On secure source coding with side information at the encoder

We consider a secure source coding problem with side informations at the decoder and the eavesdropper. The encoder has a source that it wishes to describe with limited distortion through a rate-limited link to a legitimate decoder. The message sent is also observed by the eavesdropper. The encoder aims to minimize both the distortion incurred by the legitimate decoder; and the information leakage rate at the eavesdropper. When the encoder has access to the side information (S.I.) at the decoder, we characterize the rate-distortion-information leakage rate (R.D.I.) region under a Markov chain assumption and when S.I. at the encoder does not improve the rate-distortion region as compared to the case when S.I. is absent. We then extend our setting to consider the case where the encoder and decoder obtain coded S.I. through a rate-limited helper, and characterize the R.D.I. region for several special cases under logarithmic loss distortion (log-loss). Finally, we consider the case of list or entropy constraints at the decoder and show that the R.D.I. region coincides with R.D.I. region under log-loss.