Background
Elliptic Curve Cryptography (ECC) is an approach to publickey cryptography, based on the algebraic structure of elliptic curves over finite fields. ECC requires a smaller key as compared to nonECC cryptography to provide equivalent security (a 256bit ECC security has an equivalent security attained by 3072bit 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 coefficient of x and ‘b’ is the constant of the equation
The curve is nonsingular; that is its graph has no cusps or selfintersections (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 could intersect almost 3 points when a straight line is drawn intersecting the curve. As we can see the elliptic curve is symmetric about the xaxis, this property plays a key role in the algorithm.
DiffieHellman algorithm
The DiffieHellman 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:
C
/* This program calculates the Key for two persons using the DiffieHellman 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; } 
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)) 
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.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the CS Theory Course at a studentfriendly price and become industry ready.