bn254-crypto-rust

This repository contains a from-first-principles, educational implementation of the BN254 elliptic curve and its finite-field arithmetic.
The goal is to build the entire curve step-by-step, the same way real zkSNARK libraries (arkworks, blst, gnark, halo2) do internally.

This project is not meant for production use — it is a learning journey to understand how pairing-friendly curves work under the hood.

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✅ What’s Implemented So Far

1. Base Field 𝔽ₚ (fp.rs)

We implemented the prime field Fp = ℤ/pℤ where:

p = 21888242871839275222246405745257275088548364400416034343698204186575808495617

This is the BN254 scalar field and base field used in Ethereum zkSNARKs.

Features implemented:

• Modular addition
• Modular subtraction
• Modular multiplication
• Modular negation
• Modular inverse (Extended Euclidean Algorithm)
• Modular exponentiation
• Property tests + random tests
• Operator overloading (+, -, *, -x)

The field ensures:

• All results are automatically reduced mod p
• No intermediate overflow leaks outside of the field

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2. G1 Group (g1.rs)

This is the elliptic curve group on BN254 defined over 𝔽ₚ:

E: y² = x³ + 3

Coordinate system:

We use Jacobian coordinates (X:Y:Z):

x = X / Z² y = Y / Z³

Why Jacobian?

• No field inversions during addition or doubling
• Inversions are expensive (~100× slower than multiplications)
• Jacobian avoids them → faster operations

Implemented operations:

✅ Point at infinity
✅ Affine conversion
✅ Curve validity check
✅ Point doubling
✅ Point addition (complete, handles all edge cases)
✅ Scalar multiplication (double-and-add)

Tests:

• Infinity behavior
• Affine conversion
• Curve equation validity
• Double = P + P
• Add is commutative
• Scalar multiplication