On Expected Probabilistic Polynomial-Time Adversaries: A Suggestion for Restricted Definitions and Their Benefits

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 difficulty is that the known definitions of

expected

PPT

strategies

(i.e.,

expected

PPT

interactive

machines) do not support natural results of the type presented below. To overcome this difficulty, we suggest new definitions of

expected

PPT strategies, which are more restrictive than the known definitions (but nevertheless extend the notion of

expected

PPT

non-interactive

algorithms). We advocate the conceptual adequacy of these definitions, and point out their technical advantages. Specifically, 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 definitions of

expected

PPT.

1

Security composition theorems of the type known for

strict

PPT hold for these restricted definitions of

expected

PPT, where security means simulation by normal black-box simulators.

Specifically, 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 satisfies by most known simulators and is easy to verify.