Evolving Dynamic S-Boxes Using Fractional-Order Hopfield Neural Network Based Scheme

Static substitution-boxes in fixed structured block ciphers may make the system vulnerable to cryptanalysis. However, key-dependent dynamic substitution-boxes (S-boxes) assume to improve the security and robustness of the whole cryptosystem. This paper proposes to present the construction of key-dependent dynamic S-boxes having high nonlinearity. The proposed scheme involves the evolution of initially generated S-box for improved nonlinearity based on the fractional-order time-delayed Hopfield neural network. The cryptographic performance of the evolved S-box is assessed by using standard security parameters, including nonlinearity, strict avalanche criterion, bits independence criterion, differential uniformity, linear approximation probability, etc. The proposed scheme is able to evolve an S-box having mean nonlinearity of 111.25, strict avalanche criteria value of 0.5007, and differential uniformity of 10. The performance assessments demonstrate that the proposed scheme and S-box have excellent features, and are thus capable of offering high nonlinearity in the cryptosystem. The comparison analysis further confirms the improved security features of anticipated scheme and S-box, as compared to many existing chaos-based and other S-boxes.

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Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay

Advances in Mathematical Physics ◽

10.1155/2013/657245 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 8

Author(s):

Xia Huang ◽

Zhen Wang ◽

Yuxia Li

Keyword(s):

Neural Network ◽

Neural Networks ◽

Fractional Order ◽

Hopfield Neural Network ◽

Hopfield Neural Networks ◽

Phase Portraits ◽

Chaotic Attractors ◽

Chaotic Motions ◽

Dynamical Behaviors ◽

Nonlinear Dynamics And Chaos

A fractional-order two-neuron Hopfield neural network with delay is proposed based on the classic well-known Hopfield neural networks, and further, the complex dynamical behaviors of such a network are investigated. A great variety of interesting dynamical phenomena, including single-periodic, multiple-periodic, and chaotic motions, are found to exist. The existence of chaotic attractors is verified by the bifurcation diagram and phase portraits as well.

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Dynamics analysis of fractional order three-dimensional Hopfield neural network

2010 Sixth International Conference on Natural Computation ◽

10.1109/icnc.2010.5582371 ◽

2010 ◽

Cited By ~ 13

Author(s):

Ran Zhang ◽

Dawei Qi ◽

Yanzhi Wang

Keyword(s):

Neural Network ◽

Fractional Order ◽

Three Dimensional ◽

Hopfield Neural Network ◽

Dynamics Analysis

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More Low Differential Uniformity Permutations over F22k with k Odd

Mathematical Problems in Engineering ◽

10.1155/2020/7152657 ◽

2020 ◽

Vol 2020 ◽

pp. 1-10

Author(s):

Yue Leng ◽

Jinyang Chen ◽

Tao Xie

Keyword(s):

Block Ciphers ◽

Algebraic Degree ◽

Differential Uniformity ◽

Numerical Examples ◽

High Nonlinearity ◽

Substitution Boxes ◽

Theoretical Results

Permutations with low differential uniformity, high algebraic degree, and high nonlinearity over F22k can be used as the substitution boxes for many block ciphers. In this paper, several classes of low differential uniformity permutations are constructed based on the method of choosing two permutations over F22k to get the desired permutations. The resulted low differential uniformity permutations have high algebraic degrees and nonlinearities simultaneously, which provide more choices for the substitution boxes. Moreover, some numerical examples are provided to show the efficacy of the theoretical results.

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Dynamic characteristic of a new fractional-order chaotic system based on the Hopfield Neural Network and its digital circuit implementation

Physica Scripta ◽

10.1088/1402-4896/abd904 ◽

2021 ◽

Author(s):

Feifei Yang ◽

Xingyuan Wang

Keyword(s):

Neural Network ◽

Dynamic Characteristic ◽

Fractional Order ◽

Chaotic System ◽

Hopfield Neural Network ◽

Digital Circuit ◽

Circuit Implementation

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Dynamics analysis of fractional-order Hopfield neural networks

International Journal of Biomathematics ◽

10.1142/s1793524520500837 ◽

2020 ◽

pp. 2050083

Author(s):

Iqbal M. Batiha ◽

Ramzi B. Albadarneh ◽

Shaher Momani ◽

Iqbal H. Jebril

Keyword(s):

Neural Network ◽

Lyapunov Exponents ◽

Fractional Order ◽

Hopfield Neural Network ◽

Phase Portraits ◽

Dynamics Analysis ◽

Fractional Order Systems ◽

Runge Kutta Method ◽

Intermittent Chaos ◽

Predictor Corrector

This paper proposes fractional-order systems for Hopfield Neural Network (HNN). The so-called Predictor–Corrector Adams–Bashforth–Moulton Method (PCABMM) has been implemented for solving such systems. Graphical comparisons between the PCABMM and the Runge–Kutta Method (RKM) solutions for the classical HNN reveal that the proposed technique is one of the powerful tools for handling these systems. To determine all Lyapunov exponents for them, the Benettin–Wolf algorithm has been involved in the PCABMM. Based on such algorithm, the Lyapunov exponents as a function of a given parameter and as another function of the fractional-order have been described, the intermittent chaos for these systems has been explored. A new result related to the Mittag–Leffler stability of some nonlinear Fractional-order Hopfield Neural Network (FoHNN) systems has been shown. Besides, the description and the dynamic analysis of those phenomena have been discussed and verified theoretically and numerically via illustrating the phase portraits and the Lyapunov exponents’ diagrams.

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Construction of an S-Box Based on Chaotic and Bent Functions

Symmetry ◽

10.3390/sym13040671 ◽

2021 ◽

Vol 13 (4) ◽

pp. 671

Author(s):

Zijing Jiang ◽

Qun Ding

Keyword(s):

Chaos Theory ◽

Boolean Functions ◽

Security Analysis ◽

Bent Function ◽

Bent Functions ◽

Encryption Algorithm ◽

Symmetric Encryption ◽

Differential Uniformity ◽

Strict Avalanche Criterion ◽

Independence Criterion

An S-box is the most important part of a symmetric encryption algorithm. Various schemes are put forward by using chaos theory. In this paper, a construction method of S-boxes with good cryptographic properties is proposed. The output of an S-box can be regarded as a group of Boolean functions. Therefore, we can use the different properties of chaos and Bent functions to generate a random Bent function with a high nonlinearity. By constructing a set of Bent functions as the output of an S-box, we can create an S-box with good cryptological properties. The nonlinearity, differential uniformity, strict avalanche criterion and the independence criterion of output bits are then analyzed and tested. A security analysis shows that the proposed S-box has excellent cryptographic properties.

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Chaotic Behavior Analysis of a New Incommensurate Fractional-Order Hopfield Neural Network System

Complexity ◽

10.1155/2021/3394666 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Nadjette Debbouche ◽

Adel Ouannas ◽

Iqbal M. Batiha ◽

Giuseppe Grassi ◽

Mohammed K. A. Kaabar ◽

...

Keyword(s):

Neural Network ◽

Fractional Order ◽

Complex Dynamics ◽

Chaotic Behavior ◽

Hopfield Neural Network ◽

Phase Portraits ◽

Chaotic Attractors ◽

Neural Network System ◽

The Stability ◽

Arbitrary Location

This study intends to examine different dynamics of the chaotic incommensurate fractional-order Hopfield neural network model. The stability of the proposed incommensurate-order model is analyzed numerically by continuously varying the values of the fractional-order derivative and the values of the system parameters. It turned out that the formulated system using the Caputo differential operator exhibits many rich complex dynamics, including symmetry, bistability, and coexisting chaotic attractors. On the other hand, it has been detected that by adapting the corresponding controlled constants, such systems possess the so-called offset boosting of three variables. Besides, the resultant periodic and chaotic attractors can be scattered in several forms, including 1D line, 2D lattice, and 3D grid, and even in an arbitrary location of the phase space. Several numerical simulations are implemented, and the obtained findings are illustrated through constructing bifurcation diagrams, computing Lyapunov exponents, calculating Lyapunov dimensions, and sketching the phase portraits in 2D and 3D projections.

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Stability Analysis of Fractional Order Memristor Synapse-coupled Hopfield Neural Network with Ring Structure

10.21203/rs.3.rs-973554/v1 ◽

2021 ◽

Author(s):

Leila Eftekhari ◽

Mohammad Amirian

Keyword(s):

Neural Network ◽

Fractional Order ◽

Equilibrium Points ◽

Sufficient Conditions ◽

Hopfield Neural Network ◽

Ring Structure ◽

Embedded Memory ◽

The Stability ◽

Necessary And Sufficient ◽

Memristor Synapse

Abstract A memristor is a non-linear two-terminal electrical element that incorporates memory features and nanoscale properties, enabling us to design very high-density artificial neural networks. To examine the embedded memory property, we should use mathematical frameworks like fractional calculus, which is capable of doing so. Here, we first present a fractional-order memristor synapse-coupling Hopfield neural network on two neurons and then extend the model to a neural network with a ring structure that consists of $n$ sub-network neurons. Necessary and sufficient conditions for the stability of equilibrium points are investigated, highlighting the dependency of the stability on the fractional-order value and the number of neurons. Numerical simulations and bifurcation analysis, along with Lyapunov exponents, are given in the two-neuron case that substantiates the theoretical findings, suggesting possible routes towards chaos when the fractional order of the system increases. In the $n$-neuron case also, it is revealed that the stability depends on the structure and number of sub-networks.

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