Implementation of Diffie-Hellman Algorithm

Last Updated : 03 Jul, 2024

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 is as follows:

Alice	Bob
Public Keys available = P, G	Public Keys available = P, G
Private Key Selected = a	Private Key Selected = b
Key generated = x = G^a mod P	Key generated = y = G^b mod P
Exchange of generated keys takes place
Key received = y	key received = x
Generated Secret Key = k_a = y^a mod P	Generated Secret Key = k_b = x^b mod P
Algebraically, it can be shown that k_a = k_b
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 Bob is : " << kb << endl;

    return 0;
}
// This code is contributed by Pranay Arora

/* This program calculates the Key for two persons
using the Diffie-Hellman Key exchange algorithm */
#include <math.h>
#include <stdio.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

Python

# Diffie-Hellman Code

# Power function to return value of a^b mod P
def power(a, b, p):
    if b == 1:
        return a
    else:
        return pow(a, b) % p

# Main function
def main():
    # Both persons agree upon the public keys G and P
    # A prime number P is taken
    P = 23
    print("The value of P:", P)

    # A primitive root for P, G is taken
    G = 9
    print("The value of G:", G)

    # Alice chooses the private key a
    # a is the chosen private key
    a = 4
    print("The private key a for Alice:", a)

    # Gets the generated key
    x = power(G, a, P)

    # Bob chooses the private key b
    # b is the chosen private key
    b = 3
    print("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

    print("Secret key for Alice is:", ka)
    print("Secret key for Bob is:", kb)

if __name__ == "__main__":
    main()

// 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

ElGamal Encryption Algorithm

Souvik Nandi

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