Elliptic Curve Arithmetic Library

This repository contains a custom implementation of elliptic curve arithmetic in C. It supports basic operations on elliptic curves such as verifying points, point addition, point negation, and scalar multiplication.

Table of Contents

• Overview
• Prerequisites
• Compilation
• Usage
  • Example Program
• Features
• Notes

Overview

The library provides functionalities to work with elliptic curves over finite fields defined by equations of the form:

The implementation uses GMP for arbitrary-precision arithmetic to ensure compatibility with large numbers.

Prerequisites

• GCC or any C compiler
• GMP library

Installing GMP

Ubuntu/Debian:

sudo apt update
sudo apt install libgmp-dev

macOS (using Homebrew):

brew install gmp

Windows:

Install GMP for Windows and ensure the headers and libraries are correctly linked during compilation.

Compilation

Use the following command to compile the example program:

gcc example.c EC.c -o example_program -lgmp

Usage

Example Program

An example usage of the library is provided in the example.c file. The program demonstrates:

1. Initializing curve parameters.
2. Checking if a point is on the curve.
3. Point negation.
4. Point addition (doubling).
5. Scalar multiplication.

Run the compiled example:

./example_program

Expected output:

Point P(3, 6) is on the curve.
Negation of P: (3, 91)
Point Doubling: P + P = (80, 10)
Scalar Multiplication: 3P = (80, 87)

Features

• Point Verification: Check if a given point lies on the elliptic curve.
• Point Negation: Compute the negation of a point.
• Point Addition: Add two points on the curve.
• Scalar Multiplication: Multiply a point by a scalar.

Notes

• Ensure all GMP dependencies are installed before compiling.
• This library uses affine coordinates for simplicity.
• Example values are predefined for demonstration purposes but can be replaced with custom inputs.