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Proving Algebraic Independence in Zero-Knowledge

Authors Michael A. Forbes, Andrei Staicu

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LIPIcs.ICALP.2026.93.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Complexity classes
Keywords
  • Algebraic Dependence
  • Non-Interactive Statistical Zero-Knowledge

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Abstract

A set of multivariate polynomials is algebraically independent if they exhibit no non-trivial algebraic relations, and this notion is fundamental in algebra. When these polynomials are given as algebraic circuits, deciding algebraic independence has several applications in algebraic complexity theory. Over fields of zero (or exponentially large) characteristic, this problem is known to have an efficient randomized algorithm. Over finite fields of small characteristic, a sequence of works has culminated in showing that algebraic independence admits Arthur-Merlin proofs, in particular giving the complexity bound of AM∩coAM ([Guo et al., 2019]).
We improve the complexity of deciding algebraic independence over finite fields by showing that it admits zero-knowledge proofs, in particular giving the upper bound of NISZK ⊆ AM∩coAM, the class of problems admitting non-interactive statistical zero-knowledge proofs. This is achieved by arguing that algebraically independent polynomials yield maps whose output distribution has high-entropy, while algebraically dependent polynomials yield maps with low-entropy. We can then reduce to the question of approximating entropy, which is a known NISZK-complete problem.
We also more generally show that transcendence degree, which quantifies the independence of a set of possibly dependent polynomials, can be computed in NISZK.

Cite As Get BibTex

Michael A. Forbes and Andrei Staicu. Proving Algebraic Independence in Zero-Knowledge. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 93:1-93:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.93

Author Details

Michael A. Forbes
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Andrei Staicu
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA

Funding

  • Forbes, Michael A.: Supported by NSF CAREER award 2047310.
  • Staicu, Andrei: Supported by NSF CAREER award 2047310.

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