Efficient error-correcting codes for the HQC post-quantum cryptosystem

The HQC post-quantum cryptosystem enables two parties to share noisy versions of a common secret binary string, and an error-correcting code is required to deal with the mismatch between both versions. This code is required to deal with binary symmetric channels with as large a transition parameter as possible, while guaranteeing, for cryptographic reasons, a decoding error probability of provably not more than 2^-128. This requirement is non-standard for digital communications, and modern coding techniques are not amenable to this setting. This paper explains how this issue is addressed in the last version of HQC: precisely, we introduce a coding scheme that consists of concatenating a Reed–Solomon code with the tensor product of a Reed–Muller code and a repetition code. We analyze its behavior in detail and show that it significantly improves upon the previous proposition for HQC, which consisted of tensoring a BCH and a repetition code. As additional results, we also provide a better approximation of the weight distribution for HQC error vectors, and we remark that the size of the exchanged secret in HQC can be reduced to match the protocol security which also significantly improves performance.