On the average-case complexity landscape for Tensor-Isomorphism-complete problems over finite fields
arXiv:2604.00591v1 Announce Type: cross 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 t
View PDF HTML (experimental)
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).
Comments: 45 pages
Subjects:
Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS); Probability (math.PR)
Cite as: arXiv:2604.00591 [cs.CC]
(or arXiv:2604.00591v1 [cs.CC] for this version)
https://doi.org/10.48550/arXiv.2604.00591
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Yinan Li [view email] [v1] Wed, 1 Apr 2026 07:58:53 UTC (74 KB)
Sign in to highlight and annotate this article
Conversation starters
Daily AI Digest
Get the top 5 AI stories delivered to your inbox every morning.
More about
announcestudypaper
US Department of Labor, National Science Foundation announce collaborative efforts on AI workforce, TechAccess Initiative - U.S. Department of Labor (.gov)
US Department of Labor, National Science Foundation announce collaborative efforts on AI workforce, TechAccess Initiative U.S. Department of Labor (.gov)
Quantum computers might crack today's encryption far sooner than we thought
According to a study by engineers at Caltech and the UC Department of Physics, quantum computers do not need to be nearly as powerful as previously believed to crack the most advanced cryptographic technologies. The research claims that Shor's algorithm could break RSA public-key encryption using quantum computers with just... Read Entire Article
Knowledge Map
Connected Articles — Knowledge Graph
This article is connected to other articles through shared AI topics and tags.
More in Research Papers
Quantum computers might crack today's encryption far sooner than we thought
According to a study by engineers at Caltech and the UC Department of Physics, quantum computers do not need to be nearly as powerful as previously believed to crack the most advanced cryptographic technologies. The research claims that Shor's algorithm could break RSA public-key encryption using quantum computers with just... Read Entire Article
The Indirect Method for Generating Libraries of Optimal Periodic Trajectories and Its Application to Economical Bipedal Walking
arXiv:2410.09512v2 Announce Type: replace Abstract: Trajectory optimization is an essential tool for generating efficient, dynamically consistent gaits in legged locomotion. This paper explores the indirect method of trajectory optimization, emphasizing its application in creating optimal periodic gaits for legged systems and contrasting it with the more common direct method. While the direct method provides flexibility in implementation, it is limited by its need for an input space parameterization. In contrast, the indirect method improves accuracy by computing the control input from states and costates obtained along the optimal trajectory. In this work, we tackle the convergence challenges associated with indirect shooting methods by utilizing numerical continuation methods. This is pa
Implicit Primal-Dual Interior-Point Methods for Quadratic Programming
arXiv:2604.00364v1 Announce Type: cross Abstract: This paper introduces a new method for solving quadratic programs using primal-dual interior-point methods. Instead of handling complementarity as an explicit equation in the Karush-Kuhn-Tucker (KKT) conditions, we ensure that complementarity is implicitly satisfied by construction. This is achieved by introducing an auxiliary variable and relating it to the duals and slacks via a retraction map. Specifically, we prove that the softplus function has favorable numerical properties compared to the commonly used exponential map. The resulting KKT system is guaranteed to be spectrally bounded, thereby eliminating the most pressing limitation of primal-dual methods: ill-conditioning near the solution. These attributes facilitate the solution of
Discussion
Sign in to join the discussion
No comments yet — be the first to share your thoughts!