Protocols for Univariate Sumcheck

TL;DR

This paper introduces three new approaches for univariate sumcheck protocols over roots of unity, enhancing efficiency and flexibility by combining with existing protocols and supporting round reductions.

Contribution

It presents three novel methods for univariate sumcheck, including multilinear evaluation and reductions to multivariate evaluation, compatible with Gemini and supporting round reductions.

Findings

Protocols can be combined with standard multivariate sumcheck.

Reductions enable efficient univariate sumcheck with linear prover time.

Supports round reductions from m to log(m) or O(√m).

Abstract

Three candidate approaches for univariate sumcheck over roots of unity are presented. The first takes the form of a multilinear evaluation protocol, which can be combined with the standard multivariate sumcheck protocol. The other two are reductions from univariate domain identity and univariate sumcheck to multivariate evaluation, respectively, and each can be combined with Gemini (Bootle et al., Eurocrypt 2022). Optionally, natural round reductions from $m$ to $\log(m)$ or $O(\sqrt{m})$ are supported, while retaining linear prover time.