Altered algorithms, added comments
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163
main.py
163
main.py
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@ -5,7 +5,8 @@ from itertools import combinations
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def euclid(a, b):
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"""returns the Greatest Common Divisor of a and b"""
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"""returns the Greatest Common Divisor of a and b.
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Inspired by https://en.wikipedia.org/wiki/Euclidean_algorithm#Implementations"""
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a = abs(a)
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b = abs(b)
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if a < b:
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@ -15,156 +16,136 @@ def euclid(a, b):
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return a
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def coPrime(l):
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def coPrime(x, y):
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"""returns 'True' if the values in the list L are all co-prime
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otherwise, it returns 'False'. """
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for i, j in combinations(l, 2):
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if euclid(i, j) != 1:
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return False
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if euclid(x, y) != 1:
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return False
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return True
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def extendedEuclid(a, b):
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"""return a tuple of three values: x, y and z, such that x is
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the GCD of a and b, and x = y * a + z * b"""
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if a == 0:
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return b, 0, 1
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else:
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g, y, x = extendedEuclid(b % a, a)
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return g, x - (b // a) * y, y
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def iterative_extended_euclid(a, b):
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""" returns the GCD (in old_r) of a and b as well as the "Bezout's Identity" such that
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a*old_s + b*old_t = GCD(a,b).
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Algorithm is adopted from https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Pseudocode
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"""
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old_r, r = a, b
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old_s, s = 1, 0
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old_t, t = 0, 1
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while r != 0:
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quotient = old_r // r
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old_r, r = r, old_r - (quotient * r)
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old_s, s = s, old_s - (quotient * s)
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old_t, t = t, old_t - (quotient * t)
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print(old_r, old_s, old_t)
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return old_r, old_s, old_t
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def modInv(a, m):
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"""returns the multiplicative inverse of a in modulo m as a
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positive value between zero and m-1"""
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# notice that a and m need to be co-prime to each other.
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if coPrime([a, m]):
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linearCombination = extendedEuclid(a, m)
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return linearCombination[1] % m
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positive value between zero and m-1
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Adopted from https://en.wikipedia.org/wiki/Modular_multiplicative_inverse#Extended_Euclidean_algorithm
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"""
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if coPrime(a, m):
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linear_combination = iterative_extended_euclid(a, m)
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print(linear_combination[1] % m)
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return linear_combination[1] % m
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else:
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return 0
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def extractTwos(m):
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"""m is a positive integer. A tuple (s, d) of integers is returned
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such that m = (2 ** s) * d."""
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# the problem can be reduced to counting how many '0's there are in
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# the end of bin(m). This can be done this way: m & a stretch of '1's
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# which can be represent as (2 ** n) - 1.
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assert m >= 0
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i = 0
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while m & (2 ** i) == 0:
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i += 1
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return i, m >> i
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def miller_rabin(n, k):
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# Implementation uses the Miller-Rabin Primality Test
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# The optimal number of rounds for this test is 40
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# See http://stackoverflow.com/questions/6325576/how-many-iterations-of-rabin-miller-should-i-use-for-cryptographic-safe-primes
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# for justification
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def millerRabin(n, k):
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"""
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Miller Rabin pseudo-prime test
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return True means likely a prime, (how sure about that, depending on k)
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return False means definitely a composite.
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Raise assertion error when n, k are not positive integers
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and n is not 1
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"""
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assert n >= 1
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# ensure n is bigger than 1
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assert k > 0
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# ensure k is a positive integer so everything down here makes sense
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# If number is even, it's a composite number
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if n == 2:
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return True
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# make sure to return True if n == 2
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if n % 2 == 0:
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return False
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# immediately return False for all the even numbers bigger than 2
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extract2 = extractTwos(n - 1)
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s = extract2[0]
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d = extract2[1]
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assert 2 ** s * d == n - 1
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def tryComposite(a):
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"""Inner function which will inspect whether a given witness
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will reveal the true identity of n. Will only be called within
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millerRabin"""
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x = pow(a,d,n)
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r, s = 0, n - 1
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while s % 2 == 0:
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r += 1
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s //= 2
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for _ in range(k):
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a = random.randrange(2, n - 1)
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x = pow(a, s, n)
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if x == 1 or x == n - 1:
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return None
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continue
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for _ in range(r - 1):
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x = pow(x, 2, n)
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if x == n - 1:
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break
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else:
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for j in range(1, s):
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x = pow(x,2,n)
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if x == 1:
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return False
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elif x == n - 1:
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return None
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return False
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for i in range(0, k):
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a = random.randint(2, n - 2)
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if tryComposite(a) == False:
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return False
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return True # actually, we should return probably true.
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return True
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def findAPrime(a, b, k):
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"""Return a pseudo prime number roughly between a and b,
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(could be larger than b). Raise ValueError if cannot find a
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pseudo prime after 10 * ln(x) + 3 tries. """
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(could be larger than b). """
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x = random.randint(a, b)
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for i in range(0, int(10 * math.log(x) + 3)):
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if millerRabin(x, k):
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while True:
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if miller_rabin(x, k):
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return x
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else:
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x += 1
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raise ValueError
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def newKey(a, b, k):
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""" Try to find two large pseudo primes roughly between a and b.
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Generate public and private keys for RSA encryption.
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Raises ValueError if it fails to find one"""
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try:
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p = findAPrime(a, b, k)
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while True:
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q = findAPrime(a, b, k)
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if q != p:
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break
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except:
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raise ValueError
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n = p * q
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m = (p - 1) * (q - 1)
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Generate public and private keys for RSA encryption."""
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p = findAPrime(a, b, k)
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while True:
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e = random.randint(1, m)
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if coPrime([e, m]):
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q = findAPrime(a, b, k)
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if q != p:
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break
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n = p * q
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m = (p - 1) * (q - 1) # Compute phi(n) for n=pq where p and q are prime
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# Find and e that is coprime to phi(n) to be used in the public key
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while True:
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e = random.randint(1, m)
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if coPrime(e, m):
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break
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# Let d be the modular inverse to e, to be used as private key
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d = modInv(e, m)
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return (n, e, d)
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def encrypt(message, modN, e, blockSize):
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def encrypt(message, modN, e):
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"""given a string message, public keys and blockSize, encrypt using
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RSA algorithms."""
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return pow(message, e, modN)
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def decrypt(secret, modN, d, blockSize):
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def decrypt(secret, modN, d):
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"""reverse function of encrypt"""
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return pow(secret, d, modN)
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if __name__ == '__main__':
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if __name__ == '__main__':
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n, e, d = newKey(2**40, 2 ** 41, 20)
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message = 35274764
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print("original message is {}".format(message))
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print("-"*80)
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cipher = encrypt(message, n, e, 15)
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cipher = encrypt(message, n, e)
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print("cipher text is {}".format(cipher))
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print("-"*80)
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deciphered = decrypt(cipher, n, d, 15)
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deciphered = decrypt(cipher, n, d)
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print("decrypted message is {}".format(deciphered))
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