Implementation of Diffie-Hellman Algorithm
Background
Elliptic Curve Cryptography (ECC) is an approach to public-key cryptography, based on the algebraic structure of elliptic curves over finite fields. ECC requires a smaller key as compared to non-ECC cryptography to provide equivalent security (a 256-bit ECC security has equivalent security attained by 3072-bit RSA cryptography).
For a better understanding of Elliptic Curve Cryptography, it is very important to understand the basics of the Elliptic Curve. An elliptic curve is a planar algebraic curve defined by an equation of the form
Where ‘a’ is the co-efficient of x and ‘b’ is the constant of the equation
The curve is non-singular; that is, its graph has no cusps or self-intersections (when the characteristic of the Co-efficient field is equal to 2 or 3).
In general, an elliptic curve looks like as shown below. Elliptic curves can intersect almost 3 points when a straight line is drawn intersecting the curve. As we can see, the elliptic curve is symmetric about the x-axis. This property plays a key role in the algorithm.
Diffie-Hellman algorithm
The Diffie-Hellman algorithm is being used to establish a shared secret that can be used for secret communications while exchanging data over a public network using the elliptic curve to generate points and get the secret key using the parameters.
- For the sake of simplicity and practical implementation of the algorithm, we will consider only 4 variables, one prime P and G (a primitive root of P) and two private values a and b.
- P and G are both publicly available numbers. Users (say Alice and Bob) pick private values a and b and they generate a key and exchange it publicly. The opposite person receives the key and that generates a secret key, after which they have the same secret key to encrypt.
Step by Step Explanation
Alice | Bob |
---|---|
Public Keys available = P, G | Public Keys available = P, G |
Private Key Selected = a | Private Key Selected = b |
Key generated = |
Key generated = |
Exchange of generated keys takes place | |
Key received = y | key received = x |
Generated Secret Key = |
Generated Secret Key = |
Algebraically, it can be shown that |
|
Users now have a symmetric secret key to encrypt |
Example:
Step 1: Alice and Bob get public numbers P = 23, G = 9 Step 2: Alice selected a private key a = 4 and Bob selected a private key b = 3 Step 3: Alice and Bob compute public values Alice: x =(9^4 mod 23) = (6561 mod 23) = 6 Bob: y = (9^3 mod 23) = (729 mod 23) = 16 Step 4: Alice and Bob exchange public numbers Step 5: Alice receives public key y =16 and Bob receives public key x = 6 Step 6: Alice and Bob compute symmetric keys Alice: ka = y^a mod p = 65536 mod 23 = 9 Bob: kb = x^b mod p = 216 mod 23 = 9 Step 7: 9 is the shared secret.
Implementation:
C++
/* This program calculates the Key for two persons using the Diffie-Hellman Key exchange algorithm using C++ */ #include <cmath> #include <iostream> using namespace std; // Power function to return value of a ^ b mod P long long int power( long long int a, long long int b, long long int P) { if (b == 1) return a; else return ((( long long int ) pow (a, b)) % P); } // Driver program int main() { long long int P, G, x, a, y, b, ka, kb; // Both the persons will be agreed upon the // public keys G and P P = 23; // A prime number P is taken cout << "The value of P : " << P << endl; G = 9; // A primitive root for P, G is taken cout << "The value of G : " << G << endl; // Alice will choose the private key a a = 4; // a is the chosen private key cout << "The private key a for Alice : " << a << endl; x = power(G, a, P); // gets the generated key // Bob will choose the private key b b = 3; // b is the chosen private key cout << "The private key b for Bob : " << b << endl; y = power(G, b, P); // gets the generated key // Generating the secret key after the exchange // of keys ka = power(y, a, P); // Secret key for Alice kb = power(x, b, P); // Secret key for Bob cout << "Secret key for the Alice is : " << ka << endl; cout << "Secret key for the Alice is : " << kb << endl; return 0; } // This code is contributed by Pranay Arora |
C
/* This program calculates the Key for two persons using the Diffie-Hellman Key exchange algorithm */ #include<stdio.h> #include<math.h> // Power function to return value of a ^ b mod P long long int power( long long int a, long long int b, long long int P) { if (b == 1) return a; else return ((( long long int ) pow (a, b)) % P); } //Driver program int main() { long long int P, G, x, a, y, b, ka, kb; // Both the persons will be agreed upon the // public keys G and P P = 23; // A prime number P is taken printf ( "The value of P : %lld\n" , P); G = 9; // A primitive root for P, G is taken printf ( "The value of G : %lld\n\n" , G); // Alice will choose the private key a a = 4; // a is the chosen private key printf ( "The private key a for Alice : %lld\n" , a); x = power(G, a, P); // gets the generated key // Bob will choose the private key b b = 3; // b is the chosen private key printf ( "The private key b for Bob : %lld\n\n" , b); y = power(G, b, P); // gets the generated key // Generating the secret key after the exchange // of keys ka = power(y, a, P); // Secret key for Alice kb = power(x, b, P); // Secret key for Bob printf ( "Secret key for the Alice is : %lld\n" , ka); printf ( "Secret Key for the Bob is : %lld\n" , kb); return 0; } |
Java
// This program calculates the Key for two persons // using the Diffie-Hellman Key exchange algorithm class GFG{ // Power function to return value of a ^ b mod P private static long power( long a, long b, long p) { if (b == 1 ) return a; else return ((( long )Math.pow(a, b)) % p); } // Driver code public static void main(String[] args) { long P, G, x, a, y, b, ka, kb; // Both the persons will be agreed upon the // public keys G and P // A prime number P is taken P = 23 ; System.out.println( "The value of P:" + P); // A primitive root for P, G is taken G = 9 ; System.out.println( "The value of G:" + G); // Alice will choose the private key a // a is the chosen private key a = 4 ; System.out.println( "The private key a for Alice:" + a); // Gets the generated key x = power(G, a, P); // Bob will choose the private key b // b is the chosen private key b = 3 ; System.out.println( "The private key b for Bob:" + b); // Gets the generated key y = power(G, b, P); // Generating the secret key after the exchange // of keys ka = power(y, a, P); // Secret key for Alice kb = power(x, b, P); // Secret key for Bob System.out.println( "Secret key for the Alice is:" + ka); System.out.println( "Secret key for the Bob is:" + kb); } } // This code is contributed by raghav14 |
Python3
from random import randint if __name__ = = '__main__' : # Both the persons will be agreed upon the # public keys G and P # A prime number P is taken P = 23 # A primitive root for P, G is taken G = 9 print ( 'The Value of P is :%d' % (P)) print ( 'The Value of G is :%d' % (G)) # Alice will choose the private key a a = 4 print ( 'The Private Key a for Alice is :%d' % (a)) # gets the generated key x = int ( pow (G,a,P)) # Bob will choose the private key b b = 3 print ( 'The Private Key b for Bob is :%d' % (b)) # gets the generated key y = int ( pow (G,b,P)) # Secret key for Alice ka = int ( pow (y,a,P)) # Secret key for Bob kb = int ( pow (x,b,P)) print ( 'Secret key for the Alice is : %d' % (ka)) print ( 'Secret Key for the Bob is : %d' % (kb)) |
C#
// C# implementation to calculate the Key for two persons // using the Diffie-Hellman Key exchange algorithm using System; class GFG { // Power function to return value of a ^ b mod P private static long power( long a, long b, long P) { if (b == 1) return a; else return ((( long )Math.Pow(a, b)) % P); } public static void Main() { long P, G, x, a, y, b, ka, kb; // Both the persons will be agreed upon the // public keys G and P P = 23; // A prime number P is taken Console.WriteLine( "The value of P:" + P); G = 9; // A primitive root for P, G is taken Console.WriteLine( "The value of G:" + G); // Alice will choose the private key a a = 4; // a is the chosen private key Console.WriteLine( "\nThe private key a for Alice:" + a); x = power(G, a, P); // gets the generated key // Bob will choose the private key b b = 3; // b is the chosen private key Console.WriteLine( "The private key b for Bob:" + b); y = power(G, b, P); // gets the generated key // Generating the secret key after the exchange // of keys ka = power(y, a, P); // Secret key for Alice kb = power(x, b, P); // Secret key for Bob Console.WriteLine( "\nSecret key for the Alice is:" + ka); Console.WriteLine( "Secret key for the Alice is:" + kb); } } // This code is contributed by Pranay Arora |
Javascript
<script> // This program calculates the Key for two persons // using the Diffie-Hellman Key exchange algorithm // Power function to return value of a ^ b mod P function power(a, b, p) { if (b == 1) return a; else return ((Math.pow(a, b)) % p); } // Driver code var P, G, x, a, y, b, ka, kb; // Both the persons will be agreed upon the // public keys G and P // A prime number P is taken P = 23; document.write( "The value of P:" + P + "<br>" ); // A primitive root for P, G is taken G = 9; document.write( "The value of G:" + G + "<br>" ); // Alice will choose the private key a // a is the chosen private key a = 4; document.write( "The private key a for Alice:" + a + "<br>" ); // Gets the generated key x = power(G, a, P); // Bob will choose the private key b // b is the chosen private key b = 3; document.write( "The private key b for Bob:" + b + "<br>" ); // Gets the generated key y = power(G, b, P); // Generating the secret key after the exchange // of keys ka = power(y, a, P); // Secret key for Alice kb = power(x, b, P); // Secret key for Bob document.write( "Secret key for the Alice is:" + ka + "<br>" ); document.write( "Secret key for the Bob is:" + kb + "<br>" ); // This code is contributed by Ankita saini </script> |
Output:
The value of P : 23 The value of G : 9 The private key a for Alice : 4 The private key b for Bob : 3 Secret key for the Alice is : 9 Secret Key for the Bob is : 9
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