Performance Analysis and Design of Two Edge Type LDPC Codes for the BEC Wiretap Channel

摘要

We consider transmission over a wiretap channel where both the main channeland the wiretapper's channel are Binary Erasure Channels (BEC). We propose acode construction method using two edge type LDPC codes based on the cosetencoding scheme. Using a standard LDPC ensemble with a given threshold over theBEC, we give a construction for a two edge type LDPC ensemble with the samethreshold. If the given standard LDPC ensemble has degree two variable nodes,our construction gives rise to degree one variable nodes in the code used overthe main channel. This results in zero threshold over the main channel. Inorder to circumvent this problem, we numerically optimize the degreedistribution of the two edge type LDPC ensemble. We find that the resultingensembles are able to perform close to the boundary of the rate-equivocationregion of the wiretap channel. There are two performance criteria for a coding scheme used over a wiretapchannel: reliability and secrecy. The reliability measure corresponds to theprobability of decoding error for the intended receiver. This can be easilymeasured using density evolution recursion. However, it is more challenging tocharacterize secrecy, corresponding to the equivocation of the message for thewiretapper. M\'easson, Montanari, and Urbanke have shown how the equivocationcan be measured for a broad range of standard LDPC ensembles for transmissionover the BEC under the point-to-point setup. By generalizing the method ofM\'easson, Montanari, and Urbanke to two edge type LDPC ensembles, we show howthe equivocation for the wiretapper can be computed. We find that relativelysimple constructions give very good secrecy performance and are close to thesecrecy capacity. However finding explicit sequences of two edge type LDPCensembles which achieve secrecy capacity is a more difficult problem. We poseit as an interesting open problem.