## Cryptology ePrint Archive: Report 2021/1043

Brakedown: Linear-time and post-quantum SNARKs for R1CS

Alexander Golovnev and Jonathan Lee and Srinath Setty and Justin Thaler and Riad S. Wahby

Abstract: This paper introduces Brakedown, the first built system that provides linear-time SNARKs for NP, meaning the prover incurs $O(N)$ finite field operations to prove the satisfiability of an $N$-sized R1CS instance. Brakedown’s prover is faster, both concretely and asymptotically, than prior SNARK implementations. Brakedown does not require a trusted setup and is plausibly post-quantum secure. Furthermore, it is compatible with arbitrary finite fields of sufficient size; this property is new amongst implemented arguments with sublinear proof sizes.

To design Brakedown, we observe that recent work of Bootle, Chiesa, and Groth (BCG, TCC 2020) provides a polynomial commitment scheme that, when combined with the linear-time interactive proof system of Spartan (CRYPTO 2020), yields linear-time IOPs and SNARKs for R1CS (a similar theoretical result was previously established by BCG, but our approach is conceptually simpler, and crucial for achieving high-speed SNARKs). A core ingredient in the polynomial commitment scheme that we distill from BCG is a linear-time encodable code. Existing constructions of such codes are believed to be impractical. Nonetheless, we design and engineer a new one that is practical in our context.

We also implement a variant of Brakedown that uses Reed-Solomon codes instead of our linear-time encodable codes; we refer to this variant as Shockwave. Shockwave is not a linear-time SNARK, but it provides shorter proofs and lower verification times than Brakedown (it also provides a faster prover than prior plausibly post-quantum SNARKs).

As a modest additional contribution, we observe that one can render the aforementioned SNARK zero knowledge and reduce the proof size and verifier time from $O(\sqrt{N})$ to $polylog(N)$---while maintaining a linear-time prover---by outsourcing the verifier’s work via one layer of proof composition with an existing zkSNARK as the "outer" proof system.

Category / Keywords: public-key cryptography / linear-time SNARKs, succinct arguments, proof systems

Date: received 11 Aug 2021, last revised 12 Aug 2021

Contact author: srinath at microsoft com, Justin Thaler at georgetown edu, rsw at cs stanford edu

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2021/1043

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