The dimensions of Schur squares of HRS codes
This work provides theoretical insights into the security of HRS codes against algebraic attacks, which is relevant for code-based cryptography.
The paper studies the Schur square dimensions of Hyperderivative Reed-Solomon (HRS) codes, providing lower and upper bounds and proving that under certain conditions the dimension reaches the upper bound. For specific parameters, the dimension matches that of random codes, suggesting HRS codes could resist Schur square distinguisher attacks in code-based cryptography.
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 $HRS_{t}(\{α_{1},\dots,α_{r}\},s)$ (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.