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By Michael Luby, Avi Wigderson

ISBN-10: 1933019220

ISBN-13: 9781933019222

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The expander mixing Lemma √ √ Then α0 = |A|/ n and β0 = |B|/ n, and we have |E(A, B)| = M (i, j) i∈A,j∈B = XA · M · X B     αi ri  ·  λj βj rj  = 0≤i≤n−1 = 0≤j≤n−1 λi αi βi 0≤i≤n−1 = =⇒ |E(A, B)| − d|A||B| + n d|A||B| ≤λ n 1≤i≤n−1 |αi βi | 0≤i≤n−1 ≤λ α β 2 = λ XA =λ λi αi βi 2 2 XB 2 |A||B| Another way to state the Expander Mixing Lemma is that for all A, B ⊆ U, Pr[x ∈ A, es (x) ∈ B] − Pr[x ∈ A, y ∈ B] ≤ x,s x,y λ , d where in the first random experiment x is a uniformly chosen node, s ∈R {1, .

1 One-way functions Intuitively, a one-way function is a function that is easy to compute but hard for any time-bounded adversary to invert on a random input. To gauge the success of an adversary in breaking a cryptographic function, we use the following measure. 1. (time/success ratio) The time/success ratio of an adversary for breaking a cryptographic function is T (n)/δ(n), where T (n) is the run time of the adversary and δ(n) is the success probability 35 36 Pseudo-Random Generators of the adversary with respect to inputs parameterized by n.

N}, Tj (v) m Yi (v) = 1/m · B(ei ⊕ Tj (v)) · (x v) j. j=1 The key point is that there are only 2 = O(m) possible settings for x v, and we try them all. For any x and v there is some β ∈ {0, 1} such that β = x v. , Yi (β, v) is the value obtained when β is substituted for x v in the computation of Yi (v). Consider choosing v ∈R {0, 1}n× . Since from equation (2) above, the probability that Yi (x v, v) is a good approximation for all i ∈ {1, . . 2. Hidden Bit Theorem 43 Yi (β, v) is simultaneously a good approximation for all i ∈ {1, .

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

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