International Association
for Cryptologic Research

We introduce an efficient SNARK for towers of binary fields. Adapting Brakedown (CRYPTO '23), we construct a multilinear polynomial commitment scheme suitable for polynomials over tiny fields, including that with just two elements. Our commitment scheme, unlike those of previous works, treats small-field polynomials with no embedding overhead. We further introduce binary-field adaptations of HyperPlonk (EUROCRYPT '23)'s product and permutation checks and of Lasso (EUROCRYPT '24)'s lookup. Our binary PLONKish variant captures standard hash functions—like Keccak-256 and Grøstl—extremely efficiently. With recourse to thorough performance benchmarks, we argue that our scheme can efficiently generate precisely those Keccak-256-proofs which critically underlie modern efforts to scale Ethereum.

<p>A fundamental result dating to Ligero (Des. Codes Cryptogr. '23) establishes that each fixed linear block code exhibits proximity gaps with respect to the collection of affine subspaces, in the sense that each given subspace either resides entirely close to the code, or else contains only a small portion which resides close to the code. In particular, any given subspace's failure to reside entirely close to the code is necessarily witnessed, with high probability, by a uniformly randomly sampled element of that subspace. We investigate a variant of this phenomenon in which the witness is not sampled uniformly from the subspace, but rather from a much smaller subset of it. We show that a logarithmic number of random field elements (in the dimension of the subspace) suffice to effect an analogous proximity test, with moreover only a logarithmic (multiplicative) loss in the possible prevalence of false witnesses. We discuss applications to recent noninteractive proofs based on linear codes, including Brakedown (CRYPTO '23). </p>