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In this paper we introduce a variant of the Syndrome Decoding Problem (SDP),that we call Restricted SDP (R-SDP), in which the entries of the searchedvector are defined over a subset of the underlying finite field. We prove theNP-completeness of R-SDP, via a reduction from the classical SDP, and describealgorithms which solve such new problem. We study the properties of randomcodes under this new decoding perspective, in the fashion of traditional codingtheory results, and assess the complexity of solving a random R-SDP instance.As a concrete application, we describe how Zero-Knowledge Identification(ZK-ID) schemes based on SDP can be tweaked to rely on R-SDP, and show thatthis leads to compact public keys as well as significantly reducedcommunication costs. Thus, these schemes offer an improved basis for theconstruction of code-based digital signature schemes derived fromidentification schemes through the well-know Fiat-Shamir transformation.

A New Path to Code-based Signatures via Identification Schemes with Restricted Errors