Quasi-Cyclic Stern Proof of Knowledge
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The 1993 Stern authentication protocol is a code-based zero-knowledge protocol with cheating probability equal to 2/3 based on the syndrome decoding problem which permits to obtain a proof of knowledge of a small weight vector. This protocol was improved a few years later by Véron, who proposed a variation of the scheme based on the general syndrome decoding problem which leads to better results in term of communication. A few years later, the AGS protocol introduced a variation of the Véron protocol based on quasi-cyclic matrices. The AGS protocol permits to obtain an asymptotic cheating probability of 1/2 and a strong improvement in term of communications. In the present paper we propose two new contributions. First, a Quasi-Cyclic Stern proof of knowledge construction which constitutes an adaptation of the AGS scheme in a syndrome decoding context. The main interest of this adaptation is that at the difference of the regular (non quasi-cyclic) case, the Quasi-Cyclic Stern protocol is better in terms of communication than its Véron counterpart (the AGS protocol, which can be seen as a Quasi-Cyclic Véron protocol). The difference comes from the fact that a seed related optimization is better for QC-Stern than for QC-Véron. Secondly, we also propose a general new optimization to handle random seeds in this type of protocol. Overall, the two new optimizations we propose permit to gain about 17.5 communication compared to the previous best approach for this type of protocols. Such optimizations are of great matter in the ongoing context where a new signature call for proposals has been announced by the NIST and for which such zero-knowledge approaches are a real alternative, as it was shown in the first signature call for proposals of the NIST.
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