New Preprint Proposes Two Efficient Univariate Sumcheck Techniques

Global: New Approaches to Univariate Sumcheck Highlighted in Recent Preprint

Background

A recent preprint authored by Malcom Mohamed outlines two candidate approaches for performing univariate sumcheck over roots of unity. The paper, initially submitted on May 1, 2025 and revised through January 29, 2026, proposes methods that can be integrated with existing multivariate sumcheck protocols and with the Gemini framework, respectively. Both techniques aim to reduce the number of rounds from m to log(m) while keeping prover time asymptotically linear.

First Approach: Multilinear Evaluation Protocol

The first approach adopts a multilinear evaluation protocol. By treating the univariate sumcheck as a special case of multilinear evaluation, the method can be layered onto the standard multivariate sumcheck, offering a seamless extension for protocols that already rely on multilinear checks.

Second Approach: Direct Reduction to Multilinear Evaluation

The second approach provides a direct reduction from univariate sumcheck to multilinear evaluation. This reduction enables the use of the Gemini protocol—introduced by Bootle et al. at Eurocrypt 2022—thereby leveraging its efficiency gains for univariate contexts.

Round Reduction Mechanism

Both proposals feature an exponential round reduction mechanism. Specifically, they transform an original m‑round interaction into approximately log₂(m) rounds, which can significantly lower communication overhead in large‑scale proof systems without sacrificing the linear prover runtime.

Implications for Cryptographic Proof Systems

The paper situates these contributions within the broader field of cryptographic proof systems, where sumcheck protocols serve as foundational components for zero‑knowledge proofs and verifiable computation. By improving round complexity, the approaches may facilitate more practical deployments of such systems.

Future Directions

While the preprint focuses on theoretical performance, the author notes that implementation details and empirical evaluation remain open areas for future work. Potential extensions include adapting the protocols to other algebraic settings or integrating them with emerging proof frameworks.

Access Information

The work is cataloged under the Computer Science – Cryptography and Security category on arXiv (cs.CR) and can be accessed via DOI 10.48550/arXiv.2505.00554.

This report is based on information from arXiv, licensed under Academic Preprint / Open Access. Based on the abstract of the research paper. Full text available via ArXiv.