By Andrej Bogdanov, Luca Trevisan

Average-Case Complexity is an intensive survey of the average-case complexity of difficulties in NP. The learn of the average-case complexity of intractable difficulties all started within the Nineteen Seventies, prompted via precise purposes: the advancements of the principles of cryptography and the hunt for tactics to "cope" with the intractability of NP-hard difficulties. This survey seems at either, and customarily examines the present nation of information on average-case complexity. Average-Case Complexity is meant for students and graduate scholars within the box of theoretical desktop technology. The reader also will find a variety of effects, insights, and facts concepts whose usefulness is going past the learn of average-case complexity.

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**Extra resources for Average-Case Complexity (Foundations and Trends(R) in Theoretical Computer Science)**

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R1 r2 . . rm(n) . Here, m(n) is the running time of the algorithm that computes fDn , and we assume without loss of generality (for technical reasons) that m is injective. It is easy to check that each sample is produced with the correct probability. Observe that the sampler S is efficiently invertible in the following sense: There exists an algorithm I that on input x ∈ Supp(Dn ) runs in time polynomial in n and outputs a uniformly random r ∈ {0, 1}m(n) conditioned on S(n; r) = x (meaning that S(n) outputs x when using r for its internal coin tosses).

4 Decision Versus Search and One-Way Functions In worst-case complexity, a search algorithm A for an NP-relation V is required to produce, on input x, a witness w of length poly(|x|) such that V accepts (x; w), whenever such a w exists. Abusing terminology, we sometimes call A a search algorithm for the NP-language LV consisting of all x for which such a witness w exists. Thus, when we say “a search algorithm for L” we mean an algorithm that on input x ∈ L outputs an NP-witness w that x is a member of L, with respect to an implicit NP-relation V such that L = LV .

2. 2 37 The completeness result In this section we prove the existence of a complete problem for (NP, PComp), the class of all distributional problems (L, D) such that L is in NP and D is polynomial-time computable. Our problem is the following “bounded halting” problem for non-deterministic Turing machines: BH = {(M, x, 1t ) : M is a non-deterministic Turing machine that accepts x in ≤ t steps}. 1) Note that BH is NP-complete: Let L be a language in NP and M be a non-deterministic Turing machine that decides L in time at most p(n) on inputs of length n.