矩阵秩度量码在添加-移除变换下的码距

中文摘要 AI 辅助

对于矩阵秩度量码 $\mathcal{C}$ 和整数 $i$，$\mathcal{C}$ 的第 $i$ 个子码距离是 $\mathcal{C}$ 的 $i$ 维子码的最大最小秩距离。我们研究了在添加-移除构造下子码距离如何变化，该构造保留一个余维数为 $\ell_s$ 的子码并添加一个维数为 $\ell_a$ 的空间；此构造与 Gabidulin 码一起用于 $\mathsf{MIRANDA}$ 签名方案。对于任意矩阵码，我们给出了原始码和新码的子码距离之间的不等式，并证明了 Delsarte 对偶交换 $\ell_s$ 和 $\ell_a$。对于矩阵 Gabidulin 码，其子码距离由显式公式给出，我们证明，如果 $\ell_s<\max\{m,n\}$，则在不超过 $\dim_{\mathbb{F}_q}(\mathcal{C}_s)$ 的每个索引块 $(a-1)\max\{m,n\}+1,\ldots,a\max\{m,n\}$ 内，除最后 $\ell_s$ 个子码距离外，其余子码距离与 Gabidulin 码相同，而剩余的子码距离最多减少 1；一个小例子表明这种减少可能发生。我们还用添加空间与公共子码之间的秩距离来表示新码的最小距离。对于 $\mathsf{MIRANDA}$，当 $\ell_s=0$ 时，已知区分器使用的对偶公钥码最小距离的下界是这些结果的特例；对于所提出的参数（其中 $\ell_s>0$），相同的界适用于对偶公钥码的余维数为 $\ell_s$ 的子码。

英文摘要

For a matrix rank-metric code $\mathcal{C}$ and an integer $i$, the $i$-th subcode distance of $\mathcal{C}$ is the largest minimum rank distance of an $i$-dimensional subcode of $\mathcal{C}$. We study how subcode distances change under the Add-and-Remove construction, in which a subcode of codimension $\ell_s$ is kept and a space of dimension $\ell_a$ is added; this construction is used, with Gabidulin codes, in the $\mathsf{MIRANDA}$ signature scheme. For arbitrary matrix codes, we give inequalities between the subcode distances of the original and of the new code, and we show that Delsarte duality interchanges $\ell_s$ and $\ell_a$. For matrix Gabidulin codes, whose subcode distances are given by an explicit formula, we show that, if $\ell_s<\max\{m,n\}$, then within each block of indices $(a-1)\max\{m,n\}+1,\ldots,a\max\{m,n\}$ not exceeding $\dim_{\mathbb{F}_q}(\mathcal{C}_s)$, all but the last $\ell_s$ subcode distances are the same as for the Gabidulin code, and the remaining ones decrease by at most one; a small example shows that such a decrease can occur. We also express the minimum distance of the new code in terms of the rank distance between the added space and the common subcode. For $\mathsf{MIRANDA}$, the lower bound on the minimum distance of the dual public code used by the known distinguisher when $\ell_s=0$ is a special case of these results; for the proposed parameters, where $\ell_s>0$, the same bound holds for a subcode of codimension $\ell_s$ of the dual public code.

发表机构

• Université de Rennes（雷恩大学）
• Indian Institute of Technology Mandi（印度技术学院曼迪分校）

机构由 AI 辅助整理，请以论文原文为准。