Message Transmission and Key Establishment: Conditions for Equality of Weak and Strong Capacities

Secure communication using noisy resources has been first studied in the contexts of secure message transmission (SMT) by Wyner as well as Csiszár-and-Körner, and secret key establishment (SKE) by Ahlswede-and-Csiszár as well as Maurer. The work defines secrecy (resp. secret-key (SK)) capacity as the highest achievable rate of secure transmission (resp. key establishment). Maurer and Wolf later focused on SKE and noticed that the secrecy requirement in the SK capacity definition was weak as it required only the “ratio” between the adversary’s information and the key length to be negligible. They suggested a stronger definition of the SK capacity by requiring absolute information leakage to be negligible. They provided an interesting proof for the equality of weak and strong SK capacities in the above scenarios (setups).

Followup work has since studied several setups for SKE by considering the weak SK capacity without discussing whether the results also hold for the strong definition. In this paper, we pose the question whether the equality of weak and strong SK capacities can be derived in general for all discrete memoryless communication setups. We also extend this study to message transmission and investigate the equality of weak and strong secrecy capacities. For SKE, we show that weak and strong SK capacities are equal for any setup that allows reliable transmission in any direction. For SMT, the secrecy capacities are equal when the setup allows the sender to use randomness. We furthermore provide trivial counterexamples that show these sufficient conditions are not always necessary for the equality of the capacities. Whether the conditions can be removed or relaxed by tight (necessary and sufficient) conditions remains an interesting question for future.