In this paper, we initiate the study of Extended Gabidulin codes with a Kronecker product structure and propose three enhanced variants of the Rank Quasi-Cyclic (RQC) (Melchor et.al., IEEE IT, 2018) cryptosystem. First, we establish precise bounds on the minimum rank distance of Gabidulin-Kronecker product codes under two distinct parameter regimes. Specifically, when $n_{1}=k_{1}$ and $n_{2}=m<n_{1}n_{2}$, the minimum rank distance is exactly $n_{2}-k_{2}+1$. This yields a new family of Maximum Rank Distance (MRD) codes, which are distinct from classical Gabidulin codes. For the case of $k_{1}\leq n_{1},k_{2}\leq n_{2},n_{1}n_{2}\leq m$, the minimum rank distance $d$ of Gabidulin-Kronecker product codes satisfies a tight upper and lower bound, i.e., $n_{2}-k_{2}+1 \leq d \leq (n_{1}-k_{1}+1)(n_{2}-k_{2}+1)$. Second, we introduce a new class of decodable rank-metric codes, namely Extended Gabidulin-Kronecker product (EGK) codes, which generalize the structure of Gabidulin-Kronecker product (GK) codes. We also propose a decoding algorithm that directly retrieves the codeword without recovering the error vector, thus improving efficiency. This algorithm achieves zero decoding failure probability when the error weight is within its correction capability. Third, we propose three enhanced variants of the RQC cryptosystem based on EGK codes, each offering a distinct trade-off between security and efficiency. For 128-bit security, all variants achieve significant reductions in public key size compared to the Multi-UR-AG (Bidoux et.al., IEEE IT, 2024) while ensuring zero decryption failure probability--a key security advantage over many existing rank-based schemes.

翻译：本文首次研究了具有Kronecker积结构的扩展Gabidulin码，并提出了三种增强型秩准循环（RQC）（Melchor等人，IEEE IT，2018）密码系统的变体。首先，我们在两种不同的参数体制下，为Gabidulin-Kronecker积码的最小秩距离建立了精确的界。具体而言，当$n_{1}=k_{1}$且$n_{2}=m<n_{1}n_{2}$时，最小秩距离恰好为$n_{2}-k_{2}+1$。这产生了一类新的最大秩距离（MRD）码，它们不同于经典的Gabidulin码。对于$k_{1}\leq n_{1},k_{2}\leq n_{2},n_{1}n_{2}\leq m$的情况，Gabidulin-Kronecker积码的最小秩距离$d$满足一个紧的上界和下界，即$n_{2}-k_{2}+1 \leq d \leq (n_{1}-k_{1}+1)(n_{2}-k_{2}+1)$。其次，我们引入了一类新的可解码秩度量码，即扩展Gabidulin-Kronecker积（EGK）码，它推广了Gabidulin-Kronecker积（GK）码的结构。我们还提出了一种解码算法，该算法直接恢复码字而无需恢复错误向量，从而提高了效率。当错误权重在其纠错能力范围内时，该算法实现零解码失败概率。第三，我们提出了三种基于EGK码的增强型RQC密码系统变体，每种都在安全性和效率之间提供了不同的权衡。对于128位安全性，所有变体与Multi-UR-AG（Bidoux等人，IEEE IT，2024）相比，在确保零解密失败概率的同时，实现了公钥大小的显著减少——这是相对于许多现有基于秩的方案的一个关键安全优势。