Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6969312, 11 pages

https://doi.org/10.1155/2017/6969312

## Chaotic S-Box: Intertwining Logistic Map and Bacterial Foraging Optimization

^{1}College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China^{2}Key Laboratory of Photonic and Electronic Bandgap Materials, Ministry of Education, School of Physics and Electronic Engineering, Harbin Normal University, Harbin 150025, China^{3}Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Ye Tian; moc.621@eynaitdsh

Received 16 June 2017; Revised 24 October 2017; Accepted 26 October 2017; Published 15 November 2017

Academic Editor: Maria L. Gandarias

Copyright © 2017 Ye Tian and Zhimao Lu. 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

As the unique nonlinear component of block ciphers, Substitution box (S-box) directly affects the safety of a cryptographic system. It is important and difficult to design strong S-box that simultaneously meets multiple cryptographic criteria such as bijection, nonlinearity, strict avalanche criterion (SAC), bit independence criterion (BIC), differential probability (DP), and linear probability (LP). Though many chaotic S-boxes have been proposed, the cryptographic performance of most of them needs to be further improved. A new chaotic S-box based on the intertwining logistic map and bacterial foraging optimization is designed in this paper. It firstly iterates the intertwining logistic map to construct many S-boxes and then applies a bacterial foraging optimization algorithm to find the optimal S-box. Moreover, bacterial foraging optimization algorithm considers the nonlinearity and differential uniformity as the fitness functions in the optimization process. We experiment that the proposed S-box can effectively resist multiple types of cryptanalysis attacks.

#### 1. Introduction

The dynamic developments in the multimedia industry and the Internet lead to a considerable amount of worry regarding the security of information transmitted over open or stored channels [1–3]. How to protect information from being unauthorized handled is becoming extremely crucial. Modern cryptography technique, in which block cipher algorithm is an important research direction, is an effective way to guarantee the safety of information, since then many researchers have developed a lot of block cipher algorithms. In a block cipher algorithm, Substitution box (S-box) is the only one nonlinear component [4], providing the block cipher system with necessary confusing and scrambling effect against attacks. Moreover, its cryptography security features directly determine the safety of the entire cipher performance [1]. Mathematically, an size of S-box is a nonlinear mapping , where represents the vector spaces of elements from GF, and we set in this paper.

Many papers on S-boxes have been published by scholars around the world over the past decades. In [5], Hussain and Gondal presented a design approach for S-boxes, which was an exhaustive search method; nevertheless, the performance of this procedure would become rather difficult with the increase of . Liu et al. [4] utilized near-bent Boolean functions of five variables to generate S-boxes to resist the differential attack; however, their algorithm was useful only to create an S-box of odd input bit number. Therefore, most of these approaches were inefficient and were unable to construct S-boxes that could meet multiple assessment requirements simultaneously.

Chaotic systems that satisfy the major requirements of cryptography properties such as diffusion and confusion are differentiated on the basis of their reactiveness to ergodicity, pseudorandomness, unpredictability, control parameters, and initial conditions; this makes chaotic systems particularly catch heaps of heed for cryptology [1–5]. Due to this matter, chaotic S-boxes have proved to be superior for encrypting a message.

For example, literature [6] proposed a four-step method of generating chaotic S-box based on discrete logistic map. It turned out that very simple chaotic maps and discretization procedure generated secure S-boxes. Literature [7] improved the work in [6] using bit extraction and Baker map. Furthermore, literature [8] proposed an S-box design approach based on iteration discrete chaotic that had high immunity to the differential cryptanalysis. Using three-dimensional chaotic Baker map, literature [9] constructed an S-box that approximately fulfilled all the criteria for a cryptographically strong S-box.

However, a simple chaos system also has many defects; for example, the implementation of the chaos on a computer is affected by the limited precision; the time series outputted by the simple chaotic system generally cannot reach the theoretically complete random, resulting in the problem that the pseudorandom sequence appears periodicity [10].

To cope with these problems, many complex chaotic maps based S-boxes have been presented in recent years. For example, literature [11] indicated the pseudorandomness and complexity of binary sequences produced by the Lorenz system and Chebyshev map. Literature [12] designed a new pseudorandom number generator by mixing the couple map lattice technology and the chaos iteration technology. Literature [13] proposed a secure pseudorandom number generator three-mixer. Khan et al. presented a complex chaotic S-box construction method that could provide better security in terms of resistance against various attacks by deploying the 2D Henon chaotic map and skew tent map [14]. Ahmad et al. proposed a method for synthesizing cryptographically efficient chaotic S-box, which integrated four 1D chaotic systems, namely, logistic maps and cubic maps, to modulate the normal system trajectories of the other [15]. Peng et al. designed a novel approach for dynamically generating S-boxes using a spatiotemporal chaotic system, which mapped the key to system parameters and generated the hyperchaotic sequences to construct S-boxes [16].

By increasing the complexity of the chaotic system, these complex algorithms obtained S-boxes with the higher security level to some extent. Nevertheless, the performance gap between many of these chaotic S-boxes and classic ones still exists; for example, few chaos-based S-boxes can achieve the high performance like the one used in advanced encryption standard (AES) [17].

Compared with other intelligent optimization algorithms, bacterial foraging optimization algorithm (BFO) has a group of intelligence and can carry out parallel search. Besides, it may be easy to jump out of the local minimal solution; thus, it can find solutions of higher quality. Due to these advantages, it has been widely used in some research fields. For example, Abd-Elazim and Ali proposed an optimization algorithm BFOA for controlling the damping of the power system’s electromechanical oscillations [18]. On the basis of literature [18], Ali and Abd-Elazim proposed a BFOA based Load Frequency Control (LFC) for the suppression of oscillations in power system [19]. In addition, Abd-Elazim and Ali developed an optimization algorithm BSO, which synergistically coupled the BFOA with the particle swarm optimization algorithm for the optimal design of the TCSC damping controller. Specifically, they transformed the controller design problem into an optimization problem, and the BSO was developed to find the optimal controller parameters [20]. Sur and Shukla also presented a discrete adaptive BFO algorithm, which could be applied to discrete search domains and various multidimensional problems [21]. Furthermore, to optimize a power network problem, Tripathy and Mishra proposed an improved BFO algorithm. In this work, the power network problem was formulated as a multiobjective multivariable problem, and the improved BFO was applied to solve this problem [22].

In this paper, a new scheme for designing an S-box is presented. Unlike other chaos-based algorithms that generate strong S-boxes by using the random distribution property of chaotic maps, we divide the process of designing an S-box into two steps. Firstly, we generate many S-boxes by iterating the chaotic map. Secondly, we apply a genetic algorithm, the evaluation function of which adopts the nonlinearity and differential uniformity to improve the performance of the generated S-box. We show via simulation that our scheme can generate stronger S-box. Abbreviations section shows some abbreviations of technical terms involved in this paper.

#### 2. Preliminary Work

##### 2.1. Intertwining Logistic Map

In this section, we will introduce a chaotic map, the intertwining logistic map [23], which is defined as follows:where , are the system parameters with the ranges , , , . Figures 1(a), 1(b), and 1(c) show the chaotic bifurcation diagrams of the intertwining Logistic map when , , . Figure 1(d) shows the chaotic bifurcation diagram of the logistic map. Figure 2(a) depicts the chaotic attractor diagram of the intertwining Logistic map when , , = 38.5, and . Figure 2(b) depicts the chaotic attractor diagram of the logistic map. In the intertwining logistic map system, the sequence distribution becomes more uniform, and, more importantly, empty windows are eliminated. Remarkably, comparing with a simple logistic map, the action of a intertwining logistic chaotic map is more complex, and the sequence distribution of it is more uniform.