Gabidulin码的邻近间隙及其应用
Proximity Gaps for Gabidulin Codes and Applications
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中文总结 AI 辅助
本文研究线性秩度量码的邻近间隙，证明了一般码和Gabidulin码的间隙界，并构造了紧性反例，进而改编Ligero协议得到交织Gabidulin码的IOPP和首个基于秩度量码的PCS。
中文摘要 AI 辅助
邻近间隙对于交互式邻近预言证明（IOPPs）和多项式承诺方案（PCSs）的可靠性至关重要。一个$[n,k,d]$线性码$C\subseteq\mathbb F^n$具有误差为$\epsilon$的$\delta$-邻近间隙，如果对于任意$u_0,u_1\in\mathbb F^n$，要么直线$\ell_{u_0,u_1}=\{u_0+\alpha u_1:\alpha\in\mathbb F\}$上的所有点都$\delta$-接近$C$，要么至多有$\epsilon$比例的点接近。尽管Hamming度量码的邻近间隙已被充分理解，但其秩度量对应物尽管在编码理论和密码学中有应用，却仍未得到充分探索。在这项工作中，我们研究了线性秩度量码的邻近间隙及其密码学应用。首先，我们证明$\mathbb F_{q^m}$上的每个$[n,k,d]$线性秩度量码$C$对于每个$\delta\le(d-1)/(3n)$都承认一个邻近间隙，误差至多为$q^{e+1}/q^m$，其中$e=\lfloor\delta n\rfloor$。对于Gabidulin码，我们将间隙改进为$(d-1)/(2n)$，误差为$10q^{n-1}/q^m$。这两个邻近间隙分别与一般线性Hamming度量码和Reed-Solomon（RS）码的间隙相匹配。我们通过构造一个无限族的恒定速率Gabidulin码和仿射线$\ell_{u_0,u_1}$来证明$(d-1)/(2n)$界是紧的，在这些线上有$1-o(1)$比例的点$d/(2n)$-接近该码，而$u_1$至少$3d/(4n)$-远离它。在$d/(3n)$间隙处，我们也给出了一个反例，建立了$\epsilon$的下界。作为应用，我们通过改编用于交织RS码的Ligero IOPP，为交织Gabidulin码构造了一个IOPP。然后，我们改编基于Ligero的普通多项式PCS，以获得一个$q$-线性化多项式承诺方案。据我们所知，这是第一个基于秩度量纠错码的PCS框架。
英文摘要
Proximity gaps are central to the soundness of interactive oracle proofs of proximity (IOPPs) and polynomial commitment schemes (PCSs). An $[n,k,d]$ linear code $C\subseteq\mathbb F^n$ has a $δ$-proximity gap with error $ε$ if, for every $u_0,u_1\in\mathbb F^n$, either all points on $\ell_{u_0,u_1}=\{u_0+αu_1:α\in\mathbb F\}$ are $δ$-close to $C$, or at most an $ε$ fraction are. Although proximity gaps for Hamming-metric codes are well understood, their rank-metric counterparts remain largely unexplored despite their applications in coding theory and cryptography. In this work, we study proximity gaps for linear rank-metric codes and their cryptographic applications. First, we show that every $[n,k,d]$ linear rank-metric code $C$ over $\mathbb F_{q^m}$ admits a proximity gap for every $δ\le(d-1)/(3n)$, with error at most $q^{e+1}/q^m$, where $e=\lfloorδn\rfloor$. For Gabidulin codes, we improve the gap to $(d-1)/(2n)$ with error $10q^{n-1}/q^m$. These two proximity gaps match those for general linear Hamming-metric codes and Reed--Solomon (RS) codes, respectively. We prove the $(d-1)/(2n)$ bound is tight by constructing an infinite family of constant-rate Gabidulin codes and affine lines $\ell_{u_0,u_1}$ on which a $1-o(1)$ fraction of points are $d/(2n)$-close to the code, while $u_1$ is at least $3d/(4n)$-far from it. At the $d/(3n)$ gap, we also give a counterexample establishing a lower bound on $ε$. As applications, we construct an IOPP for interleaved Gabidulin codes by adapting the Ligero IOPP for interleaved RS codes. We then adapt the Ligero-based PCS for ordinary polynomials to obtain a $q$-linearized polynomial commitment scheme. To our knowledge, this is the first PCS framework based on rank-metric error-correcting codes.
发表机构
- Shanghai Jiao Tong University（上海交通大学）
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