Paper 2025/668

(Interleaved) Extended Gabidulin Codes and Their Applications to RQC

Abstract

In this paper, we investigate the Extended Gabidulin (EG) codes and the Interleaved EG (IEG) codes, and enhance the Rank Quasi-Cyclic (RQC) encryption scheme. Our primary contribution is the development of a general decoding algorithm for (I)EG codes, for which we precisely provide the DFR, bound the decoding capacity, and estimate the decoding complexity. As the core tool, we demonstrate that the Linear Reconstruction (LR) problem derived from the decoding (I)EG codes problem can be probabilistically solved, enabling (I)EG codes to achieve arbitrarily small DFRs, decode up to the rank Gilbert-Varshamov bound (even close to the minimal distance), and decode by the Welch-Berlekamp like algorithm. An interesting and important byproduct is that we demonstrate that decoding interleaved Gabidulin codes can be achieved deterministically by solving the LR problem. We finally apply the EG codes to improve RQC (NIST PQC & Asiacrypt 2023). For 128-bit security, our optimized RQC reduces bandwidth by 69 % and 34 % compared to the original versions, respectively. The scheme also achieves at least 50% improvement in efficiency and mitigates MM algebraic attacks (as discussed in Eurocrypt 2020, Asiacrypt 2020 & 2023) as EG codes facilitate schemes operating over smaller finite fields. Overall, our scheme outperforms code-based schemes of NIST PQC Round 4 submissions, such as HQC, BIKE, and Classic McEliece, in terms of bandwidth. A conservative parameters set still remains competitive bandwidths.