Optimising SD and LSD in Presence of Non-uniform Probabilities of Revocation

Some years ago two efficient broadcast encryption schemes for stateless receivers, referred to as SD (Subset Difference Method) [NNL01] and LSD (Layered Subset Difference Method) [HS02] , were proposed. They represent one of the most suitable solution to broadcast encryption. In this paper we focus on the following issue: both schemes assume uniform probabilities of revocation of the receivers. However, in some applications, such an assumption might not hold: receivers in a certain area, due to historical and legal reasons, can be considered trustworthy, while receivers from others might exhibit more adversarial behaviours. Can we modify SD and LSD to better fit settings in which the probabilities of revocation are non-uniform? More precisely, we study how to optimise user key storage in the SD and LSD schemes in presence of non-uniform probabilities of revocation for the receivers. Indeed, we would like to give less keys to users with higher probability of revocation compared to trustworthy users. We point out that this leads to the construction of binary trees satisfying some optimality criteria. We start our analysis revisiting a similar study, which aims at minimising user key storage in LKH schemes. It was shown that such a problem is related to the well-known optimal codeword length selection problem in information theory. We discuss the approach therein pursued, pointing out that a characterisation of the properties a key assignment for LKH schemes has to satisfy, does not hold. We provide a new characterisation and give a proof of it. Then, we show that also user key storage problems of SD and LSD are related to an interesting coding theory problem, referred to as source coding with Campbell's penalties. Hence, we discuss existing solutions to the coding problem.