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 $1/Θ(q)$ fraction of inputs and (2) an average-case polynomial-time algorithm for the $4$-tensor isomorphism that succeeds in a $1/q^{Θ(1)}$ 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).

翻译：在 Grochow 与 Qiao (SIAM J. Comput., 2021) 的工作中，他们引入了复杂度类张量同构 (Tensor Isomorphism, TI)，并证明了群、代数与多项式的同构问题均为 TI-完全的。本文研究有限域上若干 TI-完全问题的平均情形算法，包括代数同构、矩阵码共轭以及 $4$-张量同构。主要结果如下：在 $q$ 阶有限域上，我们设计了 (1) 针对代数同构与矩阵码共轭的平均情形多项式时间算法，该算法在 $1/\Theta(q)$ 比例的输入上成功；以及 (2) 针对 $4$-张量同构的平均情形多项式时间算法，该算法在 $1/q^{\Theta(1)}$ 比例的输入上成功。在我们之前，具有严格平均情形分析的代数同构算法虽能在更大比例输入上成功，但其运行时间为指数级（Li--Qiao, FOCS'17; Brooksbank--Li--Qiao--Wilson, ESA'20; Grochow--Qiao--Tang, STACS'21）。这些结果揭示了 TI-完全问题平均情形复杂度的更精细图景，为基于同构问题的密码系统提供了指导。我们的主要技术贡献是将随机矩阵的谱性质引入 TI-完全问题的算法中。这不仅催生了新算法，还引出了有限域上随机矩阵理论的新问题。为解决这些问题，我们需同时扩展 Neumann 与 Praeger (J. London Math. Soc., 1998) 的生成函数方法以及 Gorodetsky 与 Rodgers (Trans. Amer. Math. Soc., 2021) 的特征和技巧。