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Dynamic phenomena of a financial hyperchaotic system and DNA sequences for image encryption

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Abstract

With advances in modern technology, the security of information, including the protection of digital images, is of particular interest. Due to the type of special storage format of images and the fateful dimension of 1D map that have a small secret key space, in this paper, a joint-based encryption technique on the pseudo-random and sophisticated character of hyperchaotic behavior and DNA coding is proposed. The entire dynamics of a financial hyperchaotic system is studied for a better selection of the sequence key using nonlinear analysis tools. A rich dynamic of this analysis reveals a plethora of phenomena such as multistability and offset boosting, which, to our knowledge, have not yet been the subject of a study on financial hyperchaotic systems. The set consisting of the pseudo-random aspect of the financial hyperchaotic system used in all stages of encryption, DNA coding (algebraic operations, complementation, and DNA rules), and the scrambling of the positions of each image pixel is exploited to reinforce the effectiveness of the confusion and diffusion of digital images. To analyze the security and robustness of the proposed algorithm, some security tests such as histogram analysis, correlation, information entropy, as well as key analysis are carried out. The values of correlation coefficients of the encrypted images using the proposed scheme are close to zero, The entropy values of the test images are overall greater than 7.99 and the key space of the is greater than 2100. Besides, differential analysis shows that the number of pixel change rate (NPCR) and unified average change intensity (UACI) for the proposed technique are greater than 99.50% and 30%, respectively. Furthermore, the quantitative analyses of occlusion and data loss attacks as well as the results of comparison with some advanced algorithms show the efficiency and security of the proposed cryptosystem.

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Authors and Affiliations

  1. Unité de Recherche de Laboratoire d’Automatique et Informatique Appliquée (URAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon

    V. R. Folifack Signing, R. L. Tagne Mogue & J. Kengne

  2. Unité de Recherche de Matière Condensée, d’Electronique et de Traitement du Signal (URMACETS), Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon

    V. R. Folifack Signing, R. L. Tagne Mogue & M. Kountchou

  3. Research Center for Nuclear Science and Technology, Institute of Geological and Mining Research, P.O. Box 4110, Yaoundé, Cameroon

    V. R. Folifack Signing, R. L. Tagne Mogue, M. Kountchou &  Saïdou

  4. Nuclear Physics Laboratory, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

    Saïdou

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  1. V. R. Folifack Signing
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Appendices

Appendix A: Preliminary study for system dynamics

1.1 A.1. Study of Dissipativity

The necessary condition to study the dissipation of the system (1) is expressed by the following equation [15, 46]. System (1) can be written as:

$$ \Big\{{\displaystyle \begin{array}{c}{f}_1\left({x}_1,{x}_2,{x}_3,{x}_4\right)={x}_3+{x}_1\left({x}_2-a\right)+{x}_4\\ {}{f}_2\left({x}_1,{x}_2,{x}_3,{x}_4\right)=1-b{x}_2-{x}_1^2\\ {}{f}_3\left({x}_1,{x}_2,{x}_3,{x}_4\right)=-{x}_1-c{x}_3\\ {}{f}_4\left({x}_1,{x}_2,{x}_3,{x}_4\right)=-d{x}_1{x}_2-k{x}_4\end{array}} $$
(A1)

The rate of volume contraction of system (A1) is computed as:

$$ \nabla V=\frac{\partial {f}_1}{\partial {x}_1}+\frac{\partial {f}_2}{\partial {x}_2}+\frac{\partial {f}_3}{\partial {x}_3}+\frac{\partial {f}_4}{\partial {x}_4}={x}_2-\left(a+b+c+k\right) $$
(A2)

1.2 A.2. Equilibrium point

To study the stability of equilibrium point amounts to determine this point which is obtained by solving the system equation \( {\dot{x}}_1={\dot{x}}_2={\dot{x}}_3={\dot{x}}_4=0 \) in order to know if it is stable or unstable. Thus, system (1) has three equilibrium points which are expressed as follows:

$$ {E}_1=\left(0,1/b,0,0\right),{E}_{2,3}=\left(\pm \vartheta, \frac{k\left( ac+1\right)}{c\left(k-d\right)},\mp \frac{\vartheta }{c},\frac{d\vartheta}{c\left(d-k\right)}\left( ca+1\right)\right) $$
(A3)

With \( \vartheta =\frac{\sqrt{dc+ kc ab+ kb+ kc}}{c\left(d-k\right)} \)

The Jacobian matrix of the system (1) around any fixed point E1, E2, or E3 is defined by:

$$ J=\left[\begin{array}{cccc}{x}_2-a& {x}_1& 1& 1\\ {}-2{x}_1& -b& 0& 0\\ {}-1& 0& -c& 0\\ {}-d{x}_2& -d{x}_1& 0& -k\end{array}\right] $$
(A4)

1.3 A.3. Lyapunov dimension

Using the Wolf algorithm [45] and for the range of the system parameters a = 1, b = 0.11, c = 1.04, d = 0.2, and k = 0.17, the four Lyapunov exponents are computed as follows: L1 = 0.0505454, L2 = 0.0166925, L3 = 0, and L1 =  − 0.679484. We find that in these conditions, the system (1) is hyperchaotic because it consists of two positive exponents L1and L2, one null Lyapunov exponent L3, and a negative Lyapunov L4 [47]. We find that, L1 + L2 + L3 = 0.0672379 positive and L1 + L2 + L3 + L4 =  − 0.6122461negative, consequently, the dimension of Kaplan Yorke [69] for this financial hyperchaotic system (1) is found as:

$$ {D}_{KY}=3+\frac{L_1+{L}_2+{L}_3}{\mid {L}_4\mid }=3.098954354 $$
(A5)

Appendix B: Algorithms of the encryption and decryption process

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Signing, V.R.F., Mogue, R.L.T., Kengne, J. et al. Dynamic phenomena of a financial hyperchaotic system and DNA sequences for image encryption. Multimed Tools Appl 80, 32689–32723 (2021). https://doi.org/10.1007/s11042-021-11180-9

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