数论算法——信息安全数学基础算法总结（python版）

/*
    Author: wsnpyo
    Update Date: 2014-11-16    Algorithm: 快速幂/Fermat, Solovay_Stassen, Miller-Rabin素性检验/Exgcd非递归版/中国剩余定理
*/
import random

def QuickPower(a, n, p): # 快速幂算法
    tmp = a
    ret = 1
    while(n > 0):
        if (n&1):
            ret = (ret * tmp) % p
        tmp = (tmp * tmp) % p
        n>>=1
    return ret

def Jacobi(n, m): # calc Jacobi(n/m)
    n = n%m
    if n == 0:
        return 0
    Jacobi2 = 1
    if not (n&1): # 若有n为偶数, 计算Jacobi2 = Jacobi(2/m)^(s) 其中n = 2^s*t t为奇数
        k = (-1)**(((m**2-1)//8)&1)
        while not (n&1):
            Jacobi2 *= k
            n >>= 1
    if n == 1:
        return Jacobi2
    return Jacobi2 * (-1)**(((m-1)//2*(n-1)//2)&1) * Jacobi(m%n, n)

def Exgcd(r0, r1): # calc ax+by = gcd(a, b) return x
    x0, y0 = 1, 0
    x1, y1 = 0, 1
    x, y = r0, r1
    r = r0 % r1
    q = r0 // r1
    while r:
        x, y = x0 - q * x1, y0 - q * y1
        x0, y0 = x1, y1
        x1, y1 = x, y
        r0 = r1
        r1 = r
        r = r0 % r1
        q = r0 // r1
    return x

def Fermat(x, T): # Fermat素性判定
        if x < 2:
                return False
        if x <= 3:
                return True
        if x%2 == 0 or x%3 == 0:
                return False
        for i in range(T):
                ran = random.randint(2, x-2) # 随机取[2, x-2]的一个整数
                if QuickPower(ran, x-1, x) != 1:
                        return False
        return True

def Solovay_Stassen(x, T): # Solovay_Stassen素性判定
    if x < 2:
        return False
    if x <= 3:
        return True
    if x%2 == 0 or x%3 == 0:
        return False
    for i in range(T): # 随机选择T个整数
        ran = random.randint(2, x-2)
        r = QuickPower(ran, (x-1)//2, x)
        if r != 1 and r != x-1:
            return False
        if r == x-1:
            r = -1
        if r != Jacobi(ran, x):
            return False
    return True

def MillerRabin(x, ran): # x-1 = 2^s*t
    tx = x-1
    s2 = tx&(~tx+1) # 取出最后一位以1开头的二进制 即2^s
    r = QuickPower(ran, tx//s2, x)
    if r == 1 or r == tx:
        return True
    while s2>1: # 从2^s -> 2^1 循环s次
        r = (r*r)%x
        if r == 1:
            return False
        if r == tx:
            return True
        s2 >>= 1
    return False

def MillerRabin_init(x, T): #Miller-Rabin素性判定
    if x < 2:
        return False
    if x <= 3:
        return True
    if x%2 == 0 or x%3 == 0:
        return False
    for i in range(T): # 随机选择T个整数
        ran = random.randint(2, x-2)
        if not MillerRabin(x, ran):
            return False
    return True

def CRT(b, m, n): # calc x = b[] % m[]
    M = 1
    for i in range(n):
        M *= m[i]
    ans = 0
    for i in range(n):
        ans += b[i] * M // m[i] * Exgcd(M//m[i], m[i])
    return ans%M

以上作为半个学期来数论学习的一个小结，也许以后难以再系统的学习数论了。略伤感咿

　　—— 多谢信息安全数学基础的老师