On the independence assumption in quasi-cyclic code-based cryptography

Cryptography based on the presumed hardness of decoding codes – i.e., code-based cryptography – has recently seen increased interest due to its plausible security against quantum attackers. Notably, of the four proposals for the NIST post-quantum standardization process that were advanced to their fourth round for further review, two were code-based. The most efficient proposals – including HQC and BIKE, the NIST submissions alluded to above – in fact rely on the presumed hardness of decoding structured codes. Of particular relevance to our work, HQC is based on quasi-cyclic codes, which are codes generated by matrices consisting of two cyclic blocks. In particular, the security analysis of HQC requires a precise un- derstanding of the Decryption Failure Rate (DFR), whose analysis relies on the following heuristic: given random “sparse” vectors e1, e2 (say, each coordinate is i.i.d. Bernoulli) multiplied by fixed “sparse” quasi-cyclic matrices A1, A2, the weight of resulting vector e1A1 + e2A2 is very concentrated around its expectation. In the documentation, the authors model the distribution of e1A1 + e2A2 as a vector with inde- pendent coordinates (and correct marginal distribution). However, we uncover cases where this modeling fails. While this does not invalidate the (empirically verified) heuristic that the weight of e1A1 + e2A2 is concentrated, it does suggest that the behavior of the noise is a bit more subtle than previously predicted. Lastly, we also discuss implications of our result for potential worst-case to average-case reductions for quasi-cyclic codes.