获取质数的三种方法(python 实现)

一.普通判断 (直接判断是否是质数)

对于数值较小的数适合,对于上千位的数判断可能要花较长的时间

#普通判断 质数
def isPrime(num):

    if num<2:
        return False

    for i in range(2,int(math.sqrt(num)+1)):
        if num%2 == 0:
            return False

    return True

二.埃拉托色尼筛子(生成范围内的质数) 介绍

该方法可以获取范围内的质数.获取较大范围内的质数花费时间较长

def primeSieve(sieveSize):

    sieve = [True] * sieveSize
    # 0 和 1 都不是质数
    sieve[0] = False
    sieve[1] = False

    for i in  range(2, int(math.sqrt(sieveSize)+1)):
        point = i * 2
        while point<sieveSize:
            sieve[point] = False
            point += 1

    primes = []

    for i in  range(sieveSize):
        if sieve[i] == True:
            primes.append(sieve[i])

    return primes

三.拉宾米勒 判断法介绍

def rabinMiller(num):

    s = num - 1
    t = 0

    while s%2 == 0:
        s  = s//2
        #用来判断除了多少次
        t += 1
    for trials in  range(5):
        a = random.randrange(2,num-1)
        v = pow(a,s,num)
        if v != 1:

            i = 0

            while v != (num - 1):
                if i == t- 1:
                    return False
                else:
                    i = i+1
                    v = (v**2)%num
    return True
#改进后判断质数的方法
def isPrime(num):
     # Return True if num is a prime number. This function does a quicker
    # prime number check before calling rabinMiller().

     if (num < 2):
         return False

     lowPrimes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101,
                  103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199,
                  211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317,
                  331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443,
                  449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577,
                  587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701,
                  709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839,
                  853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983,
                  991, 997]

     if num in lowPrimes:
         return True

     for prime in lowPrimes:
         if num % prime == 0:
             return False

     return rabinMiller(num)

def generateLargePrime(keysize=1024):
     # Return a random prime number of keysize bits in size.
     while True:
         num = random.randrange(2**(keysize-1), 2**(keysize))
         if isPrime(num):
             return num