ó ¡¼™\c@ sÜddlmZmZddlmZmZddlmZddlm Z m Z m Z ddl m Z ddlmZddlmZdd lmZmZmZmZdd lmZmZd „Zd „Zd „Zd„Zd„Zed„Z d„Z!e"d„Z#d„Z$d„Z%d„Z&d„Z'd„Z(d„Z)d„Z*d„Z+ed„Z,d„Z-d„Z.d„Z/defd „ƒYZ0d'd!„Z2d'd"„Z3d'd#d'd$„Z4d'd%„Z5d'd'd&„Z6d'S((iÿÿÿÿ(tprint_functiontdivision(tas_inttrange(tFunction(tigcdtigcdext mod_inverse(tisqrt(tSi(tisprime(t factorintttrailingttotientt multiplicity(trandinttRandomcC s°ddlm}t|ƒt|ƒ}}t||ƒdkrMtdƒ‚n|tƒ}t|ƒ}xz|jƒD]l\}}|dkr¡||c|d7>> from sympy.ntheory import n_order >>> n_order(3, 7) 6 >>> n_order(4, 7) 3 iÿÿÿÿ(t defaultdictis*The two numbers should be relatively prime( t collectionsRRRt ValueErrortintR titemsRtpow(tatnRtfactorstftpxtkxtfpxtpytkyt group_ordertordertptetexponent((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pytn_orders4     cc sŠgt|dƒjƒD]}|d|^q}d}xL||kr…x/|D]"}t|||ƒdkrMPqMqMW|V|d7}q:WdS(sJ 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 iiN(R tkeysR(R"titvRtpw((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_primitive_root_prime_iter;s1 cC sÏt|ƒ}|dkr'tdƒ‚n|dkr7dSt|ƒ}t|ƒdkrYdSt|ƒdkrd|ks‡|ddkr‹dSx*|jƒD]\}}|dkr˜Pq˜q˜Wd}xþ||kr|d7}||dkríqÁnt||ƒrÁ|SqÁWn¸d|kr'|dkr#dSdSt|jƒƒd\}}|dkr¿t|ƒ}t||dƒrr|SxJt d||dƒD].}t ||ƒdkrŠt||ƒrŠ|SqŠWnt t |ƒƒS(s Returns the smallest primitive root or None Parameters ========== p : positive integer Examples ======== >>> 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 isp is required to be positiveiiiiN( RRR tlentNoneRtis_primitive_roottlisttprimitive_rootRRtnextR*(R"Rtp1te1R'Rtg((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyR/YsD           $ cC sot|ƒt|ƒ}}t||ƒdkr=tdƒ‚n||krV||}nt||ƒt|ƒkS(s÷ 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) 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 is*The two numbers should be relatively prime(RRRR%R (RR"((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyR-›s   c C s't|dƒ}||?}x6td|dƒ}t||ƒ}|dkrPqqWt|||ƒ}t|||ƒ}d}xqt|ƒD]c} |t|||ƒ|} t| d|d| |ƒ} | ||dkrŠ|d| 7}qŠqŠWt||dd|ƒt||d|ƒ|} | S(sÀ 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 iiiÿÿÿÿi(R Rtlegendre_symbolRR( RR"tstttdtrtAtDtmR'tadmtx((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_sqrt_mod_tonelli_shanksºs"   2cC sÕ|rttt||ƒƒƒSytt|ƒƒ}t||ƒ}t|ƒ}||dkrg||S||dkr{|Sy(t|ƒ}||dkr¢||SWntk r¶nX|SWntk rÐdSXdS(sŸ 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] iN(tsortedR.t sqrt_mod_itertabsRR0t StopIterationR,(RR"t all_rootstitR8((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pytsqrt_modÚs&     c' sÃddl‰t‡fd†|Dƒƒ}t|ƒ}dg|}t}xutr¾d|}}x5|r”| r”|d8}t||ƒ\}||sii(RIttupleR+R,tTrueR0tFalse(titerst inf_iterst num_iterstcur_valtfirst_vR'R"((RIs<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_products       cc sddlm}m}ddlm}t|ƒtt|ƒƒ}}t|ƒrÛ||}|dkr|t||dƒ}nt ||dƒ}|r||kr¹x2|D] }|Vq§WqØx|D]}||ƒVqÀWqn9t |ƒ}g} g} xƒ|j ƒD]u\} } || dkr;t|| | ƒ} | sWdSnt || | ƒ} | sWdS| j | ƒ| j | | ƒqW|| |ƒ\}}}||kr×xtt | ŒD]&}||| ||||ƒ}|VqªWn=x:t | ŒD],}||| ||||ƒ}||ƒVqäWdS(sH 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] iÿÿÿÿ(tgf_crt1tgf_crt2(tZZiiN(tsympy.polys.galoistoolsRSRTtsympy.polys.domainsRURRAR t _sqrt_mod1t_sqrt_mod_prime_powerR RtappendRR(RR"tdomainRSRTRUtresR=RR(tpvRtextrxtmmR#R5tvxR8((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyR@0sF           cC sddlm}ddlm}||}||}|dkro|dkrY||ƒgSt||ƒsld S|ddkr™t||dd|ƒ}n¶|dd kr@t||dd|ƒ}|dkrìt||dd|ƒ}qOtd||d d|ƒ}d|||} t| d|ƒ|krO| }qOnt||ƒ}t||ƒ|||ƒgƒS|dkr |dkr |ddkr›d S|dkrùt ƒ} x<t d |dƒD](} | j d| ƒ| j d| ƒqÃWt | ƒS|dƒ|dƒ|d ƒ|d ƒg} d} g}x×| D]Ï}| }xK||kr|d||?}|dr‚|d|d>}n|d7}qEW||kr¬|j |ƒn|d|d>} | d|>kr6| |kr6| d||d kr|j | ƒqq6q6W|St||dƒ} | s)d S| d }|d|}d} |}xk| }|d9}||krpPn|} |d}|d||ƒd }||||}|d|}qPW| |krý||}|d||ƒd }||||}n|||gSd S( sõ 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]_ iÿÿÿÿ(R(RUiiiiiiiiN(tsympy.core.numbersRRWRUtis_quad_residueR,RR>R?tsetRtaddR.RZRY(RR"tkRRUtpkR\tsigntbR=R5R'trvRR8tnxtr1tfrRtn1tfrinv((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyRYlsŒ             *         c  szˆ|‰|ˆ}|dkr‡|d‰|ddkraˆˆd‰‡‡fd†}|ƒSˆˆ‰‡‡fd†}|ƒSnt|ƒ}|ˆ}|ddkr±d S|d‰||?}ˆdkrí||dkr d|ˆd>‰dˆd>‰‡‡‡‡fd†}|ƒS||dkrƒt|ˆ||ƒ‰ˆd krVd Sd|ˆ>‰‡‡‡‡fd†} | ƒS||dkrvt|ˆ||ƒ‰ˆd kr¹d Sd|ˆd>‰‡‡‡‡fd†} | ƒSn‰|d‰|ˆ|}t|ˆ||ƒ‰ ˆ d kr+d Sˆˆ‰ˆ||‰ˆ|ˆ‰‡‡‡‡‡‡ fd †} | ƒSd S( s~ Find solution to ``x**2 == a mod p**n`` when ``a % p == 0`` see http://www.numbertheory.org/php/squareroot.html iiic3 s,d}x|ˆkr'|V|ˆ7}q WdS(Ni((R'(tpm1tpn(s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_iter0aîsc3 s,d}x|ˆkr'|V|ˆ7}q WdS(Ni((R'(tpmRq(s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_iter0bösc3 sadˆd>}dˆ>}xB|ˆkr\|}x|ˆkrN|V||7}q0W|ˆ7}qWdS(Nii((RfR'tj(R;RpRqtpnm1(s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_iter1s c3 sptƒ}x`ˆD]X}d}xI|ˆkrg|ˆ>|}||krZ|j|ƒ|Vn|ˆ7}qWqWdS(Ni(RdRe(R5R8R'R=(R;RqtpnmR\(s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_iter2s    c3 sttƒ}xdˆD]\}d}xM|ˆkrk|ˆ>|ˆ}||kr^|j|ƒ|Vn|ˆ7}qWqWdS(Ni(RdRe(R5R8R'R=(R;RqRvR\(s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_iter3's    c3 s~tƒ}ˆˆ}xdˆD]\}d}xM|ˆkru||ˆ}||krh|j|ƒ||Vn|ˆ7}q)WqWdS(Ni(RdRe(R5RsR_R'R=(R;R"RqRxtpnrtres1(s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_iter4<s      N(R R,RY( RR"RRrRtRR8ta1RwRyRzR}(( R;R"RsRpRqRxRvR{R\R|s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyRXás^                    cC sèt|ƒt|ƒ}}|dkr4tdƒ‚n||ksL|dkrY||}n|dksq|dkrutSt|ƒsÊ|dr¤t||ƒdkr¤tSt||ƒ}|dkrÃtStSnt||dd|ƒdkS(sÕ 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)]). 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 is p must be > 0iiiiÿÿÿÿN( RRRKR t jacobi_symbolRLRER,R(RR"R8((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyRcJs    cC sÖt|ƒt|ƒt|ƒ}}}|dkrAtdƒ‚n|dkr\tdƒ‚n|dkrwtdƒ‚n|dkr|dkr“tS|dkS|dkr­tS|dkrÆt||ƒSt|||ƒS(s  Returns True if ``x**n == a (mod m)`` has solutions. References ========== .. [1] P. Hackman "Elementary Number Theory" (2009), page 76 is m must be > 0sn must be >= 0sa must be >= 0ii(RRRLRKRct_is_nthpow_residue_bign(RRR;((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pytis_nthpow_residuens &         cC s‡t|ƒdkrRx9t|ƒjƒD]%\}}t||||ƒs%tSq%WtSt|ƒ}|t||ƒ}t |||ƒdkS(s>Returns True if ``x**n == a (mod m)`` has solutions for n > 2.iN( R/R,R Rt#_is_nthpow_residue_bign_prime_powerRLRKR RR(RRR;tprimetpowerRRf((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyR€Šs cC sÕ||rm|dkr/t||t||ƒƒS|d@r=tSt|ƒ}|tdt|d|ƒƒdkS|t||ƒ;}|sŠtSt||ƒ}||r§tSt||ƒ}t||||||ƒSdS(sVReturns True/False if a solution for ``x**n == a (mod(p**k))`` does/doesn't exist.iiN(R€RRKR tminRRLR‚(RRR"RftctmuRs((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyR‚™s    $ cC smt|ƒ}g}x.|jƒD] \}}|j|g|ƒqWx#|D]}t|||tƒ}qJW|S(N(R Rtextendt _nthroot_mod1RL(R5tqR"RR(RiR#tqx((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt _nthroot_mod2²s  cC s²t|ƒ}t|ƒs-t|||ƒ}n|d}|d|dksQt‚d}x(||dkr|d7}||}qZWt| |ƒd|}||} d| |} t|| |ƒ} t|||ƒ} t||||ƒ} t|| | ƒ}t|| ||ƒ}t||ƒd}| ||}|g}t||d||ƒ} |}x3t|dƒD]!}|| |}|j|ƒqoW|r¨|j ƒ|St |ƒS(s« Root of ``x**q = s mod p``, ``p`` prime and ``q`` divides ``p - 1`` References ========== .. [1] A. M. Johnston "A Generalized qth Root Algorithm" ii( R/R RŒtAssertionErrorRRt discrete_logRRZtsortR…(R5RŠR"RCR3R8RRftf1tzR=Rlts1thR6tg2tg3R\thxR'((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyR‰¼s:       c C s¾ddlm}t|ƒt|ƒt|ƒ}}}|dkrRt|||ƒSt|||ƒshdSt|ƒdkr‰tdƒ‚n|d|dkr°t||||ƒS|}|d}d}||kró||||f\}}}}nxl|rat ||ƒ\}} t |||ƒ} || |ƒd} | ||} || }}|| }}qöW|dkr‰|r€|g} qº|} n1|dkr¥t|||ƒSt||||ƒ} | S(s¶ 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 iÿÿÿÿ(Ris,Not Implemented for m without primitive rootiiN( RbRRRERR,R/tNotImplementedErrorR‰tdivmodR( RRR"RCRtpatpbRiRŠR8R†R\((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt nthroot_modês<&   !      cC s]t|ƒ}tƒ}x5t|ddƒD]}|jt|d|ƒƒq*Wtt|ƒƒS(sÁ Returns the list of quadratic residues. Examples ======== >>> from sympy.ntheory.residue_ntheory import quadratic_residues >>> quadratic_residues(7) [0, 1, 2, 4] ii(RRdRReRR?R.(R"R8R'((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pytquadratic_residues*s  cC slt|ƒt|ƒ}}t|ƒ s2|dkrAtdƒ‚n||}|sUdSt||ƒrhdSdS(s 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 isp should be an odd primeiiiÿÿÿÿ(RR RRc(RR"((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyR4<s# cC s™t|ƒt|ƒ}}|dks0|d r?tdƒ‚n|dksW||krd||}n|szt|dkƒS|dks’|dkr–dSt||ƒdkr¯dSd}|dkrå| }|ddkrå| }qånx˜|dkrxD|ddkr:|dkr:|dL}|dd kr÷| }q÷q÷W||}}|ddkrr|ddkrr| }n||;}qèW|dkr•d}n|S( sµ 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 Mul, 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 iis#n should be an odd positive integeriiiii(ii(RRRR(R;RRu((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyRjs89         tmobiuscB seZdZed„ƒZRS(s¥ Möbius 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 Möbius 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" cC s©|jr*|jtk r6tdƒ‚q6n tdƒ‚|jrFtjS|tjkr\tjS|j r¥t |ƒ}t d„|j ƒDƒƒr”tj Stjt|ƒSdS(Nsn should be a positive integersn should be an integercs s|]}|dkVqdS(iN((RHR'((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pys ÷s(t is_integert is_positiveRKRt TypeErrortis_primeR t NegativeOnetOnet is_IntegerR tanytvaluestZeroR+(tclsRR((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pytevalês     (t__name__t __module__t__doc__t classmethodR©(((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyRÂs'cC st||;}||;}|dkr)|}nd}x2t|ƒD]$}||krR|S|||}q<Wtdƒ‚dS(s‡ Trial multiplication algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. The algorithm finds the discrete logarithm using exhaustive search. This naive method is used as fallback algorithm of ``discrete_log`` when the group order is very small. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_trial_mul >>> _discrete_log_trial_mul(41, 15, 7) 3 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). isLog does not existN(R,RR(RRRiR!R=R'((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_discrete_log_trial_mulüs     c C sø||;}||;}|dkr2t||ƒ}nt|ƒd}tƒ}d}x,t|ƒD]}|||<|||}q^Wt||ƒ}t|||ƒ}|}x>t|ƒD]0}||krÖ||||S|||}q´Wtdƒ‚dS(se Baby-step giant-step algorithm for computing the discrete logarithm of ``a`` to the base ``b`` modulo ``n``. The algorithm is a time-memory trade-off of the method of exhaustive search. It uses `O(sqrt(m))` memory, where `m` is the group order. Examples ======== >>> from sympy.ntheory.residue_ntheory import _discrete_log_shanks_steps >>> _discrete_log_shanks_steps(41, 15, 7) 3 See Also ======== discrete_log References ========== .. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). isLog does not existN(R,R%RtdictRRRR( RRRiR!R;tTR=R'R‘((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_discrete_log_shanks_steps#s$      i cC s||;}||;}|dkr2t||ƒ}ntƒ}|dk rW|j|ƒnx&t|ƒD]}|jd|dƒ}|jd|dƒ} t|||ƒt|| |ƒ|} | d} | dkr÷|| |} |} | d|}n[| dkr0| | |} |||} | | |}n"|| |} |d|} | }x't|ƒD]}| d} | dkrš|| |} | d|} nU| dkrÓ| | |} |||}| | |} n|| |} |d|}| d} | dkr$|| |} |d|}nU| dkr]| | |} | | |} |||}n|| |} | d|} | d} | dkr®|| |} |d|}nU| dkrç| | |} | | |} |||}n|| |} | d|} | | kr_| ||}yCt||ƒ| ||}t|||ƒ||dkr_|SWntk rsnXPq_q_WqdWtdƒ‚dS(sg 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). iiis&Pollard's Rho failed to find logarithmN( R,R%RtseedRRRRR(RRRiR!tretriestrseedtprngR'taatbatxaR†txbtabtbbRuR8R#((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_discrete_log_pollard_rhoQs€     &                cC sdddlm}||;}||;}|dkrBt||ƒ}nt|ƒ}dgt|ƒ}xÄt|jƒƒD]°\}\}} x›t| ƒD]} t ||||ƒ} t |t | |ƒ||| d|ƒ} t ||||ƒ} t || | |t ƒ}||c||| 7>> 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). i(tcrtiN( tmodularR½R,R%R R+RGRRRRRŽRK(RRRiR!R½RtlR'tpitriRutgjtajtbjtcjR7t_((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyt_discrete_log_pohlig_hellman¸s     %+ 8cC sÉt|ƒt|ƒt|ƒ}}}|dkrDt||ƒ}n|dkr_t|ƒ}n|dkr~t||||ƒS|r¶|dkr£t||||ƒSt||||ƒSt||||ƒS(s 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] http://mathworld.wolfram.com/DiscreteLogarithm.html .. [2] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., & Vanstone, S. A. (1997). ièI¥ÔèN(RR,R%R R®R±R¼RÇ(RRRiR!t prime_order((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pyRŽés&    N(7t __future__RRtsympy.core.compatibilityRRtsympy.core.functionRRbRRRtsympy.core.powerRtsympy.core.singletonR t primetestR tfactor_R R R RtrandomRRR%R*R/R-R>RLRERRRR@RYRXRcRR€R‚RŒR‰R›RœR4RRR,R®R±R¼RÇRŽ(((s<lib/python2.7/site-packages/sympy/ntheory/residue_ntheory.pytsD" +  B  4 " < u i $    . @  . X: ' .g 1