Scribe : Constantinos Daskalakis Notes for Lecture 18 1 Basic Definitions

In the previous lecture we defined the notion of a randomness extractor as follows: Definition 1 A function Ext : {0, 1} n ×{0, 1} d → {0, 1} m is a (k,)-extractor if, for every random variable X of range {0, 1} n-which we will call a source-of min-entropy H ∞ (X) ≥ k, the following holds for the statistical difference of Ext(X, U d) and U m : · the min-entropy of a random variable X of a finite range A is defined as H ∞ (X) = min a∈A 1 Pr[X=a] · the statistical difference ||·|| SD of two random variables Y and Z of a finite range A is defined as: ||Y − Z|| SD = max The general goal is to keep the " high-quality " input randomness-parameter d of the model-as small as possible, while maintaining the output randomness-parameter m-as close to the sum k + d as possible. Last time, we showed some negative results asserting that the best tradeoff we can hope for is something of the flavor: m = k + d − 2 log 1 + Θ(1) and d = log (n − k) + 2 log 1 − Θ(1) Today we will give a construction of randomness extractors. But first we will motivate our pursuit by giving an interesting application. Suppose that A L is a randomized algorithm for a language L ⊆ {0, 1} l , which uses m bits of randomness and has error probability ≤ 1 4. A common way to boost the probability of success of A L is to execute it t times on independent randomness and output the majority answer. In this case, the random bits that are needed are t · m and the probability of error is bounded by e −Ω(t) .