CapstoneProject-EC-ElGamal

This repository is designed to help understand elliptic curves and cryptosystems at a basic level. It includes implementations of elliptic curves, ElGamal encryption, elliptic curve ElGamal encryption, and the baby-step giant-step algorithm for attacks. Most of the code is available for both real numbers and finite fields.

The El-Gamal cryptosystem provides a secure encryption and decryption cryptosystem based on the fundamental principles of the Diffie-Hellman key exchange protocol.
Main code is el_gamal_ec_pointmsg.py. This code demonstrates how the El-Gamal encryption system is applied on an elliptic curve. When selecting the elliptic curve, only the finite field F_p is used instead of the extended field F_p^n (different methods are used for the p^n case). It operates with large prime numbers used in El-Gamal encryption, but may experience a loss of performance with very large prime numbers.

An $\textbf{elliptic curve}$ over a field $F$ of characteristic not equal to 2 or 3 is defined as a non-singular curve $E$ given by the equation: $E: y^2 = x^3 + ax + b$
where $x,y,a,b \in F$ and $\Delta = 4a^3 + 27b^2 \neq 0$. Additionally, the point at infinity, denoted by $\mathcal{O}$ is included as part of the curve.
The elliptic curves of the above form are called $\textbf{(short) Weierstrass Form}$ elliptic curves. The codes for elliptic curves are use this form and finite fields $F_p$ but not $F_{p^n}$. You can use large primes to construct an elliptic curve but may experience a loss of performance with very large prime numbers. Here the operations:
1: Elliptic Curve points and graph
2: Addition
3: Scalar multiplication
4: Scalar multiplication with double and add algorithm
5: Order of an element and its graph