By Michael Luby, Avi Wigderson
ISBN-10: 1933019220
ISBN-13: 9781933019222
Read or Download Pairwise Independence and Derandomization (Foundations and Trends(R) in Theoretical Computer Science) PDF
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Extra resources for Pairwise Independence and Derandomization (Foundations and Trends(R) in Theoretical Computer Science)
Example text
M} |Ci | ≤ ≤ m. 1 of the previous section, we can approxi1 mate f (F ) with N = 4m 2 log( δ ) trials. In each trial we will 1. Choose index i ∈ {1, . . , m} with probability |Ci |/|U |. ) 2. Choose a ∈R Ci . This step takes O(r) time. 3. See if (i, a) ∈ G. This can be done in the obvious way in time O(rm). 4. The value produced by the trial is |U | if (i, a) ∈ G, and 0 otherwise. The overall estimate is the average of the values produced by the N trials. 1 this is guaranteed to be an -good relative estimate of f (F ) with probability at least 1 − δ.
Let f (F ) = |{a ∈ {0, 1}r : M (F, a) = 1}|. It is #P-complete to compute f . The following fpras for approximating f is due to Karpinski-Luby [26]. We design two different fpras algorithms A0 and A1 ; A0 is used in the case when F does not contain the constant term 1 and A1 is used in the case when F contains the term 1. Note that the term 1 corresponds to the product of the empty set of variables, and is satisfied by all assignments to y. The analyses of the two algorithms are very similar. The running time of A0 is O(rm2 ln(1/δ)/ 2 ) and the running time of A1 is O(rm3 ln(1/δ)/ 2 ).
N do: Compute Yi (β, v) = 1/m · m j=1 B(ei ⊕ Tj (v)) · β L ← L ∪ { bit(Y1 (β, v)), . . , bit(Yn (β, v)) }. j. From the above analysis, it follows that x ∈ L with probability at least 1/2, where this probability is over the random choice of v. As long as the running time T for computing B is large compared to n (which it is in our use of the Hidden Bit Technical Theorem to prove the Hidden Bit Theorem), the running time of S B is O(n3 T /δ 4 ). , it is certainly not the case that any bit of the input to f is hidden if f is a one-way function.
Pairwise Independence and Derandomization (Foundations and Trends(R) in Theoretical Computer Science) by Michael Luby, Avi Wigderson
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