#pairing #crypto #cryptography


Pairing cryptography with the Barreto-Naehrig curve

13 releases

0.5.11 Jan 23, 2022
0.5.10 Dec 8, 2021
0.5.9 Feb 18, 2021
0.5.7 Jul 24, 2020
0.4.4 Jun 14, 2020

#94 in Cryptography

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bn Crates.io Build status

This is a pairing cryptography library written in pure Rust. It makes use of the Barreto-Naehrig (BN) curve construction from [BCTV2015] to provide two cyclic groups G1 and G2, with an efficient bilinear pairing:

e: G1 × G2 → GT

Security warnings

This library, like other pairing cryptography libraries implementing this construction, is not resistant to side-channel attacks.


Add the bn crate to your dependencies in Cargo.toml...

bn = "0.4.2"

...and add an extern crate declaration to your crate root:

extern crate bn;


  • Fr is an element of Fr
  • G1 is a point on the BN curve E/Fq : y^2 = x^3 + b
  • G2 is a point on the twisted BN curve E'/Fq2 : y^2 = x^3 + b/xi
  • Gt is a group element (written multiplicatively) obtained with the pairing function over G1 and G2.


Joux's key agreement protocol

In a typical Diffie-Hellman key exchange, relying on ECDLP, a three-party key exchange requires two rounds. A single round protocol is possible through the use of a bilinear pairing: given Alice's public key aP1 and Bob's public key bP2, Carol can compute the shared secret with her private key c by e(aP1, bP2)c.

(See examples/joux.rs for the full example.)

// Generate private keys
let alice_sk = Fr::random(rng);
let bob_sk = Fr::random(rng);
let carol_sk = Fr::random(rng);

// Generate public keys in G1 and G2
let (alice_pk1, alice_pk2) = (G1::one() * alice_sk, G2::one() * alice_sk);
let (bob_pk1, bob_pk2) = (G1::one() * bob_sk, G2::one() * bob_sk);
let (carol_pk1, carol_pk2) = (G1::one() * carol_sk, G2::one() * carol_sk);

// Each party computes the shared secret
let alice_ss = pairing(bob_pk1, carol_pk2).pow(alice_sk);
let bob_ss = pairing(carol_pk1, alice_pk2).pow(bob_sk);
let carol_ss = pairing(alice_pk1, bob_pk2).pow(carol_sk);

assert!(alice_ss == bob_ss && bob_ss == carol_ss);


Licensed under either of

at your option.

Copyright 2016 Zcash Electric Coin Company. The Zcash Company promises to maintain the "bn" crate on crates.io under this MIT/Apache-2.0 dual license.



Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.


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