Identity-Based Cryptosystems and Quadratic Residuosity

Three approaches are currently used for devising identitybased encryption schemes. They respectively build on pairings, quadratic residues (QR), and lattices. Among them, the QR-based scheme proposed by Cocks in 2001 is notable in that it works in standard RSA groups: its security relies on the standard quadratic residuosity assumption. But it has also a number of deficiencies, some of them have been subsequently addressed in follow-up works. Currently, one of the main limitations of Cocks’ scheme resides in its apparent lack of structure. This considerably restricts the range of possible applications. For example, given two Cocks ciphertexts, it is unknown how to evaluate of a function thereof. Cocks’ scheme is believed to be non-homomorphic. This paper disproves this conjecture and proposes a constructive method for computing over Cocks ciphertexts. The discovery of the hidden algebraic structure behind Cocks encryption is at the core of the method. It offers a better understanding of Cocks’ scheme. As a further illustration of the importance of the knowledge of the underlying structure, this paper shows how to anonymize Cocks ciphertexts without increasing their size or sacrificing the security. Finally and of independent interest, this paper presents a simplified version of the abstract identity-based cryptosystem with short ciphertexts of Boneh, Gentry, and Hamburg.