Shorter Signatures from Proofs of Knowledge for the SD, MQ, PKP and RSD Problems
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The MPC in the head introduced in [IKOS07] has established itself as an important paradigm in order to design efficient digital signatures. In particular, it has been leveraged in the Picnic scheme [CDG+ 20] that is currently considered in the third round of NIST Post-Quantum Standardization process. In addition, it has 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. In this paper, our contributions are twofold. Firstly, we introduce a new PoK with Helper for the Syndrome Decoding (SD) problem. This construction relies on ideas from [BGKM22] and [FJR21] and improve the latter using a new technique that can be seen as performing the cut and choose with a meet in the middle approach. Secondly and most importantly, we introduce a new paradigm to design PoK that brings 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 new approach does not suffer from the performance overhead inherent to the PoK with Helper paradigm hence offers different trade-offs between signature sizes and performances. Interestingly, our new approach is quite generic and can be applied to many problems and their associated PoK. In order to demonstrate this versatility, we provide new PoK related to the SD, MQ, PKP and RSD problems. In practice, these PoK lead to shorter signatures for the aforementioned problems. Indeed, considering (public key + signature), we get sizes below 12 kB for our signature related to the SD problem, below 8 kB for our signature related to the MQ problem, below 9 kB for our signature related to the PKP problem and below 7 kB for our signature related to the RSD problem.
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