Secret Sharing Over a Gaussian Broadcast Channel: Optimal Coding Scheme Design and Deep Learning Approach at Short Blocklength

Consider a secret sharing model where a dealer shares a secret with several participants through a Gaussian broadcast channel such that predefined subsets of participants can reconstruct the secret and all other subsets of participants cannot learn any information about the secret. Our first contribution is to show that, in the asymptotic blocklength regime, it is optimal to consider coding schemes that rely on two coding layers, namely, a reliability layer and a secrecy layer, where the reliability layer is a channel code for a compound channel without any security constraint. Our second contribution is to design such a two-layer coding scheme at short blocklength. Specifically, we design the reliability layer via an autoencoder, and implement the secrecy layer with hash functions. To evaluate the performance of our coding scheme, we empirically evaluate the probability of error and information leakage, which is defined as the mutual information between the secret and the unauthorized sets of users channel outputs. We empirically evaluate this information leakage via a neural network-based mutual information estimator. Our simulation results demonstrate a precise control of the probability of error and leakage thanks to the two-layer coding design.