On the equivalence of two post-quantum cryptographic families

TL;DR

This paper establishes a direct polynomial-time equivalence between two fundamental post-quantum cryptographic problems, MLD and MQ, revealing a deep connection that impacts the understanding of their security foundations.

Contribution

It provides the first polynomial-time reduction and isomorphism between MLD and MQ, unifying two major classes of post-quantum cryptographic primitives.

Findings

Polynomial-time reduction from MLD to MQ

Polynomial-time reduction from MQ to MLD

Demonstration of a polynomial-time isomorphism between MQ and MLD

Abstract

The Maximum Likelihood Decoding Problem (MLD) is known to be NP-hard and its complexity is strictly related to the security of some post-quantum cryptosystems, that is, the so-called code-based primitives. Analogously, the Multivariate Quadratic System Problem (MQ) is NP-hard and its complexity is necessary for the security of the so-called multivariate-based primitives. In this paper we present a closed formula for a polynomial-time reduction from any instance of MLD to an instance of MQ, and viceversa. We also show a polynomial-time isomorphism between MQ and MLD, thus demonstrating the direct link between the two post-quantum cryptographic families.