# A Novel Method for Constructing the S-Box Based on Spatiotemporal Chaotic Dynamics
## 1 Introduction  
With the development of communication technology [1; 2; 3; 4], encryption algorithms for data transmission security have attracted extensive attention. Block encryption algorithms play an important role in modern cryptographic systems. The substitution box (S-box) has been widely employed in many block cryptosystems; for instance, Data Encryption Standard (DES), International Data Encryption Algorithm (IDEA), and Advanced Encryption Standard (AES). S-box provides a chaotic effect for the cryptosystem, and its security strength determines the security strength of the whole cryptosystem. Hence, S-box is an important nonlinear component for the security of cryptographic schemes. Currently, strong S-boxes have received intensive attention. Many S-box construction methods have been proposed [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. In the literature, S-boxes should resist differential and linear attacks [5; 6; 7; 8; 9]. A new construction method for S-box was proposed in [10], which relies on exhaustive search, but the large values of \(n\) in the construction method result in it being time consuming. Recently, the theory of chaos has been broadly used for designing S-boxes due to the inherent features of the chaotic map; for instance, random-like behaviors and sensitivity to initial conditions. [PERSON] and [PERSON] [11] constructed secure S-boxes using exponential and logistic chaotic maps. [PERSON] et al. [12] proposed a chaos-based approach to the design of cryptographically-secure substitutions. [PERSON] et al. [13] designed \(8\times 8\) S-boxes using logistic and Baker chaotic maps. [PERSON] et al. [14] presented an extended method for constructing S-boxes by the use of the Chebyshev map and the 3D Baker map. An effective and dynamic method for constructing the S-box was developed by using the tent map in [15]. [PERSON] and [PERSON] [16] developed a generation method for S-boxes by using the random-like behaviors of theLorenz system. [PERSON] et al. [17] proposed an efficient S-box generation method by the use of a nonlinear chaotic algorithm. In [18], the Lorenz system and the Rossler system were employed to synthesize new S-boxes. [PERSON] [19] developed a new method of generating S-box by the use of chaotic tent map and composition method. [PERSON] et al. [20] designed S-boxes by using 3D four-wing autonomous chaos system. [PERSON] and [PERSON] [21] proposed an efficient method for generating S-boxes by the use of the chaotic sine map. In [22], the logistic-sine map was applied to construct S-boxes. In [23], S-boxes were generated by using the logistic-tent map and linear fractional transformation. [PERSON] et al. [24] presented a novel technique for the construction of strong S-boxes based on chaotic Lorenz systems. In [25], a novel method to construct strong bijective substitution-boxes based on a 5D hyperchaotic system was presented. However, in these aforementioned chaotic S-box construction schemes, the output Bit Independence Criterion (BIC) and ability to resist linear password attacks were not ideal. Moreover, the applied chaotic systems are low dimensional systems. For the low dimensional chaotic system, the dynamical behaviors may degrade under finite precision computation in modern computers. This degradation can lead to periodic trajectories that affect the randomness of chaotic sequences. Actually, [26; 27] have successfully cracked the encryption schemes that are based on the low dimensional chaotic system. To overcome the above drawbacks, we present a novel and efficient S-box construction method by using a spatiotemporal chaotic system. It can improve the BIC property of the S-box and its ability to resist linear password attacks. The spatiotemporal chaotic system contains multi-dimensional Lyapunov exponents that can resist the degradation under finite precision computation in modern computers. We summarize the three contributions of this work as follows:  
1.