Semantic security and anonymity are the two main properties that an identity-based encryption scheme can satisfy. Such properties can be de ned in either an adaptive or a selective scenario, which di er on the moment where the attacker chooses the identity/ies that are the target of the attack. There are well-known separations between selective and adaptive semantic security on the one hand, and between selective and adaptive anonymity on the other hand. In this paper we investigate the relations between these selective and adaptive notions, for identity-based encryption schemes enjoying at the same time some security and anonymity properties. On the negative side, we prove that there is a separation between selective and adaptive anonymity even for schemes which enjoy adaptive semantic security. On the positive side, we prove that selective semantic security and adaptive anonymity imply adaptive semantic security.

The work of Javier Herranz is supported by a Ramón y Cajal grant, partially funded by the European Social Fund (ESF) of the Spanish MICINN Ministry. Carla Ràfols holds an FPI grant of the Spanish MICINN Ministry. The work of both these authors is partially supported by the Spanish MICINN Ministry under project MTM2009-07694. The work of Fabien Laguillaumie is supported by the French ANR-07- TCOM-013-04 PACE Project