#elliptic-curve #finite-fields #math

no-std ark-bw6-761

The BW6-761 pairing-friendly elliptic curve

6 releases (3 breaking)

new 0.5.0-alpha.0 Jun 20, 2024
0.4.0 Jan 17, 2023
0.4.0-alpha.2 Dec 28, 2022
0.4.0-alpha.1 Nov 29, 2022
0.2.0 Mar 25, 2021

#1446 in Cryptography

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131,482 downloads per month
Used in 10 crates (5 directly)


14K SLoC


The arkworks ecosystem consist of Rust libraries for designing and working with zero knowledge succinct non-interactive arguments (zkSNARKs). This repository contains efficient implementations of the key algebraic components underlying zkSNARKs: finite fields, elliptic curves, and polynomials.

This library is released under the MIT License and the Apache v2 License (see License).

WARNING: This is an academic proof-of-concept prototype, and in particular has not received careful code review. This implementation is NOT ready for production use.

Directory structure

This repository contains several Rust crates:

  • ark-ff: Generic abstractions for, and implementations of various kinds of finite fields
  • ark-ec: Generic abstractions for prime-order groups, and implementations of various kinds of (pairing-friendly and standard) elliptic curves
  • ark-poly: Interfaces for univariate, multivariate, and multilinear polynomials, and FFTs over finite fields
  • ark-serialize: Efficient interfaces for serialization and point compression for finite fields and elliptic curves

In addition, the curves repository contains concrete implementations of popular elliptic curves; see here for details.

Build guide

The library compiles on the stable toolchain of the Rust compiler (v 1.51+). To install the latest version of Rust, first install rustup by following the instructions here, or via your platform's package manager. Once rustup is installed, install the Rust toolchain by invoking:

rustup install stable

After that, use cargo, the standard Rust build tool, to build the libraries:

git clone https://github.com/arkworks-rs/algebra.git
cd algebra
cargo build --release


This library comes with comprehensive unit and integration tests for each of the provided crates. Run the tests with:

cargo test --all


To run the benchmarks, install the nightly Rust toolchain, via rustup install nightly, and then run the following command:

cargo +nightly bench

Assembly backend for field arithmetic

The ark-ff crate contains (off-by-default) optimized assembly implementations of field arithmetic that rely on the adcxq, adoxq and mulxq instructions. These are available on most x86_64 platforms (Broadwell onwards for Intel and Ryzen onwards for AMD). Using this backend can lead to a 30-70% speedup in finite field and elliptic curve arithmetic. To build with this backend enabled, run the following command:

RUSTFLAGS="-C target-feature=+bmi2,+adx" cargo [test/build/bench] --features asm

To enable this in the Cargo.toml of your own projects, enable the asm feature flag:

ark-ff = { version = "0.4", features = [ "asm" ] }

Note that because inline assembly support in Rust is currently unstable, using this backend requires using the Nightly compiler at the moment.


The crates in this repository are licensed under either of the following licenses, at your discretion.

Unless you explicitly state otherwise, any contribution submitted for inclusion in this library by you shall be dual licensed as above (as defined in the Apache v2 License), without any additional terms or conditions.


This work was supported by: a Google Faculty Award; the National Science Foundation; the UC Berkeley Center for Long-Term Cybersecurity; and donations from the Ethereum Foundation, the Interchain Foundation, and Qtum.

An earlier version of this library was developed as part of the paper "ZEXE: Enabling Decentralized Private Computation".


This library implements the BW6_761 curve generated in [EG20]. The name denotes that it is a curve generated using the Brezing--Weng method, and that its embedding degree is 6. The main feature of this curve is that the scalar field equals the base field of the BLS12_377 curve.

Curve information:

  • Base field: q = 6891450384315732539396789682275657542479668912536150109513790160209623422243491736087683183289411687640864567753786613451161759120554247759349511699125301598951605099378508850372543631423596795951899700429969112842764913119068299
  • Scalar field: r = 258664426012969094010652733694893533536393512754914660539884262666720468348340822774968888139573360124440321458177
  • valuation(q - 1, 2) = 1
  • valuation(r - 1, 2) = 46

G1 curve equation: y^2 = x^3 + ax + b, where

  • a = 0,
  • b = -1,

G2 curve equation: y^2 = x^3 + Ax + B

  • A = 0
  • B = 4


~76K SLoC