On the average-case complexity landscape for Tensor-Isomorphism-complete problems over finite fields

TL;DR

This paper develops average-case polynomial-time algorithms for Tensor Isomorphism problems over finite fields, revealing their complexity landscape and introducing spectral methods from random matrix theory.

Contribution

It introduces the first average-case polynomial algorithms for TI-complete problems over finite fields, utilizing spectral properties of random matrices.

Findings

Algorithms succeed on a 1/Θ(q) fraction of inputs for algebra isomorphism and matrix code conjugacy.

An algorithm succeeds on a 1/q^{Θ(1)} fraction of inputs for 4-tensor isomorphism.

The work connects tensor isomorphism complexity with spectral properties of random matrices.

Abstract

In Grochow and Qiao (SIAM J. Comput., 2021), the complexity class Tensor Isomorphism (TI) was introduced and isomorphism problems for groups, algebras, and polynomials were shown to be TI-complete. In this paper, we study average-case algorithms for several TI-complete problems over finite fields, including algebra isomorphism, matrix code conjugacy, and $4$-tensor isomorphism. Our main results are as follows. Over the finite field of order $q$, we devise (1) average-case polynomial-time algorithms for algebra isomorphism and matrix code conjugacy that succeed in a $1/\Theta(q)$ fraction of inputs and (2) an average-case polynomial-time algorithm for the $4$-tensor isomorphism that succeeds in a $1/q^{\Theta(1)}$ fraction of inputs. Prior to our work, algorithms for algebra isomorphism with rigorous average-case analyses ran in exponential time, albeit succeeding on a larger fraction of inputs (Li--Qiao, FOCS'17; Brooksbank--Li--Qiao--Wilson, ESA'20; Grochow--Qiao--Tang, STACS'21). These results reveal a finer landscape of the average-case complexities of TI-complete problems, providing guidance for cryptographic systems based on isomorphism problems. Our main technical contribution is to introduce the spectral properties of random matrices into algorithms for TI-complete problems. This leads to not only new algorithms but also new questions in random matrix theory over finite fields. To settle these questions, we need to extend both the generating function approach as in Neumann and Praeger (J. London Math. Soc., 1998) and the characteristic sum method of Gorodetsky and Rodgers (Trans. Amer. Math. Soc., 2021).