任意取两个不等质数p,q，记N=pq,构造Zn，并且证明Zn为群
设计思路
- 利用筛法，求出一定范围内的素数，并将这些素数存在一个list当中
- 从list中随机选出p,q,若p==q则重新选择
- 计算出N，并根据N，构造出Zn*
- 判断是否Zn*中每个元素都含有逆元
- 根据群的性质，判断Zn*是否为群
下面贴出代码
import random
import math
from random import choice
def getRandomPrime(Nmax):
isPrime = [0]*(Nmax+1)
primeList = []
isPrime[2] = 1
for i in range(3,Nmax,2):
isPrime[i] = 1
for i in range(3,int(math.sqrt((Nmax+1)))+1,2):
if(isPrime[i] == 1):
for j in range(i*i,Nmax+1,2*i):
isPrime[j] = 0
for i in range(2,Nmax+1):
if (isPrime[i]==1):
primeList.append(i)
#使用全局变量，将while里面的取值，取出
global a,b
while True:
p,q = (random.sample(primeList,2))
if(p!=q):
a,b = p,q
break
return a,b
def isCoprime(a,b):
flag = 0
if(b>a):
a,b = b,a
while(a%b):
tmp = b
b = a%b
a = tmp
if(b==1):
flag = 1
return flag
def getZnStar(Nmax):
znStar = []
p,q = getRandomPrime(Nmax)
print("p is %d, q is %d"%(p,q))
N = p*q
print("N is %d"%N)
for i in range(1,N):
if(isCoprime(N,i)==1):
znStar.append(i)
print("Zn_star is " + str(znStar))
# print (type(len(znStar)))
return znStar,N,p,q
def isEveryInverse(znStar,N):
count = 0
size = len(znStar)
for i in range(0,size):
for j in range(0,size):
if((znStar[i] * znStar[j]) % N == 1):
count = count + 1
break
if(count==size):
print("Every element has its inverse")
return 1
else:
print("Not every element has its inverse")
return 0
def isGroup(znStar,N):
closure_flag = 0
exchange_flag = 0
unit_flag = 0
size = len(znStar)
a,b = random.sample(range(0,size),2)
c = choice(range(0,size))
for i in range(0,size):
if((znStar[a] * znStar[b]) % N == znStar[i]):
closure_flag = 1
break
#print(type(N))
if ((((znStar[a]*znStar[b])%N)*znStar[c])%N == (((znStar[b]*znStar[c])%N)*znStar[a])%N):
exchange_flag=1
if(znStar[0] == 1):
unit_flag = 1
if(closure_flag == 1 and exchange_flag == 1 and unit_flag == 1 and isEveryInverse(znStar,N)==1):
print ("znStar is a group")
else:
print ("znStar is not a group")
def Number(znStar,p,q):
size = len(znStar)
if (size == (p-1)*(q-1)):
print ("The size of ZnStar is (p-1)*(q-1)")
else:
print("The size of ZnStar is (p-1)*(q-1)")
if __name__=="__main__":
znStar,N,p,q = getZnStar(100)
isGroup(znStar,N)
Number(znStar,p,q)