# MPF Problem over Modified Medial Semigroup Is NP-Complete

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Matrix Power Function

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

- 1.
- The direct MPF value computation is easy;
- 2.
- The MPF problem is polynomially equivalent to a certain hard problem with not known polynomial time algorithm.

- Alice chooses two secret circulant matrices X and Y at random of size m. Using these matrices she computes the MPF value $A={}^{X}{W}^{Y}$ and sends it to Bob;
- Bob chooses two secret circulant matrices U and V at random of size m. Using these matrices he computes the MPF value $B={}^{U}{W}^{V}$ and sends it to Alice;
- Alice and Bob compute the same secret key in the following way:$${K}_{A}={}^{X}{B}^{Y}={}^{X}{\left({}^{U}{W}^{V}\right)}^{Y}={}^{U}{\left({}^{X}{W}^{Y}\right)}^{V}={K}_{B}=K.$$

**Remark**

**1.**

## 3. Modified Medial Semigroup as Platform Semigroup of MPF

**Remark**

**2.**

**Remark**

**3.**

**Definition**

**9.**

**Definition**

**10.**

## 4. Proof of NP-Completeness

**Definition**

**11.**

**Remark**

**4.**

**Definition**

**12.**

**Definition**

**13.**

**Theorem**

**1.**

- (a)
- Every relation in S is satisfied when all the variables are 0 (0-valid clause);
- (b)
- Every relation in S is satisfied when all the variables are 1 (1-valid clause);
- (c)
- Every relation in S is definable by a CNF formula in which each conjunct has at most one negated variable (dual Horn clause);
- (d)
- Every relation in S is definable by a CNF formula in which each conjunct has at most one unnegated variable (Horn clause);
- (e)
- Every relation in S is definable by a CNF formula having at most two literals in each conjunct (bijunctive clause);
- (f)
- Every relation in S is the set of solutions of a system of linear equation over the two element field $\{0,1\}$ (affine clause).

**Definition**

**14.**

**Theorem**

**2.**

**Proof.**

- (c’)
- For all pairs $({\overrightarrow{x}}_{1},{\overrightarrow{y}}_{1})$ and $({\overrightarrow{x}}_{2},{\overrightarrow{y}}_{2})$, satisfying System (19) and Equation (20), the pair $({\overrightarrow{x}}_{1}\vee {\overrightarrow{x}}_{2},{\overrightarrow{y}}_{1}\vee {\overrightarrow{y}}_{2})$ is a solution of System (19) and Equation (20);
- (d’)
- For all pairs$({\overrightarrow{x}}_{1},{\overrightarrow{y}}_{1})$ and $({\overrightarrow{x}}_{2},{\overrightarrow{y}}_{2})$, satisfying System (19) and Equation (20), the pair $({\overrightarrow{x}}_{1}\wedge {\overrightarrow{x}}_{2},{\overrightarrow{y}}_{1}\wedge {\overrightarrow{y}}_{2})$ is a solution of System (19) and Equation (20);
- (e’)
- For all pairs $({\overrightarrow{x}}_{1},{\overrightarrow{y}}_{1})$, $({\overrightarrow{x}}_{2},{\overrightarrow{y}}_{2})$ and $({\overrightarrow{x}}_{3},{\overrightarrow{y}}_{3})$, satisfying System (19) and Equation (20), the pair $(({\overrightarrow{x}}_{1}\vee {\overrightarrow{x}}_{2})\wedge ({\overrightarrow{x}}_{1}\vee {\overrightarrow{x}}_{3})\wedge ({\overrightarrow{x}}_{2}\vee {\overrightarrow{x}}_{3}),({\overrightarrow{y}}_{1}\vee {\overrightarrow{y}}_{2})\wedge ({\overrightarrow{y}}_{1}\vee {\overrightarrow{y}}_{3})\wedge ({\overrightarrow{y}}_{2}\vee {\overrightarrow{y}}_{3}))$ is a solution of System (19) and Equation (20).

**Remark**

**5.**

**Remark**

**6.**

**Theorem**

**3.**

**Proof.**

**Proof.**

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

**Theorem**

**5.**

**Example**

**1.**

**Theorem**

**6.**

**Proof.**

**Remark**

**10.**

## 5. Conclusions

- The proof of NP-Completeness of author’s constructed MPF in previous Symmetry journal publication is presented. It is a new evidence, that this type of MPF can be considered for construction of a non-commuting cryptography primitive as a conjectured OWF.
- The proof is based on two main approaches: we prove that certain GSAT is NP-Complete using modified Schaefer criteria, and, using this result, we prove that this GSAT is a sub-problem of the considered MPF problem. Hence this type of MPF problem is NP-Complete.
- It is a new step to prove that KAP presented in our previous publication mentioned above has a provable security property.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MPF | Matrix power function |

OWF | one-way function |

MQ problem | Multivariate quadratic problem |

MMQ problem | Matrix MQ problem |

BMMQ problem | Binary matrix MQ problem |

SBMMQ problem | Singular binary matrix MQ problem |

CSBMMQ problem | Constrained singular binary matrix MQ problem |

GSAT problem | General satisfiability problem |

NP-Commplete problem | Non-deterministic polynomial complete problem |

CNF | Conjuntive normal form |

## References

- Mihalkovich, A.; Sakalauskas, E. Asymmetric cipher based on MPF and its security parameters evaluation. In Proceedings of the Lithuanian Mathematical Society, Klaipeda, Lithuania, 11–12 June 2012; VU Matematikos ir Informatikos Institutas: Vilnius, Lithuania, 2012. Ser. A. Volume 53, pp. 72–77. [Google Scholar]
- Mihalkovich, A.; Sakalauskas, E.; Venckauskas, A. New asymmetric cipher based on matrix power function and its implementation in microprocessors efficiency investigation. Elektron. Elektrotech.
**2013**, 19, 119–122. [Google Scholar] [CrossRef] - Sakalauskas, E.; Listopadskis, N.; Tvarijonas, P. Key Agreement Protocol (KAP) Based on Matrix Power Function. In Advanced Studies in Software and Knowledge Engineering; International Book Series “Information Science and Computing”; World Scientific: Singapore, 2008; pp. 92–96. [Google Scholar]
- Sakalauskas, E.; Luksys, K. Matrix Power S-Box Construction. IACR Cryptology ePrint Archive 2007. Available online: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.78.2327&rep=rep1&type=pdf (accessed on 26 October 2018).
- Sakalauskas, E.; Mihalkovich, A. New asymmetric cipher of non-commuting cryptography class based on matrix power function. Informatica
**2014**, 25, 283–298. [Google Scholar] [CrossRef] - Sakalauskas, E. The multivariate quadratic power problem over Zn is NP-Complete. Inf. Technol. Control
**2012**, 41, 33–39. [Google Scholar] [CrossRef] - Sakalauskas, E.; Mihalkovich, A. Improved Asymmetric Cipher Based on Matrix Power Function Resistant to Linear Algebra Attack. Informatica
**2017**, 28, 517–524. [Google Scholar] [CrossRef] - Sakalauskas, E.; Mihalkovich, A.; Venčkauskas, A. Improved asymmetric cipher based on matrix power function with provable security. Symmetry
**2017**, 9, 9. [Google Scholar] [CrossRef] - Shor, P.W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev.
**1999**, 41, 303–332. [Google Scholar] [CrossRef] - Sakalauskas, E. Enhanced Matrix Power Function for Cryptographic Primitive Construction. Symmetry
**2018**, 10, 43. [Google Scholar] [CrossRef] - Garey, M.R.; Johnson, D.S. Computers and Intractability; WH Freeman: New York, NY, USA, 2002. [Google Scholar]
- Patarin, J.; Goubin, L. Trapdoor one-way permutations and multivariate polynomials. In Proceedings of the International Conference on Information and Communications Security, Beijing, China, 11–14 November 1997; Springer: Berlin, Germany, 1997; pp. 356–368. [Google Scholar]
- Schaefer, T.J. The complexity of satisfiability problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, San Diego, CA, USA, 1–3 May 1978; ACM: New York, NY, USA, 1978; pp. 216–226. [Google Scholar]
- Davis, P.J. Circulant Matrices; Wiley: New York, NY, USA, 1970. [Google Scholar]
- Sakalauskas, E.; Mihalkovich, A. Candidate One-Way Function Based on Matrix Power Function with Conjugation Constraints. In Proceedings of the Conference proceedings Bulgarian Cryptography Days 2012, Sofia, Bulgaria, 20–21 September 2012; pp. 29–37. [Google Scholar]
- Liu, J.; Zhang, H.; Jia, J. A linear algebra attack on the non-commuting cryptography class based on matrix power function. In Proceedings of the International Conference on Information Security and Cryptology, Beijing, China, 4–6 November 2016; Springer: Berlin, Germany, 2016; pp. 343–354. [Google Scholar]
- Chrislock, J.L. On medial semigroups. J. Algebra
**1969**, 12, 1–9. [Google Scholar] [CrossRef] - Dechter, R.; Pearl, J. Structure identification in relational data. Artif. Intell.
**1992**, 58, 237–270. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Sakalauskas, E.; Mihalkovich, A.
MPF Problem over Modified Medial Semigroup Is NP-Complete. *Symmetry* **2018**, *10*, 571.
https://doi.org/10.3390/sym10110571

**AMA Style**

Sakalauskas E, Mihalkovich A.
MPF Problem over Modified Medial Semigroup Is NP-Complete. *Symmetry*. 2018; 10(11):571.
https://doi.org/10.3390/sym10110571

**Chicago/Turabian Style**

Sakalauskas, Eligijus, and Aleksejus Mihalkovich.
2018. "MPF Problem over Modified Medial Semigroup Is NP-Complete" *Symmetry* 10, no. 11: 571.
https://doi.org/10.3390/sym10110571