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# Implementation of Diffie-Hellman Algorithm

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 Coefficient 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 is as follows:

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 ``#include ` `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`

## 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

 `# Diffie-Hellman Code`  `def` `prime_checker(p):``    ``# Checks If the number entered is a Prime Number or not``    ``if` `p < ``1``:``        ``return` `-``1``    ``elif` `p > ``1``:``        ``if` `p ``=``=` `2``:``            ``return` `1``        ``for` `i ``in` `range``(``2``, p):``            ``if` `p ``%` `i ``=``=` `0``:``                ``return` `-``1``            ``return` `1`  `def` `primitive_check(g, p, L):``    ``# Checks If The Entered Number Is A Primitive Root Or Not``    ``for` `i ``in` `range``(``1``, p):``        ``L.append(``pow``(g, i) ``%` `p)``    ``for` `i ``in` `range``(``1``, p):``        ``if` `L.count(i) > ``1``:``            ``L.clear()``            ``return` `-``1``        ``return` `1`  `l ``=` `[]``while` `1``:``    ``P ``=` `int``(``input``(``"Enter P : "``))``    ``if` `prime_checker(P) ``=``=` `-``1``:``        ``print``(``"Number Is Not Prime, Please Enter Again!"``)``        ``continue``    ``break` `while` `1``:``    ``G ``=` `int``(``input``(f``"Enter The Primitive Root Of {P} : "``))``    ``if` `primitive_check(G, P, l) ``=``=` `-``1``:``        ``print``(f``"Number Is Not A Primitive Root Of {P}, Please Try Again!"``)``        ``continue``    ``break` `# Private Keys``x1, x2 ``=` `int``(``input``(``"Enter The Private Key Of User 1 : "``)), ``int``(``    ``input``(``"Enter The Private Key Of User 2 : "``))``while` `1``:``    ``if` `x1 >``=` `P ``or` `x2 >``=` `P:``        ``print``(f``"Private Key Of Both The Users Should Be Less Than {P}!"``)``        ``continue``    ``break` `# Calculate Public Keys``y1, y2 ``=` `pow``(G, x1) ``%` `P, ``pow``(G, x2) ``%` `P` `# Generate Secret Keys``k1, k2 ``=` `pow``(y2, x1) ``%` `P, ``pow``(y1, x2) ``%` `P` `print``(f``"\nSecret Key For User 1 Is {k1}\nSecret Key For User 2 Is {k2}\n"``)` `if` `k1 ``=``=` `k2:``    ``print``(``"Keys Have Been Exchanged Successfully"``)``else``:``    ``print``(``"Keys Have Not Been Exchanged Successfully"``)`

## C

 `/* This program calculates the Key for two persons``using the Diffie-Hellman Key exchange algorithm */``#include ``#include ` `// 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;``}`

## 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

 ``

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```

This article is contributed by Souvik Nandi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.GeeksforGeeks.org or mail your article to contribute@GeeksforGeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.