Message transmission and key establishment: General equality for 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). Focusing on SKE, Maurer and Wolf later noticed that the secrecy requirement in the SKE study 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 SKE in various setups 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 setups with discrete memoryless resources. We first extract the requirements that have been implicit in Maurer-and-Wolf's proof and then investigate whether these required conditions can be removed. Our results show that weak and strong SK capacities are equal for any discrete memoryless communication setup. We also extend this study to message transmission and prove the equality of weak and strong secrecy capacities in any discrete memoryless setup.