Our paper – together with Valerio Cini, Russell W. F. Lai, Giulio Malavolta and Sri Aravinda Krishnan Thyagarajan – titled Lattice-Based SNARKs: Publicly Veriﬁable, Preprocessing, and Recursively Composable will be presented at CRYPTO’22. A pre-print is available and here’s the abstract:
A succinct non-interactive argument of knowledge (SNARK) allows a prover to produce a short proof that certifies the veracity of a certain NP-statement. In the last decade, a large body of work has studied candidate constructions that are secure against quantum attackers. Unfortunately, no known candidate matches the efficiency and desirable features of (pre-quantum) constructions based on bilinear pairings.
In this work, we make progress on this question. We propose the first lattice-based SNARK that simultaneously satisfies many desirable properties: It (i) is tentatively post-quantum secure, (ii) is publicly-verifiable, (iii) has a logarithmic-time verifier and (iv) has a purely algebraic structure making it amenable to efficient recursive composition. Our construction stems from a general technical toolkit that we develop to translate pairing-based schemes to lattice-based ones. At the heart of our SNARK is a new lattice-based vector commitment (VC) scheme supporting openings to constant-degree multivariate polynomial maps, which is a candidate solution for the open problem of constructing VC schemes with openings to beyond linear functions. However, the security of our constructions is based on a new family of lattice-based computational assumptions which naturally generalises the standard Short Integer Solution (SIS) assumption.
In this post, I want to give you a sense of our new family of assumptions, the k-M-ISIS family of assumptions, and its variants. Meanwhile, Russell has written a post focusing on building the SNARK and Aravind has written about the nice things that we can do with our lattice-based SNARKs.