# bn [](https://crates.io/crates/bn)
This is a [pairing cryptography](https://en.wikipedia.org/wiki/Pairing-based_cryptography) library written in pure Rust. It makes use of the Barreto-Naehrig (BN) curve construction from [[BCGTV13]](http://eprint.iacr.org/2013/507) to provide two cyclic groups **G<sub>1</sub>** and **G<sub>2</sub>**, with an efficient bilinear pairing:
*e: G<sub>1</sub> × G<sub>2</sub> → G<sub>T</sub>*
**This code is still in early development and should not be used in production software.**
## Usage
Add the `bn` crate to your dependencies in `Cargo.toml`...
```toml
[dependencies]
bn = "0.1.*"
```
...and add an `extern crate` declaration to your crate root:
```rust
extern crate bn;
```
## API
* `Scalar` is an element of F<sub>r</sub>
* `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`.
### Examples
#### 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 *a*P<sub>1</sub> and Bob's public key *b*P<sub>2</sub>, Carol can compute the shared secret with her private key *c* by *e*(*a*P<sub>1</sub>, *b*P<sub>2</sub>)<sup>c</sup>.
(See `examples/joux.rs` for the full example.)
```rust
// Generate private keys
let alice_sk = Scalar::random(rng);
let bob_sk = Scalar::random(rng);
let carol_sk = Scalar::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) ^ &alice_sk;
let bob_ss = pairing(&carol_pk1, &alice_pk2) ^ &bob_sk;
let carol_ss = pairing(&alice_pk1, &bob_pk2) ^ &carol_sk;
assert!(alice_ss == bob_ss && bob_ss == carol_ss);
```
## License
Licensed under either of
* MIT license, ([LICENSE-MIT](LICENSE-MIT) or http://opensource.org/licenses/MIT)
* Apache License, Version 2.0 ([LICENSE-APACHE](LICENSE-APACHE) or http://www.apache.org/licenses/LICENSE-2.0)
at your option.
Copyright 2016 [Zcash Electric Coin Company](https://z.cash/). The Zcash Company promises to maintain the "bn" crate on crates.io under this MIT/Apache-2.0 dual license.
### Authors
* [Sean Bowe](https://github.com/ebfull)
### Contribution
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.