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 Xⁿ and wishes to transmit this source loss-lessly to Bob at a rate R_x. A helper, say Helen, observes a correlated source Yⁿ and transmits at a rate R_y to Bob. A passive eavesdropper can observe the coded output of Alice. The equivocation ¿ is measured by the conditional entropy H(Xⁿ 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 Wⁿ and Zⁿ available at Bob and Eve, respectively. For this model, we provide a complete characterization of the rate-equivocation region when Yⁿ ¿ Xⁿ ¿ (Wⁿ, Zⁿ) forms a Markov chain.