A Framework for Distributed Source Coding with Encryption: a New Strong Converse and More

We are interested in investigating the security of encryption schemes for distributed sources coding based on the general framework for distributed sources coding with encryption which Oohama and Santoso proposed in ITW 2021. In particular, we aim to derive the necessary condition, i.e., the strong converse, to guarantee secrecy even when the secret keys are correlated. In this paper, we propose a new metric for the secrecy criterion, defined as the maximum value of mutual information between joint ciphertexts and joint plaintexts from all encryption nodes, where the maximum is taken over all possible distributions of correctly decryptable joint plaintexts. Under our new secrecy criterion, we successfully formulate the rate region, which serves as both necessary and sufficient conditions to have secure transmission even under correlated secret keys. Furthermore, we show that the perfect security under our new secrecy criterion acts as the necessary condition for having the perfect secrecy under the ordinary mutual information in the case that all correctly decryptable joint plaintexts have non-zero probabilities of occurrence.