wgddlmZddlmZmZmZmZmZmZm Z ddl m Z ddl m Z ddlmZmZmZmZddlmZddlmZdd lmZmZdd lmZdd lmZdd lmZdd l m!Z!ddl"m#Z#ddl$m%Z%m&Z&ddl'm(Z(dZ)dZ*dZ+dZ,d9dZ-dZ.dZ/d:dZ0e1fdZ2dZ3dZ4dZ5dZ6dZ7dZ8d Z9d:d!Z:d;d"Z;ed#d$d%&d'Z<ed(d$d%&d)Z=ed*d$d%&d+Z>dd0ZBd 1. a and n should be relatively prime Returns ======= int : the order of ``a`` modulo ``n`` Raises ====== ValueError If `n \le 1` or `\gcd(a, n) \neq 1`. If ``a`` or ``n`` is not an integer. Examples ======== >>> from sympy.ntheory import n_order >>> n_order(3, 7) 6 >>> n_order(4, 7) 3 See Also ======== is_primitive_root We say that ``a`` is a primitive root of ``n`` when the order of ``a`` modulo ``n`` equals ``totient(n)`` r%n should be an integer greater than 1*The two numbers should be relatively prime)r ValueErrorrritemspowrint) ana_orderpepepe_orderfactorsorderpxexxs b/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/sympy/ntheory/residue_ntheory.pyn_orderr3s0X !9fQiqAAv@AA AAAv 1ayA~EFFG! ""$ &1 TEQQZ'AE" q5QGAJmmo FBAx2r6)2.Aq&2rN q&  gu% & w<c#Kdk(rdydzdvrdnd}dkr dk(rdn|nWtd z jDcgc] }d z |z }}t|D]tfd |DsntddD]$}t d z |d k(st |&ycc}ww) a  Generates the primitive roots for a prime ``p``. Explanation =========== The primitive roots generated are not necessarily sorted. However, the first one is the smallest primitive root. Find the element whose order is ``p-1`` from the smaller one. If we can find the first primitive root ``g``, we can use the following theorem. .. math :: \operatorname{ord}(g^k) = \frac{\operatorname{ord}(g)}{\gcd(\operatorname{ord}(g), k)} From the assumption that `\operatorname{ord}(g)=p-1`, it is a necessary and sufficient condition for `\operatorname{ord}(g^k)=p-1` that `\gcd(p-1, k)=1`. Parameters ========== p : odd prime Yields ====== int the primitive roots of ``p`` Examples ======== >>> from sympy.ntheory.residue_ntheory import _primitive_root_prime_iter >>> sorted(_primitive_root_prime_iter(19)) [2, 3, 10, 13, 14, 15] References ========== .. [1] W. Stein "Elementary Number Theory" (2011), page 44 N)r)rc3>K|]}t|dk7ywrNr$).0pwgr)s r2 z-_primitive_root_prime_iter..s2"3q"a=A%2s)rkeysrangeallrr$)r)g_minivkrBs` @r2_primitive_root_prime_iterrK[sV AvQ&AaE2vbAe#,QU#3#8#8#: ;aa!e\ ; ;ua A222  G 1a^ q1ua=A aA,  B9&B>=)B>'B>c#K|dk(rt|Ed{y|dz}t|D]Rtd|z |z |z}td||D]*|k7s fdtd||z|DEd{,Ty7k7 w)am Generates the primitive roots of `p^e`. Explanation =========== Let ``g`` be the primitive root of ``p``. If `g^{p-1} \not\equiv 1 \pmod{p^2}`, then ``g`` is primitive root of `p^e`. Thus, if we find a primitive root ``g`` of ``p``, then `g, g+p, g+2p, \ldots, g+(p-1)p` are primitive roots of `p^2` except one. That one satisfies `\hat{g}^{p-1} \equiv 1 \pmod{p^2}`. If ``h`` is the primitive root of `p^2`, then `h, h+p^2, h+2p^2, \ldots, h+(p^{e-2}-1)p^e` are primitive roots of `p^e`. Parameters ========== p : odd prime e : positive integer Yields ====== int the primitive roots of `p^e` Examples ======== >>> from sympy.ntheory.residue_ntheory import _primitive_root_prime_power_iter >>> sorted(_primitive_root_prime_power_iter(5, 2)) [2, 3, 8, 12, 13, 17, 22, 23] rNr7rc3.K|] }z|zywN)r@mrBrJs r2rCz3_primitive_root_prime_power_iter..sFaA F)rKr$rE)r)r*p2trBrJs @@r2 _primitive_root_prime_power_iterrTsD Av-a000 T+A. GASAE2&&",A1b!_ G6F5AqD"3EFFF G G 1Gs(BBABB:B; BBc#\Kt||D]}|dzdk(r||||zzyw)a0 Generates the primitive roots of `2p^e`. Explanation =========== If ``g`` is the primitive root of ``p**e``, then the odd one of ``g`` and ``g+p**e`` is the primitive root of ``2*p**e``. Parameters ========== p : odd prime e : positive integer Yields ====== int the primitive roots of `2p^e` Examples ======== >>> from sympy.ntheory.residue_ntheory import _primitive_root_prime_power2_iter >>> sorted(_primitive_root_prime_power2_iter(5, 2)) [3, 13, 17, 23, 27, 33, 37, 47] r7rN)rT)r)r*rBs r2!_primitive_root_prime_power2_iterrVs;:.a 3 q5A:Gad(N s*,cBt|}|dkr td|dkr|dz S|dzdk(}|s|}n |dzr|dz}nyt|rd}n t|d}|sy|\}}t|sy|s,|rt t ||St t ||S|r(td|dD]}||zs t||s|cSt t|}|dk(st||dz |dzdk7r|St|dz|D]}||zs t||s|cSy)a Returns a primitive root of ``p`` or None. Explanation =========== For the definition of primitive root, see the explanation of ``is_primitive_root``. The primitive root of ``p`` exist only for `p = 2, 4, q^e, 2q^e` (``q`` is an odd prime). Now, if we know the primitive root of ``q``, we can calculate the primitive root of `q^e`, and if we know the primitive root of `q^e`, we can calculate the primitive root of `2q^e`. When there is no need to find the smallest primitive root, this property can be used to obtain a fast primitive root. On the other hand, when we want the smallest primitive root, we naively determine whether it is a primitive root or not. Parameters ========== p : integer, p > 1 smallest : if True the smallest primitive root is returned or None Returns ======= int | None : If the primitive root exists, return the primitive root of ``p``. If not, return None. Raises ====== ValueError If `p \le 1` or ``p`` is not an integer. Examples ======== >>> from sympy.ntheory.residue_ntheory import primitive_root >>> primitive_root(19) 2 >>> primitive_root(21) is None True >>> primitive_root(50, smallest=False) 27 See Also ======== is_primitive_root References ========== .. [1] W. Stein "Elementary Number Theory" (2011), page 44 .. [2] P. Hackman "Elementary Number Theory" (2009), Chapter C r%p should be an integer greater than 1r7rNr6) rr"rrnextrVrTrEis_primitive_rootrKr$)r)smallestp_evenqr*rPrHrBs r2primitive_rootr_sK| q AAv@AAAv1u UaZF   Q qDqz  1a 1qz  9!Q?@ @4Q:;; q!Q A1u*1a0  ' *+AAvQAq!t$) 1q5!_ q5&q!,Hr4c$ttcdkr tdztdk7r tddkrdz k(Sdzr}n dzrdz}nyt|r"|dz t |dz j }ndt |d}|sy|\}}t|sy||dz z|dz ztt |dz j }|j|tfd|DS) a; Returns True if ``a`` is a primitive root of ``p``. Explanation =========== ``a`` is said to be the primitive root of ``p`` if `\gcd(a, p) = 1` and `\phi(p)` is the smallest positive number s.t. `a^{\phi(p)} \equiv 1 \pmod{p}`. where `\phi(p)` is Euler's totient function. The primitive root of ``p`` exist only for `p = 2, 4, q^e, 2q^e` (``q`` is an odd prime). Hence, if it is not such a ``p``, it returns False. To determine the primitive root, we need to know the prime factorization of ``q-1``. The hardness of the determination depends on this complexity. Parameters ========== a : integer p : integer, ``p`` > 1. ``a`` and ``p`` should be relatively prime Returns ======= bool : If True, ``a`` is the primitive root of ``p``. Raises ====== ValueError If `p \le 1` or `\gcd(a, p) \neq 1`. If ``a`` or ``p`` is not an integer. Examples ======== >>> from sympy.functions.combinatorial.numbers import totient >>> from sympy.ntheory import is_primitive_root, n_order >>> is_primitive_root(3, 10) True >>> is_primitive_root(9, 10) False >>> n_order(3, 10) == totient(10) True >>> n_order(9, 10) == totient(10) False See Also ======== primitive_root rrXr!rYr7Fr6c3DK|]}t|zdk7ywr>r?)r@primer& group_orderr)s r2rCz$is_primitive_root..s$Is1kU*A.!3Is ) rr"rrrrDrsetaddrF)r&r)r^r-rPr*rcs`` @r2r[r[Ost !9fQiDAqAv@AA AA 1ayA~EFF AvAEz1u  Q qDqz!e AE"'') 1a 1qz!a%j!a%( iA&++-. A II IIr4ct|dz }||z }|dzdk(rd}n+|dzdvrd}n! td|dz }t||dk(rn t|||}t|||}d}t |D]?}|t|||z|z} t| d |dz |z z|} | |z|dz k(s8|d |zz }At||dzd z|t||d z|z|z} | S) a Returns the square root in the case of ``p`` prime with ``p == 1 (mod 8)`` Assume that the root exists. Parameters ========== a : int p : int prime number. should be ``p % 8 == 1`` Returns ======= int : Generally, there are two roots, but only one is returned. Which one is returned is random. Examples ======== >>> from sympy.ntheory.residue_ntheory import _sqrt_mod_tonelli_shanks >>> _sqrt_mod_tonelli_shanks(2, 17) in [6, 11] True References ========== .. [1] Carl Pomerance, Richard Crandall, Prime Numbers: A Computational Perspective, 2nd Edition (2005), page 101, ISBN:978-0387252827 r r<r6)r7r6rr7)r rrr$rE) r&r)srSdADrPrHadmr1s r2_sqrt_mod_tonelli_shanksrosB !a%A QA2v{  Q& 1q5!Aa|r! Aq! A Aq! A A 1XAq! nq #q1q519~q) 7a!e  AIA  AAz1c!QT1o-1A Hr4c|rtt||Stt|}|dz}d}t||D]}||kr|cS||kDr||z cS|}|S)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] r7N)sorted sqrt_mod_iterabsr)r&r) all_rootshalfpr1rs r2sqrt_modrwsr<mAq)** F1IA FE A 1a  u9H Yq5LA  Hr4c #Kt|tt|}}g}g}t}t|j D]T\}}||zrt |||}nt |||}t}|sy|j||j||zVt|dk(rt||dEd{yt|t\} } } ||D]} |t| || | | t y7@w)a Iterate over solutions to ``x**2 = a mod p``. Parameters ========== a : integer p : positive integer domain : integer domain, ``int``, ``ZZ`` or ``Integer`` Examples ======== >>> from sympy.ntheory.residue_ntheory import sqrt_mod_iter >>> list(sqrt_mod_iter(11, 43)) [21, 22] See Also ======== sqrt_mod : Same functionality, but you want a sorted list or only one solution. Nrr)rrsrrr#_sqrt_mod_prime_power _sqrt_mod1rappendlenmapr r r) r&r)domainrIpv_productr/r0rxmmr*rjvxs r2rrrrs0 !9c&)nqA A BHA,$$& B r6&q"b1B Ar2&BH    "b&  1v{vqt$$$2r?AqA, 8BRQ267 7 8 %sB:C?<C==AC?c||z}||z}|dk(rx|dzdk7ry|dkrttd|dSd}td|D]}|dz|z |z dzs|d|dz zz }d|dz z}t|||z ||z|z||z |zgSt||dk7ry|dzdk(rt ||dzdz|}nT|dzdk(r@t ||dzdz|}t |d|||zk7r&|t d|dz dz|z|z}n t ||}|dkDrp|}t|j dz D]'} |dz}td|z|} ||dz|z | zz |z})||dz zr td|z|} ||dz|z | zz |z}t|||z gS)aS Find the solutions to ``x**2 = a mod p**k`` when ``a % p != 0``. If no solution exists, return ``None``. Solutions are returned in an ascending list. 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]_ r7r8rNr6rYr<)listrErqrr$ro bit_lengthr) r&r)rJpkrvnxhresr/_frinvs r2ryryFs6 AB BAAv q5A: 6aQ( (  1+ #BAb A%Q26]" # !a%Lq"q&1q5B,!a%2 >??a|q1uz!a!e\1% Q!!a!e\1% sAq>QU "AA!|Q//!3C&q!,1uq||~)* 2AQB1S5"%E#q&1*e++r1C 2 A;1S5"%E#q&1*e++r1C 3S/ ""r4cz|z}|dk(rtddzdzzSt|\}}|dzdk(ryt||z }|y|dzfd|DS)a< Find solution to ``x**2 == a mod p**n`` when ``a % p == 0``. If no solution exists, return ``None``. Parameters ========== a : integer p : prime number, p must divide a n : positive integer References ========== .. [1] http://www.numbertheory.org/php/squareroot.html rrr7Nc3^K|]$}t|zzz zD]}|&ywrNrE)r@rr1rPr'r)pns r2rCz_sqrt_mod1..s6 D"U2ad7BAE %C DA DA Ds*-)rEr ry)r&r)r'rvrrPrs `` @@r2rzrzs" AB BAAvQAQ1 -.. !Q>> from sympy.ntheory import is_quad_residue >>> is_quad_residue(21, 100) True Indeed, ``pow(39, 2, 100)`` would be 21. >>> is_quad_residue(21, 120) False That is, for any integer ``x``, ``pow(x, 2, 120)`` is not 21. 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, sqrt_mod rz p must be > 0r7r6TrhFrir)rr"r rrrr#r )r&r)rSa_rvjr/r0s r2is_quad_residuers;f !9fQiqA1u))FA1uA ! A!q&\ " A1uqA  a Q q5AEq! ABw'!*Av A,$$& B r6a}!RVBQw2rNEB1ur2!+  r4cF|zttt|c}|dkr tddkr tddk(r |dk(rydk(Sdk(rydk(rydk(r t|Stfdt |j DS) z Returns True if ``x**n == a (mod m)`` has solutions. References ========== .. [1] P. Hackman "Elementary Number Theory" (2009), page 76 rz m must be > 0zn must be >= 0rFTr7c3@K|]\}}t||ywrN)#_is_nthpow_residue_bign_prime_power)r@r)r*r&r's r2rCz$is_nthpow_residue..'s(11a31aA>1s)rr"rrFrr#)r&r'rPs`` r2is_nthpow_residuer s AAQiF1IGAq!Av))1u)**Av 6Av AvAvAvq!$$ 1$Q<--/1 11r4cP||zdk(r5|t||z}|syt||\}}||zry||z}||zdk(r5|dk7r5||dz z|dz z}t||t||zt||dk(S|dzrytt |dz|}|td|zdk(S)z Returns True if `x^n = a \pmod{p^k}` has solutions for `n > 2`. Parameters ========== a : positive integer n : integer, n > 2 p : prime number k : positive integer rTFr7r)r$r rminr )r&r'r)rJmufcs r2rr+s a%1* SAYq! 2 6 R a%1* Av AJA 1a3q!9nc!Qi0A551u IaL1 a A s1ay=A r4c ntt|}|}t|jD]\}}|dz ||zz}||zdk(r||z}||zdk(r|t | |z} d| z|z} t |t |||t |||z|} t|D]/} t || |t || | z|dz z|z|z}| |z} 1|g} t ||dz |z|}|}t|dz D]} ||z|z}| j||r| j| St| S)aE Root of ``x**q = s mod p``, ``p`` prime and ``q`` divides ``p - 1``. Assume that the root exists. Parameters ========== s : integer q : integer, n > 2. ``q`` divides ``p - 1``. p : prime number all_roots : if False returns the smallest root, else the list of roots Returns ======= list[int] | int : Root of ``x**q = s mod p``. If ``all_roots == True``, returned ascending list. otherwise, returned an int. Examples ======== >>> from sympy.ntheory.residue_ntheory import _nthroot_mod1 >>> _nthroot_mod1(5, 3, 13, False) 7 >>> _nthroot_mod1(13, 4, 17, True) [3, 5, 12, 14] References ========== .. [1] A. M. Johnston, A Generalized qth Root Algorithm, ACM-SIAM Symposium on Discrete Algorithms (1999), pp. 929-930 rr) rZrKrr#r discrete_logr$rEr{sortr)rjr^r)rtrBrvqxr0rzr1rSrrrhxs r2 _nthroot_mod1rIsgH ' *+A AA,$$& B Ur2v "fk "HA"fk faRn  UrM C1aL#a2q/ :r AAq! SQBqDAENA66:A "HA   #C AA!|QA B 1q5\daZ 2   s8Or4cnt||||sgS||z}|dk(rdg}n|dz |zdk(rt|||d}nu|}|dz }d}||kr ||||f\}}}}|r.t||\} } t|| ||z|z} || }}|| }}|r.|dk(r|g}n#|dk(rt ||d}nt|||d}|dk(r|S|||zz}t } |D]} t| |dz ||z|z}|}|dk7rSt ||}t|dz D]$}||z}t| |||z }||z}| |z |z} &| j| s| h}t|dz D]T}||z}t }|D]<}t|||||zk7r||vs|j||||zz|z}||vr!>|}V| |z} t| S)a, Root of ``x**n = a mod p**k``. Parameters ========== a : integer n : integer, n > 2 p : prime number k : positive integer Returns ======= list[int] : Ascending list of roots of ``x**n = a mod p**k``. If no solution exists, return ``[]``. rrT)rtr7) rrdivmodr$rwrdrrErerq)r&r'r)rJa_mod_p base_rootspapbbr^pcr tot_rootsrootdiffnew_basem_invrtmp roots_in_base new_rootsk_s r2_nthroot_mod_prime_powerrs[& /q!Q : !eG!|S a%1 "7AqDA  U  7!"B!3 GRB2rNEArAr1 '!+ABAQG  7!J 1W!'1=J&wAFJAvAIAI24Q"1$q( 194OE1q5\ /A $8,q0u s h.  / MM$ !FM1q5\ *A E '?B2q(+q8|; I-! b) HM2h>I-? !*  *"M1I/20 ) r4cl||z}t|t|t|}}}|dkr td|dkr td|dk(r|r|gS|S|dk(r t|||Sg}g}t|j D]M\}}t ||||}|s |rgcSdcS|j ||z|j t|Ot|t\} } } tttt|D chc]} t| || | | tc} } |r| S| r| dSycc} w)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 Returns ======= list[int] | int | None : solutions to ``x**n = a mod p``. The table of the output type is: ========== ========== ========== all_roots has roots Returns ========== ========== ========== True Yes list[int] True No [] False Yes int False No None ========== ========== ========== Raises ====== ValueError If ``a``, ``n`` or ``p`` is not integer. If ``n`` or ``p`` is not positive. 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 References ========== .. [1] P. Hackman "Elementary Number Theory" (2009), page 76 rzn should be positivezp should be positiver7Nr)rr"rwrr#rr{rqr r r}r%rr)r&r'r)rtbase prime_powerr^r*rPESrrets r2 nthroot_modrsTh AAQiF1I!qA1u/001u/00Avs&Q&Av1i(( DK! ""$'1,Q1a8 "2 , ,1a4  F9%& ' k2&GAq! S$+TN4 #1k1aB?45 6C  1v  4s?D1ct|}t|dzdzDchc]}t|d|}}t|Scc}w)z Returns the list of quadratic residues. Examples ======== >>> from sympy.ntheory.residue_ntheory import quadratic_residues >>> quadratic_residues(7) [0, 1, 2, 4] r7r)rrEr$rq)r)rHrvs r2quadratic_residuesr%sD q A$Q!VaZ01!Q11A1 !9 2sAz~The `sympy.ntheory.residue_ntheory.legendre_symbol` has been moved to `sympy.functions.combinatorial.numbers.legendre_symbol`.z1.13z%deprecated-ntheory-symbolic-functions)deprecated_since_versionactive_deprecations_targetc ddlm}|||S)a2 Returns the Legendre symbol `(a / p)`. .. deprecated:: 1.13 The ``legendre_symbol`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.legendre_symbol` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. 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.functions.combinatorial.numbers 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 r)legendre_symbol)%sympy.functions.combinatorial.numbersr)r&r)_legendre_symbols r2rr5sZZ Aq !!r4zzThe `sympy.ntheory.residue_ntheory.jacobi_symbol` has been moved to `sympy.functions.combinatorial.numbers.jacobi_symbol`.c ddlm}|||S)a Returns the Jacobi symbol `(m / n)`. .. deprecated:: 1.13 The ``jacobi_symbol`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.jacobi_symbol` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. 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.functions.combinatorial.numbers 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 r) jacobi_symbol)rr)rPr'_jacobi_symbols r2rrfsFV !Q r4zlThe `sympy.ntheory.residue_ntheory.mobius` has been moved to `sympy.functions.combinatorial.numbers.mobius`.cddlm}||S)a Mobius function maps natural number to {-1, 0, 1} .. deprecated:: 1.13 The ``mobius`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.mobius` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. 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.functions.combinatorial.numbers 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" r)mobius)rr)r'_mobiuss r2rrsdH 1:r4Ncz||z}||z}||}d}t|D]}||k(r|cS||z|z}td)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). rLog does not exist)rEr")r'r&rr.r1rHs r2_discrete_log_trial_mulrs_6FAFA } A 5\ 6H EAI ) **r4c||z}||z}| t||}t|dz}i}d}t|D]}|||<||z|z}t|| |}|}t|D]}||vr ||z||zcS||z|z}t d)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). rr)r3rrEr$r") r'r&rr.rPTr1rHrs r2_discrete_log_shanks_stepsr s4FAFA }1  U aA A A 1X! EAI Ar1 A A 1X 6q51Q4<  EAI ) **r4c||z}||z}| t||}t|}t|D]}|d|dz }|d|dz } t|||t|| |z|z} | dz} | dk(r|| z|z} |} | dz|z}n0| dk(r| | z|z} ||z|z} | | z|z}n|| z|z} |dz|z} | }t|D] }| dz} | dk(r|| z|z} | dz|z} n.| dk(r| | z|z} ||z|z}| | z|z} n|| z|z} |dz|z}| dz} | dk(r|| z|z} |dz|z}n.| dk(r| | z|z} | | z|z} ||z|z}n|| z|z} | dz|z} | dz} | dk(r|| z|z} |dz|z}n.| dk(r| | z|z} | | z|z} ||z|z}n|| z|z} | dz|z} | | k(s| |z |z} t ||| |z z|z}t||||z |zdk(r|ccS t d#t $rYwxYw)ag Pollard's Rho algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. It is a randomized algorithm with the same expected running time as ``_discrete_log_shanks_steps``, but requires a negligible amount of memory. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_pollard_rho >>> _discrete_log_pollard_rho(227, 3**7, 3) 7 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). rr6rz&Pollard's Rho failed to find logarithm)r3rrEr$rZeroDivisionErrorr")r'r&rr.retriesrseedrrHaabaxarxbabbbrrvr*s r2_discrete_log_pollard_rhor7s4FAFA }1 uoG 7^@ Q " Q " B]SB] *Q . F 6R!BBq&E!B !Vb1Br'U"Br'U"BR!Bq&E!BBu- AQAAvVaZ1f%a"Wq[2g&2g&VaZ1f%QAAvVaZ1f%a"Wq[2g&2g&VaZ1f%QAAvVaZ1f%a"Wq[2g&2g&VaZ1f%Rx"W%q%(BG4u>)s/,G33 G?>G?cdgt|z}t|D])\}}||zdk(s||xxdz cc<||z}||zdk(r+|dk7ry|S)zTry to factor n with respect to a given factorbase. Upon success a list of exponents with repect to the factorbase is returned. Otherwise None.rrN)r| enumerate)r' factorbaser-rHr)s r2_discrete_log_is_smoothrskc#j/!G*%1!eqj AJ!OJQA!eqj Av Nr4c t|}ddlm}m}m}||z}||z}t |d||||||zzdd|||z zz} d| z| z} t t| } t| } |dz } |}t|D]@}|dk(r ||z |zcSt|| }|r|Dcgc]}||z c}|gz}n||z|z}Btddg| z}d}d}|d| zkro|| kri|d| }tt|||| }||dz }7|dz }d}||gz }| }t| D]U}|||z}|dkDr0||+t| dzD]}||||||zz |z||<n|||<||dkDsN|| k(sT|}W|| k(s||t||d |}t|| dzD]}|||z|z||<|||<t| D]H}||dkDr4||/||}t| dzD]}||||||zz |z||<||dkDsHn)||| z |z}t||||k(r|Std|d| zkr|| kritdcc}w) a Index Calculus algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. The group order must be given and prime. It is not suitable for small orders and the algorithm might fail to find a solution in such situations. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_index_calculus >>> _discrete_log_index_calculus(24570203447, 23859756228, 2, 12285101723) 4519867240 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). r)rexplogg?rr<zIndex Calculus failedNr6ri) rmathrrrr%rrr|rErr"r$)r'r&rr.rrrrrBmaxrlfordermoabxr1 relationarv relationsrJkkrelationindexrHrirrinvrbis r2_discrete_log_index_calculusrsN4uoG##FAFA CdCFSQ[022Q3s1v;5FH IJA a%!)Cjm$J ZBAgG C 5\ 2 !8AI& &+C< ,56qU6!>> 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). rrrT) modularrr3rr|rr#rEr$r)r'r&rr. order_factorsrlrHpirrajbjcjrkrs r2_discrete_log_pohlig_hellmanr s#6FAFA }1 !%( c-  A !4!4!67 8Br AQQ1q))5BQK+?CBQ Q'BaRT2B aDBQJ D   ]%8%8%:;62rB;Q ?DAq H>> from sympy.ntheory import discrete_log >>> discrete_log(41, 15, 7) 3 References ========== .. [1] https://mathworld.wolfram.com/DiscreteLogarithm.html .. [2] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). r)rrrirY lJ))rrrrrr#rEr$rrrrrr)r'r&rr. prime_orderrrr-r/kxpykyrr)r*rHrs r2rr;s @QiF1I!qA }l((* %FBAv=BK26)K"$q&GBK#BF+113 %B=BK2%K"$GBK  % %mmo FB RVOE  MMO )DAqA1X q%1*a(A-Q;561   1u#$q5 a  )en  t|&q!Q66  T#a&SV$ % %E R 7/1a? ? ] "-aAu= =(Aq%88 '1a FFr4c t|}tt|}tdkr td|z}z|z}|dk(rt| Sdk(r3g}|dk(r|jd|zdzr|jd|St d|zdk(rJt |}|z||z}dzrz dzt fdtdz|z DSt}tdzd|z|zz d|zzD]3}|jfdtd|z|z d|zzD5t |S)ah Find the solutions to `a x^2 + b x + c \equiv 0 \pmod{n}`. Parameters ========== a : int b : int c : int n : int A positive integer. Returns ======= list[int] : A sorted list of solutions. If no solution exists, ``[]``. Examples ======== >>> from sympy.ntheory.residue_ntheory import quadratic_congruence >>> quadratic_congruence(2, 5, 3, 7) # 2x^2 + 5x + 3 = 0 (mod 7) [2, 6] >>> quadratic_congruence(8, 6, 4, 15) # No solution [] See Also ======== polynomial_congruence : Solve the polynomial congruence rr rr7c3.K|] }|z zywrNrO)r@rHrr's r2rCz'quadratic_congruence..sFaq1ukFrQrYc3(K|] }|z ywrNrO)r@rr's r2rCz'quadratic_congruence..sGQ1q5Gs) rr"rr{rrrqrrrdupdate)r&rrr'rootsinv_arrHs ` ` r2quadratic_congruencer s~D q Aq Aq Aq AAv@AAFAFAFAAv QB**Av 6 LLO EQ; LLO  1Q3{aq!  U  U  q5 FA aF=A1+EFFF %C 1a4!A#a%<1Q /H G"3AaCQ!A"FGGH #;r4c|js tdt|}|js td|jt k(s td|j 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) is_polynomialr"r is_univariater~r all_coeffs)expr polynomials r2 _valid_exprrsc    @AAdJ  # #>??    "QRR  ""r4c t|}|Dcgc]}||z }}t|}|dk(r tg||S|dk(rtdg||S|ddk(r)d|dzt|k(rt |d |dz |dSt ||Scc}w)a Find the solutions to a polynomial congruence equation modulo m. Parameters ========== expr : integer coefficient 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] See Also ======== sympy.polys.galoistools.gf_csolve : low level solving routine used by this routine r6r7rrriT)rr|r sumrr)rrP coefficientsnumranks r2polynomial_congruencers2t$L'34C!G4L4 | D qy#5\5155 qy#A8 8a88A!L$4 4L8I IL,,dQh4@@ \1 %%5s B c 6|dkr td|dks |dks||kDryt|}|jDcgc]\}}t||||}}}t |jDcgc] \}}||z c}}|ddScc}}wcc}}w)ayCompute ``binomial(n, m) % k``. Explanation =========== Returns ``binomial(n, m) % k`` using a generalization of Lucas' Theorem for prime powers given by Granville [1]_, in conjunction with the Chinese Remainder Theorem. The residue for each prime power is calculated in time O(log^2(n) + q^4*log(n)log(p) + q^4*p*log^3(p)). Parameters ========== n : an integer m : an integer k : a positive integer Examples ======== >>> from sympy.ntheory.residue_ntheory import binomial_mod >>> binomial_mod(10, 2, 6) # binomial(10, 2) = 45 3 >>> binomial_mod(17, 9, 10) # binomial(17, 9) = 24310 0 References ========== .. [1] Binomial coefficients modulo prime powers, Andrew Granville, Available: https://web.archive.org/web/20170202003812/http://www.dms.umontreal.ca/~andrew/PDF/BinCoeff.pdf rzk is required to be positiverF)check)r"rr#_binomial_mod_prime_powerr)r'rPrJ factorisationr)r*residuesrAs r2 binomial_modr sB 1uJ=>>  1uAQqaLMBOBUBUBWX$!Q)!Q15XHX =#6#6#89%!R2985 QRS TTY9s B2B ctfddfdtdgfdfd}d}||||z }}}dx} x} x} } |r||z} ||z||zzz}|zt|t|c}\}}\}}||z| zz} | | z } | dz k\r| | z } || t|zz}|z}| dz } |rtdk(rd k\rdnd | }t| |z|zzS) aCompute ``binomial(n, m) % p**q`` for a prime ``p``. Parameters ========== n : positive integer m : a nonnegative integer p : a prime q : a positive integer (the prime exponent) Examples ======== >>> from sympy.ntheory.residue_ntheory import _binomial_mod_prime_power >>> _binomial_mod_prime_power(10, 2, 3, 2) # binomial(10, 2) = 45 0 >>> _binomial_mod_prime_power(17, 9, 2, 4) # binomial(17, 9) = 24310 6 References ========== .. 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