Extended Galbraith’s test on the anonymity of IBE schemes from higher residuosity

At PKC 2019, Clear and McGoldrick presented the first identity-based encryption (IBE) scheme that is group homomorphic for addition modulo a poly-sized prime e. Assuming that deciding solvability of a special system of multivariate polynomial equations is hard, they proved that their scheme for e> 2 is anonymous. In this paper, we review the classical Galbraith’s test on the anonymity of the first pairing-free IBE scheme due to Cocks. With the eye of the reciprocity law for F_q[x] , we can have a profound understanding of the test and naturally extend it to give a practical attack on the anonymity of the Clear–McGoldrick IBE scheme. Furthermore, we believe that our technique plays a crucial role in anonymizing IBE schemes from higher residuosity.