Recovering S-Box Design Structures and Quantifying Distances Between S-Boxes Using Deep Learning

At ASIACRYPT’19, Bonnetain et al. demonstrated that an S-box can be distinguished from a permutation chosen uniformly at random by quantifying the distances between their behaviors. In this study, we extend this approach by proposing a deep learning-based method to quantify distances between two different S-boxes and evaluate similarities in their design structures. First, we introduce a deep learning-based framework that trains a neural network model to recover the design structure of a given S-box based on its cryptographic table. We then interpret the decision-making process of our trained model to analyze which coefficients in the table play significant roles in identifying S-box structures. Additionally, we investigate the inference results of our model across various scenarios to evaluate its generalization capabilities. Building upon these insights, we propose a novel approach to quantify distances between structurally different S-boxes. Our method effectively assesses structural similarities by embedding S-boxes using the deep learning model and measuring the distances between their embedding vectors. Furthermore, experimental results confirm that this approach is also applicable to structures that the model has never seen during training. Our findings demonstrate that deep learning can reveal the underlying structural similarities between S-boxes, highlighting its potential as a powerful tool for S-box reverse-engineering.