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 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 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.
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
|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|
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.
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 student-friendly price and become industry ready.
- Java Implementation of Deffi-Hellman Algorithm between Client and Server
- Implementation of a Back-off Algorithm for CSMA/CD
- TCP Server-Client implementation in C
- UDP Server-Client implementation in C
- Hamming code Implementation in Java
- Approaches to Information Security Implementation
- Hamming code Implementation in C/C++
- RSA Algorithm in Cryptography
- Computer Network | Leaky bucket algorithm
- HMAC Algorithm in Computer Network
- Probabilistic shortest path routing algorithm for optical networks
- RSA Algorithm using Multiple Precision Arithmetic Library
- RC4 Encryption Algorithm
- Back-off Algorithm for CSMA/CD
- Algorithm for Dynamic Time out timer Calculation
- RC5 Encryption Algorithm
- Cristian's Algorithm
- How to solve RSA Algorithm Problems?
- ElGamal Encryption Algorithm
- Berkeley's Algorithm