Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms

Authors Joshua A. Grochow , Youming Qiao , Gang Tang

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Joshua A. Grochow
  • Departments of Computer Science and Mathematics, University of Colorado Boulder, Boulder, CO, USA
Youming Qiao
  • Center for Quantum Software and Information, University of Technology Sydney, Ultimo, NSW 2007, Australia
Gang Tang
  • Center for Quantum Software and Information, University of Technology Sydney, Ultimo, NSW 2007, Australia


We thank the anonymous reviewers for their careful reading and helpful suggestions.

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Joshua A. Grochow, Youming Qiao, and Gang Tang. Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 38:1-38:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms f, g ∈ 𝔽_q[x_1, … , x_n], and decides whether f and g are isomorphic in time q^O(n) for most f. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow & Qiao, ITCS 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Algebraic algorithms
  • Computing methodologies → Combinatorial algorithms
  • polynomial isomorphism
  • trilinear form equivalence
  • algebra isomorphism
  • average-case algorithms
  • tensor isomorphism complete
  • symmetric and alternating bilinear maps


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