Reliable and perfectly secret communication over the generalized Ozarow-Wyner's wire-tap channel

In a typical secure communication system, messages undergo two different encodings: an error-correcting code is applied at the physical layer to ensure correct reception by the addressee (integrity), while at an upper protocol layer cryptography is leveraged to enforce secrecy with respect to eavesdroppers (confidentiality). All constructive solutions proposed so far to concurrently achieve both integrity and confidentiality at the physical layer, aim at meeting the secrecy capacity of the channel, i.e., at maximizing the rate of the code while guaranteeing an asymptotically small information leakage. In this paper, we propose a viable encoding scheme that, to the best of our knowledge, is the first one to guarantee both perfect secrecy (i.e., no information leakage) and reliable communication over the generalized Ozarow-Wyner's wire-tap channel. To this end, we first introduce a metric called uncertainty rate that, similarly to the equivocation rate metric, captures the amount of information leaked by a coding scheme in the considered threat model, but it is simpler to apply in the context of linear codes. Based on this metric, we provide an alternative and simpler proof of the known result that no linear error-correcting code alone can achieve perfect secrecy. Finally, we propose a constructive solution combining secret sharing and linear error-correcting codes, and we show that our solution provides the desired combination of reliable and perfectly secret communication. The provided solution, other than being supported by thorough analysis, is viable in practical communication systems.