A Novel S-Box Generation Methodology Based on the Optimized GAN Model

S-boxes can be the core component of block ciphers, and how to efficiently generate S-boxes with strong cryptographic properties appears to be an important task in the design of block ciphers. In this work, an optimized model based on the generative adversarial network (GAN) is proposed to generate 8-bit S-boxes. The central idea of this optimized model is to use loss function constraints for GAN. More specially, the Advanced Encryption Standard (AES) S-box is used to construct the sample dataset via the affine equivalence property. Then, three models are respectively built and cross-trained to generate 8-bit S-boxes based on three extended frameworks of GAN, i.e., Deep Convolution Generative Adversarial Networks (DCGAN), Wasserstein Generative Adversarial Networks (WGAN), and Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP). Besides, an optimized model based on WGAN-GP referred to as WGP-IM is also proposed, which adds the loss function constraints to the generator network of the WGAN-GP model, including bijection loss, differential uniformity loss, and nonlinearity loss. In this case, 8-bit S-boxes can be generated with cross-training. Experimental results illustrate that the WGP-IM model can generate S-boxes with excellent cryptographic properties. In particular, the optimal differential uniformity of the generated S-boxes can be reduced to 8, and the nonlinearity can be up to 104. Compared with previous S-box generation methods, this new method is simpler and it can generate S-boxes with excellent cryptographic properties.