We ask whether an Identity Based Encryption (IBE) system can be built from simpler public-key primitives. We show that there is no black-box construction of IBE from Trapdoor Permutations (TDP) or even from Chosen Ciphertext Secure Public Key Encryption (CCA-PKE). These black-box separation results are based on an essential property of IBE, namely that an IBE system is able to compress exponentially many public-keys into a short public parameters string.

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This paper initiates the study of SAT instances of bounded diameter. The diameter of an ordered CNF formula is defined as the maximum difference between the index of the first and the last occurrence of a variable. We study the complexity of the satisfiability, the counting and the maximization problems for formulas of bounded diameter. We investigate the relation between the diameter of a formula, and the tree-width and the path-width of its corresponding incidence graph, and show that under highly parallel and efficient transformations, diameter and path-width are equal up to a constant factor.Our main technical contribution is that the computational complexity of SAT, Max-SAT, #SAT grows smoothly with the diameter (as a function of the number of variables). Our main focus is in providing space efficient and highly parallel algorithms, while the running time of our algorithms matches previously known results. Among others, we show NL-completeness of SAT and NC 2 algorithms for Max-SAT, #SAT when diameter is O(log n). Given the tree decomposition of a formula, we further improve on the space efficiency to decide SAT as asked by Alekhnovich and Razborov [1].