Fp2

An efficient, flexible and constant time Rust implementation of finite fields $\mathbb{F}_{p}$ and $\mathbb{F}_{p^2}$ where $p \equiv 3 \pmod 4$. Used currently for various Rust implementations of isogeny-based cryptographic protocols.

Motivation

These two macros have ended up being stuck inside every rust crypto thing I've written recently for isogeny-based crypto. The idea of this repository is to dedicate a central place to work on them to avoid there being many related but incompatible versions throughout my projects.

Usage

The base field can be defined using the macro define_fp_core using the modulus as input:

// Fp251: a finite field element GF(p) with p = 3 mod 4.
// Contents are opaque, all functions are constant-time.
fp2::define_fp_core!(
    typename = Fp251,
    modulus = [0xFFFFFFFFFFFFFFFFu64, 0xFFFFFFFFFFFFFFFFu64, 0xFFFFFFFFFFFFFFFFu64,
);

For the extension field, it can be generated directly from the modulus as with the base field:

fp2::define_fp2_from_modulus!(
    typename = Fp251Ext,
    base_typename = Fp251,
    modulus = [0xFFFFFFFFFFFFFFFFu64, 0xFFFFFFFFFFFFFFFFu64, 0xFFFFFFFFFFFFFFFFu64,
);

Or given a type for the base field the extension can be generated directly, which would allow users to supply their own GF(p) arithmetic to extend:

// Fp251Ext: a finite field element GF(p^2) with modulus x^2 + 1.
// Contents are opaque, all functions are constant-time.
fp2::define_fp2_from_type!(
    typename = Fp251Ext,
    base_field = Fp251,
);

The easiest way to generate macro parameters is to generate the above code snippets with the sage file scripts/gen_fp.sage.

Tests

Tests can be run:

cargo test --features test-utils

Benchmarks

Benchmarks can be run with:

RUSTFLAGS="-C target-cpu=native" cargo bench