MURAVE: A New Rank Code-Based Signature with MUltiple RAnk VErification

Abstract

We propose a new rank metric code-based signature scheme constructed via the Schnorr approach. Our scheme is designed in a way to avoid leakage of the information on the support for the secret key used in the signature generation. We define some new problems in rank metric code-based cryptography: the Rank Support Basis Decomposition problem and the Advanced Rank Support Basis Decomposition problem. We also discuss their hardness and solving complexity. Furthermore, we give a proof in the $\mathsf{EUF}\text {-}\mathsf{CMA}$ security model, by reducing the security of our scheme to the Rank Syndrome Decoding problem, the Ideal LRPC Codes Indistinguishability problem and the Decisional Rank Support Basis Decomposition problem. We analyze the practical security for our scheme against the known attacks on rank metric signature schemes. Our scheme is efficient in terms of key size (5.33 KB) and of signature sizes (9.69 KB) at 128-bit classical security level.

Appendix A Rank Support Recovery Algorithm

Let $f=(f_1,\ldots ,f_d) \in E_{m,d,d}$, $e=(e_1,\ldots ,e_r) \in E_{m,r,r}$ and $\textit{\textbf{s}}=(s_1,\ldots ,s_n) \in \mathbb {F}_{q^m}^n$ such that $S:=\langle s_1,\ldots ,s_n \rangle = \langle f_1 e_1,\ldots ,f_d e_r \rangle$. Given $\textit{\textbf{f}}$, $\textit{\textbf{s}}$ and r as input, the Rank Support Recovery Algorithm will output a vector space E which satisfies $E = \langle e_1,\ldots ,e_r \rangle$. Denote $S_i := f_i^{-1}.S$ and $S_{ i,j} := S_i \cap S_j$.