The dimensions of Schur squares of HRS codes

TL;DR

This paper studies the dimensions of Schur squares of Hyperderivative Reed-Solomon (HRS) codes, providing bounds and exact values under certain conditions, with implications for cryptography.

Contribution

It derives bounds and exact dimensions for Schur squares of HRS codes, advancing understanding of their algebraic structure and cryptographic resistance.

Findings

Lower and upper bounds for Schur square dimensions of HRS codes are established.

Exact dimension formulas are proved for specific parameter ranges.

HRS codes can resist Schur square distinguishers in cryptographic applications.

Abstract

The Schur square of linear codes over a finite field has emerged as a fundamental operation in both classical and quantum coding theory. In this paper, we investigate the Schur square problem of Hyperderivative Reed-Solomon (HRS) codes. By solving certain special determinants, we first give a lower bound and an upper bound for the dimensions of Schur squares of HRS codes, and then prove that when $p\geq t\geq 2s$ and $t\leq \frac{r+2s-1}{2}$, the dimension of the Schur square of the HRS code $HRS_{t}(\{\alpha_{1},\dots,\alpha_{r}\},s)$ (with length $rs$ and dimension $t$) reaches the upper bound $(2t-2s+1)s$. In particular, when $p \ge t=2s$ and $r\geq t+1$, the dimension of the Schur square equals $\frac{t(t+1)}{2}$ which is the dimension of the Schur squares of random codes with high probability. As an application in code-based cryptography, HRS codes with specific parameter settings might resist the attack of Schur square distinguisher.