Liyan Liu , Cheng Zhang and Yingqian Zhang
Constructing S-Box Using Fractional-Order Nonlinear Coupled Map and Elementary Cellular Automata
Abstract: A novel substitution box (S-box) construction method is proposed by using a fractional-order nonlinear coupled map lattices chaotic system combined with an elementary cellular automaton. An original S-box is produced by utilizing chaos sequences from the fractional-order nonlinear coupled map lattices spatiotemporal chaotic system. The S-box elements are then rearranged by applying different state values of various cells in the elementary cellular automaton. These chaotic sequences with independent properties are subsequently used to perturb the S-box’s elements. Comparative results with previous schemes indicate that the constructed S-box demonstrates enhanced security performance and could be effectively utilized in the development of block encryption algorithms.
Keywords: Elementary Cellular Automaton , Fractional-Order Nonlinear Coupled Map Lattices , S-Box
1. Introduction
Substitution boxes (S-boxes) are of vital importance in many block cipher algorithms, including the data encryption standard, international data encryption algorithm, and advanced encryption standard. As the sole nonlinear component in these algorithms, S-boxes provide confusion and diffusion, which are essential for enhancing security.
A variety of design methods for S-boxes have emerged in recent years [1-18]. One notable approach is to construct S-boxes by employing spatiotemporal chaotic systems, which have garnered extensive attention, due to their superior dynamic behaviors compared to traditional chaotic systems. These behaviors include longer cycles, higher positive Lyapunov exponents, and enhanced randomness. For instance, Yuan et al. [1] designed an algorithm to generate S-boxes based on spatiotemporal chaos. Similarly, Peng et al. [2] dynamically designed S-boxes utilizing a spatiotemporal chaotic system. Liu et al. [4] designed S-boxes using nonlinear coupled map lattices (NCML) and coupled map lattices (CML) [5]. In [6], the dynamic S-box values were initialized using the logistic dynamic coupled map lattice spatio-temporal chaos system. However, these spatiotemporal chaos systems are fundamentally rooted in classical integer-order logistic maps. In recent years, fractional-order spatiotemporal chaotic systems [19,20] have gained popularity, due to their unique characteristics, such as a broader state amplitude range for ergodicity and a wider parameter range, leading to an expanded key space for encryptions. This study introduces the fractional-order nonlinear coupled map lattices (FONCML) spatiotemporal chaotic system [19] as a novel approach to designing S-boxes. To enhance the robustness of our S-box design, we combine the FONCML spatiotemporal chaotic system with another dynamic system instead of relying on it alone. Cellular automata [21], recognized as intelligent algorithms, have attracted widespread interest [22,23]. The elementary cellular automaton (ECA) [24], a simple one-dimensional cellular automaton demonstrates excellent dynamic performance due to its discrete nature in both time and space, with calculations occurring in the binary domain. This characteristic mitigates issues of dynamic degradation when utilized in cryptographic systems. In addition, the properties of pseudo-randomness and long periodicity of the generated sequence can be ensured if suitable iteration rules are chosen. Due to the unpredictable nature of input and iteration rules, mathematical analysis becomes exceedingly challenging, which can augment the complexity of S-box construction. For example, Zhao et al. [12] designed S-boxes using Lorenz and skew tent systems in conjunction with ECA, achieving good security.
Building upon this analysis we explore an innovative approach to constructing S-boxes by employing both the FONCML spatiotemporal chaos system and ECA. The key results of our research are as below.
· The FONCML spatiotemporal chaotic system is utilized to construct S-boxes due to its high dimensional characteristics, broad state amplitude for ergodicity and wide parameter range. These attributes enhance encryption performance and security of S-boxes.
· The using of ECA complicates the mathematical analysis of the constructed S-boxes.
· The S-box constructed in this study exhibits greater nonlinearity compared to those designed using only spatiotemporal chaos systems [4,5].
· The combination of the FONCML spatiotemporal system and ECA ensures that the reliance on the FONCML spatiotemporal system alone reduced, thereby improving the security performance of the S-boxes.
· Compared to previous studies, our S-box demonstrates superior performance. The S-box construction method using both the FONCML spatiotemporal system and ECA is highly recommended.
The layout of the remaining sections of this paper is as described below. Section 2 outlines the FONCML system and ECA. An innovative S-box construction method is introduced in Section 3. A performance analysis of the constructed S-box is provided in Section 4. Section 5 concludes the study.
2. Preliminaries
2.1 FONCML Spatiotemporal System
The FONCML spatiotemporal system [19] is described as
(1)
[TeX:] $$\begin{equation} \begin{aligned} x_{t t+1}(l)=(1-\varepsilon) & \left(x_{t t}(l)+\beta \mu x_{t t}(l)\left(1-x_{t t}(l)\right)\right) \\ & +\varepsilon / 2\left\{\begin{array}{l} \left(x_{t t}(g)+\beta \mu x_{t t}(g)\left(1-x_{t t}(g)\right)\right) \\ +\left(x_{t t}(s)+\beta \mu x_{t t}(s)\left(1-x_{t t}(s)\right)\right) \end{array}\right\} \end{aligned} \end{equation}$$The system consists of N lattices. In (1), tt represents time, ε denotes the coupling coefficient [TeX:] $$\begin{equation} (0 \leq \varepsilon \leq 1) \end{equation}$$, [TeX:] $$\begin{equation} \beta=r^\alpha / \Gamma(1+\alpha) \end{equation}$$, α represents a fraction, [TeX:] $$\begin{equation} \Gamma(\cdot) \end{equation}$$ is the Eulerian gamma-function and r is a fraction used for segmentation. The variables l, g, and s correspond to lattices [TeX:] $$\begin{equation} (1 \leq l,g,s \leq N) \end{equation}$$ and meet the following Arnold cat map:
(2)
[TeX:] $$\begin{equation} \left[\begin{array}{l} g \\ s \end{array}\right]=\left[\begin{array}{cc} 1 & b \\ a & a b+1 \end{array}\right]\left[\begin{array}{l} l \\ l \end{array}\right](\bmod N), \end{equation}$$which a and b represent the parameters.
The FONCML system exhibits exceptional chaotic properties when parameters are set to appropriate values. Compared to the integer-order spatiotemporal chaotic systems previously mentioned, the FONCML spatiotemporal chaotic system has a higher percentage of lattices that exhibit chaotic behavior and a larger parameter range conducive to chaos. In addition, its bifurcation diagrams do not feature periodic windows. These novel characteristics make the FONCML spatiotemporal chaotic system particularly suitable for cryptography [19].
2.2 ECA
ECA is made of a linear arrangement of cells, with two possible states represented as {0,1}. Each cell has a half-diameter of 1, with the neighboring cells as its left and right neighbors. In ECA, for a cell, the states of its neighbors and its current state determine its next state, as expressed by where t represents time, [TeX:] $$\begin{equation} c(c \in[1, M]) \end{equation}$$ is the index of a cell, At(c) is the state of cell c at time t, ir represents an iteration rule of ECA and f denotes a Boolean function.
(3)
[TeX:] $$\begin{equation} A_{t+1}(c)=f_{i r}\left(A_t(c-1), A_t(c), A_t(c+1)\right), \end{equation}$$ECA’s boundary conditions are periodic, as expressed in the following equation.
(4)
[TeX:] $$\begin{equation} \left\{\begin{array}{l} c+1=1, c=M \\ c-1=M, c=1 . \end{array}\right. \end{equation}$$Fig. 1 illustrates the iterative process when ECA employs a cyclic boundary condition.
According to (3), f is a Boolean function defined over a binary field, where the input consists of three state values and the output includes one state value. This means that the input comprises three binary numbers, whereas the output contains one. The possibility of 23=8 inputs and two outputs yields 28=256 distinct ECA rules. The iteration rules of ECA can be categorized into five types based on the nature of their results: invalid, periodic, fixed point, global chaos and local chaos rules [25,26]. Among these, the iteration results under global chaos rules exhibit long period and chaotic characteristics. The study in [26] identifies 34 types of global chaos rules, as depicted in Fig. 2.
Table 1 lists all output values corresponding to any input values of the function f169. Because the Boolean function f of ECA has multiple inputs that yield the same output, it is classified as a nonlinear and irreversible function. However, it can generate correct results if iteration rules and the initial value of the ECA are determined. When the input and iteration rules are unknown, mathematical becomes quite challenging. This characteristic is relevant to S-box construction, as it complicates mathematical analysis.analysis
3. S-Box Construction Method
This section describes our development of an innovative S-box construction method utilizing the FONCML system and ECA. To construct the S-box, we first set the parameters of N=100, ϵ=0.8, μ=10, r=0.25, α=0.95, M=200. The specific steps are as below:
Step 1. Compute (1) and obtain a matrix [TeX:] $$\begin{equation} x_{t t}(j)(t t \in[1,2000], j \in[1, N]) \end{equation}$$.
Step 2. Construct a new sequence y as follows:
(5)
[TeX:] $$\begin{equation} y(i)=\bmod \left(\text { floor }\left(x_{200+i}(66) \times 10^{14}\right), 512\right),(i \in[1,256]) . \end{equation}$$Step 3. Obtain an index sequence z(1×256) of the sequence y(1×256) whose elements are sorted in ascending order.
Step 4. Convert each value in the sequence xtt(66) into an 8-bit unsigned integer.
Step 5. Convert these unsigned integers into binary numbers and obtain a binary sequence u.
Step 6. Extract M bits from the 660th in the sequence u and use them as values from A_1 (1) to A_1 (M).
Step 7. Compute (3) and obtain a matrix [TeX:] $$$$.
Step 8. Extract 60 to 67 columns from the matrix At>(i) and convert the binary number of each row into a decimal number; then, add 1to achieve a new sequence v(10000×1).
Step 9. Swap the values in sequence z using (6).
Step 10. Reconvert z(1×256) into a matrix S(16×16).
Step 11. Compute u and w using (7) and (8), respectively.
(7)
[TeX:] $$\begin{equation} \begin{gathered} u=\operatorname{Mod}(\lfloor(F O N C M L(\operatorname{Mod}(S((g-1) \times 16+l, 100)+1,600+(g-1) \times 16 \\ \left.\left.\left.+l) \times 10^{14}\right)\right\rfloor, 16\right)+1, \end{gathered} \end{equation}$$
(8)
[TeX:] $$\begin{equation} w=\operatorname{Mod}\left(\left[\left(\operatorname{FONCML}(m, n) \times 10^{14}\right)\right], 16\right)+1, \end{equation}$$where [TeX:] $$\begin{equation} g, l \in[1,16] \end{equation}$$, m and n meet the following Arnold cat map.
(9)
[TeX:] $$\begin{equation} \left[\begin{array}{l} m \\ n \end{array}\right]=\left[\begin{array}{cc} 1 & b \\ a & a b+1 \end{array}\right]\left[\begin{array}{l} g \\ l \end{array}\right](\bmod N) . \end{equation}$$Step 12. Swap the values in matrix S using (10).
The final result, S, is the constructed S-box.
4. Performance Analysis
The following six criteria are commonly used to evaluate S-boxes [4,5]. The constructed S-box using the proposed method is showcased in Table 2.
Table 2.
| 230 | 223 | 89 | 72 | 27 | 200 | 99 | 28 | 171 | 169 | 203 | 91 | 240 | 232 | 215 | 114 |
| 207 | 177 | 229 | 145 | 20 | 74 | 111 | 216 | 151 | 124 | 63 | 75 | 32 | 77 | 175 | 110 |
| 127 | 113 | 93 | 23 | 246 | 79 | 253 | 224 | 82 | 199 | 15 | 78 | 21 | 248 | 170 | 122 |
| 178 | 54 | 33 | 135 | 19 | 35 | 14 | 42 | 80 | 119 | 38 | 237 | 65 | 58 | 43 | 115 |
| 244 | 183 | 150 | 139 | 9 | 143 | 76 | 12 | 6 | 140 | 123 | 31 | 100 | 105 | 17 | 172 |
| 57 | 221 | 71 | 103 | 161 | 61 | 156 | 92 | 157 | 40 | 37 | 222 | 11 | 13 | 142 | 152 |
| 195 | 16 | 197 | 56 | 166 | 120 | 44 | 208 | 242 | 204 | 256 | 249 | 206 | 102 | 10 | 168 |
| 69 | 49 | 5 | 2 | 59 | 87 | 220 | 238 | 132 | 159 | 191 | 133 | 164 | 160 | 188 | 217 |
| 3 | 192 | 34 | 243 | 245 | 213 | 117 | 198 | 189 | 88 | 193 | 22 | 233 | 252 | 154 | 190 |
| 210 | 162 | 66 | 109 | 180 | 155 | 7 | 26 | 70 | 98 | 62 | 128 | 50 | 146 | 186 | 112 |
| 118 | 247 | 236 | 107 | 225 | 108 | 48 | 29 | 138 | 136 | 24 | 121 | 241 | 211 | 167 | 149 |
| 96 | 41 | 126 | 25 | 184 | 187 | 90 | 251 | 81 | 94 | 227 | 30 | 163 | 125 | 51 | 83 |
| 4 | 97 | 254 | 84 | 64 | 174 | 228 | 8 | 173 | 231 | 18 | 147 | 239 | 53 | 95 | 55 |
| 106 | 137 | 205 | 250 | 181 | 235 | 47 | 226 | 1 | 129 | 234 | 39 | 45 | 60 | 36 | 194 |
| 131 | 209 | 67 | 144 | 86 | 158 | 182 | 185 | 141 | 218 | 46 | 176 | 148 | 68 | 85 | 214 |
| 73 | 116 | 134 | 219 | 104 | 153 | 202 | 165 | 196 | 130 | 212 | 52 | 101 | 179 | 201 | 255 |
4.1 Property of the Bijective
If an S-box satisfies the formula [TeX:] $$\begin{equation} h w\left(\sum_{i=1}^n a_i f_i\right)=2^{n-1} \end{equation}$$, in which [TeX:] $$\begin{equation} h w(\cdot) \end{equation}$$ expresses the Hamming weight, [TeX:] $$\begin{equation} a_i \in\{0,1\} \end{equation}$$ and fi is a Boolean function, then the S-box is bijective. Based on the proposed method, the constructed S-box demonstrates bijective performance.
4.2 Nonlinearity
Nonlinearity is a critical criterion for assessing the security of a cryptosystem, as the nonlinear structure significantly influences its overall safety. The formula for determining nonlinearity Nf is expressed as
(11)
[TeX:] $$\begin{equation} N_f=2^{n-1}\left(1-2^{-n} \max _{\omega \in G F\left(2^n\right)}\left|S_{(f)}(\omega)\right|\right), \end{equation}$$in which the definition of the S(f)(ω) of f(x) is
(12)
[TeX:] $$\begin{equation} S_{(f)}(\omega)=\sum_{\omega \in G F\left(2^n\right)}(-1)^{f(x) \oplus x \cdot \omega}, \end{equation}$$where x⋅ω is the dot product of x and ω.
Table 3 lists the results of the nonlinearity analysis. For the constructed S-box, the smallest nonlinearity value was 102, the largest nonlinearity value was 108, and the mean was 105.75, indicating commendable nonlinearity. The constructed S-box showed nonlinearity comparable to other S-boxes [12,14-18] and outperformed those that were constructed using spatiotemporal chaotic system alone [4,5].
Table 3.
| S-boxes | Nonlinearity | ||
|---|---|---|---|
| Max | Min | Average | |
| Proposed method | 108 | 102 | 105.75 |
| Liu et al. [4] | 108 | 102 | 104.50 |
| Liu and Lei [5] | 108 | 102 | 104.25 |
| Javeed et al. [7] | 110 | 106 | 107.50 |
| Jiang and Ding [9] | 108 | 104 | 106.75 |
| Zhao et al. [12] | 108 | 104 | 106.00 |
| Zhang et al. [14] | 112 | 104 | 106.50 |
| Hua et al. [15] | 108 | 102 | 105.25 |
| Liu et al. [16] | 108 | 100 | 105.00 |
| Aslam et al. [17] | 108 | 104 | 106.00 |
| Zhou et al. [18] | 108 | 104 | 105.25 |
4.3 Strict Avalanche Criterion
When an input bit has its value complemented, each output bit has a 50% chance of being changed, indicating compliance with the strict avalanche criterion (SAC). An independent matrix is often employed to represent the SAC. In fact, a well-constructed S-box possesses a dependence matrix with elements close to 0.5.
The formula for estimating the dependence matrix offsets is given as
(1)
[TeX:] $$\begin{equation} S(f)=1 / n \sum_{1 \leq i \leq n} \sum_{1 \leq j \leq n}\left|1 / 2-P_{i, j}(f)\right|, \end{equation}$$where [TeX:] $$\begin{equation} P_{i, j}(f)=2^{-n} \sum_{x \in B^n} f_j(x) \oplus f_j\left(x \oplus e_i\right), e_i=\left[\delta_{i, 1} \delta_{i, 2} \cdots \delta_{i, n}\right]^T \end{equation}$$, [TeX:] $$\begin{equation} [\cdot]^T \end{equation}$$ indicates a matrix transpose, and [TeX:] $$\begin{equation} \delta_{i, j}= \begin{cases}1, & i=j \\ 0, & i \neq j\end{cases} \end{equation}$$.
Table 4.
| 0.5000 | 0.4688 | 0.5313 | 0.4375 | 0.4531 | 0.5000 | 0.5000 | 0.4688 |
| 0.4375 | 0.5469 | 0.3906 | 0.4219 | 0.5000 | 0.5625 | 0.4063 | 0.5156 |
| 0.5313 | 0.4531 | 0.5313 | 0.5156 | 0.5313 | 0.5469 | 0.5313 | 0.4531 |
| 0.5000 | 0.5313 | 0.5000 | 0.5156 | 0.4844 | 0.5313 | 0.4844 | 0.5156 |
| 0.3906 | 0.5000 | 0.5156 | 0.5000 | 0.5156 | 0.5156 | 0.6094 | 0.5469 |
| 0.5938 | 0.4688 | 0.5469 | 0.4844 | 0.5156 | 0.5313 | 0.4531 | 0.4375 |
| 0.4688 | 0.5313 | 0.5313 | 0.3750 | 0.5000 | 0.4844 | 0.4375 | 0.6094 |
| 0.5469 | 0.5156 | 0.4844 | 0.5469 | 0.5156 | 0.3750 | 0.5000 | 0.4531 |
Table 4 outlines the dependence matrix of the constructed S-box. All values fall between 0.6094 and 0.3750, and an average of 0.4968 is near the desired value of 0.5, indicating that the S-box is compliant with the SAC. Comparisons of various S-boxes about SAC are outlined in Table 5, showing that the constructed S-box displays superior SAC characteristics.
Table 5.
| S-boxes | SAC average |
|---|---|
| Proposed method | 0.4968 |
| Liu et al. [4] | 0.4980 |
| Liu and Lei [5] | 0.5083 |
| Javeed et al. [7] | 0.4997 |
| Jiang and Ding [9] | 0.4975 |
| Zhao et al. [12] | 0.5014 |
| Zhang et al. [14] | 0.5060 |
| Hua et al. [15] | 0.5351 |
| Liu et al. [16] | 0.5002 |
| Aslam et al. [17] | 0.4983 |
| Zhou et al. [18] | 0.5070 |
4.4 Output Bits Independence Criterion
The output bits independence criterion (BIC) asserts that any pair of output bits should be mutually independent when an input bit is complemented. When the value of [TeX:] $$\begin{equation} f_m \oplus f_n \end{equation}$$ fulfills the SAC and is highly nonlinear, the S-box possesses the BIC property, where [TeX:] $$\begin{equation} f_m, f_n(m \neq n) \end{equation}$$ are Boolean functions and represent two distinct output bits of the S-box.
For the constructed S-box, its BIC-nonlinearity values are listed in Table 6, its BIC-SAC values are listed in Table 7, and its mean values for BIC-nonlinearity and BIC-SAC were 103.57 and 0.5001, respectively. The comparisons of various S-boxes for BIC are summarize in Table 8, showing that the constructed S-box are consistent with other S-boxes [4,5,7,9,12,15,16]. Thus, the constructed S-box demonstrates a strong BIC property.
Table 6.
| - | 98 | 100 | 104 | 108 | 104 | 104 | 104 |
| 98 | - | 104 | 100 | 102 | 106 | 104 | 100 |
| 100 | 104 | - | 102 | 102 | 104 | 108 | 106 |
| 104 | 100 | 102 | - | 104 | 104 | 104 | 100 |
| 108 | 102 | 102 | 104 | - | 106 | 100 | 106 |
| 104 | 106 | 104 | 104 | 106 | - | 108 | 104 |
| 104 | 104 | 108 | 104 | 100 | 108 | - | 104 |
| 104 | 100 | 106 | 100 | 106 | 104 | 104 | - |
Table 7.
| - | 0.4805 | 0.5117 | 0.4883 | 0.5176 | 0.4941 | 0.4922 | 0.5137 |
| 0.4805 | - | 0.5039 | 0.5117 | 0.4980 | 0.5020 | 0.5195 | 0.5117 |
| 0.5117 | 0.5039 | - | 0.5254 | 0.5020 | 0.4941 | 0.5078 | 0.5039 |
| 0.4883 | 0.5117 | 0.5254 | - | 0.4844 | 0.4941 | 0.4863 | 0.5176 |
| 0.5176 | 0.4980 | 0.5020 | 0.4844 | - | 0.4922 | 0.4941 | 0.4785 |
| 0.4941 | 0.5020 | 0.4941 | 0.4941 | 0.4922 | - | 0.4902 | 0.4785 |
| 0.4922 | 0.5195 | 0.5078 | 0.4863 | 0.4941 | 0.4902 | - | 0.5078 |
| 0.5137 | 0.5117 | 0.5039 | 0.5176 | 0.4785 | 0.4785 | 0.5078 | - |
Table 8.
| S-boxes | BIC-SAC | BIC-nonlinearity |
|---|---|---|
| Proposed method | 0.5001 | 103.57 |
| Liu et al. [4] | 0.5079 | 104.64 |
| Liu and Lei [5] | 0.5008 | 103.07 |
| Javeed et al. [7] | 0.5048 | 104.64 |
| Jiang and Ding [9] | 0.5022 | 103.57 |
| Zhao et al. [12] | 0.5001 | 103.79 |
| Zhang et al. [14] | - | 105.60 |
| Hua et al. [15] | 0.5000 | 103.21 |
| Liu et al. [16] | 0.5038 | 103.00 |
| Aslam et al. [17] | 0.5002 | 106.00 |
| Zhou et al. [18] | 0.5039 | 102.72 |
4.5 Equiprobable Input/Output XOR Distribution
The ability of an S-box to resist differential attacks can be analyzed by examining the imbalances in its input/output XOR distribution table. This analysis typically utilizes the differential approximation probability (DP) and the differential approach table to assess the input/output XOR distribution. DP is defined as
(14)
[TeX:] $$\begin{equation} D P_h=\max _{\Delta x \neq 0, \Delta y}\left(\#\{x \in X \mid h(x) \oplus h(x \oplus \Delta x)=\Delta y\} / 2^n\right), \end{equation}$$where the set X includes every possible input, and 2n indicates the quantity of elements in the set X. A well-constructed S-box will have a DPh value that is small as possible.
For the constructed S-box, its differential approach table is delineated in Table 9. The maximum DP value observed to be 12, indicating that the S-box has a high level of resistance towards differential attacks. Comparisons of maximum DP values for various DPh are provided in Table 10. The constructed S-box shows a maximum DPh value consistent with most other S-boxes, confirming its robust DP feature.
Table 9.
| 6 | 8 | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 6 | 6 | 8 | 6 |
| 4 | 6 | 6 | 6 | 8 | 8 | 10 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 6 | 6 |
| 8 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 6 | 6 | 6 | 8 | 6 | 6 | 6 | 6 |
| 6 | 6 | 8 | 8 | 8 | 6 | 4 | 8 | 6 | 8 | 8 | 8 | 6 | 6 | 8 | 6 |
| 8 | 6 | 12 | 6 | 6 | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 6 | 8 | 6 |
| 6 | 6 | 8 | 6 | 6 | 8 | 6 | 6 | 8 | 6 | 6 | 8 | 8 | 6 | 6 | 6 |
| 6 | 8 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 6 | 6 | 6 | 6 | 8 | 6 | 6 |
| 6 | 6 | 8 | 6 | 6 | 8 | 6 | 6 | 6 | 8 | 6 | 6 | 6 | 6 | 8 | 8 |
| 6 | 6 | 6 | 12 | 8 | 8 | 8 | 10 | 10 | 6 | 6 | 6 | 6 | 6 | 8 | 6 |
| 6 | 6 | 6 | 6 | 6 | 8 | 8 | 6 | 6 | 6 | 6 | 8 | 8 | 6 | 6 | 8 |
| 8 | 8 | 6 | 6 | 10 | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 6 | 8 |
| 6 | 6 | 10 | 6 | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 4 |
| 8 | 8 | 6 | 8 | 8 | 6 | 6 | 6 | 6 | 8 | 6 | 6 | 8 | 6 | 6 | 8 |
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 8 | 8 | 6 | 8 | 8 | 8 | 6 |
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 6 | 8 | 6 | 8 | 6 | 6 | 6 | 4 |
| 6 | 8 | 6 | 6 | 6 | 8 | 6 | 6 | 10 | 6 | 6 | 4 | 6 | 6 | 6 | 0 |
Table 10.
| S-boxes | MaxDP |
|---|---|
| Proposed method | 0.0469 |
| Liu et al. [4] | 0.0469 |
| Liu and Lei [5] | 0.0469 |
| Zhao et al. [12] | 0.0391 |
| Aslam et al. [17] | 0.0380 |
| Zhou et al. [18] | 0.0390 |
4.6 Linear Approximation Probability
An event’s largest imbalance value is determined by the linear approximation probability (LP) defined as
(15)
[TeX:] $$\begin{equation} L P=\max _{\alpha_1, \alpha_2 \neq 0}\left(\#\left\{x \in X \mid x \cdot \alpha_1=h(x) \cdot \alpha_2\right\} / 2^n-1 / 2\right), \end{equation}$$where a1 represents the input mask, and a2 is the output mask. Here, the input bits parity selected by a1 matches the output bits parity selected by a2.
Table 11 lists the comparative results for the maximum LP of different S-boxes. For maximum LP, the constructed S-box is comparable to other S-boxes. Accordingly, it demonstrates strong LP performance.
5. Conclusion
An innovative S-box construction method is developed by utilizing the FONCML spatiotemporal chaotic system and ECA in this study. The FONCML system is characterized by broad range of state amplitude for ergodicity and wide parameter range, which contribute to the development of S-boxes with strong encryption capabilities and high security. In addition, ECA adds complexity, making mathematical analysis of S-boxes challenging. The integration of the FONCML system and ECA enhances the overall complexity of the S-box. During the construction phase of the S-box, chaos sequences created by the FONCML system are utilized to establish its initial value. Different state values for various cells in the ECA are then employed to rearrange the elements of the S-box. A perturbing operation is finally executed utilizing independent chaotic sequences. Performance analysis indicates that this S-box construction method is effective. The resulting S-box demonstrates greater nonlinearity than those produced solely with spatiotemporal chaotic system. Compared to previous S-boxes, the obtained S-box using the proposed method exhibits superior safety properties. Consequently, the constructed S-box is suitable for block encryption algorithms. Furthermore, this S-box construction method could benefit the design of other cryptographic algorithms that utilize nonlinear transformation at their core. Although this construction method has yet to be mathematically proven, further research will continue in the future.
Biography
Biography
Biography
Yingqian Zhang
https://orcid.org/0000-0001-9568-0392
He received his Ph.D. in computer application from Dalian University of Technology in 2014 and is currently a professor at the School of Electrical Engineering and Artificial Intelligence at Xiamen University Malaysia and the School of Information Science and Technology at Xiamen University Tan Kah Kee College. His research interests include nonlinear dynamics, cryptography, and image processing.
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