Paper 2024/1972

RoK, Paper, SISsors – Toolkit for Lattice-based Succinct Arguments

Abstract

Lattice-based succinct arguments allow to prove bounded-norm satisfiability of relations, such as $f(\overrightarrow{s})=\overrightarrow{t}modq$ and $\Vert \overrightarrow{s}\Vert \le \beta$, over specific cyclotomic rings ${\mathcal{O}}_{\mathcal{K}}$, with proof size polylogarithmic in the witness size. However, state-of-the-art protocols require either 1) a super-polynomial size modulus $$ due to a soundness gap in the security argument, or 2) a verifier which runs in time linear in the witness size. Furthermore, construction techniques often rely on specific choices of $$ which are not mutually compatible. In this work, we exhibit a diverse toolkit for constructing efficient lattice-based succinct arguments: (i) We identify new subtractive sets for general cyclotomic fields $$ and their maximal real subfields $$, which are useful as challenge sets, e.g. in arguments for exact norm bounds. (ii) We construct modular, verifier-succinct reductions of knowledge for the bounded-norm satisfiability of structured-linear/inner-product relations, without any soundness gap, under the vanishing SIS assumption, over any $$ which admits polynomial-size subtractive sets. (iii) We propose a framework to use twisted trace maps, i.e. maps of the form $$, to embed $$-inner-products as $$-inner-products for some structured subrings $$ whenever the conductor has a square-free odd part. (iv) We present a simple extension of our reductions of knowledge for proving the consistency between the coefficient embedding and the Chinese Remainder Transform (CRT) encoding of $$ over any cyclotomic field $$ with a smooth conductor, based on a succinct decomposition of the CRT map into automorphisms, and a new, simple succinct argument for proving automorphism relations. Combining all techniques, we obtain, for example, verifier-succinct arguments for proving that $$ satisfying $$ has binary coefficients, without soundness gap and with polynomial-size modulus $$.