MPF Problem over Modified Medial Semigroup Is NP-Complete

Eligijus Sakalauskas,Aleksejus Mihalkovich

doi:10.3390/sym10110571

Abstract

This paper is a continuation of our previous publication of enhanced matrix power function (MPF) as a conjectured one-way function. We are considering a problem introduced in our previous paper and prove that tis problem is NP-Complete. The proof is based on the dual interpretation of well known multivariate quadratic (MQ) problem defined over the binary field as a system of MQ equations, and as a general satisfiability (GSAT) problem. Due to this interpretation the necessary constraints to MPF function for cryptographic protocols construction can be added to initial GSAT problem. Then it is proved that obtained GSAT problem is NP-Complete using Schaefer dichotomy theorem. Referencing to this result, GSAT problem by polynomial-time reduction is reduced to the sub-problem of enhanced MPF, hence the latter is NP-Complete as well.

Highlights

It is very natural to look for a new conjectured one-way functions (OWFs) for cryptographic applications in connection with new challenges caused by quantum cryptanalysis
It is thought that OWF security based on the NP-Complete problem is not vulnerable to the quantum cryptanalysis, while the cryptosystems based on conjectured OWFs such as factoring and discrete logarithm problems are vulnerable due to [9]
That constrained singular binary matrix MQ problem (CSBMMQ) problem is a sub-problem of CSMMQN0 problem, when semiring N0 is homomorphically mapped to the field Z2

Summary

Introduction

It is very natural to look for a new conjectured one-way functions (OWFs) for cryptographic applications in connection with new challenges caused by quantum cryptanalysis. Some results were published considering the security of presented primitives in [6,7,8] The security of these primitives is based on the complexity of MPF inversion named as MPF problem. In [6] the NP-Completeness of a more general problem named as multivariate quadratic power problem is presented. In [10] our efforts were directed toward the increasing expectable complexity of MPF problem by choosing more complicated algebraic structures for MPF definition but at the same time preserving the necessary properties for the cryptographic primitives construction. In this paper we present a proof of NP-Completeness of sub-problem of enhanced MPF problem previously considered in [10]. The proof is based on proving that this GSAT is NP-Complete and on polynomial-time reduction from GSAT to the sub-problem of enhanced MPF problem

Matrix Power Function

Modified Medial Semigroup as Platform Semigroup of MPF

Proof of NP-Completeness

Conclusions