On the mean number of encryptions for tree-based broadcast encryption schemes

The challenge of stateless-receiver broadcast encryption lies in minimizing storage and the number of encryptions while maintaining system security. Tree-based key distribution schemes offer the best known trade-off between the two parameters. Examples include the complete subtree scheme [D. Wallner, et al., Internet draft, http://www.ietf.org/ID.html [10]; C.K. Wong, et al., in: Proc. SIGCOMM, 1998, pp. 68–79 [11]], the subset difference scheme [D. Naor, et al., in: CRYPTO 2001, Lecture Notes in Comput. Sci., vol. 2139, 2001, pp. 41–62 [7]], and the layered subset difference scheme [D. Halevy, A. Shamir, in: CRYPTO 2002, Lecture Notes in Comput. Sci., vol. 2442, 2002, pp. 47–60 [5]]. We introduce generating functions for this family of schemes, which lead to analysis of the mean number of encryptions over all privileged sets of users. We also derive the mean number of encryptions when the number of privileged users is fixed. We expect that the techniques introduced as well as the results in this work will find applications in related areas.