Linear-Time Verification of Rings and Fields

2026-08-07

AI Summary

This work addresses the problem of efficiently determining whether a finite algebraic structure defined by two $n \times n$ operation tables forms a ring or a field. We present the first deterministic $O(n^2)$-time algorithm that unifies the verification of both rings and fields through elementary algebraic analysis and combinatorial validation techniques, without invoking the Classification of Finite Simple Groups (CFSG). Our approach achieves, for the first time, linear-time deterministic recognition of ring structures and establishes the optimal time complexity for this problem in the deterministic setting. By circumventing reliance on advanced group-theoretic machinery, the method significantly reduces the theoretical prerequisites for such algebraic verification tasks.

Abstract

We consider the following problems: Given two $n \times n$ tables defining binary operations $+$ and $\cdot$ on a set $S$ of $n$ elements, decide whether $(S,+,\cdot)$ forms a ring or, respectively, a field. Recently, Dudek, Fischer, Gokaj, Jin, Künnemann, Mao, and Redzic (STOC 2026) obtained the following two (near-)optimal results: (1) A randomized $O(n^2\log(1/δ))$-time algorithm for verifying rings. (2) A deterministic $O(n^2)$-time algorithm for verifying fields. Their algorithms build on machinery of Evra, Gadot, Klein, and Komargodski (FOCS 2024), which relies on Classification of Finite Simple Groups (CFSG). In this work, we give a deterministic $O(n^2)$-time algorithm for ring verification, resolving the deterministic complexity of this problem. As a corollary, we also obtain a deterministic $O(n^2)$-time algorithm for field verification. Our algorithms are elementary and avoid CFSG machinery entirely.

Problem

• ring verification
• field verification
• algebraic structure
• deterministic algorithm
• computational complexity

Innovation

• deterministic algorithm
• ring verification
• field verification
• elementary method
• linear-time complexity