Distributed encryption and the Slepian-Wolf theorem

Starting with Shannon's perfect secrecy theorem, information theory has provided a tool for quantifying the amount of secrecy available for a fixed transmission setup. In this paper, we derive tight upper bounds on the secrecy offered for a broad class of problems consisting of distributed encryption and joint decryption. In particular, we consider the scenario where correlated messages must be encrypted independently, transmitted through perfect channels to a common receiver, and jointly decrypted. The main result we present is a theorem stating that the secrecy of a distributed system does not depend on whether dependent messages are encrypted jointly or separately, as long as joint decryption is allowed. This result is analogous to the Slepian-Wolf theorem for data compression, which states that asymptotically, distributed data compression performs as well as joint compression. We show that the equivalence of distributed and joint encryption is true whether the system uses a key or not, and consequently we demonstrate that if the system does not rely on a key, no secrecy can be offered. We briefly discuss how the results obtained may be applied to the problem of drone or sensor networks, as well as parallel encryption