Proximity gaps improve codes for cryptography and error detection

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.