**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 have an 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 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 could intersect atmost 3 points when a straight line is drawn intersecting the curve. As we can see that 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 received the key and from 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:**

`/* 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; ` `} ` |

*chevron_right*

*filter_none*

**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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