Programmability in the Generic Ring and Group Models

The programmability has long been used as a tool to prove security of schemes in the random oracle model (ROM) even in the cases where schemes do not seem to have a security proof in the standard model [3, 8, 10]. On the other hand, it seems that a similar property has never been studied in the generic models, i.e., the generic ring and group models, respectively the GRM and the GGM. This work introduces this study. We start by proposing the use of the GRM and the GGM in simulationbased security proofs, instead of the classical two-step approach: (1) find an efficient reduction R from a problem P to an adversary breaking the scheme in some sense, and (2) use the GRM/GGM to find a lower bound in the complexity of solving P. We observe that in such a model the simulator can choose the outputs for the generic operation oracle in a similar fashion as the programmability property of the ROM. We introduce four models named programmable and non-programmable for the GGM and analogously for the GRM. We show that in the programmable generic models it is possible to turn around the negative result by Nielsen [21], regarding the non-committing encryption in the presence of an adversary who corrupts the receiver. We illustrate our idea by proving that the Goldwasser-Micali encryption scheme is a non-committing encryption scheme regarding corruption of the receiver in the programmable GRM. Whereas, for the programmable GGM, we show that the popular ElGamal encryption scheme is also non-committing despite the corruptions of the receiver and the sender. In both schemes the attack exposes the secret key.