On Expected Probabilistic Polynomial-Time Adversaries: A suggestion for restricted de nitions and their bene ts

This paper concerns the possibility of developing a coherent theory of security when feasibility is associated with expected probabilistic polynomial-time (expected PPT). The source of diiculty is that the known deenitions of expected PPT strategies (i.e., expected PPT interactive machines) do not support natural results of the type presented below. To overcome this diiculty, we suggest new deenitions of expected PPT strategies, which are more restrictive than the known deenitions (but nevertheless extend the notion of expected PPT non-interactive algorithms). We advocate the conceptual adequacy of these deenitions, and point out their technical advantages. Speciically, identifying a natural subclass of black-box simulators, called normal, we prove the following two results: 1. Security proofs that refer to all strict PPT adversaries (and are proven via normal black-box simulators), extend to provide security with respect to all adversaries that satisfy the restricted deenitions of expected PPT. 2. Security composition theorems of the type known for strict PPT hold for these restricted deenitions of expected PPT, where security means simulation by normal black-box simu-lators. Speciically, a normal black-box simulator is required to make an expected polynomial number of steps, when given oracle access to any strategy, where each oracle call is counted as a single step. This natural property is satisses by most known simulators and is easy to verify.