Code-based Signatures from New Proofs of Knowledge for the Syndrome Decoding Problem

In this paper, we study code-based signatures constructed from Proof of Knowledge (PoK). This line of work can be traced back to Stern who introduces the first efficient PoK for the syndrome decoding problem in 1993. Afterward, different variations were proposed in order to reduce signature's size. In practice, obtaining a smaller signature size relies on the interaction of two main considerations: (i) the underlying protocol and its soundness error and (ii) the type of optimizations which are compatible with a given protocol. Over the years, different variations were proposed to improve the Stern scheme such as the Veron scheme (with public key a noisy codeword rather than a syndrome), the AGS scheme which is a 5-pass protocol with cheating probability asymptotically equal to 1/2 and more recently the FJR approach which permits to decrease the cheating probability to 1/N but induces a performance overhead. Overall the length of the signature depends on a trade-off between: the scheme in itself, the possible optimizations and the cost of the implementation. The recent approaches which increase the cost of the implementation opens the door to many different type of trade-offs. In this paper we propose three new schemes and different trade-offs, which are all interesting in themselves, since depending on potential future optimizations a scheme may eventually become more efficient than another. All the schemes we propose use a trusted helper: a first scheme permits to get a 1/2 cheating probability, a second scheme permits to decrease the cheating probability in 1/N but with a different approach than the recent FJR scheme and at last a third scheme propose a Veron-like adaptation of the FJR scheme in which the public key is a noisy codeword rather than a syndrome. We provide an extensive comparison table which lists various trade-offs between our schemes and previous ones.