Generic-group identity-based encryption: A tight impossibility result

Following the pioneering work of Boneh and Franklin (CRYPTO'01), the challenge of constructing an identity-based encryption scheme based on the Diffie-Hellman assumption remained unresolved for more than 15 years. Evidence supporting this lack of success was provided by Papakonstantinou, Rackoff and Vahlis (ePrint'12), who ruled out the existence of generic-group identity-based encryption schemes supporting an identity space of sufficiently large polynomial size. Nevertheless, the breakthrough result of Döttling and Garg (CRYPTO'17) settled this long-standing challenge via a non-generic construction. We prove a tight impossibility result for generic-group identity-based encryption, ruling out the existence of any non-trivial construction: We show that any scheme whose public parameters include npp group elements may support at most npp identities. This threshold is trivially met by any generic-group public-key encryption scheme whose public keys consist of a single group element (e.g., ElGamal encryption). In the context of algebraic constructions, generic realizations are often both conceptually simpler and more efficient than non-generic ones. Thus, identifying exact thresholds for the limitations of generic groups is not only of theoretical significance but may in fact have practical implications when considering concrete security parameters.