Construction of dynamical S-boxes based on image encryption approach

Abstract

Many cryptographic schemes have been implemented, using the connection between chaos and cryptography. However, the present study introduces a new algorithm that, as a standard format, can be used in cyberspace. The substitution box (S-box) is one of the most successful examples of these strategies. However, most S-box design schemes contain a fixed point, which may cause a security issue for cryptographic primitives. A dynamic S-box is constructed based on a piecewise map to achieve a low automatic correlation. The introduced piecewise map has an attractive behavior by displaying the ergodic behavior and fractal features in its chaotic domain. The ergodic property is of great interest in diffusion and confusion processes for encryption. We have examined the performance of the introduced S-box. The applied attacks (nonlinearity, SAC, BIC, LP, and DP) results are close to the optimal value. Then, an image encryption algorithm based on a designated S-box is proposed, and the ability of the encrypted image to resist the attacks has been examined with statistical analysis. The proposed S-box based on the piecewise map has a large and safe keyspace for encryption.

One parameter families

As an example, some of these maps are given below:

$$\begin{aligned} \varPhi _{2}=\frac{4\alpha ^{2}x(1-x)}{1+4(\alpha ^{2}-1)x(1-x)},\nonumber \\ \varPhi _{3}=\frac{\alpha ^{2}x(4x-3)^{2}}{\alpha ^{2}x(4x-3)^{2}+(1-x)(4x-1)^{2}}. \end{aligned}$$

(22)

where $\alpha$ is the control parameter that can potentially be used as the secret keys for secure communication. F substitutes with Chebyshev polynomial of type one $T_{N}(x)$. It is shown that these maps have an attractive property; that is, for even values of N the $\varPhi _N(x,\alpha )$ maps have only a fixed point attractor $x=1$ provided that their parameter belongs to interval $(N,\infty )$ while at $\alpha \ge N$ they bifurcate to the chaotic regime without having any period doubling or period-n-tupling scenario and remain chaotic for all $\alpha \in {(0,N)}$, but for odd values of N, these maps have only fixed point attractor at $x=0$ for $\alpha \in (\frac{1}{N}, N)$, again they bifurcate to a chaotic regime at $\alpha \ge \frac{1}{N}$, and remain chaotic for $\alpha \in (0, \frac{1}{N})$, finally they bifurcate at $\alpha =N$ to have $x=1$ as fixed point attractor for all $\alpha \in (\frac{1}{N}, \infty )$. Here, in this paper we are concerned with their conjugate maps which are defined as:

$$\begin{aligned} \tilde{\varPhi }_N(x,\alpha )=h\circ \varPhi _N(x,\alpha )\circ h^{-1}=\frac{1}{\alpha ^{2}}\tan ^{2}(N \arctan \sqrt{x_n}), \end{aligned}$$

(23)

Conjugacy means that invertible map $h(x)=\frac{1-x}{x}$ maps $I=[0,1]$ into $[0,\infty )$. We have derived analytically their invariant measure for arbitrary values of the parameter $\alpha$ and any integer values of N:

$$\begin{aligned} \mu _{\varPhi _N(x,\alpha )}(x,\beta )=\frac{1}{\pi }\frac{\sqrt{\beta }}{\sqrt{x(1-x)}(\beta +(1-\beta )x)}\;. \end{aligned}$$

(24)

with $\beta >0$ is the invariant measure of the maps $\varPhi _N^{(1,2)}(x,\alpha )$ provided that we choose the parameter $\alpha$ in the following form:

$$\begin{aligned} \alpha =\left\{ \begin{array}{l} \frac{\varSigma _{k=0}^{\left[ \frac{(N-1)}{2}\right] }C_{2k+1}^{N}\beta ^{-k}}{\varSigma _{k=0}^{\left[ \frac{N}{2}\right] }C_{2k}^{N}\beta ^{-k}}\quad \quad for \quad \quad odd \; values \; of \;N \\ \\ \frac{\beta \varSigma _{k=0}^{\left[ \frac{(N)}{2}\right] }C_{2k}^{N}\beta ^{-k}}{\varSigma _{k=0}^{\left[ \frac{(N-1)}{2}\right] }C_{2k+1}^{N}\beta ^{-k}} \quad \quad for \quad \quad even \;values \; of \;N \\ \end{array}\right. \end{aligned}$$

(25)

where the symbol $[\;\;]$ means the greatest integer part. Studies based on invariant measure analysis can be useful for confirming the ergodic behavior of a map. An ergodic system has ‘convergent’ qualities over time; variances are finite and a non-time-dependent process.