\t||dd|t||d||} | S)z Returns the square root in the case of ``p`` prime with ``p == 1 (mod 8)`` References ========== .. [1] R. Crandall and C. Pomerance "Prime Numbers", 2nt Ed., page 101 rr4r)rrlegendre_symbolr!r ) r"r)srLdrADrNr1Zadmxr,r,r-_sqrt_mod_tonelli_shankss     (rWFcCs|rtt||Sztt|}t||}t|}||dkrH||WS||dkrZ|WSz$t|}||dkr|||WWSWntk rYnX|WSWntk rYdSXdS)a Find a root of ``x**2 = a mod p``. Parameters ========== a : integer p : positive integer all_roots : if True the list of roots is returned or None Notes ===== If there is no root it is returned None; else the returned root is less or equal to ``p // 2``; in general is not the smallest one. It is returned ``p // 2`` only if it is the only root. Use ``all_roots`` only when it is expected that all the roots fit in memory; otherwise use ``sqrt_mod_iter``. Examples ======== >>> from sympy.ntheory import sqrt_mod >>> sqrt_mod(11, 43) 21 >>> sqrt_mod(17, 32, True) [7, 9, 23, 25] r4N)sorted sqrt_mod_iterabsrr? StopIteration)r"r) all_rootsitrSr,r,r-sqrt_mods&       r^cgsxtdd|D}t|}dg|}d}d|}}|rX|sX|d8}t||\}||<q2|sl|sl|rtd}nqt|Vq(dS)z Cartesian product generator Notes ===== Unlike itertools.product, it works also with iterables which do not fit in memory. See https://bugs.python.org/issue10109 Author: Fernando Sumudu with small changes css|]}tt|VqdSN)r enumerate)r0r]r,r,r-rHDsz_product..NTrrF)tupler;r?)ZitersZ inf_itersZ num_itersZcur_valZfirst_vr1r)r,r,r-_product7s   rbccs^t|tt|}}t|rv||}|dkr>> from sympy.ntheory.residue_ntheory import sqrt_mod_iter >>> list(sqrt_mod_iter(11, 43)) [21, 22] rrN) rrZr _sqrt_mod1_sqrt_mod_prime_powerr rrappendr rbr )r"r)domainresrVr@r6pvr%exrxmmr*rQZvxrSr,r,r-rYXs@          rYcCs$||}||}|dkr|dkr,t|gS||dksTt||dd|dksTdS|ddkrvt||dd|}n|ddkrt||dd|}|dkrt||dd|}qtd||dd|}d|||}t|d||kr|}n t||}tt|t||gS|dkr |dkrV|ddkr:dS|dkrt}td|dD]"} |d| |d | qVt|Stdtdtdtd g} d} g}| D]} | } | |kr| d|| ?}|dr| d| d>} | d7} q| |kr|| | d|d>}|d| >kr||kr|d||dkr||q|St ||d} | sldS| d} | d|}d} |}| }|d9}||krq|} |d}t d| |d}| |||} | d|}q| |kr||}t d| |d}| |||} | || gSdS) a Find the solutions to ``x**2 = a mod p**k`` when ``a % p != 0`` Parameters ========== a : integer p : prime number k : positive integer Examples ======== >>> from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power >>> _sqrt_mod_prime_power(11, 43, 1) [21, 22] References ========== .. [1] P. Hackman "Elementary Number Theory" (2009), page 160 .. [2] http://www.numbertheory.org/php/squareroot.html .. [3] [Gathen99]_ rr4Nr9r:rrO) r r!rWrXrIr rJr=rerdr)r"r)kpkrgsignbrVrQr1rvr#rSnxr1frr%Zn1Zfrinvr,r,r-rds  $               rdc s||}|dkrh|d|ddkrLdfdd}|Sfdd}|St|}|}|ddkrdS|d||?}dkr||dkrd|d>dd>fd d }|S||dkr2t|||dkrdSd|>fd d } | S||dkrt|||dkr^dSd|d>fd d} | Snn|d||}t||| dkrdS||| fdd} | SdS)z~ Find solution to ``x**2 == a mod p**n`` when ``a % p == 0`` see http://www.numbertheory.org/php/squareroot.html rr4rc3s d}|kr|V|7}qdSNrr,r1)pm1pnr,r-_iter0asz_sqrt_mod1.._iter0ac3s d}|kr|V|7}qdSrwr,rx)pmrzr,r-_iter0bsz_sqrt_mod1.._iter0bNc3sFdd>}d>}|krB|}|kr8|V||7}q |7}qdS)Nrr4r,)ror1j)rNryrzpnm1r,r-_iter1*s  z_sqrt_mod1.._iter1c3sNt}D]>}d}|kr |>|}||kr>|||V|7}qq dSrwrIrJrQrSr1rV)rNrzpnmrgr,r-_iter29s  z_sqrt_mod1.._iter2c3sRt}D]B}d}|kr |>|}||krB|||V|7}qq dSrwrr)rNrzrrgr,r-_iter3Is z_sqrt_mod1.._iter3c3sZt}}D]B}d}|kr||}||krJ||||V|7}qqdSrwr)rQr|rjr1rV)rNr)rzrpnrres1r,r-_iter4^s   z_sqrt_mod1.._iter4)rrd) r"r)r#r{r}r@rSZa1rrrrr,) rNr)r|ryrzrrrrgrr-rcs^                 rccCst|t|}}|dkr"td||ks2|dkr:||}|dksJ|dkrNdSt|s|drpt||dkrpdSt||}|d krdSdSt||dd|dkS) a Returns True if ``a`` (mod ``p``) is in the set of squares mod ``p``, i.e a % p in set([i**2 % p for i in range(p)]). Examples ======== If ``p`` is an odd prime, an iterative method is used to make the determination: >>> from sympy.ntheory import is_quad_residue >>> sorted(set([i**2 % 7 for i in range(7)])) [0, 1, 2, 4] >>> [j for j in range(7) if is_quad_residue(j, 7)] [0, 1, 2, 4] See Also ======== legendre_symbol, jacobi_symbol rz p must be > 0rr4r:TrOFN)rrr jacobi_symbolr^r!)r"r)rSr,r,r-is_quad_residuels rcCs||}t|t|t|}}}|dkr4td|dkrDtd|dkr`|dkrXdS|dkS|dkrldS|dkrxdS|dkrt||St|||S)z Returns True if ``x**n == a (mod m)`` has solutions. References ========== .. [1] P. Hackman "Elementary Number Theory" (2009), page 76 rz m must be > 0zn must be >= 0rFTr4)rrr_is_nthpow_residue_bignr"r#rNr,r,r-is_nthpow_residues"  rcCsvt|dkst||dkrHt|D]\}}t||||s&dSq&dSt|}t|t||}t||t|dkS)z=Returns True if `x^n = a \pmod{n}` has solutions for `n > 2`.NrFT)r>rrr#_is_nthpow_residue_bign_prime_powerrrr!)r"r#rNrFpowerr@ror,r,r-rsrcCs||rR|dkr"t||t||S|d@r.dSt|}|tdt|d|dkS|t||;}|shdSt||}||r~dSt||}t||||||SdS)zRReturns True/False if a solution for `x^n = a \pmod{p^k}` does/does not exist.r4rTFN)rr!rminrr)r"r#r)rocmur|r,r,r-rs  rcCsJt|}g}|D]\}}||g|q|D]}t|||d}q2|S)NF)rrextend _nthroot_mod1)rQrMr)r@r6rrr*Zqxr,r,r- _nthroot_mod2srcCsDt|}t|st|||}n|d}|d|dks:td}||dkr\|d7}||}q>t| |d|}||} d| |} t|| |} t|||} t||||} t|| | }t|| ||}t||d}| ||}|g}t||d||} |}t|dD]}|| |}||q |r<| |St |S)z Root of ``x**q = s mod p``, ``p`` prime and ``q`` divides ``p - 1`` References ========== .. [1] A. M. Johnston "A Generalized qth Root Algorithm" rr) r>rrAssertionErrorrr! discrete_logr resortr)rQrMr)r\rCrSr@rof1zrVrus1hrLg2Zg3rghxr1r,r,r-rs:         rcsddlmt}i}|D]$\}}t}|dkrH|||n||D]} || |} | dkr|} t| |} td|D]"} | |9} | || | | | } q~|| qP|}| h}|t ||kr||9}t}|D]8}|||dkrq||kr||||||}qq|}q||B}qP|tkr8gS||t ||<q g}g|D]"\}} || t |qXt fddt |DS)a Helper function for _nthroot_mod_composite and polynomial_congruence. Parameters ========== m : positive integer prime_modulo_method : function to calculate the root of the congruence equation for the prime divisors of m diff_method : function to calculate derivative of expression at any given point expr_val : function to calculate value of the expression at any given point rcrtrcsh|]}t|dqS)r)r=r/rrNr,r- Isz_help..)Zsympy.ntheory.modularrrrrIupdaterr rJr!rer=rXr)rNZprime_modulo_methodZ diff_methodZexpr_valr@ddr)r*Z tot_rootsrootdiffZppowZm_invr~new_baseZ roots_in_baseZ new_rootsror"rVyr,rr-_helpsL         rcs*t|fddfddfddS)zG Find the solutions to ``x**n = a mod m`` when m is not prime. cst|dS)NT) nthroot_modr2r"r#r,r-Qz(_nthroot_mod_composite..cst|d|||S)NrrErr))r#r,r-rRrcst|||Sr_rErrr,r-rSr)rrr,rr-_nthroot_mod_compositeLs    rc CsV||}t|t|t|}}}|dkr8t|||St|sLt|||S||dkr^dgSt|||sv|rrgSdS|d|dkrt||||S|}|d}d}||kr||||f\}}}}|rt||\}}t|||} t| |d} | ||} ||}}|| }}q|dkr.|r(|g} n|} n$|dkrDt|||St||||} | S)a Find the solutions to ``x**n = a mod p``. Parameters ========== a : integer n : positive integer p : positive integer all_roots : if False returns the smallest root, else the list of roots Examples ======== >>> from sympy.ntheory.residue_ntheory import nthroot_mod >>> nthroot_mod(11, 4, 19) 8 >>> nthroot_mod(11, 4, 19, True) [8, 11] >>> nthroot_mod(68, 3, 109) 23 r4rNr) rr^rrrrdivmodr!r) r"r#r)r\paZpbrrrMrSrrgr,r,r-rVs@            rz list[int])returncs.tfddtddD}t|S)z Returns the list of quadratic residues. Examples ======== >>> from sympy.ntheory.residue_ntheory import quadratic_residues >>> quadratic_residues(7) [0, 1, 2, 4] csh|]}t|dqS)r4rEr/r2r,r-rsz%quadratic_residues..r4r)rr rX)r)rSr,r2r-quadratic_residuess rcCsZt|t|}}t|r"|dkr*td||}|s:dSt||dd|dkrVdSdS)a Returns the Legendre symbol `(a / p)`. For an integer ``a`` and an odd prime ``p``, the Legendre symbol is defined as .. math :: \genfrac(){}{}{a}{p} = \begin{cases} 0 & \text{if } p \text{ divides } a\\ 1 & \text{if } a \text{ is a quadratic residue modulo } p\\ -1 & \text{if } a \text{ is a quadratic nonresidue modulo } p \end{cases} Parameters ========== a : integer p : odd prime Examples ======== >>> from sympy.ntheory import legendre_symbol >>> [legendre_symbol(i, 7) for i in range(7)] [0, 1, 1, -1, 1, -1, -1] >>> sorted(set([i**2 % 7 for i in range(7)])) [0, 1, 2, 4] See Also ======== is_quad_residue, jacobi_symbol r4zp should be an odd primerrrO)rrrr!)r"r)r,r,r-rPs#rPcCst|t|}}|dks"|ds*td|dks:||krB||;}|sRt|dkS|dksb|dkrfdSt||dkrxdSd}|dkr|ddkr|dkr|dL}|ddkr| }q||}}|d|dkrdkrnn| }||;}q||S) a Returns the Jacobi symbol `(m / n)`. For any integer ``m`` and any positive odd integer ``n`` the Jacobi symbol is defined as the product of the Legendre symbols corresponding to the prime factors of ``n``: .. math :: \genfrac(){}{}{m}{n} = \genfrac(){}{}{m}{p^{1}}^{\alpha_1} \genfrac(){}{}{m}{p^{2}}^{\alpha_2} ... \genfrac(){}{}{m}{p^{k}}^{\alpha_k} \text{ where } n = p_1^{\alpha_1} p_2^{\alpha_2} ... p_k^{\alpha_k} Like the Legendre symbol, if the Jacobi symbol `\genfrac(){}{}{m}{n} = -1` then ``m`` is a quadratic nonresidue modulo ``n``. But, unlike the Legendre symbol, if the Jacobi symbol `\genfrac(){}{}{m}{n} = 1` then ``m`` may or may not be a quadratic residue modulo ``n``. Parameters ========== m : integer n : odd positive integer Examples ======== >>> from sympy.ntheory import jacobi_symbol, legendre_symbol >>> from sympy import S >>> jacobi_symbol(45, 77) -1 >>> jacobi_symbol(60, 121) 1 The relationship between the ``jacobi_symbol`` and ``legendre_symbol`` can be demonstrated as follows: >>> L = legendre_symbol >>> S(45).factors() {3: 2, 5: 1} >>> jacobi_symbol(7, 45) == L(7, 3)**2 * L(7, 5)**1 True See Also ======== is_quad_residue, legendre_symbol rr4z#n should be an odd positive integerrrl)r:rmr9r:)rrrr)rNr#r~r,r,r-rs,9     rc@seZdZdZeddZdS)mobiusa Mobius function maps natural number to {-1, 0, 1} It is defined as follows: 1) `1` if `n = 1`. 2) `0` if `n` has a squared prime factor. 3) `(-1)^k` if `n` is a square-free positive integer with `k` number of prime factors. It is an important multiplicative function in number theory and combinatorics. It has applications in mathematical series, algebraic number theory and also physics (Fermion operator has very concrete realization with Mobius Function model). Parameters ========== n : positive integer Examples ======== >>> from sympy.ntheory import mobius >>> mobius(13*7) 1 >>> mobius(1) 1 >>> mobius(13*7*5) -1 >>> mobius(13**2) 0 References ========== .. [1] https://en.wikipedia.org/wiki/M%C3%B6bius_function .. [2] Thomas Koshy "Elementary Number Theory with Applications" cCsz|jr|jdk r"tdntd|jr.tjS|tjkr>tjS|jrvt |}t dd| Drhtj Stjt |SdS)NTzn should be a positive integerzn should be an integercss|]}|dkVqdSrDr,r/r,r,r-rH]szmobius.eval..) is_integerZ is_positiver TypeErrorZis_primerZ NegativeOneZOneZ is_IntegerranyvaluesZZeror;)clsr#r"r,r,r-evalPs   z mobius.evalN)__name__ __module__ __qualname____doc__ classmethodrr,r,r,r-r(s'rNcCsV||;}||;}|dkr|}d}t|D] }||kr<|S|||}q(tddS)a Trial multiplication algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. The algorithm finds the discrete logarithm using exhaustive search. This naive method is used as fallback algorithm of ``discrete_log`` when the group order is very small. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_trial_mul >>> _discrete_log_trial_mul(41, 15, 7) 3 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). NrLog does not exist)r r)r#r"rrr(rVr1r,r,r-_discrete_log_trial_mulbs rc Cs||;}||;}|dkr"t||}t|d}i}d}t|D]}|||<|||}q>t||}t|||}|}t|D],}||kr||||S|||}qztddS)ae Baby-step giant-step algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. The algorithm is a time-memory trade-off of the method of exhaustive search. It uses `O(sqrt(m))` memory, where `m` is the group order. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_shanks_steps >>> _discrete_log_shanks_steps(41, 15, 7) 3 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). Nrr)r.rr rr!r) r#r"rrr(rNTrVr1rr,r,r-_discrete_log_shanks_stepss$      r c Cs||;}||;}|dkr"t||}t|}t|D]}|d|d}|d|d} t|||t|| ||} | d} | dkr|| |} |} | d|}nJ| dkr| | |} |||} | | |}n|| |} |d|} | }t|D]}| d} | dkr"|| |} | d|} nH| dkrR| | |} |||}| | |} n|| |} |d|}| d} | dkr|| |} |d|}nH| dkr| | |} | | |} |||}n|| |} | d|} | d} | dkr || |} |d|}nH| dkr:| | |} | | |} |||}n|| |} | d|} | | kr| ||}zBt||| ||}t|||||dkr|WSWntk rYnXq2qq2tddS)ag Pollard's Rho algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. It is a randomized algorithm with the same expected running time as ``_discrete_log_shanks_steps``, but requires a negligible amount of memory. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_pollard_rho >>> _discrete_log_pollard_rho(227, 3**7, 3) 7 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). Nrr:rz&Pollard's Rho failed to find logarithm)r.rr r!rr)r#r"rrr(retriesZrseedrr1ZaabaZxarxbabZbbr~rSr*r,r,r-_discrete_log_pollard_rhos|                            rcCsddlm}||;}||;}|dkr.t||}t|}dgt|}t|D]\}\}} t| D]n} t||||} t|t | |||| d|} t||||} t || | |d}||||| 7<qdqP|dd|D|\}}|S)a Pohlig-Hellman algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. In order to compute the discrete logarithm, the algorithm takes advantage of the factorization of the group order. It is more efficient when the group order factors into many small primes. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_pohlig_hellman >>> _discrete_log_pohlig_hellman(251, 210, 71) 197 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). rrNrTcSsg|]\}}||qSr,r,)r0pirir,r,r-r3Isz0_discrete_log_pohlig_hellman..) Zmodularrr.rr;r`rr r!rr)r#r"rrr(rr@lr1rrr~ZgjZajZbjcjrRr+r,r,r-_discrete_log_pohlig_hellmans    "rcCst|t|t|}}}|dkr.t||}|dkr>t|}|dkrTt||||S|r||dkrnt||||St||||St||||S)a Compute the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. This is a recursive function to reduce the discrete logarithm problem in cyclic groups of composite order to the problem in cyclic groups of prime order. It employs different algorithms depending on the problem (subgroup order size, prime order or not): * Trial multiplication * Baby-step giant-step * Pollard's Rho * Pohlig-Hellman Examples ======== >>> from sympy.ntheory import discrete_log >>> discrete_log(41, 15, 7) 3 References ========== .. [1] https://mathworld.wolfram.com/DiscreteLogarithm.html .. [2] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). NilJ))rr.rrrrr)r#r"rrr(Z prime_orderr,r,r-rMs rc Cst|}t|}t|}t|}||}||}||}|dkrNt|| |S|dkrg}|ddkrp|d|||ddkr|d|St|rt||}||9}||9}|ddkr||}||d||}t||dd}t}|D]} || |d|qt|St||d||d||dd}t}|D]<} td|| |d||} | D]} || |qnqJt|S)z Find the solutions to ``a x**2 + b x + c = 0 mod p. Parameters ========== a : int b : int c : int p : int A positive integer. rr4rr9T)r\) rr rerrr^rIrJrX) r"rrrr)rootsZinv_arRrrgr1rr~r,r,r-quadratic_congruence~sF       &rcCsg}t|}td|D]f}d}td|dD],}|t|t||d||||}q,||d}||dkr||q|S)zA helper function used by polynomial_congruence. It returns the root of a polynomial modulo prime number by naive search from [0, p). Parameters ========== coefficients : list of integers p : prime number rrrO)r;r r!rre) coefficientsr)rrankr1f_valcoeffr,r,r-_polynomial_congruence_primes *   rcCsbd}t|}td|dD]>}||s(q|t|||d|||d|||}q||S)zA helper function used by polynomial_congruence. It returns the derivative of the polynomial evaluated at the root (mod p). Parameters ========== coefficients : list of integers p : prime number root : integer rrr4r;r r!)rrr)rrrr,r,r- _diff_polys  rcCsXt|}d}td|dD](}|t|||d||||}q||d}||S)zA helper function used by polynomial_congruence. It returns value of the polynomial at root (mod p). Parameters ========== coefficients : list of integers p : prime number root : integer rrrOr)rrr)rrrr,r,r- _val_polys  rcCs@|stdt|}|js&td|jtks8td|S)z} return coefficients of expr if it is a univariate polynomial with integer coefficients else raise a ValueError. z%The expression should be a polynomialz#The expression should be univariatez6The expression should should have integer coefficients)Z is_polynomialrr Z is_univariaterfr Z all_coeffs)exprZ polynomialr,r,r- _valid_exprs rcst|fddDt}|dkr8tfS|dkrPtd fSddkrddtkrtd |ddStrtStfd d fd d fd d S)a Find the solutions to a polynomial congruence equation modulo m. Parameters ========== coefficients : Coefficients of the Polynomial m : positive integer Examples ======== >>> from sympy.ntheory import polynomial_congruence >>> from sympy.abc import x >>> expr = x**6 - 2*x**5 -35 >>> polynomial_congruence(expr, 6125) [3257] csg|] }|qSr,r,)r0num)rNr,r-r3sz)polynomial_congruence..r:r4rrrOTcs t|Sr_)rr2rr,r-r#rz'polynomial_congruence..cs t||Sr_)rrrr,r-r$rcs t||Sr_)rrrr,r-r%r)r)rr;rsumrrrr)rrNrr,)rrNr-polynomial_congruences      r)F)F)N)N)NrN)N)NN)G __future__rZsympy.core.functionrZsympy.core.numbersrrrZsympy.core.powerrZsympy.core.singletonrZ sympy.polysr Zsympy.polys.domainsr Zsympy.polys.galoistoolsr r r Z primetestrZfactor_rrrrrZsympy.utilities.miscrZsympy.core.randomrr itertoolsrrr.r8r>r<rWr^rbrrYrdrcrrrrrrrrrrrPrrrrrrrrrrrrrr,r,r,r-s\        5B< 4! 9ri) /9 B.R: ' . e 1 13