Coding constructions for blacklisting problems without computational assumptions

Abstract

We consider the broadcast exclusion problem: how to transmit a message over a broadcast channel shared by N = 2n users so that all but some specified coalition of k excluded users can understand the contents of the message. Using error-correcting codes, and avoiding any computational assumptions in our constructions, we construct natural schemes that completely avoid any dependence on n in the transmission overhead. Specifically, we construct: (i) (for illustrative purposes,) a randomized scheme where the server’s storage is exponential (in n), but the transmission overhead is O(k), and each user’s storage is O(kn); (ii) a scheme based on polynomials where the transmission overhead is O(kn) and each user’s storage is O(kn); and (iii) a scheme using algebraic-geometric codes where the transmission overhead is O(k2) and each user is required to store O(kn) keys. In the process of proving these results, we show how to construct very good cover-free set systems and combinatorial designs based on algebraic-geometric codes,which may be of independent interest and application. Our approach also naturally extends to solve the problem in the case where the broadcast channel may introduce errors or lose information.