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Interactive Proofs for Batch Polynomial Evaluation

Authors Gal Arnon , Alessandro Chiesa , Giacomo Fenzi , Eylon Yogev

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ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Theory of computation → Cryptographic protocols
Keywords
  • interactive proofs
  • polynomial evaluation

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Abstract

Polynomials are a fundamental mathematical object underlying virtually all of theoretical computer science. In proof systems, a common task for the verifier is to evaluate a polynomial of degree d at m distinct points. The best known algorithm for this problem performs O((m + d) ⋅ log²(m + d)) field operations.
We present a concretely efficient MA protocol for this problem in which the verifier runs in linear time: the prover sends a single message consisting of d - 1 field elements, and the verifier performs only O(m + d) field operations. We further extend our protocol to handle the more general setting of evaluating multiple polynomials at multiple points, and for this problem, we construct an AMA protocol.
Our protocols improve the verifier time in several interactive proofs. Most notable are the sumcheck protocol over a large summation domain and protocols that rely on polynomial quotienting. In particular, by a straightforward application of our results, we reduce the verifier’s runtime in the STIR protocol (CRYPTO 2024) to match that of WHIR (EUROCRYPT 2025), despite WHIR being highly optimized for verification time.
As an additional application, we show that any univariate polynomial commitment scheme (PCS) can be transformed, in a black-box manner, into a new scheme that efficiently supports batch openings at multiple points. In particular, opening m points incurs only a constant overhead compared to opening a single point.

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Gal Arnon, Alessandro Chiesa, Giacomo Fenzi, and Eylon Yogev. Interactive Proofs for Batch Polynomial Evaluation. In 7th Conference on Information-Theoretic Cryptography (ITC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 385, pp. 3:1-3:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITC.2026.3

Author Details

Gal Arnon
  • Bocconi University, Milan, Italy
Alessandro Chiesa
  • EPFL, Lausanne, Switzerland
Giacomo Fenzi
  • EPFL, Lausanne, Switzerland
Eylon Yogev
  • Bar-Ilan University, Ramat Gan, Israel

Funding

Alessandro Chiesa and Giacomo Fenzi are supported in part by the Ethereum Foundation.
  • Arnon, Gal: Gal Arnon is supported by the European Research Council (ERC) under the EU’s Horizon 2020 research and innovation programme (Grant agreement No. 101019547), and Stellar Foundation Grant.
  • Yogev, Eylon: Eylon Yogev is supported by the Israel Science Foundation (Grant No. 2302/22), European Research Union (ERC, CRYPTOPROOF, 101164375).

Acknowledgements

This work was done in part while Gal Arnon and Giacomo Fenzi were visiting the Simons Institute for the Theory of Computing, and Giacomo Fenzi was visiting Succinct.

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