B = 15
privatekey = 0
#brute force O(n)
for i in range(n):
if pow(2, i, n) == B:
privatekey = i
break
print("brute force result: ", privatekey)
#shank algorithm O(sqrt(n))
terminate = False
m = int(math.sqrt(n - 1)) + 1
for i in range(0, m):
t1 = pow(a, m * i, n)
#print(t1)
for j in range(0, m):
t2 = B * commons.modInverse(pow(a, j), n) % n
#print("\t",t2)
if t1 == t2:
#print("two tables have ",t1," value")
print("shank algorithm result: ", m * i + j)
terminate = True
break
if terminate == True:
break
e = random.randint(1, totient)
print("public key: ",e)
#-------------------------------
#find multiplicative inverse of e mod totient
#brute force
"""d=0
for i in range(totient):
if (e*i) % totient == 1:
d = i
break
"""
d = commons.modInverse(e, totient)
print("private key: ",d)
print("key generation is complete in ",time.time() - starttime," seconds\n")
publickey = e
privatekey = d
#--------------------------------
print("-------------------------")
print("message encryption")
print("-------------------------")
m = 11
y = pow(g, x, p)
print("private key: ",x)
print("public key: ",y)
print("public params: p=",p,", q=",q,", g=",g,"")
#-------------------------------
print("signing")
k = 10 #random key
h = 123
r = pow(g, k, p) % q
s = commons.modInverse(k, q) * (h + x*r) % q
print("signature: (r=",r,", s=",s,")")
#-------------------------------
print("verification")
h = 123
#Bob knows public key -> y
#Also, he knows public params -> p, q, g
#Also, he knows h, (r, s) pair
w = commons.modInverse(s, q)
u1 = h * w % q
u2 = r * w % q
y = pow(g, x, p)
print("private key: ", x)
print("public key: ", y)
print("public params: p=", p, ", q=", q, ", g=", g, "")
#-------------------------------
print("signing")
k = 10 #random key
h = 123
r = pow(g, k, p) % q
s = commons.modInverse(k, q) * (h + x * r) % q
print("signature: (r=", r, ", s=", s, ")")
#-------------------------------
print("verification")
h = 123
#Bob knows public key -> y
#Also, he knows public params -> p, q, g
#Also, he knows h, (r, s) pair
w = commons.modInverse(s, q)
u1 = h * w % q
u2 = r * w % q
print("digital signature")
print("-----------------------")
print("signing")
hash = 100
k = random.randint(1, p - 1)
while commons.gcd(p - 1, k) != 1:
k = random.randint(1, p - 1)
#print("random key: ",k)
r = pow(g, k, p)
s = (hash - x * r) * commons.modInverse(k, p - 1) % (p - 1)
print("signature: (r=", r, ", s=", s, ")")
print("verification")
hash = 100
checkpoint1 = pow(g, hash, p)
checkpoint2 = (pow(y, r, p) * pow(r, s, p)) % p
print("checkpoint1: ", checkpoint1)
print("checkpoint2: ", checkpoint2)
if checkpoint1 == checkpoint2:
print("signature is valid")