Paper Talk: “Extracting Randomness: How and Why. A survey,” by Noam Nisan

1. Paper Talk: “Extracting Randomness: How and Why. A survey,” by Noam Nisan Albert Boggess

2. Randomized Algorithms • Can solve problems for which there are no known deterministic algorithms • Often simpler than deterministic equivalents

3. Generating Randomness • Pseudo-random number generators are not sufficient • True randomness? • Physical source can provide some true randomness • Dispersers and extractors

4. Dispersers and Extractors • The goal is to “convert a somewhat random distribution into an almost random distribution” by adding a small number of truly random bits. • Can be represented as either graphs or functions

5. Definitions • Probability distribution X over finite space A: • X(a) ≥ 0 for all a in A • ∑aX(a) = 1 • Statistical distance between two probability distributions: • d(X, Y) = (½)∑a|X(a) – Y(a)| • X is e-close to Y if d(X, Y) ≤ e • Min-Entropy of a distribution: • H∞(X) = mina{-log2(X(a))}

6. Extractors and Dispersers Graph • A type of bipartite graph where: • Left set [N] contains N = 2n vertices and right set [M] contains M = 2m vertices. Typically n > m. • Vertices are numbered by binary integers: • N = {1…N} = {0,1}n • M = {1…M} = {0,1}m • All vertices in the left set have the same degree D = 2d.

7. Extractors and Dispersers Graph • Given a graph G = ([N], [M], E), the neighbor set of a vertex a in [N] is defined as T(a) = {z in [M] | (a,z) is in E}. • For a probability distribution X, T(X) is the probability distribution induced on [M] by choosing an a in [N] according to X, and then choosing a random neighbor z in T(a).

8. Dispersers and Extractors Graph • Disperser: • G = ([N], [M], E) is a (k, e)-disperser if for all A in [N] where |A| ≥ K = 2k, |T(A)| ≥ (1 – e)M. • Extractor: • G = ([N], [M], E) is a (k, e)-extractor if for any distribution X with H∞(X) ≥ k, T(x) is e-close to uniform on [M]. • Any (k, e)-extractor is a (k, e)-disperser.

9. Dispersers and Extractors Function • Given integer sets [N], [M],and [D], the function is defined as G : [N] x [D]  [M]. • T(x) = {z = G(x, y) | x is in [N], y is in [D]} • T(X) is the distribution of G(x, y) induced by choosing x according to distribution X and y uniformly in [D].

10. Extractors and Dispersers Function • Disperser: • G : [N] x [D]  [M] is called a (k, e)-disperser if for any A in [N] where |A| ≥ K = 2k, |T(A)| ≥ (1 – e)M. • Extractor: • G : [N] x [D]  [M] is called a (k, e)-extractor if for any distribution X on [N] where H∞(X) ≥ k, T(X) is e-close to uniform.

11. Goals • Minimize k, the amount of required “randomness” of the distribution on [N]. • Minimize e, the error. • Minimize d, the number of truly random bits required.

12. Construction: Hashing • Hash functions can be used to construct extractors. • If H is a family of hash functions h : [N]  [L], then we say that H has collision error ∂ if P(h(a) = h(b)) ≤ (1 + ∂)/L • Given a family of hash functions H where h : [N]  [L], the extractor defined by H is G(x, h) = h(x)ºh. Therefore D = |H| and M = DL.

13. Construction: Hashing • Given a family of hash functions H which map [N] to [L] and have collision error ∂, the extractor defined from H is a (k, e)-extractor where K = 2k = O(L/∂) and e = O(√∂).

14. Construction: Hashing • Universal Hashing: • H = {h : [N]  [L]} where P(h(a) = x AND h(b) = y) = 1/L2 • Universal hash functions have 0 collision error. • For all 1 ≤ i ≤ n, there are universal hash function families of size |H| = poly(N). • d = O(n) • k = m – d + O(loge-1) • d is much too high.

15. Construction: Hashing • Tiny families of hash functions: • We don’t require 0 collision error, just small collision error. • For all 1 ≤ i ≤ L ≤ N and e > 0, there exist families of hash functions that map [N] to [L] that have collision error e, and size |H| = poly(n,e,L). • This translates to (k, e)-extractors with D = poly(n,e,M) and k = m – d + O(loge-1)

16. Composing Extractors • Can be composed by using the output of one extractor as the input of another. G1(x1, G2(x2 ,y)). • Only holds if X and X are independent. • (X1,X2) is a block-wise source if: • H∞(X1) ≥ k1 • H∞(X2 | X1 = x1) ≥ k2

17. Reference Noam Nisan, “Extracting Randomness: How and Why A survey,” Proceedings of Computation Complexity 1996.

18. Known Results