Compact Post-Quantum Signatures from Proofs of Knowledge leveraging Structure for the PKP, SD and RSD Problems

Abstract: The MPC-in-the-head introduced in [IKOS07] has established itself as an important paradigm to design efficient digital signatures. It has been leveraged in the Picnic scheme [CDG+ 20] that reached the third round of the NIST PQC Standardization process. It has also been used in [Beu20] to introduce the Proof of Knowledge (PoK) with Helper paradigm. This construction permits to design shorter signatures but induces a non negligible performance overhead as it uses cut-and-choose. In this paper, we introduce the PoK leveraging structure paradigm along with its associated challenge space amplification technique. Our new approach to design PoK brings some improvements over the PoK with Helper one. Indeed, we show how one can substitute the Helper in these constructions by leveraging the underlying structure of the considered problem. This approach does not suffer from the performance overhead inherent to the PoK with Helper paradigm hence offers different trade-offs between security, signature sizes and performances. We also present four new post-quantum signature schemes. The first one is based on a new PoK with Helper for the Syndrome Decoding problem. It relies on ideas from [BGKM22] and [FJR21] and improve the latter using a new technique that can be seen as performing some cut-and-choose with a meet in the middle approach. The three other signatures are based on our new PoK leveraging structure approach and as such illustrate its versatility. We provide new PoK related to the Permuted Kernel Problem (PKP), Syndrome Decoding (SD) problem and Rank Syndrome Decoding (RSD) problem. In practice, these PoK lead to comparable or shorter signatures than existing ones. Indeed, considering (public key + signature), we get sizes below 9kB for our signature related to the PKP problem, below 15kB for our signature related to the SD problem and below 7kB for our signature related to the RSD problem.