cosmian_bls12_381

This crate provides an implementation of the BLS12-381 pairing-friendly elliptic curve construction.

Forked by Cosmian

July 2021

• In order to serialize/deserialize Gt elements and thanks to Aurore Guillevic, Cosmian added this implementation. Also, this eprint 2019-077 from Michael Scott confirmes the Gt deserialization verifications.
• Also, 2 issues on bls12_381 github are still pending:
  • https://github.com/zkcrypto/bls12_381/issues/10
  • https://github.com/zkcrypto/bls12_381/pull/12

Novembre 2021

• Hence no there is no plan to add Gt serialization, the crate is pusblished as cosmian_bls12_381 in 0.4.0 version.

  • A Github-PR will be made soon on bls12_381 original crate.

• But there is a remaining problem:

  Cosmian dit not update last version of bls12_381 (version 0.5) since it includes a dependency problem. Indeed, bls12_381-0.5 uses ff-0.10 which uses bitvec-0.22. The problem comes from the dependency funty-1.2.0 which did a breaking change (exposing another struct named BITS, confusing with same struct in bitvec crate). A comment on the funty crate has been made on July 2021 (followed by multiple same posts): https://github.com/myrrlyn/funty/issues/3#issuecomment-877833781

• This implementation has not been reviewed or audited. Use at your own risk.
• This implementation targets Rust 1.47 or later.
• This implementation does not require the Rust standard library.
• All operations are constant time unless explicitly noted.

Features

• groups (on by default): Enables APIs for performing group arithmetic with G1, G2, and GT.
• pairings (on by default): Enables some APIs for performing pairings.
• alloc (on by default): Enables APIs that require an allocator; these include pairing optimizations.
• nightly: Enables subtle/nightly which tries to prevent compiler optimizations that could jeopardize constant time operations. Requires the nightly Rust compiler.
• endo (on by default): Enables optimizations that leverage curve endomorphisms. Deprecated, will be removed in a future release.

Documentation

Curve Description

BLS12-381 is a pairing-friendly elliptic curve construction from the BLS family, with embedding degree 12. It is built over a 381-bit prime field GF(p) with...

• z = -0xd201000000010000
• p = (z - 1)²(z⁴ - z² + 1) / 3 + z
  • = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab
• q = z⁴ - z² + 1
  • = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001

... yielding two source groups G₁ and G₂, each of 255-bit prime order q, such that an efficiently computable non-degenerate bilinear pairing function e exists into a third target group G_T. Specifically, G₁ is the q-order subgroup of E(F_p) : y² = x³ + 4 and G₂ is the q-order subgroup of E'(F_p²) : y² = x³ + 4(u + 1) where the extension field F_p² is defined as F_p(u) / (u² + 1).

BLS12-381 is chosen so that z has small Hamming weight (to improve pairing performance) and also so that GF(q) has a large 2³² primitive root of unity for performing radix-2 fast Fourier transforms for efficient multi-point evaluation and interpolation. It is also chosen so that it exists in a particularly efficient and rigid subfamily of BLS12 curves.

Curve Security

Pairing-friendly elliptic curve constructions are (necessarily) less secure than conventional elliptic curves due to their small "embedding degree". Given a small enough embedding degree, the pairing function itself would allow for a break in DLP hardness if it projected into a weak target group, as weaknesses in this target group are immediately translated into weaknesses in the source group.

In order to achieve reasonable security without an unreasonably expensive pairing function, a careful choice of embedding degree, base field characteristic and prime subgroup order must be made. BLS12-381 uses an embedding degree of 12 to ensure fast pairing performance but a choice of a 381-bit base field characteristic to yield a 255-bit subgroup order (for protection against Pollard's rho algorithm) while reaching close to a 128-bit security level.

There are known optimizations of the Number Field Sieve algorithm which could be used to weaken DLP security in the target group by taking advantage of its structure, as it is a multiplicative subgroup of a low-degree extension field. However, these attacks require an (as of yet unknown) efficient algorithm for scanning a large space of polynomials. Even if the attack were practical it would only reduce security to roughly 117 to 120 bits. (This contrasts with 254-bit BN curves which usually have less than 100 bits of security in the same situation.)

Alternative Curves

Applications may wish to exchange pairing performance and/or G₂ performance by using BLS24 or KSS16 curves which conservatively target 128-bit security. In applications that need cycles of elliptic curves for e.g. arbitrary proof composition, MNT6/MNT4 curve cycles are known that target the 128-bit security level. In applications that only need fixed-depth proof composition, curves of this form have been constructed as part of Zexe.

Acknowledgements

Please see Cargo.toml for a list of primary authors of this codebase.

License

Licensed under either of

• Apache License, Version 2.0, (LICENSE-APACHE or http://www.apache.org/licenses/LICENSE-2.0)
• MIT license (LICENSE-MIT or http://opensource.org/licenses/MIT)

at your option.

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.