## Abstract

The newly introduced family of complex chaotic maps has different behavior concerning the well-known complex chaotic maps. These complex maps have excellent properties such as invariant measures and fractal features. The present study introduces a new algorithm that, as a standard format, can be used in cyberspace. To achieve a low automatic correlation, S-box is constructed based on a complex map. Then, an image encryption algorithm based on a designated S-box is proposed. The complex and ergodic nature of the introduced hierarchy proposed the large and safe Keyspace for encryption. Theoretical and experimental results have indicated that the proposed algorithm is high sensitivity to key, and robust to attacks.

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## Schwarzian derivative

### Schwarzian derivative

Study the critical points of introduced complex chaotic maps clarify that, \({\varPhi }_{N}(z,c)\) have \((N-1)\) critical points in its interval. By choosing the odd (even numbers), *N*, both \(z=0\) and \(z=1\) belong to one of the n-cycles (only \(z=1\) belong to one of the n-cycles of \({\varPhi }_{N}(z,c)\)). The Schwarzian derivative function \({\varPhi }_{N}(z,c)\) at point \({\mathbf{z}}\) in direction \(\lambda \) is given by:

Since \(D_{\lambda }(T_N\left( \sqrt{\frac{2+z}{4}}\right) ^{2})\) can be written as:

with \(0\le {z_1}<{z_2}<{z_3}<\cdots <z_{N-1}\le {1}\), then we have:

It means the map has an \(N+1\) attracting periodic orbits (Devaney 2008). The new introduced family of complex chaotic maps have a different behavior respect to the well known complex chaotic maps (Isa et al. 2016). We study the behavior of the composition of map (\({\varPhi }^{(n)}\)) by taking the derivative of it, concerning its possible n periodic points of an n-cycle (\(z_{2}={\varPhi }(z_{1},c,z_{3}={\varPhi }(z_{2},c),\ldots , z_{1}={\varPhi }(z_{n},c)\)).

since for \(z_{k}\in [0,1]\), we have: \(min(c^{2}+(1-c^{2}z_{k}))=min(1,c^{2})\) then:

It is definitely greater than one for \(\frac{1}{N}< c < N \), therefore maps do not have any kind of n-cycle or periodic orbits in the interval \( \frac{1}{N}< c < N \). It means that introduced families are ergodic, in this interval. It follows the values of \(\mid D_{\lambda }{\varPhi }^{(n)}\mid \) at *n* periodic points of the n-cycle belong to the interval [0, 1], varies between \({(Nc)}^{n}\) and \({(\frac{N}{c})}^{n}\) for \(c<\frac{1}{N}\) and between \((\frac{N}{c})^{n}\) and \((Nc )^{n}\) for \(c>N\), respectively.

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Sani, R.H., Behnia, S. & Akhshani, A. Creation of S-box based on a hierarchy of Julia sets: image encryption approach.
*Multidim Syst Sign Process* (2021). https://doi.org/10.1007/s11045-021-00786-9

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### Keywords

- Digital image encryption
- S-box
- Ergodic dynamical systems
- Julia sets