群的构造

1.demo：

from Crypto.Util import number
import random
# 群生成算法，输入安全参数n，返回[p, q, g]，分别是模p，群的阶q，生成元g
def GroupGen(n):
    p = number.getStrongPrime(2 * n)
    q = (p - 1) / 2
    q =(int)(q)
    while True:
        x = number.getRandomRange(1, p)
        if number.GCD(x, p) == 1 and x != 1 and x != p-1:
            break
    g = x * x % p
    return [p, q, g]

# 模指数运算，输入底a，指数p，模n，返回a^p % n
def ModExp(a, p, n):
    if p == 0:
        return 1
    result = a % n
    remainders = []
    while p != 1:
        remainders.append(p & 1)
        p = p >> 1
    while remainders:
        rem = remainders.pop()
        result = ((a ** rem) * result ** 2) % n
    return result


# 群指数运算，输入模p，群的阶q，生成元g，指数i，返回g^i
def GroupExp(p, q, g, i):
    return ModExp(g, i%q, p)

# GCD算法，输入整数a和b，返回d = gcd(a, b)
def GCD(a, b):
    return number.GCD(a, b)


# 扩展GCD算法，输入整数a和b，返回[d, X, Y]，满足d = gcd(a, b)，Xa + Yb = d
def eGCD(a, b):
    if b == 0:
        return (a, 1, 0)
    [x1, y1, x2, y2] = [1, 0, 0, 1]
    while b != 0:
        [q, r] = [(int)(a/b), a%b]
        [a, b] = [b, r]
        x = x1 - q * x2
        [x1, x2] = [x2, x]
        y = y1 - q * y2
        [y1, y2] = [y2, y]
    return [a, x1, y1]


# 求逆元算法，输入元素a，模n，返回a的逆元
def GroupInverse(a, n):
    [d, X, Y] = eGCD(a, n)
    if d != 1:
        return 0
    return X % n

# 群内乘法，输入群元素a和b，模p，返回a * b % p
def GroupTimes(a, b, p):
    return a * b % p

# 随机生成一个群元素，输入模p，群的阶q，生成元g，返回群中的一个随机元素
def GroupPickupElement(p, q, g):
    i = number.getRandomRange(1, q)
    return GroupExp(p, q, g, i)


[p, q, g] = GroupGen(512)

print(g)

print(GroupExp(p, q, g, 5))
print(g ** 5 % p)

res = GroupExp(p, q, g, q-20)
print(res)
print(g ** 20 % p)


print(GroupTimes(res, GroupInverse(res, p), p))
print(GroupExp(p, q, g, 0))
print(GroupExp(p, q, g, q))

res = GroupExp(p, q, g, q+5)
print(GroupTimes(res, GroupInverse(g ** 5 % p, p), p))