**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; }

**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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