Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on Univariate Sumcheck

Malcom Mohamed

TL;DR

The paper addresses efficient verification of multilinear extension evaluations by coupling univariate and multivariate sumcheck techniques through a novel mlex-to-unex adaptor. It introduces a square/non-square folding strategy with generalized CRT interpolation, enabling a univariate proof system that preserves completeness and achieves tight soundness with Schwartz–Zippel checks, while maintaining linear prover time and logarithmic verifier effort. A key contribution is natural round reductions, allowing early stopping of the multivariate phase and compatibility with single-round verifiers like Aurora, reducing communication and domain overhead. The work offers practical improvements over kernel interpolation and quotient-based approaches, delivering asymptotically optimal prover time $O(2^m)$ and a streamlined, round-efficient adaptor for unex-based sumchecks. This provides a flexible, scalable framework for integrating univariate and multivariate sumchecks in SNARK-like proofs and related cryptographic protocols.

Abstract

This note describes a univariate polynomial interactive oracle proof for multilinear extension evaluation. Unlike prior protocols, (1) the verifier here is given a univariate extension oracle for the same vector of which the multilinear extension is getting evaluated and (2) the prover only has linear complexity. For these reasons, the protocol is well-suited for combining multivariate and univariate sumcheck techniques.

Notes on Univariate Sumcheck

TL;DR

The paper addresses efficient verification of multilinear extension evaluations by coupling univariate and multivariate sumcheck techniques through a novel mlex-to-unex adaptor. It introduces a square/non-square folding strategy with generalized CRT interpolation, enabling a univariate proof system that preserves completeness and achieves tight soundness with Schwartz–Zippel checks, while maintaining linear prover time and logarithmic verifier effort. A key contribution is natural round reductions, allowing early stopping of the multivariate phase and compatibility with single-round verifiers like Aurora, reducing communication and domain overhead. The work offers practical improvements over kernel interpolation and quotient-based approaches, delivering asymptotically optimal prover time and a streamlined, round-efficient adaptor for unex-based sumchecks. This provides a flexible, scalable framework for integrating univariate and multivariate sumchecks in SNARK-like proofs and related cryptographic protocols.

Abstract

This note describes a univariate polynomial interactive oracle proof for multilinear extension evaluation. Unlike prior protocols, (1) the verifier here is given a univariate extension oracle for the same vector of which the multilinear extension is getting evaluated and (2) the prover only has linear complexity. For these reasons, the protocol is well-suited for combining multivariate and univariate sumcheck techniques.
Paper Structure (17 sections, 1 theorem, 12 equations, 1 table)

This paper contains 17 sections, 1 theorem, 12 equations, 1 table.

Table of Contents

  1. Introduction
  2. Reducing $\mathsf{mlex}$ to $\mathsf{unex}$.
  3. Round reductions.
  4. Preliminaries
  5. Notations
  6. Notations
  7. Polynomial Interactive Oracle Proofs and Reductions
  8. The multivariate sumcheck protocol.
  9. Evaluating $\mathsf{mlex}$ via $\mathsf{unex}$
  10. Square Evaluation Folding
  11. Completeness.
  12. Soundness.
  13. Prover time.
  14. Comparisons
  15. Round-Reduced Sumcheck
  16. ...and 2 more sections

Key Result

proposition 1

proto:mlex2unex is a PIOP for the relation with perfect completeness and the following other properties:

Theorems & Definitions (4)

  • proposition 1
  • proof
  • remark 1
  • remark 2