# Improved Asymmetric Cipher Based on Matrix Power Function with Provable Security

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## Abstract

**:**

## 1. Introduction

## 2. Our Previous Work

## 3. Previous Asymmetric Cipher Protocol

- platform semigroup ${\mathit{M}}_{\mathit{S}}$ and power ring ${\mathit{M}}_{\mathit{R}}$;
- the base matrix Q;
- two non-commuting matrices ${Z}_{1}$ and ${Z}_{2}$.

- computes a secret matrix U as a product of two polynomials of ${Z}_{1}$ and ${Z}_{2}$ i.e., U = ${\mathit{P}}_{a1}({Z}_{1})\xb7{\mathit{P}}_{a2}({Z}_{2})$;
- computes matrices $X{Z}_{1}{X}^{-1}={A}_{1},X{Z}_{2}{X}^{-1}={A}_{2},{}^{X}{Q}^{U}=E$.

- Bob chooses randomly a non-singular matrix Y in ${\mathit{M}}_{\mathit{R}}$;
- He selects two sets of coefficients in numerical ring $\mathit{R}$ to define two polynomials ${\mathit{P}}_{b1}(\xb7)$ and ${\mathit{P}}_{b2}(\xb7)$ and computes a secret matrix V = ${\mathit{P}}_{b1}({Z}_{1})$·${\mathit{P}}_{b2}({Z}_{2})$. Then he takes matrices ${A}_{1}$ and ${A}_{2}$ and computes a matrix ${\mathit{P}}_{b1}({A}_{1})$·${\mathit{P}}_{b2}({A}_{2})$ = $XV{X}^{-1}=W$;
- He raises matrix ${}^{X}{Q}^{U}$ to the obtained power matrix $W=XV{X}^{-1}$ on the left and obtains ${}^{XV}{Q}^{U}$ since $WX=XV$;
- He raises the result matrix to the power matrix Y on the right and obtains ${}^{XV}{Q}^{UY}$ = K and converts it to a bit string. One of the possible ways to do this is to write all the elements of matrix K in a string of the form$${k}_{11}{k}_{12}\dots {k}_{1m}{k}_{21}{k}_{22}\dots {k}_{2m}\dots {k}_{mm}$$
- Bob computes the ciphertext C = $K\oplus M$, where ⊕ is bitwise sum modulo 2 of all entries of bitstings K and M;
- Bob computes three matrices $({Y}^{-1}{Z}_{1}Y={B}_{1},{Y}^{-1}{Z}_{2}Y={B}_{2},{}^{V}{Q}^{Y}=F)$ which we denote by encryptor ε and sends it to Alice together with C.

- Using given matrices ${B}_{1}$ and ${B}_{2}$ Alice computes ${\mathit{P}}_{a1}({B}_{1})$·${\mathit{P}}_{a2}({B}_{2})$ = ${Y}^{-1}UY$, since U = ${\mathit{P}}_{a1}({Z}_{1})$·${\mathit{P}}_{a2}({Z}_{2})$;
- Alice raises matrix ${}^{V}{Q}^{Y}$ to the power ${Y}^{-1}UY$ on the right and then raises the result matrix to the power X on the left and hence obtains a matrix $K={}^{XV}{Q}^{UY}$ and converts it to a bitstring.
- Alice can now decrypt a ciphertext C using encryption key K and relation$$M=K\oplus C=K\oplus K\oplus M.$$

**Definition**

**1.**

**Example**

**1.**

## 4. Improvements of the Asymmetric Cipher Protocol

- Size of the Sylow group ${\mathbf{\Gamma}}_{p,n}$ p;
- Parameter n, which defines the multiplicative semigroup ${\mathit{Z}}_{n}$;
- The prime factor ${p}_{1}$ of the parameter n;
- Generator of the Sylow group ${\mathbf{\Gamma}}_{p,n}$ γ;
- Idempotent $j\in {\mathit{Z}}_{n}$;

## 5. Security Analysis

**Definition**

**2.**

**Proposition**

**1.**

**Proposition**

**2.**

**Corollary**

**1.**

**Proposition**

**3.**

**Corollary**

**2.**

**S**to define the platform semigroup and hence ${\mathrm{ld}}_{g}Q$ can be obtained easily if $\mathit{S}=\mathit{G}$.

**Example**

**2.**

**Example**

**3.**

## 6. Discussion

## Author Contributions

## Conflicts of Interest

## References

- 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.; Luksys, K. Matrix power function and its application to block cipher s-box construction. Int. J. Innov. Comput.
**2012**, 8, 2655–2664. [Google Scholar] - Sakalauskas, E.; Tvarijonas, P.; Raulynaitis, A. Key agreement protocol (KAP) using conjugacy and discrete logarithm problems in group representation level. Informatica
**2007**, 18, 115–124. [Google Scholar] - Luksys, K.; Sakalauskas, E.; Venčkauskas, A. Implementation analysis of matrix power cipher in embedded systems. Elektron. Elektrotech.
**2012**, 2, 95–98. [Google Scholar] [CrossRef] - Vitkus, P.; Sakalauskas, E.; Listopadskis, N.; Vitkiene, R. Microprocessor realization of key agreement protocol (KAP) based on matrix power function. Elektron. Elektrotech.
**2012**, 117, 33–36. [Google Scholar] - Myasnikov, A.; Shpilrain, V.; Ushakov, A. Group-Based Cryptography; Birkhäuser Verlag: Basel, Switzerland, 2008. [Google Scholar]
- Jacobs, K. A Survey of Modern Mathematical Cryptology. University of Tennessee Honors Thesis Projects: Knoxville, TN, USA, April 2011; Available online: http://trace.tennessee.edu/cgi/viewcontent.cgi?article=2422&context=utk_chanhonoproj (accessed on 5 December 2016).
- Ottaviani, V.; Zanoni, A.; Regoli, M. Conjugation as Public Key Agreement Protocol in Mobile Cryptography. In Proceedings of the 2010 International Conference on Security and Cryptography, University of Piraeus, Athens, Greece, 26–28 July 2010; pp. 1–6.
- Sracic, M. Quantum Circuits for Matrix Multiplication. July 2011. Available online: https://www.math.ksu.edu/reu/sumar/QuantumAlgorithms.pdf (accessed on 5 December 2016).
- Hall, M. The Theory of Groups; Macmillan: New York, NY, USA, 1959. [Google Scholar]
- Sakalauskas, E. The multivariate quadratic power problem over Zn is NP-Complete. Inf. Technol. Control
**2012**, 41, 33–39. [Google Scholar] [CrossRef] - Wegman, M.N.; Carter, J.L. New hash functions and their use in authentication and set equality. J. Comput. Syst. Sci.
**1981**, 22, 265–279. [Google Scholar] [CrossRef] - Vaudenay, S. Decorrelation: A theory for block cipher security. J. Cryptol.
**2003**, 16, 249–286. [Google Scholar] [CrossRef] - Sakalauskas, E.; Mihalkovich, A. Candidate One-Way Function Based on Matrix Power Function with Conjugation Constraints. In Proceedings of the Bulgarian Cryptography Days 2012, Sofia, Bulgaria, 20–21 September 2012; pp. 29–37.
- Patarin, J.; Goubin, L. Trapdoor One-Way Permutations and Multivariate Polynomials. In Proceedings of the First International Conference (ICICS’97), Beijing, China, 11–14 November 1997; pp. 356–368.
- Mihalkovich, A.; Toldinas, J.; Venčkauskas, A. The Analysis of the Performance of Matrix Power Asymmetric Cipher Protocol. In Proceedings of the GV-Global Virtual Conference, Žilina, Slovakia, 6–10 April 2015; EDIS-Publishing Institution of the University of Žilina: Žilina, Slovakia, 2015; pp. 149–153. [Google Scholar]

p | n | ${\mathit{p}}_{1}$ | γ | j |
---|---|---|---|---|

5 | 33 | 11 | 4 | 12 |

7 | 87 | 29 | 7 | 30 |

13 | 159 | 53 | 10 | 54 |

17 | 309 | 103 | 13 | 207 |

19 | 573 | 191 | 25 | 192 |

23 | 141 | 47 | 4 | 48 |

29 | 177 | 59 | 4 | 60 |

31 | 933 | 311 | 7 | 312 |

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**MDPI and ACS Style**

Sakalauskas, E.; Mihalkovich, A.; Venčkauskas, A.
Improved Asymmetric Cipher Based on Matrix Power Function with Provable Security. *Symmetry* **2017**, *9*, 9.
https://doi.org/10.3390/sym9010009

**AMA Style**

Sakalauskas E, Mihalkovich A, Venčkauskas A.
Improved Asymmetric Cipher Based on Matrix Power Function with Provable Security. *Symmetry*. 2017; 9(1):9.
https://doi.org/10.3390/sym9010009

**Chicago/Turabian Style**

Sakalauskas, Eligijus, Aleksejus Mihalkovich, and Algimantas Venčkauskas.
2017. "Improved Asymmetric Cipher Based on Matrix Power Function with Provable Security" *Symmetry* 9, no. 1: 9.
https://doi.org/10.3390/sym9010009