**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 */` `#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 primitve 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 primitve 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 primitve 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))` |

## 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 primitve 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

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.

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