Chaotic neural networks have received a great deal of attention these last years. In this paper we establish a precise correspondence between the so-called chaotic iterations and a particular class of artificial neural networks: global recurrent multi-layer perceptrons. We show formally that it is possible to make these iterations behave chaotically, as defined by Devaney, and thus we obtain the first neural networks proven chaotic. Several neural networks with different architectures are trained to exhibit a chaotical behavior.
Today, another field of research in which chaotic neural networks have received a lot of attention is data security. In fact, chaotic cryptosystems are an appealing alternative to classical ones due to properties such as sensitivity to initial conditions or topological transitivity. Thus chaotic ANNs have been considered to build ciphering methods, hash functions, digital watermarking schemes, pseudo-random number generators, etc. In @cite_4 such a cipher scheme based on the dynamics of Chua's circuit is proposed. More precisely, a feed-forward MLP with two hidden layers is built to learn about 1500 input-output vector pairs, where each pair is obtained from the three nonlinear ordinary differential equations modeling the circuit. Hence, the proposed chaotic neural network is a network which is trained to learn a true chaotic physical system. In the cipher scheme the ANN plays the role of chaos generator with which the plain-text will be merged. Untrained neural networks have been also considered to define block ciphering @cite_13 or hash functions @cite_17 . The background idea is to exploit the inherent properties of the ANNs architecture such as diffusion and confusion.
@cite_13 @cite_4 @cite_17
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In this paper, the neural network composed of a chaotic neuron layer and a linear neuron layer is used to construct a block cipher that transforms the data from the plaintext form into the unintelligible form under the control of the key. Among them, the chaotic neuron layer realizes data diffusion, the linear neuron layer realizes data confusion, and the two layers are repeated for several times to strengthen the cipher. The decryption process is symmetric to the encryption process. Theoretical analysis and experimental results show that the block cipher has good computing security and is more suitable for image encryption. It is expected to attract more researchers in this field. Chaotic systems are sensitive to initial conditions, system parameters and topological transitivity and these properties are also remarkable for cryptanalysts. Noise like behavior of chaotic systems is the main reason of using these systems in cryptology. However some properties of chaotic systems such as synchronization, fewness of parameters etc. cause serious problems for cryptology. In this paper, to overcome disadvantages of chaotic systems, the dynamics of Chua's circuit namely x , y and z were modeled using Artificial Neural Network (ANN). ANNs have some distinctive capabilities like learning from experiences, generalizing from a few data and nonlinear relationship between inputs and outputs. The proposed ANN was trained in different structures using different learning algorithms. To train the ANN, 24 different sets including the initial conditions of Chua's circuit were used and each set consisted of about 1800 input-output data. The experimental results showed that a feed-forward Multi Layer Perceptron (MLP), trained with Bayesian Regulation backpropagation algorithm, was found as the suitable network structure. As a case study, a message was first encrypted and then decrypted by the chaotic dynamics obtained from the proposed ANN and a comparison was made between the proposed ANN and the numerical solution of Chua's circuit about encrypted and decrypted messages. An algorithm for constructing a one-way novel Hash function based on two-layer chaotic neural network structure is proposed. The piecewise linear chaotic map (PWLCM) is utilized as transfer function, and the 4-dimensional and one-way coupled map lattices (4D OWCML) is employed as key generator of the chaotic neural network. Theoretical analysis and computer simulation indicate that the proposed algorithm presents several interesting features, such as high message and key sensitivity, good statistical properties, collision resistance and secure against meet-in-the-middle attacks, which can satisfy the performance requirements of Hash function.