Pairwise Independence and Derandomization (Foundations and by Michael Luby, Avi Wigderson PDF

By Michael Luby, Avi Wigderson

ISBN-10: 1933019220

ISBN-13: 9781933019222

ISBN-10: 193301976X

ISBN-13: 9781933019765

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Pseudo-random generators 45 generator that stretches by 1 bit. The reduction should describe a TM M with the property that if A is an adversary for distinguishing g with time/success ratio S(n) then M A is an adversary for inverting f with time/success ratio S(n)c for some constant c > 0. The simple construction of a pseudo-random generator given in the previous exercise was one of the motivating forces behind the work of Goldreich-Levin [15]. The reduction from an arbitrary one-way function to a pseudo-random generator can be found in H˚ astad-ImpagliazzoLevin-Luby [19].

The definition of the complexity class #P, and the realization of its importance, are due to Valiant [38]. Examples of f ∈ #P are the following: • If x is the description of a graph then f (x) is the number of perfect matchings in the graph, else f (x) = 0. • If x is the description of a graph then f (x) is the number of Hamiltonian tours in the graph, else f (x) = 0. 47 48 Deterministic Counting • If x is the description of a DNF boolean formula then f (x) is the number of truth assignments that satisfy the formula, else f (x) = 0.

6 The Karp-Pippenger-Sisper generator The Karp-Pippenger-Sipser [25] generator uses the explicit construction of expanders: The set {0, 1}r is identified with the nodes of a 2r node k-regular expander with λ ≤ k 9/10 . The seed to the generator G is a string z ∈R {0, 1}r , and G(z) produces y1 , . . , yk , which are the k neighbors of z in the expander graph. Thus, this scheme uses exactly r random bits. 8. If x ∈ L then Pr[{y1 , . . , yk } ⊆ W x ] ≤ 2k −1/10 . Proof. Let A ⊆ {0, 1}r be the set of nodes z with the property that all neighbors of z are in W x .

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Pairwise Independence and Derandomization (Foundations and Trends in Theoretical Computer Science) by Michael Luby, Avi Wigderson


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