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The main approaches currently used to construct identity-based encryption (IBE) schemes are based on bilinear mappings, quadratic residues and lattices. Among them, the most attractive approach is the one based on quadratic residues, due to the fact that the underlying security assumption is a well-understood hard problem. The first such IBE scheme was constructed by Cocks, and some of its deficiencies were addressed in subsequent works. In this paper, we focus on two constructions that address the anonymity problem inherent in Cocks’ scheme, and we tackle some of their incomplete theoretical claims. More precisely, we rigorously study Clear et al.’s and Zhao et al.’s schemes and give accurate probabilities of successful decryption and identity detection in the non-anonymized version of the schemes. Furthermore, in the case of Zhao et al.’s scheme, we give a proper description of the underlying security assumptions.

Generalized Galbraith’s Test: Characterization and Applications to Anonymous IBE Schemes