A new path to code-based signatures via identification schemes with restricted errors

In this paper, we introduce a new variant of the syndrome decoding problem (SDP), called restricted SDP (R-SDP), in which the entries of the solution vector live in a fixed subset of the underlying finite field. We prove the NP-completeness of R-SDP via a reduction from SDP and show how this new problem can be employed in identification schemes. We revisit some concepts of classical coding theory in light of this new setting, provide several bounds such as the Gilbert-Varshamov, the Singleton, and the Plotkin bound, and study the behavior of random codes. Resulting from this initial work, several proposals have arisen; in particular, the code-based digital signature scheme named CROSS, which is currently competing within NIST's additional call for post-quantum digital signatures, is based on R-SDP.