A new steganographic algorithm based on coupled chaotic maps and a new chaotic S-box

Abstract

The art of concealing information by embedding it in a seemingly “innocent” message is called steganography. An appropriate system for steganography is essential to guarantee the safety of the transfer file and, moreover, the size of the attached file is of great importance. Ergodic dynamical systems with confusion guarantee an acceptable level of security for cryptographic systems. Here, we suggest a new steganography algorithm based on a measurable dynamical system and a new chaotic S-box. We use two different chaotic maps at the same time to create the new S-box aimed at providing adequate key space and high security for the encryption. In the encryption stage, the message is encrypted with a new S-box. The capability of S-box and encryption has been confirmed by performance analysis. Our goal in adding this encryption step is to increase security and complicate the process to access the steganographic stage secret message. The pixel position of the cover color image is determined by using chaotic maps in the proposed algorithm, in which, a secret information bit can be hidden. By considering the key role in the security of cryptographic systems, an entropy calculation is presented to determine the chaotic area of the proposed system. The problems of the low security against some existing tests as well as the small key space may lead to the steganography failure, which can be fixed by the proposed algorithm. The contributions of the suggested steganography algorithm are as follows: (1) It allows us to use an ergodic coupled system that provides ample key space. (2) It uses an encryption step with new, well-performance S-boxes that provides high security. (3) The performance of the steganographic design performs better than previous works.

Appendices

Appendix A: The ratio of polynomials of degree N

${{\varPhi }}_{N}^{(1,2)}(x,\alpha )$ are Chebyshev polynomials of type 1 and type 2 [74]. By considering:

$$ \begin{array}{@{}rcl@{}} {{\varPhi}}^{(1,2)} (x,\alpha ) = \frac{{{\alpha^{2}}{{F}^{2}}}}{{1 + ({\alpha^{2}} - 1){{F}^{2}}}} \end{array} $$

(19)

where F is $({T_{N}}(\sqrt x ))$ or $({U_{N}}(\sqrt x ))$. These chaotic maps are defined in the interval [0, 1]. Considering the conjugate or isomorphic maps of this equation, we could extract the invariant measure and calculate the KS-entropy. The invertible map

$$h(x) = \frac{1 - x}{x}$$

(which maps I = [0, 1] into $[0,\infty )$) transforms maps ${{\varPhi }}_{N}^{(1,2)}(x,\alpha )$ into $\tilde {{\varPhi }}_{N}^{(1,2)}(x,\alpha )$, defined as [24]:

$$ \begin{array}{@{}rcl@{}} \left\{ \begin{array}{l} \tilde {{\varPhi}}_{N}^{(1)}(x,\alpha ) = \frac{1}{{{\alpha^{2}}}}{\tan^{2}}(N arc tan\sqrt x )\\ \tilde {{\varPhi}}_{N}^{(2)}(x,\alpha ) = \frac{1}{{{\alpha^{2}}}}{\cot^{2}}(N arc tan\frac{1}{{\sqrt x }}) \end{array} \right. \end{array} $$

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this is mean of conjugacy.

Appendix B: Proof of the ergodicity

The introduced pair coupled maps (2) at the synchronized state x = y, by considering the $[0,\infty )$ domain, is reduced to:

$$ \begin{array}{@{}rcl@{}} X={((1 - \varepsilon ){\alpha^{p}} + \varepsilon {\beta^{p}})^{\frac{1}{p}}}{\tan^{2}}(N arctan(\sqrt x )). \end{array} $$

(21)

The definition of the invariant measure for the introduced hierarchy of chaotic maps (1) is [17]:

$$ \begin{array}{@{}rcl@{}} \mu (X,Y) = \int {dx\int {dy\delta (X - F(x,y)) \times } } \delta (Y - F(y,x)) \times \mu (y,x) \end{array} $$

(22)

By considering the following invariant measure for the hierarchy of chaotic maps (19) [25]:

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{l} \mu (x)=\frac{1}{\pi }\frac{{\sqrt b }}{{\sqrt {x(1 - x)} (b + (1 - b )x)}}\\ \\ \alpha = \frac{{\sum\nolimits_{k = 0}^{[\frac{{(N - 1)}}{2}]} {C_{2k + 1}^{N}{b^{- k}}} }}{{\sum\nolimits_{k = 0}^{[\frac{{(N)}}{2}]} {C_{2k}^{N}{b^{- k}}} }} \end{array}\right. . \end{array} $$

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Simply one could find the following relation [4]:

$$ {((1 - \varepsilon ){\alpha^{p}} + \varepsilon {\beta^{p}})^{\frac{1}{p}}} = {\left( {\frac{{\sum\limits_{K = 0}^{\left[ {\frac{N}{2}} \right]} {C_{2K}^{N}{b^{- K}}} }}{{\sum\limits_{K = 0}^{\left[ {\frac{{N - 1}}{2}} \right]} {C_{2K + 1}^{N}{b^{- K}}} }}} \right)^{2}} $$

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Appendix C: KS-entropy

The definition of KS-entropy is [17]:

$$ \begin{array}{@{}rcl@{}} h(\mu ,{{\varPhi}} ) = \int {dx\int {dy\mu (x,y)} } \ln \left| {\frac{{\partial (X,Y)}}{{\partial (x,y)}}} \right| \end{array} $$

(25)

By considering the invariant measure, the KS-entropy for (2) can calculate, which leads to

$$ \begin{array}{@{}rcl@{}} h_{ks}=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\ln|\frac{\partial X}{\partial(x)}|x=y+ \frac{\partial X}{\partial(y)}|x=y|+\lim\limits_{n\rightarrow\infty}\frac{1}{n}\\ \ln|\frac{\partial Y}{\partial(x)}|x=y+\frac{\partial Y}{\partial(y)}|x=y|. \end{array} $$

(26)

The KS-entropy for pair coupled maps is computed in [4]:

$$ \begin{array}{@{}rcl@{}} h_{ks}=\ln \left| {(1 - \varepsilon )\alpha^{p} - \varepsilon \beta^{p}} \right| + \ln \left| {(1 - \varepsilon )\alpha^{p} + \varepsilon \beta^{p}} \right| + H, \end{array} $$

(27)

where $H=2{h_{KS}}(X = {((1 - \varepsilon )\alpha ^{p} + \varepsilon \beta ^{p})^{\frac {1}{p}}}ta{n^{2}}(N arc tan(\sqrt x )))$ is KS-entropy for one-dimensional map. The equality of KS-entropy and Lyapunov exponent for ergodic maps is implied by Birkohf ergodic theorem [17].

$$ h_{KS}=\lambda $$

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