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Syndrome Decoding in the Head: Shorter Signatures from Zero-Knowledge Proofs

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13508))

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Abstract

Zero-knowledge proofs of knowledge are useful tools to design signature schemes. The ongoing effort to build a quantum computer urges the cryptography community to develop new secure cryptographic protocols based on quantum-hard cryptographic problems. One of the few directions is code-based cryptography for which the strongest problem is the syndrome decoding (SD) for random linear codes. This problem is known to be NP-hard and the cryptanalysis state of the art has been stable for many years. A zero-knowledge protocol for this problem was pioneered by Stern in 1993. Since its publication, many articles proposed optimizations, implementation, or variants.

In this paper, we introduce a new zero-knowledge proof for the syndrome decoding problem on random linear codes. Instead of using permutations like most of the existing protocols, we rely on the MPC-in-the-head paradigm in which we reduce the task of proving the low Hamming weight of the SD solution to proving some relations between specific polynomials. Specifically, we propose a 5-round zero-knowledge protocol that proves the knowledge of a vector x such that \(y=Hx\) and \({\text {wt}}(x)\le w\) and which achieves a soundness error closed to 1/N for an arbitrary N.

While turning this protocol into a signature scheme, we achieve a signature size of 11–12 KB for 128-bit security when relying on the hardness of the SD problem on binary fields. Using larger fields (like \(\mathbb {F}_{2^8}\)), we can produce fast signatures of around 8 KB. This allows us to outperform Picnic3 and to be competitive with SPHINCS+, both post-quantum signature candidates in the ongoing NIST standardization effort. Moreover, our scheme outperforms all the existing code-based signature schemes for the common “signature size \(+\) public key size” metric.

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Notes

  1. 1.

    More cryptanalysis of the SD problem over \(\mathbb {F}_{256}\) would be welcome to get more confidence in the choice of the parameters. Such research is out of the scope of present article.

  2. 2.

    We did not include “Sig 3” from [BGKM22] since it is similar to [FJR21] with slight differences (message decoding setting) which do not improve the scheme.

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Acknowledgements

This work has been supported by the European Union’s H2020 Programme under grant agreement number ERC-669891.

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Correspondence to Thibauld Feneuil .

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Feneuil, T., Joux, A., Rivain, M. (2022). Syndrome Decoding in the Head: Shorter Signatures from Zero-Knowledge Proofs. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13508. Springer, Cham. https://doi.org/10.1007/978-3-031-15979-4_19

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