By Andrej Bogdanov, Luca Trevisan

Average-Case Complexity is a radical survey of the average-case complexity of difficulties in NP. The research of the average-case complexity of intractable difficulties all started within the Nineteen Seventies, encouraged via certain functions: the advancements of the rules of cryptography and the hunt for tactics to "cope" with the intractability of NP-hard difficulties. This survey appears to be like at either, and usually examines the present kingdom of data on average-case complexity. Average-Case Complexity is meant for students and graduate scholars within the box of theoretical laptop technology. The reader also will find a variety of effects, insights, and evidence options whose usefulness is going past the examine of average-case complexity.

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**Example text**

Abusing notation, we also denote the class whose only member is the uniform ensemble by U. 3). The converse does not hold unless P = P#P . In fact, PComp = PSamp if and only if P = P#P . Distributional complexity classes: A distributional complexity class is a collection of distributional decision problems. For a class of languages C and a class of ensembles D, we use (C, D) to denote the distributional complexity class consisting of all problems (L, D), where L ∈ C and D ∈ D. In this study we focus on the distributional complexity classes (NP, PSamp), (NP, PComp), and (NP, U).

3. Average-case complexity and one-way functions 49 Observe that the search-to-decision reduction only applies to decision algorithms that succeed on most instances. For the argument to achieve non-trivial parameters, the fraction of instances on which the decision algorithm fails must be smaller than 1/m(n)2 . 3 Average-case complexity and one-way functions If every problem is easy-on-average for the uniform ensemble, can oneway functions exist? The above arguments show that in the case for oneway permutations, the answer is no.

Using standard error reduction be repetition, the constants 21 and 14 can be amplified to 1 − exp(−(n/δ)O(1) ) and exp(−(n/δ)O(1) ), respectively. Finally, we define heuristic search algorithms: Such algorithms are allowed to output incorrect witnesses on a small fraction of inputs. 4. (randomized heuristic search) We say A is a randomized heuristic search algorithm for (L, D), where L ∈ NP, if for every n, on input x in the support of Dn and parameter δ > 0, A runs in time polynomial in n and 1/δ, and Prx∼Dn x ∈ L and PrA [A(x; n, δ) is not a witness for x] > 1/4 ≤ δ.