On the Independence Assumption in Quasi-Cyclic Code-Based Cryptography

• URL: http://arxiv.org/abs/2501.02626v1
• Date: Sun, 05 Jan 2025 18:43:08 GMT
• Title: On the Independence Assumption in Quasi-Cyclic Code-Based Cryptography
• Authors: Maxime Bombar, Nicolas Resch, Emiel Wiedijk,
• Abstract summary: Most efficient proposals rely on the presumed hardness of decoding structured codes. HQC is based on quasi-cyclic codes, which are codes generated by matrices consisting of two cyclic blocks. We discuss implications of our result for potential worst-case to average-case reductions for quasi-cyclic codes.
• Score: 3.298620737848292
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• Abstract: 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 understanding of the Decryption Failure Rate (DFR), whose analysis relies on the following heuristic: given random ``sparse'' vectors $e_1,e_2$ (say, each coordinate is i.i.d. Bernoulli) multiplied by fixed ``sparse'' quasi-cyclic matrices $A_1,A_2$, the weight of resulting vector $e_1A_1+e_2A_2$ is very concentrated around its expectation. In the documentation, the authors model the distribution of $e_1A_1+e_2A_2$ as a vector with independent 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 $e_1A_1+e_2A_2$ 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.

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