Average-case algorithms for testing isomorphism of polynomials, algebras, and multilinear forms

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 $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$...

Full description

Saved in:

Bibliographic Details
Published in:	Groups, complexity, cryptology Vol. 14, Issue 1
Main Authors:	Grochow, Joshua A., Qiao, Youming, Tang, Gang
Format:	Journal Article
Language:	English
Published:	Episciences 01.01.2022
Subjects:	computer science - computational complexity computer science - data structures and algorithms
ISSN:	1869-6104, 1869-6104
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Description
Summary:	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 $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, 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.
ISSN:	1869-6104 1869-6104
DOI:	10.46298/jgcc.2022.14.1.9431