#### Title

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

#### Publication Date

8-28-2020

#### Description

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.

#### Journal

Lipics Leibniz internatioLipics Leibniz International Proceedings In Informaticsnal proceedings in Informatics

#### Volume

173

#### Department

Mathematics

#### DOI

10.4230/LIPIcs.ESA.2020.26

#### Recommended Citation

Brooksbank, Peter A.; Li, Yinan; Qiao, Youming; and Wilson, James B.. "Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice." (2020) .