# Perfectly Secure Shannon Cipher Construction Based on the Matrix Power Function

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

## 3. Shannon Cipher Construction Based on the Matrix Power Function (MPF)

**K**, the message space

**M**and the ciphertext space

**C**.

**Definition**

**1.**

- $Gen$ is a function of secret key K generation at random and uniformly distributed in
**K**. - $Enc$ is the encryption function which takes as an input a key K in
**K**and a message M in**M**and produces as output a ciphertext C in**C**.$$C=Enc(K,M).$$ - $Dec$ is a decryption function that takes as input a key K in
**K**and a ciphertext C in**C**and produces a message M in**M**.$$M=Dec(K,C).$$The Shannon cipher is defined over (**K**,**M**,**C**) and with this notation we can write:$$\begin{array}{c}\hfill Enc:\mathbf{K}\times \mathbf{M}\to \mathbf{C},\\ \hfill Dec:\mathbf{K}\times \mathbf{C}\to \mathbf{M}.\end{array}$$

**M**, however, it is not assumed that M is uniformly distributed over

**M**. The key K is uniformly distributed in

**K**and is independent of M, while ciphertext $C=Enc(K,M)$ is a random variable distributed over the ciphertext space

**C**.

**M**consists of $n\times n$ matrices M and ciphertext space

**C**of $n\times n$ matrices C and both spaces are denoted by ${Z}_{3}^{n\times n}$.

**K**consists of two matrices X and Y composing a vector valued symmetric key $K=(X,Y)$, where $X=\left\{{x}_{ij}\right\}$, ${x}_{ij}\in {Z}_{3}$ and $Y=\left\{{y}_{ij}\right\}$, ${y}_{ij}\in {Z}_{3\setminus 0}$. Then the key space

**K**is a direct product of the spaces ${Z}_{3}^{n\times n}\times {Z}_{3\setminus 0}^{n\times n}$. The additional requirement is that the matrix Y is an invertible matrix.

## 4. Security Analysis

**M**and ${C}_{0}=Enc(K,{M}_{0})$ is in

**C**. Referencing to [5] the following Lemma can be formulated.

**Lemma**

**1.**

**M**is perfectly secret if and only if for every probability distribution over

**M**, every message $M\in \mathbf{M}$, and every ciphertext $C\in \mathbf{C}$

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

#### The Theorem of Perfect Security

**Theorem**

**1.**

**K**, the probability distribution of M over

**M**is arbitrary, the distributions of K and M over

**K**and

**M**are independent and given the encryption algorithm $Enc$, the distribution of C over

**C**is fully determined by the distributions over

**K**and

**M**, then the Shannon cipher in Equation (6) based on MPF is perfectly secure.

**Proof.**

## 5. Conclusions and Discussions

## Author Contributions

## Funding

## Conflicts of Interest

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${\mathit{z}}_{1}$ | ${\mathit{z}}_{2}$ | ${\mathit{z}}_{1}\xb7{\mathit{z}}_{2}$ |
---|---|---|

1 | 1 | 1 |

1 | 2 | 2 |

2 | 1 | 2 |

2 | 2 | 1 |

w | u | ${\mathit{w}}^{\mathit{u}}$ |
---|---|---|

1 | 1 | 1 |

1 | 2 | 1 |

2 | 1 | 2 |

2 | 2 | 4 |

4 | 1 | 4 |

4 | 2 | 2 |

${\mathit{v}}_{1}$ | ${\mathit{v}}_{2}$ | ${\mathit{v}}_{1}\xb7{\mathit{v}}_{2}$ |
---|---|---|

1 | 1 | 1 |

1 | 2 | 2 |

1 | 4 | 4 |

2 | 1 | 2 |

2 | 2 | 4 |

2 | 4 | 1 |

4 | 1 | 4 |

4 | 2 | 1 |

4 | 4 | 2 |

${\mathit{c}}_{10}$ | ${\mathit{m}}_{0}$ | $-{\mathit{m}}_{0}$ | ${\mathit{x}}_{\mathit{i}\mathit{j}}$ |
---|---|---|---|

0 | 0 | 0 | 0 |

0 | 1 | 2 | 2 |

0 | 2 | 1 | 1 |

1 | 0 | 0 | 1 |

1 | 1 | 2 | 0 |

1 | 2 | 1 | 2 |

2 | 0 | 0 | 2 |

2 | 1 | 2 | 1 |

2 | 2 | 1 | 0 |

${\mathit{z}}^{\mathit{y}}$ | ${\left({\mathit{z}}^{\mathit{y}}\right)}^{-1}$ |
---|---|

1 | 1 |

2 | 4 |

4 | 2 |

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**MDPI and ACS Style**

Sakalauskas, E.; Dindienė, L.; Kilčiauskas, A.; Lukšys, K.
Perfectly Secure Shannon Cipher Construction Based on the Matrix Power Function. *Symmetry* **2020**, *12*, 860.
https://doi.org/10.3390/sym12050860

**AMA Style**

Sakalauskas E, Dindienė L, Kilčiauskas A, Lukšys K.
Perfectly Secure Shannon Cipher Construction Based on the Matrix Power Function. *Symmetry*. 2020; 12(5):860.
https://doi.org/10.3390/sym12050860

**Chicago/Turabian Style**

Sakalauskas, Eligijus, Lina Dindienė, Aušrys Kilčiauskas, and Kȩstutis Lukšys.
2020. "Perfectly Secure Shannon Cipher Construction Based on the Matrix Power Function" *Symmetry* 12, no. 5: 860.
https://doi.org/10.3390/sym12050860