The equivalence of the random oracle model and the ideal cipher model has been studied in a long series of results. Holenstein, Künzler, and Tessaro (STOC, 2011) have recently completed the picture positively, assuming that, roughly speaking, equivalence is indifferentiability from each other. However, under the stronger notion of reset indifferentiability this picture changes significantly, as Demay et al. (EUROCRYPT, 2013) and Luykx et al. (ePrint, 2012) demonstrate.
We complement these latter works in several ways. First, we show that any simulator satisfying the reset indifferentiability notion must be stateless and pseudo-deterministic. Using this characterization we show that, with respect to reset indifferentiability, two ideal models are either equivalent or incomparable, that is, a model cannot be strictly stronger than the other model. In the case of the random oracle model and the ideal cipher model, this implies that the two are incomparable. Finally, we examine weaker notions of reset indifferentiability that, while not being able to allow composition in general, allow composition for a large class of multi-stage games. Here we show that the seemingly much weaker notion of 1-reset indifferentiability proposed by Luykx et al. is equivalent to reset indifferentiability. Hence, the impossibility of coming up with a reset indifferentiable construction transfers to the setting where only one reset is permitted, thereby re-opening the quest for an achievable and meaningful notion in between the two variants.