Coding for the Wiretap Channel
Abstract: We consider code design for Wyner’s wiretap channel. Optimal coding schemes for this channel require an overall code that is capacity achieving for the main channel, partitioned into smaller subcodes, all of which are capacity achieving for the wiretapper’s channel. To accomplish this we introduce two edge type low density parity check (LDPC) ensembles for the wiretap channel. For the scenario when the main channel is error free and the wiretapper’s channel is a binary erasure channel (BEC) we find secrecy capacity achieving code sequences based on standard LDPC code sequences for the BEC. However, this construction does not work when there are also erasures on the main channel. For this case we develop a method based on linear programming to optimize two edge type degree distributions. Using this method we find code ensembles that perform close to the secrecy capacity of the binary erasure wiretap channel (BEC- WT). We generalize a method of M ́easson, Montanari, and Urbanke in order to compute the conditional entropy of the message at the wire- tapper. This conditional entropy is a measure of how much information is leaked to the wiretapper. We apply this method to relatively simple ensembles and find that they show very good secrecy performance.Based on the work of Kudekar, Richardson, and Urbanke, which showed that regular spatially coupled codes are capacity achieving for the BEC, we construct a regular two edge type spatially coupled ensem- ble. We show that this ensemble achieves the whole capacity-equivocation region for the BEC-WT.We also find a coding scheme using Arıkans polar codes. These codes achieve the whole capacity-equivocation region for any symmetric binary input wiretap channel where the wiretapper’s channel is degraded with respect to the main channel.
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