Security and Communication Networks

Volume 2017 (2017), Article ID 5101934, 16 pages

https://doi.org/10.1155/2017/5101934

## A Novel Construction of Substitution Box Involving Coset Diagram and a Bijective Map

^{1}Department of Mathematics, University of Education Lahore, Jauharabad Campus, Jauharabad, Pakistan^{2}Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan^{3}Department of Mathematics, Government College University Faisalabad, Faisalabad, Pakistan^{4}Department of Basic Sciences, University of Engineering and Technology, Taxila, Punjab, Pakistan^{5}Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan^{6}Department of Information Technology, University of Education Lahore, Jauharabad Campus, Jauharabad, Pakistan

Correspondence should be addressed to Abdul Razaq

Received 15 August 2017; Accepted 10 October 2017; Published 20 November 2017

Academic Editor: Zheng Yan

Copyright © 2017 Abdul Razaq et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The substitution box is a basic tool to convert the plaintext into an enciphered format. In this paper, we use coset diagram for the action of on projective line over the finite field to construct proposed S-box. The vertices of the cost diagram are elements of which can be represented by powers of , where is the root of irreducible polynomial over . Let denote the elements of which are of the form of even powers of . In the first step, we construct a matrix with the elements of in a specific order, determined by the coset diagram. Next, we consider defined by to destroy the structure of . In the last step, we apply a bijective map on each element of the matrix to evolve proposed S-box. The ability of the proposed S-box is examined by different available algebraic and statistical analyses. The results are then compared with the familiar S-boxes. We get encouraging statistics of the proposed box after comparison.

#### 1. Introduction

In secure communication, the role of the nonlinear component for block ciphers (substitution box) is of significant importance. The concept of substitution box was given by Shannon in 1949 [1]. In order to create confusion during the process of enciphering the digital data, substitution box plays a central role [2]. If the S-box is not good, it means one has to compromise on the quality of encryption. The strength of the S-box affirms the capability of block ciphers. Several attempts have been made to increase the quality of the S-box. In order to assess the properties of well-known S-boxes, the cryptographers have drawn attention to the literature. Different techniques have been developed to inspect the statistical and algebraic structure of S-boxes. These analyses include linear approximation probability (LP) method, bit independence criterion (BIC), majority logic criterion (MLC), strict avalanche criterion (SAC), nonlinearity method, and differential approximation probability (DP) method.

In this paper, we establish a novel technique to construct substitution boxes by coset diagrams and bijective maps.

#### 2. Coset Diagrams for Modular Group

The modular group, denoted by , has a finite presentation , where and are linear fractional transformations which map to and , respectively. Coset diagrams ([3–7]) are the graphical representation of the action of on , where is a prime. Since the order of is three, its three cycles are represented by triangles. The vertices of the triangles, which are elements of , are permuted anticlockwise by . Any two of vertices of the triangles are joined by an edge which represents . The heavy dots are used to denote fixed points of and , if they exist.

Consider the action of modular group on (Figure 1). The permutation representations of and can be calculated by and .The action of is not possible on , because image of 0 under does not belong to . Therefore, we choose for the action of instead of .