Secure Source Coding with a Helper

We consider a secure lossless source coding problem with a rate-limited helper. In particular, Alice observes an i.i.d. source X-n and wishes to transmit this source losslessly to Bob at a rate R-x. A helper, say Helen, observes a correlated source Y-n and transmits at a rate R-y to Bob. A passive eavesdropper can observe the coded output of Alice. The equivocation Delta is measured by the conditional entropy H (X-n vertical bar J(x))/n, where J(x) is the coded output of Alice. We first completely characterize the rate-equivocation region for this secure source coding model, where we show that Slepian-Wolf type coding is optimal. We next study two generalizations of this model and provide single-letter characterizations for the respective rate-equivocation regions. In particular, we first consider the case of a two-sided helper where Alice also has access to the coded output of Helen. We show that for this case, Slepian-Wolf type coding is suboptimal and one can further decrease the information leakage to the eavesdropper by utilizing the side-information at Alice. We finally generalize this result to the case when there are both secure and insecure rate-limited links from Helen and additional uncoded side informations W-n and Z(n) available at Bob and Eve, respectively. For this model, we provide a complete characterization of the rate-equivocation region when Y-n -> X-n -> (W-n, Z(n)) forms a Markov chain.