NP-completeness of Certain Sub-classes of the Syndrome Decoding Problem

TL;DR

This paper investigates whether the NP-completeness of the Syndrome Decoding problem persists when considering specific subclasses with constrained parameters, which is crucial for cryptographic security assumptions.

Contribution

It analyzes the complexity of Syndrome Decoding for subclasses with particular constraints, providing insights into their computational hardness.

Findings

Certain subclasses remain NP-complete under specific constraints

The complexity depends on the relationship between target weight, dimension, and length

Implications for cryptographic security based on Syndrome Decoding

Abstract

The problem of Syndrome Decoding was proven to be NP-complete in 1978 and, since then, quite a few cryptographic applications have had their security rely on the (provable) difficulty of solving some instances of it. However, in most cases, the instances to be solved follow some specific constraint: the target weight is a function of the dimension and length of the code. In these cases, is the Syndrome Decoding problem still NP-complete? This is the question that this article intends to answer.