Performance Evaluation of Secure Communication Systems on Turing Machines

Communication systems need to satisfy pre-specified requirements on spectral efficiency and security. Physical layer security is a concept that unifies both and connects them with entropic quantities. This paper presents a frame-work based on Turing machines to address the question of deciding whether or not a communication system meets these requirements. It is known that the class of Turing solvable problems coincides with the class of algorithmically solvable problems so that this framework allows for effective verification of such performance requirements. In this paper, it is shown that certain scenarios such as the wire-tap channel or compound channel are Turing computable so that the corresponding communication protocols can be effectively verified. A condition is presented and proved under which a communication scenario is not Turing computable. The channel with an active jammer falls into this category when deterministic codes are used. Interestingly, random codes robustify the channel with an active jammer in the sense that it becomes Turing computable and the corresponding communication protocol effectively verifiable.