def crt(a1, a2, m1, m2):
M = m1 * m2
M1 = M // m1
M2 = M // m2
M1_inv = ext_gcd(M1, m1)[1]
M2_inv = ext_gcd(M2, m2)[1]
if (M1_inv < 0): M1_inv = M1_inv + m1
if (M2_inv < 0): M2_inv = M2_inv + m2
n = (a1 * M1 * M1_inv + a2 * M2 * M2_inv) % M
return n
def euler(n):
φ = 1
for _ in range(2, n):
if ext_gcd(_, n)[0] == 1:
φ = φ + 1
return φ
# print(euler(15))
def all_primitive_roots(p):
root = primitive_root(p)
euler_ = euler(p)
root_list = []
for _ in range(2, euler_):
# 指数与euler(m)互素
if ext_gcd(_, euler_)[0] == 1:
root_list.append(_)
# print(_, end=',')
return root_list
def ord_of_am(a, m):
# a,m需互素
if ext_gcd(a, m)[0] == 1:
euler_ = euler(m)
for _ in range(1, euler(m)):
# 指数整除euler(m)
if euler_ % _ == 0:
mod = exp_mod(a, _, m)
if mod == 1:
return _
return euler(m)
def gen_key(p, q):
n = p * q
fy = (p - 1) * (q - 1) # 計算與n互質的整數個數 尤拉函式
e = 3889 # 選取e 一般選取65537
# generate d
a = e
b = fy
r, x, y = ext_gcd(a, b)
#print(x) # 計算出的x不能是負數，如果是負數，說明p、q、e選取失敗，一般情況下e選取65537
d = x
# 返回： 公鑰 私鑰
return (n, e), (n, d)
def gen_key(p, q, e):
n = p * q
fy = (p - 1) * (q - 1) # 计算与n互质的整数个数 欧拉函数
#e = 3889 # 选取e 一般选取65537
# generate d
a = e
b = fy
r, x, y = ext_gcd(a, b)
print(x) # 计算出的x不能是负数，如果是负数，说明p、q、e选取失败，一般情况下e选取65537
d = x
# 返回： 公钥 私钥
return (n, e), (n, d)
def decrypt(c, prikey):
d = prikey[0]
p = prikey[1]
c1 = c[0]
c2 = c[1]
#m = c2*(c1^d)^-1 mod p
c1_d = exp_mod(c1, d, p)
if (c1_d < 0): c1_d += p
c1_dinv = ext_gcd(c1_d, p)[1]
if (c1_dinv < 0): c1_dinv += p
m = ((c2 % p) * c1_dinv) % p
return m
def gen_key(p, q):
n = p * q
euler = (p - 1) * (q - 1)
e = random.randint(3, euler)
a = e
b = euler
r, x, y = ext_gcd(a, b)
while r != 1:
e = random.randint(3, euler)
a = e
b = euler
r, x, y = ext_gcd(a, b)
print('r', r)
print('euler', euler)
print('x', x)
if x < 0:
print("x is negative")
x = x - (x // (b // r)) * (b // r)
print('x', x)
d = x
# error happens when x is negative
print('n', n, 'e', e, 'd', d)
return (n, e), (n, d)
def is_prime_Fermat(n, t=1):
flag = True
for _ in range(t):
b = 2
while ext_gcd(b, n)[0] != 1:
b = random.randint(3, n - 1)
r = exp_mod(b, n - 1, n)
if r != 1:
flag = False
if flag:
# print("根据Fermat素性检验，%d为素数"%n)
return 1
else:
# print("根据Fermat素性检验，%d为合数"%n)
return 0
def gen_key(p, q):
n = p * q
fy = (p - 1) * (q - 1) # 计算与n互质的整数个数 欧拉函数
e = 65537 # 选取e 一般选取65537
# generate d
a = e
b = fy
r, x, y = ext_gcd(a, b)
# 计算出的x不能是负数，如果是负数，说明p、q、e选取失败，不过可以把x加上fy，使x为正数，才能计算。
if x < 0:
x = x + fy
d = x
# 返回： 公钥 私钥
print("n:", n)
print("d:", d)
return (n, e), (n, d)
def gen_key(p, q):
n = p * q
phi = (p - 1) * (q - 1) # 算phi(n)
# 選e
while (True):
e = int(input("請輸入e (e與phi(n)必須互質)\n"))
if (gcd(e, phi) != 1):
print("e與phi(n)沒有互質，請重新輸入")
continue
else:
break
r, x, y = ext_gcd(e, phi) # 用擴展歐幾里得算 d = e^-1 mod phi(n)
d = x
if (d < 0): d += phi
return (n, e), (n, d) #回傳公鑰<n,e>、私鑰<n,d>
d = -1
# do until d is > 0
while (d < 0):
print('find p')
p = makePrime(1024)
print('find q')
q = makePrime(1024)
n = p * q
r = (p - 1) * (q - 1)
print('find d')
e = 65537
a, d, b = ext_gcd(e, r)
# write publicKey(n,e) to text file
with open('./publicKey.txt', 'wb') as f:
f.write(str(n))
f.write('\n')
f.write(str(e))
# write privateKey(n,d) to text file
with open('./privateKey.txt', 'wb') as f:
f.write(str(n))
f.write('\n')
f.write(str(d))
print('finish')