def KeyGenerator(bit=1024):
# Generate 2 large prime p, q, (p - 1) % q == 0
print("Generate...")
flag = False
while not flag:
c = 0
q = prime.PrimeGenerator(160) # q is 160 bits
# (p - 1) % q == 0 --> p = q * k + 1
# p is 1024 bits (default) --> k is (1024 - 160) bits
while 1:
k = random.getrandbits(bit - 160)
p = k * q + 1
if (p.bit_length() == bit and prime.MillerRabinTest(p)):
flag = True
break
c += 1
# 可能因為 q 的關係導致很難找到 p ，所以如果一段時間找不到 p ，就換一個 q 試試看
if (c >= 100):
break
# Generate g, such that multiplicative order modulo p of g is q
h = 2
while util.SQandMU(h, (p - 1) // q, p) == 1:
h = random.randrange(2, p - 1)
g = util.SQandMU(h, (p - 1) // q, p)
priKey = random.randrange(1, q) # Grnerate private key
pubKey = util.SQandMU(g, priKey, p) # Generate public key = g ^ priKey % q
return (p, q, g), pubKey, priKey
def Verify(m, p, q, g, pubKey, r, s):
# 驗證簽章
assert r >= 0 and r < q and s >= 0 and s < q
w = util.ModInv(s, q)
u1 = (util.H(m) * w) % q
u2 = (r * w) % q
v = ((util.SQandMU(g, u1, p) * util.SQandMU(pubKey, u2, p)) % p) % q
return v % q == r % q
def test(a):
x = util.SQandMU(a, u, p)
if x == 1 or x == p - 1:
return False
for i in range(t - 1):
x = util.SQandMU(a, (1 << i) * u, p)
if x == p - 1:
return False
return True
def DecryptionCRT(n, p, q, key, message):
xp = message % p
xq = message % q
dp = key % (p - 1)
dq = key % (q - 1)
yp = util.SQandMU(xp, dp, p)
yq = util.SQandMU(xq, dq, q)
cp = util.ModInv(q, p)
cq = util.ModInv(p, q)
return (round(q * cp) * yp + round(p * cq) * yq) % n
def Sign(m, p, q, g, priKey):
# 簽章
r, s = 0, 0
while 1:
k = random.randrange(1, q)
r = util.SQandMU(g, k, p) % q
if (r == 0):
continue
# 發生R=0的情況，試另一組k
dk = util.ModInv(k, q)
s = (dk * (util.H(m) + priKey * r)) % q
if (s != 0):
break
return r, s
def Encryption(n, key, message):
return util.SQandMU(message, key, n)