Using Reed-Solomon codes in the $\left( U\mid U+V\right)$ construction and an application to cryptography

TL;DR

This paper introduces a modified Reed-Solomon code construction that surpasses traditional decoding limits at low rates and applies it to enhance cryptographic schemes, specifically in code-based cryptography.

Contribution

It presents a novel code construction using a (U|U+V) scheme with Koetter-Vardy decoding, improving decoding radius and security in cryptographic applications.

Findings

Achieves decoding radius beyond Guruwami-Sudan limit at low rates

Proposes a cryptographic scheme resistant to known structural attacks

Demonstrates comparable or improved performance over standard Reed-Solomon codes

Abstract

In this paper we present a modification of Reed-Solomon codes that beats the Guruwami-Sudan $1-\sqrt{R}$ decoding radius of Reed-Solomon codes at low rates $R$. The idea is to choose Reed-Solomon codes $U$ and $V$ with appropriate rates in a $\left( U\mid U+V\right)$ construction and to decode them with the Koetter-Vardy soft information decoder. We suggest to use a slightly more general version of these codes (but which has the same decoding performances as the $\left( U\mid U+V\right)$-construction) for code-based cryptography, namely to build a McEliece scheme. The point is here that these codes not only perform nearly as well (or even better in the low rate regime) as Reed-Solomon codes, their structure seems to avoid the Sidelnikov-Shestakov attack which broke a previous McEliece proposal based on generalized Reed-Solomon codes.