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.

We show that the problem of code construction for multiple access channel (MAC) resolvability can be reduced to the simpler problem of code construction for source resolvability. Specifically, we propose a MAC resolvability code construction that relies on a combination of multiple source resolvability codes, used in a black-box manner, and leverages randomness recycling implemented via distributed hashing and block-Markov coding. Since explicit source resolvability codes are known, our results also yield the first explicit coding schemes that achieve the entire MAC resolvability region for any discrete memoryless multiple-access channel with binary input alphabets.

We consider secret sharing where a dealer wants to share a secret with several participants such that predefined subsets of participants can reconstruct the secret and all other subsets of participants cannot learn any information about the secret. To this end, the dealer and the participants have access to samples of correlated random variables and a one-way (from the dealer to the participants), authenticated, public, and rate-limited communication channel. For this problem, we propose the first constructive and low-complexity coding scheme able to handle arbitrary access structures. Our construction relies on a vector quantization coupled with distribution approximations with polar codes to handle the reliability constraints, followed by universal hashing to handle the security constraints. We stress that our coding scheme does not require symmetry or degradation assumptions on the correlated random variables, and does not need a pre-shared secret among the participants and dealer. Our result is also optimal in the special case of rate-unlimited public communication when all the participants are needed to reconstruct the secret.

We show that the problem of code construction for multiple access channel resolvability can be reduced to the simpler problem of code construction for source resolvability. Specifically, we propose a multiple access channel resolvability coding scheme that involves randomness recycling, implemented via distributed hashing, and block-Markov encoding, where each encoding block is obtained as a combination of several source resolvability codes. Our construction is independent of the way the source resolvability codes are implemented and yields explicit coding schemes that achieve the multiple access channel resolvability region for an arbitrary discrete memoryless multiple access channel whose input alphabets are binary.