We propose a new Lee-metric code-based signature scheme (called LMQCS) based on 2-quasi-cyclic codes (2QC-codes) and assuming the hardness of the Lee metric syndrome decoding problem for 2QC-codes (2QC-SDP). Furthermore, we also give a detailed security analysis and a brief security proof of the LMQCS signature scheme. Based on the complexities of solving the underlying 2QC-SDP problem, the public key size and signature size of the LMQCS signature scheme are 2217 bytes and 4434 bytes respectively at 128-bit security level.

MDS matrices are excellent candidate for providing diffusion properties in block ciphers. While MDS matrices over F2(k)are more commonly used, the paper [12] suggested to consider the use of MDS matrices over Galois ringGR(2n,k). In this paper, we explore some constructions of MDS matrices overGR(2n,k) with special properties (e.g, Hadamard, almost orthogonal) from given MDS matrices over F2(k). We show that 2r× 2renabling Hadamard (1, -1) -matrices do not exist forr≥ 4. We also give an algorithm to construct almost orthogonal MDS matrices overGR(2n,k).

We propose a rank metric code-based signature scheme based on the rank syndrome decoding problem, and analyze its security. We also provide necessary conditions for being MRD codes. Then, we provide parameters for the constructed signature scheme based on random linear codes constructed from Cauchy matrix. We also compare the signature scheme with those code-based signature schemes submitted to the NIST call for Post-Quantum Cryptography Standardization. The comparison shows that our signature scheme performs better than those schemes in terms of public key size and secret key size. The public key size and secret key size of our signature scheme are more than 3 times and 40 times smaller than those of the signature schemes submitted to the NIST call for Post-Quantum Cryptography Standardization.