eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-26
26:1
26:15
10.4230/LIPIcs.ESA.2020.26
article
Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice
Brooksbank, Peter A.
1
Li, Yinan
2
Qiao, Youming
3
Wilson, James B.
4
Department of Mathematics, Bucknell University, Lewisburg, PA, USA
CWI and QuSoft, Amsterdam, The Netherlands
Centre for Quantum Software and Information, University of Technology Sydney, Ultimo, Australia
Department of Mathematics, Colorado State University, Fort Collins, CO, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol173-esa2020/LIPIcs.ESA.2020.26/LIPIcs.ESA.2020.26.pdf
Alternating Matrix Spaces
Average-case Algorithm
p-groups of Class 2nd Exponent p
Magma