Add diffie hellman key exchange (#615)

* [Add] Find order algorithm and its supplemental

* [Add] Find primitive root algorithm

* [Edit] Add 'find_' in front of primitive root algorithm file

* [Add/Fix] Supplemental & exception handling for the case n = 1

* [Fix] Exception handling for the case a == n == 1 in findOrder function

* [Delete] Algorithms in my other branch

* [Add] Diffie-Hellman Key Exchange Illustration

* [Add] Description for Diffie-Hellman Key exchange

* [Add] Supplemental for DH Key Exchange

* [Add] Test and its operating environment

* [Edit] Add link to README.md file

* [Edit] For not primitive root case, code raise ValueError now

* [Edit] Change test cases according to immediate before commit

* [Edit] For not primitive root case, code return False now, for consistency (return Bool only)

* Change test cases according to immediate before commit

* [Edit] If optional parameter is given, additional explanation would arise

Author: D-Sinus

README.md

@@ -225,6 +225,7 @@ If you want to uninstall algorithms, it is as simple as:
     - [recursive_binomial_coefficient](algorithms/maths/recursive_binomial_coefficient.py)
     - [find_order](algorithms/maths/find_order_simple.py)
 	  - [find_primitive_root](algorithms/maths/find_primitive_root_simple.py)
+	  - [diffie_hellman_key_exchange](algorithms/maths/diffie_hellman_key_exchange.py)
 - [matrix](algorithms/matrix)
     - [sudoku_validator](algorithms/matrix/sudoku_validator.py)
     - [bomb_enemy](algorithms/matrix/bomb_enemy.py)

algorithms/maths/__init__.py

@@ -17,3 +17,4 @@
 from .cosine_similarity import *
 from .find_order_simple import *
 from .find_primitive_root_simple import *
+from .diffie_hellman_key_exchange import *

algorithms/maths/diffie_hellman_key_exchange.py

@@ -0,0 +1,190 @@
+import math
+from random import randint
+
+"""
+Code from /algorithms/maths/prime_check.py,
+written by 'goswami-rahul' and 'Hai Honag Dang'
+"""
+def prime_check(n):
+    """Return True if n is a prime number
+    Else return False.
+    """
+
+    if n <= 1:
+        return False
+    if n == 2 or n == 3:
+        return True
+    if n % 2 == 0 or n % 3 == 0:
+        return False
+    j = 5
+    while j * j <= n:
+        if n % j == 0 or n % (j + 2) == 0:
+            return False
+        j += 6
+    return True
+
+
+"""
+For positive integer n and given integer a that satisfies gcd(a, n) = 1,
+the order of a modulo n is the smallest positive integer k that satisfies
+pow (a, k) % n = 1. In other words, (a^k) ≡ 1 (mod n).
+Order of certain number may or may not be exist. If so, return -1.
+"""
+def find_order(a, n):
+    if ((a == 1) & (n == 1)):
+        return 1
+        """ Exception Handeling :
+        1 is the order of of 1 """
+    else:
+        if (math.gcd(a, n) != 1):
+            print ("a and n should be relative prime!")
+            return -1
+        else:
+            for i in range(1, n):
+                if (pow(a, i) % n == 1):
+                    return i
+            return -1
+
+"""
+Euler's totient function, also known as phi-function ϕ(n),
+counts the number of integers between 1 and n inclusive,
+which are coprime to n.
+(Two numbers are coprime if their greatest common divisor (GCD) equals 1).
+Code from /algorithms/maths/euler_totient.py, written by 'goswami-rahul'
+"""
+def euler_totient(n):
+    """Euler's totient function or Phi function.
+    Time Complexity: O(sqrt(n))."""
+    result = n;
+    for i in range(2, int(n ** 0.5) + 1):
+        if n % i == 0:
+            while n % i == 0:
+                n //= i
+            result -= result // i
+    if n > 1:
+        result -= result // n;
+    return result;
+
+"""
+For positive integer n and given integer a that satisfies gcd(a, n) = 1,
+a is the primitive root of n, if a's order k for n satisfies k = ϕ(n).
+Primitive roots of certain number may or may not be exist.
+If so, return empty list.
+"""
+
+def find_primitive_root(n):
+    if (n == 1):
+        return [0]
+        """ Exception Handeling :
+        0 is the only primitive root of 1 """
+    else:
+        phi = euler_totient(n)
+        p_root_list = []
+        """ It will return every primitive roots of n. """
+        for i in range (1, n):
+            if (math.gcd(i, n) != 1):
+                continue
+                """ To have order, a and n must be
+                relative prime with each other. """
+            else:
+                order = find_order(i, n)
+                if (order == phi):
+                    p_root_list.append(i)
+                else:
+                    continue
+        return p_root_list
+
+
+"""
+Diffie-Hellman key exchange is the method that enables
+two entities (in here, Alice and Bob), not knowing each other,
+to share common secret key through not-encrypted communication network.
+This method use the property of one-way function (discrete logarithm)
+For example, given a, b and n, it is easy to calculate x
+that satisfies (a^b) ≡ x (mod n).
+However, it is very hard to calculate x that satisfies (a^x) ≡ b (mod n).
+For using this method, large prime number p and its primitive root a must be given.
+"""
+
+def alice_private_key(p):
+    """Alice determine her private key
+    in the range of 1 ~ p-1.
+    This must be kept in secret"""
+    return randint(1, p-1)
+
+
+def alice_public_key(a_pr_k, a, p):
+    """Alice calculate her public key
+    with her private key.
+    This is open to public"""
+    return pow(a, a_pr_k) % p
+
+
+def bob_private_key(p):
+    """Bob determine his private key
+    in the range of 1 ~ p-1.
+    This must be kept in secret"""
+    return randint(1, p-1)
+
+
+def bob_public_key(b_pr_k, a, p):
+    """Bob calculate his public key
+    with his private key.
+    This is open to public"""
+    return pow(a, b_pr_k) % p
+    
+
+def alice_shared_key(b_pu_k, a_pr_k, p):
+    """ Alice calculate secret key shared with Bob,
+    with her private key and Bob's public key.
+    This must be kept in secret"""
+    return pow(b_pu_k, a_pr_k) % p
+
+
+def bob_shared_key(a_pu_k, b_pr_k, p):
+    """ Bob calculate secret key shared with Alice,
+    with his private key and Alice's public key.
+    This must be kept in secret"""
+    return pow(a_pu_k, b_pr_k) % p
+
+
+def diffie_hellman_key_exchange(a, p, option = None):
+    if (option != None):
+        option = 1
+        """ Print explanation of process
+        when option parameter is given """
+    if (prime_check(p) == False):
+        print("%d is not a prime number" % p)
+        return False
+        """p must be large prime number"""
+    else:
+        try:
+            p_root_list = find_primitive_root(p)
+            p_root_list.index(a)
+        except ValueError:
+            print("%d is not a primitive root of %d" % (a, p))
+            return False
+            """ a must be primitive root of p """
+        
+        a_pr_k = alice_private_key(p)
+        a_pu_k = alice_public_key(a_pr_k, a, p)
+        
+        
+        b_pr_k = bob_private_key(p)
+        b_pu_k = bob_public_key(b_pr_k, a, p)
+        
+        if (option == 1):
+            print ("Private key of Alice = %d" % a_pr_k)
+            print ("Public key of Alice = %d" % a_pu_k)
+            print ("Private key of Bob = %d" % b_pr_k)
+            print ("Public key of Bob = %d" % b_pu_k)
+
+        """ In here, Alice send her public key to Bob,
+        and Bob also send his public key to Alice."""
+
+        a_sh_k = alice_shared_key(b_pu_k, a_pr_k, p)
+        b_sh_k = bob_shared_key(a_pu_k, b_pr_k, p)
+        print ("Shared key calculated by Alice = %d" % a_sh_k)
+        print ("Shared key calculated by Bob = %d" % b_sh_k)
+        
+        return (a_sh_k == b_sh_k)

tests/test_maths.py

@@ -19,6 +19,7 @@
     cosine_similarity,
     find_order,
     find_primitive_root,
+    alice_private_key, alice_public_key, bob_private_key, bob_public_key, alice_shared_key, bob_shared_key, diffie_hellman_key_exchange
 )
 
 import unittest
@@ -326,6 +327,7 @@ def test_cosine_similarity(self):
         self.assertAlmostEqual(cosine_similarity(vec_a, vec_b), -1)
         self.assertAlmostEqual(cosine_similarity(vec_a, vec_c), 0.4714045208)
 
+
 class TestFindPrimitiveRoot(unittest.TestCase):
     """[summary]
     Test for the file find_primitive_root_simple.py
@@ -353,6 +355,20 @@ def test_find_order_simple(self):
         self.assertEqual(-1, find_order(128, 256))
         self.assertEqual(352, find_order(3, 353))
 
+
+class TestDiffieHellmanKeyExchange(unittest.TestCase):
+    """[summary]
+    Test for the file diffie_hellman_key_exchange.py
+
+    Arguments:
+        unittest {[type]} -- [description]
+    """
+    def test_find_order_simple(self):
+        self.assertFalse(diffie_hellman_key_exchange(3, 6))
+        self.assertTrue(diffie_hellman_key_exchange(3, 353))
+        self.assertFalse(diffie_hellman_key_exchange(5, 211))
+        self.assertTrue(diffie_hellman_key_exchange(11, 971))
+
 if __name__ == "__main__":
     unittest.main()