International Association
for Cryptologic Research

This paper surveys the results obtained so far in designing identity-based encryption (IBE) schemes based on the quadratic residuosity assumption (QRA). We begin by describing the r-st such scheme due to Cocks, and then we advance to the novel idea of Boneh, Gentry and Hamburg. Major improvements of the Boneh-Gentry-Hamburg scheme are then recalled. The recently revealed algebraic torus structures of the Cocks scheme allows for a better understanding of this scheme, as well as for new applications of it such as homomorphic and anonymous variants of it.