## 2 Security Requirement and S-Box Design Methods
### Combination Method
#### 2.2.2 Other S-Boxes Proposal  
Aside from the S-boxes used in block ciphers, there exist a large number of S-box construction proposals in the literature. We have reviewed, analyzed and categorized all the S-box proposals that we have collected into random generation, heuristic, mathematical function and combination methods. Random Generation  
There are several S-box constructions that generate dynamic S-boxes which depend on a key. Among others are the proposal in [45; 46; 47; 48]. In particular, their construction begins with an initial S-box, then a string of keys is generated to permute the content of the initial S-box. Thus, a new S-box is constructed. Various methods are used to generate the keys such as random generation, a pseudo-random number generator (PRNG) or a round key obtained from the RC4 algorithm [49]. [PERSON] et al. [50] also proposed an S-box construction based on random generation. The initial S-box is represented as eight component functions. The first three component functions are deterministically constructed, such that the highest nonlinearity for each Boolean function is found. Then, the other five component functions are generated randomly to complete the S-box. However, the S-box construction using random generation has received less attention among researchers. This might be because of the unfavorable cryptographic properties exhibited by this approach. Heuristic  
The heuristic approach has received more attention from researchers, especially on chaos-based S-box construction. This might be because chaos is a stochastic process in nonlinear dynamic systems, thus satisfying the need for nonlinearity in block ciphers. Among the popular chaotic maps used are the Chebyshev map [51; 52], Baker map [53; 54], Lorenz systems [55; 56], logistic map [57; 58], neural network [59; 60], piecewise linear chaotic map [61; 62], tent map [63; 64] and Van Der Pol oscillator [65]. In general, chaos-based S-box construction starts with the selection of an initial value in selected chaotic map. Then, they iterate the chaotic map and use their defined technique to obtain an integer that lies between 0 and 255. This procedure is repeated to generate another integer until all 256 elements of the \(8\times 8\) S-box are fulfilled. Furthermore, there are several S-box proposals that we categorized as heuristic approaches, such as the S-box construction using hill climbing [43; 66], Latin square [67], analytical approach [68], genetic algorithm [69; 70], simulated annealing [71; 72], cellular automata [73; 74], ant colony optimization [75], artificial immune system [76] and bee waggle dance [77]. Nature-inspired systems, such as genetic algorithms, work in reverse, i.e., [70], gradient descent [43] (i.e., modified hill climbing), artificial immune system [76] and bee waggle dance [77] show encouraging developments in S-box construction using the heuristic approach. Mathematical Function  
The main criterion for categorizing S-box proposals as mathematical functions is that designers use the same transformation as AES [8]. This includes either revising the transformation [78; 79; 80] itself by using different irreducible polynomials or by adding another affine transformations [81; 82; 83; 84] in their construction. Moreover, some S-box proposals using mathematical functions other than AES were found in the literature, such as linear fractional transformation or projective general linear group [85; 86; 87], trace-representation polynomial function [88], gray code encode [89], semi-fields pseudo-extensions [90] and using finite field power functions [91]. All these S-box constructions provide encouraging results and can be categorized among cryptographically strong S-boxes. [PERSON] et al. proposed 2-step tweaking [28] and 4-step tweaking [92] in order to avoid linear redundancy and preserve the involution properties of an S-box, respectively.