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The environment outputs 1 if the adversary succeeds and 0 otherwise. , break the environment. More formally, Definition 33 We say that an interactive PPT machine E, called environment, is hard for the language L, the generator GenL , and T (n)-adversaries, if for all interactive probabilistic machine A with running time bounded by T (n), every z ∈ {0, 1}poly(n) P r[x ← GenL , A(z), E (x) = 1] is negligible as a function of n, where GenL is a sampling machine that chooses an element x ∈ L ∩ {0, 1}n according to some distribution.

8 In fact, we show that straight-line concurrent T (n)-simulatable arguments are “safe” for any environment that is hard for time T (n). The Composition Theorem We start by formally defining the notion of an environment. We see an environment as a system that an adversary is trying to break. The environment outputs 1 if the adversary succeeds and 0 otherwise. , break the environment. More formally, Definition 33 We say that an interactive PPT machine E, called environment, is hard for the language L, the generator GenL , and T (n)-adversaries, if for all interactive probabilistic machine A with running time bounded by T (n), every z ∈ {0, 1}poly(n) P r[x ← GenL , A(z), E (x) = 1] is negligible as a function of n, where GenL is a sampling machine that chooses an element x ∈ L ∩ {0, 1}n according to some distribution.

In order to obtain a notion that is robust under composition we restrict our attention to T (n)-simulatable proofs, where T (n) is a class of function that is closed under composition with any polynomial. Definition 28 Let T (n) be a class of functions that is closed under composition with any polynomial. We say that an interactive proof (or argument) (P, V ) for the language L ∈ N P, with the witness relation RL , is T (n)-simulatable if there for every PPT machine V ∗ exists a probabilistic simulator S with running time bounded by T (n) such that the following two ensembles are computationally indistinguishable (when the distinguishing gap is a function in n = |x|) 38 CHAPTER 3.

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