By Andrej Bogdanov, Luca Trevisan

Average-Case Complexity is an intensive survey of the average-case complexity of difficulties in NP. The examine of the average-case complexity of intractable difficulties all started within the Nineteen Seventies, prompted by means of specific functions: the advancements of the principles of cryptography and the hunt for ways to "cope" with the intractability of NP-hard difficulties. This survey seems at either, and usually examines the present country of information on average-case complexity. Average-Case Complexity is meant for students and graduate scholars within the box of theoretical machine technological know-how. The reader also will find a variety of effects, insights, and facts thoughts whose usefulness is going past the learn of average-case complexity.

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**Example text**

If (NP, U) ⊆ AvgBPP (respectively, HeurBPP), then every problem in (NP, U) has an errorless (respectively, heuristic) randomized search algorithm. 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?

Computable ensemble) We say that an ensemble D = {Dn } is polynomial-time computable if there is an algorithm that, given an integer n and a string x, runs in time polynomial in n and computes fDn (x). Observe that if {Dn } is a computable ensemble, then in particular the function Dn (x) is computable in time polynomial in n. We let PSamp denote the class of polynomial-time samplable ensembles, and PComp denote the class of polynomial-time computable ensembles. 1 We stress, however, that the results that we prove about samplable ensembles remain true even if we adopt more relaxed definitions of samplability.

3, that is, assume that there is a polynomial p and an ε > 0 such that for every n, p(n) PrDn [tA (x; n) ≥ t] ≤ ε . t Then define the algorithm A that on input x and parameters n, δ simulates A(x; n) for (p(n)/δ)1/ε steps. If the simulation halts within the required number of steps, then A (x; n, δ) gives the same output as A(x; n); otherwise A (x; n, δ) outputs ⊥. It is easy to see that A satisfies the definition of an errorless heuristic scheme. Suppose now that A is an errorless heuristic scheme for (L, D).