Perfectly Secure Shannon Cipher Construction Based on the Matrix Power Function

Eligijus Sakalauskas,Kȩstutis Lukšys,Lina Dindienė,Aušrys Kilčiauskas

doi:10.3390/sym12050860

Eligijus Sakalauskas, Kȩstutis Lukšys + Show 2 more

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https://doi.org/10.3390/sym12050860

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Abstract

A Shannon cipher can be used as a building block for the block cipher construction if it is considered as one data block cipher. It has been proved that a Shannon cipher based on a matrix power function (MPF) is perfectly secure. This property was obtained by the special selection of algebraic structures to define the MPF. In an earlier paper we demonstrated, that certain MPF can be treated as a conjectured one-way function. This property is important since finding the inverse of a one-way function is related to an N P -complete problem. The obtained results of perfect security on a theoretical level coincide with the N P -completeness notion due to the well known Yao theorem. The proposed cipher does not need multiple rounds for the encryption of one data block and hence can be effectively parallelized since operations with matrices allow this effective parallelization.

Highlights

The modern design of block ciphers is based on the confusion–diffusion paradigm introduced by Claude Shannon ([1])
A direct implementation of the above paradigm is a substitution–permutation network (SPN), which is used for the block cipher construction when it is realized in multiple rounds, each of which uses a different sub-key derived from the original key
In order to increase the security of the Data Encryption Standard (DES), which is only 64 bits key length, the Tripple DES (TDES) algorithm was adopted by the ANSI committee X9.F.1 in 1998

Summary

Introduction

The modern design of block ciphers is based on the confusion–diffusion paradigm introduced by Claude Shannon ([1]). This paper presents a Shannon cipher based on the matrix power function defined over the certainly-selected algebraic structures. The proof that Shannon cipher based on the MPF defined over the certainly-selected algebraic structures is perfectly secure is presented. The main trend of the block cipher construction used the number of rounds for one data block encryption to achieve a good confusion and diffusion, providing a required level of security. These rounds are performed sequentially and there is no ability to parallelize computations. The proposed Shannon cipher can be effectively realized in multiprocessor computation devices

Mathematical Background

Security Analysis

Conclusions and Discussions