A new finding is proposed for multi-fractional order of neural networks by multi-time delay (MFNNMD) to obtain stable chaotic synchronization. Moreover, our new result proved that chaos synchronization of two MFNNMDs could occur with fixed parameters and initial conditions with the proposed control scheme called sliding mode control (SMC) based on the time-delay chaotic systems. In comparison, the fractional-order Lyapunov direct method (FLDM) is proposed and is implemented to SMC to maintain the systems’ sturdiness and assure the global convergence of the error dynamics. An extensive literature survey has been conducted, and we found that many researchers focus only on fractional order of neural networks (FNNs) without delay in different systems. Furthermore, the proposed method has been tested with different multi-fractional orders and time-delay values to find the most stable MFNNMD. Finally, numerical simulations are presented by taking two MFNNMDs as an example to confirm the effectiveness of our control scheme.

This research proposes the idea of double encryption, which is the combination of chaos synchronization of non-identical multi-fractional-order neural networks with multi-time-delays (FONNSMD) and symmetric encryption. Symmetric encryption is well known to be outstanding in speed and accuracy but less effective. Therefore, to increase the strength of data protection effectively, we combine both methods where the secret keys are generated from the third part of the neural network systems (NNS) and used only once to encrypt and decrypt the message. In addition, a fractional-order Lyapunov direct function (FOLDF) is designed and implemented in sliding mode control systems (SMCS) to maintain the convergence of approximated synchronization errors. Finally, three examples are carried out to confirm the theoretical analysis and find which synchronization is achieved. Then the result is combined with symmetric encryption to increase the security of secure communication, and a numerical simulation verifies the method’s accuracy.

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In this paper, we explore the pairing-based cryptography on elliptic curve. The security of protocols using composite order bilinear pairing on elliptic curve depends on the difficulty of factoring the number N. Here, we show how to construct composite ordinary pairing-friendly elliptic curve having the subgroup of composite order N by using Cocks-Pinch Method. We also introduce dual system encryption to transform Identity-Based Encryption (IBE) scheme built over prime-order bilinear, to composite order bilinear groups. The new Identity-Based Encryption (IBE) is secured since it uses the Dual System Encryption methodology which guaranteed full security of the new IBE system.

The purpose is to present a method for synchronizing a recurrent neural networks system between integer and fractional-order order delay by active sliding mode control . The Active Sliding Mode Control (ASMC) scheme is used to solve the synchronization problem between the integer-order delayed recurrent neural networks system via active sliding mode control (IoDRNNASM) systems and the fractional-order delay recurrent neural networks system via active sliding mode control (FoDRNNASM) system based on the Lyapunov direct fractional method (LDFM). To explore the behavior of the IoDRNNASM systems and the FoDRNNASM systems, we performed the technique of numerical simulations using MATLAB software to prove the feasibility and strength of the archived outcomes. This concept can also be enhanced with the implementation of double encryption using RSA encryption to secure communication. Because we expected in the future that this enhanced concept will strengthen and increase the network security capabilities that will provide powerful protection in secure communications.