Abstract Polynomial commitments schemes are a powerful tool that enables one party to commit to a polynomial p of degree d , and prove that the committed function evaluates to a certain value z at a specified point u , i.e. $$p(u) = z$$ p ( u ) = z , without revealing any additional information about the polynomial. Recently, polynomial commitments have been extensively used as a cryptographic building block to transform polynomial interactive oracle proofs (PIOPs) into efficient succinct arguments. In this paper, we propose a lattice-based polynomial commitment that achieves succinct proof size and verification time in the degree d of the polynomial. Extractability of our scheme holds in the random oracle model under a natural ring version of the BASIS assumption introduced by Wee and Wu (EUROCRYPT 2023). Unlike recent constructions of polynomial commitments by Albrecht et al. (CRYPTO 2022), and by Wee and Wu, we do not require any expensive preprocessing steps, which makes our scheme particularly attractive as an ingredient of a PIOP compiler for succinct arguments. We further instantiate our polynomial commitment, together with the PIOP (EUROCRYPT 2020), to obtain a publicly-verifiable trusted-setup succinct argument for Rank-1 Constraint System (R1CS). Performance-wise, we achieve $$17$$ 17 MB proof size for $$2^{20}$$ 2 20 constraints, which is $$15$$ 15 X smaller than currently the only publicly-verifiable lattice-based SNARK proposed by Albrecht et al.

We propose a 2-round blind signature protocol based on the random oracle heuristic and the hardness of standard lattice problems (Ring/Module-SIS/LWE and NTRU) with a signature size of 20 KB. The protocol is round-optimal and has a transcript size that can be as small as 60 KB. This blind signature is around 4 times shorter than the most compact lattice-based scheme based on standard assumptions of del Pino and Katsumata (Crypto 2022) and around 2 times shorter than the scheme of Agrawal et al. (CCS 2022) based on their newly-proposed one-more-ISIS assumption. We also propose a "keyed-verification'' blind signature scheme in which the verifier and the signer need to share a secret key. This scheme has a smaller signature size of only 48 bytes, but further work is needed to explore the efficiency of its signature generation protocol.

We present a novel lattice-based zero-knowledge proof system for showing that (arbitrary-sized) committed integers satisfy additive and multiplicative relationships. The proof sizes of our schemes are between two to three orders of magnitude smaller than in the lattice proof system of Libert et al. (CRYPTO 2018) for the same relations. Because the proof sizes of our protocols grow linearly in the integer length, our proofs will eventually be longer than those produced by quantum-safe succinct proof systems for general circuits (e.g. Ligero, Aurora, etc.). But for relations between reasonably-sized integers (e.g. $512$-bit), our proofs still result in the smallest zero-knowledge proof system based on a quantum-safe assumption. Of equal importance, the run-time of our proof system is at least an order of magnitude faster than any other quantum-safe scheme.