The automorphism groups of random linear codes

2026-07-26

AI Summary

This work investigates the automorphism group structure of random q-ary linear codes to provide theoretical underpinnings for the security of cryptographic schemes based on the Linear Code Equivalence (LCE) problem, such as the LESS signature scheme. Employing probabilistic methods, combinatorial analysis, linear algebra over finite fields, and asymptotic techniques, the paper establishes—for the first time—a rigorous proof that high-dimensional random linear codes possess a trivial automorphism group with high probability whenever $\min\{k, n-k\} \geq (2+\varepsilon)\log_q n$. This result furnishes a sufficient condition for the triviality of the automorphism group, thereby filling a significant theoretical gap at the intersection of coding theory and cryptography and strengthening the foundational security assumption underlying LCE-based cryptosystems.

Abstract

The study of automorphism groups of linear codes is a fundamental topic in coding theory. The matching codewords framework is currently a standard tool for analyzing the security of cryptographic schemes based on the hardness of the Linear Code Equivalence (LCE) problem, such as the LESS signature scheme. This framework often relies on the assumption that $q$-ary random codes have trivial automorphism groups. However, this assumption has not been formally proved in the literature. In this paper, we prove that with high probability, $k$-dimensional random codes $\mathcal{C} \subseteq \mathbb{F}_q^n$ have a trivial automorphism group as $n$ goes to infinity as long as $\min\{k, n-k\} \geq (2+\varepsilon)\log_q n$, for any $\varepsilon >0$.

Problem

• automorphism groups
• random linear codes
• Linear Code Equivalence
• coding theory
• trivial automorphism

Innovation

• automorphism groups
• random linear codes
• Linear Code Equivalence
• coding theory
• cryptographic security