Elliptic Curve Implementation

A C# implementation of elliptic curve arithmetic operations over finite fields, primarily focused on the mathematical fundamentals that underpin elliptic curve cryptography (ECC).

Overview

This project provides a generic implementation of elliptic curves using the Weierstrass form: y² = x³ + ax + b over a finite field of prime characteristic p. It includes core operations like:

• Point addition
• Point doubling
• Scalar multiplication
• Point validation
• Point negation

Usage Example

// Create an elliptic curve y² = x³ + 2x + 3 (mod 17)
var curve = new EllipticCurve(
    a: 2,
    b: 3, 
    prime: 17,
    basePoint: new Point<EllipticCurve>(3, 6),
    order: 19
);

// Verify if a point lies on the curve
var point = new Point<EllipticCurve>(3, 6);
bool isValid = curve.IsOnCurve(point); // true

// Perform point addition
var sum = EllipticCurve.Add(curve, point1, point2);

// Scalar multiplication 
var result = EllipticCurve.ScalarMultiply(curve, point, k);

Features

• Generic Implementation: Supports different curve types through the IEllipticCurve interface
• Core Operations:
  • Point addition and doubling
  • Scalar multiplication using double-and-add algorithm
  • Point validation
  • Point negation
• Mathematical Utilities:
  • Modular arithmetic operations
  • Extended Euclidean algorithm for modular inverse
• Test Coverage: Comprehensive test suite verifying curve operations

Mathematical Background

The implementation uses the standard formulas for elliptic curve arithmetic:

• Point Addition: When P₁ = (x₁, y₁) ≠ ±P₂ = (x₂, y₂):

  • λ = (y₂ - y₁)/(x₂ - x₁)
  • x₃ = λ² - x₁ - x₂
  • y₃ = λ(x₁ - x₃) - y₁

• Point Doubling: When P = (x, y) and P is doubled:

  • λ = (3x² + a)/(2y)
  • x₃ = λ² - 2x
  • y₃ = λ(x - x₃) - y

All operations are performed modulo the prime characteristic p.

Security Note

This implementation is primarily for educational purposes and demonstrates the mathematical concepts behind ECC. For production cryptographic purposes, use established cryptographic libraries.

License

MIT