Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice

Authors Peter A. Brooksbank, Yinan Li, Youming Qiao, James B. Wilson

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Peter A. Brooksbank
  • Department of Mathematics, Bucknell University, Lewisburg, PA, USA
Yinan Li
  • CWI and QuSoft, Amsterdam, The Netherlands
Youming Qiao
  • Centre for Quantum Software and Information, University of Technology Sydney, Ultimo, Australia
James B. Wilson
  • Department of Mathematics, Colorado State University, Fort Collins, CO, USA


The authors would like to acknowledge László Babai and Xiaorui Sun for discussions, and Joshua A. Grochow for discussions and comments on improving the presentation. The authors acknowledge the Hausdorff Institute for Mathematics, the University of Auckland, the Santa Fe Institute, and the TACA workshop at the University of Colorado, Boulder and Colorado State University, where this research was carried out by (subsets of) the authors.

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Peter A. Brooksbank, Yinan Li, Youming Qiao, and James B. Wilson. Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Motivated by testing isomorphism of p-groups, we study the alternating matrix space isometry problem (AltMatSpIso), which asks to decide whether two m-dimensional subspaces of n×n alternating (skew-symmetric if the field is not of characteristic 2) matrices are the same up to a change of basis. Over a finite field 𝔽_p with some prime p≠2, solving AltMatSpIso in time p^O(n+m) is equivalent to testing isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. The latter problem has long been considered a bottleneck case for the group isomorphism problem. Recently, Li and Qiao presented an average-case algorithm for AltMatSpIso in time p^O(n) when n and m are linearly related (FOCS '17). In this paper, we present an average-case algorithm for AltMatSpIso in time p^O(n+m). Besides removing the restriction on the relation between n and m, our algorithm is considerably simpler, and the average-case analysis is stronger. We then implement our algorithm, with suitable modifications, in Magma. Our experiments indicate that it improves significantly over default (brute-force) algorithms for this problem.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Algebraic algorithms
  • Computing methodologies → Combinatorial algorithms
  • Mathematics of computing → Graph algorithms
  • Alternating Matrix Spaces
  • Average-case Algorithm
  • p-groups of Class 2nd Exponent p
  • Magma


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