smooth质数

p-1光滑

当 p 是 N 的因数，并且 p - 1 是光滑的时候，可能可以使用 Pollard's p − 1 算法来分解 N，但是也不是完全可以成功的。
参考 https://www.cnblogs.com/lordtianqiyi/articles/17069448.html

p+1光滑

当 p 是 n 的因数，并且 p + 1 是光滑的时候，可能可以使用 Williams's p + 1 算法来分解 N，但是也不是完全可以成功的。
参考 https://www.cnblogs.com/lordtianqiyi/articles/17069456.html

python primefac 库分解大整数

脚本使用

# Williams's p+1 算法 分解大整数
import primefac
re = primefac.williams_pp1(n=7941371739956577280160664419383740967516918938781306610817149744988379280561359039016508679365806108722198157199058807892703837558280678711420411242914059658055366348123106473335186505617418956630780649894945233345985279471106888635177256011468979083320605103256178446993230320443790240285158260236926519042413378204298514714890725325831769281505530787739922007367026883959544239568886349070557272869042275528961483412544495589811933856131557221673534170105409)
# pollard's p-1算法分解大整数
res = primefac.pollard_pm1(n=1224542620373200232525165378018470774334801515191193204875465445916504222883935188890019876845076388385357911730689547124547346252951158814249284724565588433721828377715469374541007756509231399263095022024229078845538543233785364809917406108015271780070196140628158427541531563472532516237632553353655535922926443707617182025475004547531104052989085765070550150028833424395972178427807901747932614235844448614858629761183210600428438018388051958214596857405813088470933109693499438012040822262549119751099008671892966082341548512112435591881692782766559736840448702039918465573051130405935280702181505538733234675792472428666968900055706926735800561218167237812066851519973807203332801575980055838563085817664973968944323258406789203078387708964307931318918136664885818917720073433998810127482159223895026085726623747340692196977140382318293090736558135980651252533606603312148824142669800602887109353065489282386215179238458743567166284295855288783740314247952124965482197632971993708775190564519250754150756867653033527903848903210074426177258586450311109023467944412194124015505951966140443860862968311560843608415723549525497729679097936310538451467530605937684408079363677707513923579164067808729408365886209340192468399685190639)

命令行使用

This is primefac version 2.0.11.

USAGE:
    primefac [-vs|-sv] [-v|--verbose] [-s|--summary|--summarize] [-t=NUMBER]
             [-r=NUMBER] [-m=[prb][,p-1][,p+1][,ecm][,siqs]] rpn

The value "rpn" is an expression in reverse Polish notation, also called
    postfix notation, and is evaluated using integer arithmetic.  The values
    that remain on the stack after evaluation are then factored in sequence.

    "-t" specifies the largest prime to use for trial division.  The default
    value for this parameter is 1000.  Using "-t=inf" will make primefac use
    trial division exclusively.

    "-r" is the number of iterations of Pollard's rho algorithm to do before
    calling a cofactor "difficult".  Default == 42,000.  Use "-r=inf" to use
    Pollard's rho algorithm exclusively once trial division is completed.

    If verbosity is invoked, then we provide progress reports and also state
    which algorithms produced which factors during the multifactor phase.

    If the summary and verbosity flags are absent, then the output should be
    identical to the output of the GNU factor command, modulo permutation of
    the factors.  If the verbosity flag is invoked, then we provide progress
    reports, turn on the summary flag, and state which methods yielded which
    factors during the multifactor phase.

    If the summary flag is present, then the output is modified by including
    a single newline between each item's output, before the first, and after
    the last.  Each item's output is also modified by printing a second line
    of data summarizing the results by indicating the number of digits (base
    10) in the input, the number of digits (base 10) in each factor, and the
    factors' multiplicities.  For example:

        >>> user@computer:~$ primefac  -s   24 ! 1 -   7 f
        >>> 
        >>> 620448401733239439359999: 991459181683 625793187653
        >>> Z24  =  P12 x P12  =  625793187653 x 991459181683
        >>> 
        >>> 5040: 2 2 2 2 3 3 5 7
        >>> Z4  =  P1^4 x P1^2 x P1 x P1  =  2^4 x 3^2 x 5 x 7
        >>> 
        >>> user@computer:~$

    Note that primes in the ordinary output lines are listed in the order in
    which they were found, while primes in the summary lines are reported in
    strictly-increasing order.
    
    The -v and -s flags may be collapsed into a single flag, -vs or -sv, but
    recall that -v implies -s.

    The "-m" flag controls what methods are used during the difficult-factor
    phase.  The "prb" and "ecm" options can be provided several times to use
    multiple concurrent instances of these methods.  Concurrent applications
    of the p-1, p+1, or SIQS methods confers no benefit over a single usage,
    so repeated listings of those methods are ignored.

    This program can be imported into Python scripts as a module; however, I
    recommend importing them from the module "labmath" instead.  This module
    is available via pip (https://pypi.org/project/labmath/).

缺点

使用primefac有一些问题：
比如res的值久久求不出来，但我们要是使用脚本来跑的话，2min内就能跑出结果
经过少量实践初步验证，如果已知p-1或者p+1是光滑数，还是套用上面给的算法更好。有的时候p-1可能因子太多而形成指数爆炸，primefac针对这一点处理的不够好。

__EOF__

本文作者：_TLSN
本文链接：https://www.cnblogs.com/lordtianqiyi/articles/17069463.html
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