Construction of S-box based on chaotic piecewise map: Watermark application

Abstract

The spread of using communication technologies has made it necessary to take certain measures to prevent illegal copying. Watermarking techniques are of great potential value in this regard. The present study introduces a watermarking scheme as a block cypher. We use entropy for sorting the blocks. The sorted blocks after finite ridgelet transformation are hosting the logo. The logo’s information encrypts before transferring into the blocks by the proposed S-box. We introduce a piecewise non-linear chaotic map for generating the S-box. The ergodic nature of introducing a map is proved by the invariant measure. The ergodic nature of the map is providing an excellent confusion property for encryption. The ability of the watermarked image to resist the attacks is an exam with statistical analysis(PSNR and MSE). The performance of generated S-box is studying with corresponding attacks (Non-linearity, SAC, BIC, LP, and DP). The results are close to the optimal value.

Appendices

Appendix A: Detail of derivation of invariant measure

In this appendix we try to obtain the invariant measure of Piecewise non-linear Chaotic Maps. Denoting the left hand side of Eq.1 by X and inverting it, $F=\frac {X}{\alpha ^{2} +(\alpha ^{2} -1)X}$, then, taking derivative of F with respect to x, we obtain:

$$ \frac{dF}{dx}=\frac{\alpha^{2}}{(\alpha^{2} +(\alpha^{2} -1)X)^{2}} $$

(13)

Now, by denoting x in 0 ≤ x ≤ p₁ by x₁, x in p₁ ≤ x ≤ p₂ by x₂ and x in p₂ ≤ x ≤ 1 by x₃, solving it for x₁, x₂ and x₃ then, taking derivative of x₁, x₂ and x₃ with respect to X, we obtain:

$$ dx_{1}=\frac{p_{1}\alpha^{2}dx}{(\alpha^{2}-X(\alpha^{2} -1))^{2}},\quad dx_{2}=\frac{-p_{2}\alpha^{2}dx}{(\alpha^{2}-X(\alpha^{2} -1))^{2}},\quad dx_{3}=\frac{p_{3}\alpha^{2}dx}{(\alpha^{2}-X(\alpha^{2} -1))^{2}}. $$

(14)

Now, by considering the FP (4),

$$ dx\mu (x)=\mu (x_{1})dx_{1}+\mu (x_{2})dx_{2}+\mu (x_{3})dx_{3} $$

we obtain:

$$ dx\mu (x)=\frac{\alpha^{2}dx}{(\alpha^{2}-X(\alpha^{2} -1))^{2}}(p_{1}\mu (x_{1})dx_{1}-p_{2}\mu (x_{2})dx_{2}+p_{3}\mu (x_{3})dx_{3}) $$

(15)

considering the following anatz for the invariant measure

$$ \mu(x)=\frac{1}{A+Bx} $$

(16)

Equation (15) reduced to the following equation

$$ \frac{1}{A+Bx}=\frac{\alpha^{2}}{(\alpha^{2}-X(\alpha^{2} -1))^{2}}(p_{1}\frac{1}{A+B\frac{XP_{1}}{\alpha^{2}-X(\alpha^{2} -1)}} $$

$$ -(p_{2}\frac{1}{A+B\frac{Xp_{2}}{\alpha^{2}-X(\alpha^{2} -1)}}+(p_{3}\frac{1}{A+B\frac{Xp_{3}}{\alpha^{2}-X(\alpha^{2} -1)}}) $$

(17)

which leads to

$$ B=\frac{1}{p_{3}} \quad A=\frac{p_{3}-p_{1}-p_{2}}{p_{3}} $$

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Appendix B: The analysis of the S-boxes

• Non-linearity Non-linearity is the most significant feature of the S-box. The non-linearity of a non-linear alternating box indicates the non-linear order of the Boolean functions of the box and it is equal to 112 for the Lorentz replacement box. Non-linearity has been defined through using Walsh Spectrum [25].

  $$ N_{f}=2^{n-1}(1-2^{-n}\max\limits_{\omega\in GF(2^{n})}\mid S_{(f)}(\omega)\mid) $$

  (19)

  In the above equation, S_(f)(ω) is the Walsh spectrum for f(x) and is given by:

  $$ S_{(f)}(\omega)=\sum\limits_{\omega \in GF(2^{n})}^{}(-1)^{f(x)\bigoplus\omega .x} $$

  (20)

  In this regard, ω belongs to GF(2ⁿ), and x and ω are multiplied by a dot product. The non-linearity of one proposed S-box is given in Table 2.

• Strict Avalanche Criterion To check whether a given cryptographic transformation satisfies the strict avalanche criterion (SAC), we examined our S-box (Table 1). The ideal value is 0.5 [68]. The dependence matrix of the generated S-box is presented in Tables 6 and 2.

  Table 6 The dependence matrix for the proposed S-box

• Bit Dependent Critica Webster and Tavers introduced the bit dependent critica (BDP) to analyze the S-box [68]. In this technique, it is tried to understand whether the set of vectors produced with the switch bit of plaintext is autonomous from all avalanche variable seta or not. The ideal value for satisfying is 0.5. The test results are presented in Tables 2 and 7.

  Table 7 The BIC-SAC matrix for the proposed S-box

• Linear Approximation Probability We could compute the masking of all inputs by considering the two masks Γx and Γy. The maximum value of masking is known as a maximum linear approximation. If we represent the input value by x and the corresponding output S(x), the linear approximation probability would be defined as:

  $$ LP=\max\limits_{{\varGamma} x,{\varGamma} y \neq 0}\vert\frac{\sharp \lbrace x\vert x.{\varGamma} x=S(x).{\varGamma} y \rbrace}{2{n}}-\frac{1}{2}\vert $$

  (21)

  The smaller the LP value, the stronger the ability to fight against linear transmission attacks, and vise versa. The results of this analysis are shown in Table 2.

• Differential Approximation Probability

  The differential Approximation Probability (DP) is the maximum probability of y output, when the input is x, and it is used to represent the XOR distribution of input and output of the boolean function. It is given by:

  $$ DP=\max\limits_{\Delta x \neq 0,{\Delta} y}\vert\ \frac{\sharp \lbrace x \in X \vert f(x) \bigoplus f(x \bigoplus {\Delta} x)= {\Delta} y \rbrace}{2^{n}}-\frac{1}{2}\vert $$

  (22)

  The smaller the DP value, the stronger the ability to fight against linear transmission attacks. The results of the DP analysis are shown in Table 2.