Design of S-boxes based on neural networks*

In this paper, we present a framework for the design of S-boxes used in ciphers based on neural networks. It can yield S-boxes with different input and output length. The designed n × n S-boxes satisfy the desired cryptographic properties of non-linearity, completeness, strict avalanche, and output bits independence criteria. We propose a four layer topology, where the number of neurons, located at the input layer, is two times the number of input bits of the designed S-box and also, the number of neurons, located at the first hidden layer, is as equal as input layer neurons, while its second hidden layer included n/2 neurons, and its output layer included n neurons. The input value of the designed S-boxes consists of n-bit input vector and constant n-bit initial value (IV). We apply a Sigmoid nonlinear function as the activation function of our scheme. The values of weights were obtained through error back propagation learning algorithm, while a training set is used for learning. The used training set consists some different pairs of plaintexts and ciphertexts with AES's S-box. We also implement an 8 × 8 S-box based on neural networks with the essential security criteria. The results indicate that the proposed scheme can yield S-boxes with the desired cryptographic properties.