We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. More precisely, a distribution X over binary strings of length n is called a δ-source if it assigns probability at most 2 −δn to any string of length n, and for any δ> 0 we construct the following poly(n)-time computable functions: 2-source disperser: D: ({0, 1} n) 2 → {0, 1} such that for any two independent δ-sources X1, X2 we have that the support of D(X1, X2) is {0, 1}. Bipartite Ramsey graph: Let N = 2 n. A corollary is that the function D is a 2-coloring of the edges of KN,N (the complete bipartite graph over two sets of N vertices) such that any induced subgraph of size N δ by N δ is not monochromatic. 3-source extractor: E: ({0, 1} n) 2 → {0, 1} such that for any three independent δ-sources X1, X2, X3 we have that E(X1, X2, X3) is (o(1)-close to being) an unbiased random bit. No previous explicit construction was known for either of these, for any δ < 1/2 and these results constitute major progress to long-standing open problems. A component in these results is a new construction of condensers that may be of independent

Abstract. We give an introduction to the area of “randomness extraction” and survey the main concepts of this area: deterministic extractors, seeded extractors and multiple sources extractors. For each one we briefly discuss background, definitions, explicit constructions and applications. 1

We construct an explicit disperser for affine sources over F n 2 with entropy k = 2 log0.9 n = n o(1). This is a polynomial time computable function D: F n 2 → {0, 1} such that for every affine space V of F n 2 that has dimension at least k, D(V) = {0, 1}. This improves the best previous construction of Ben-Sasson and Kopparty (STOC 2009) that achieved k = Ω(n 4/5). Our technique follows a high level approach that was developed in Barak, Kindler, Shaltiel, Sudakov

Getting the deterministic complexity closer to the best known randomized complexity is an important goal in algorithms and communication protocols. In this work, we investigate the case where instead of one input, the algorithm\protocol is given multiple inputs sampled independently from an arbitrary unknown distribution. We show that in this case a strong and generic derandomization result can be obtained by a simple argument. Our method relies on extracting randomness from “same-source ” product distributions, which are distributions generated from multiple independent samples from the same source. The extraction process succeeds even for arbitrarily low min-entropy, and is based on the order of the values and not on the values themselves (This may be seen as a generalization of the classical method of Von-Neumann [23] extended by Elias [7] for extracting randomness from a biased coin). The tools developed in the paper are generic, and can be used elsewhere. We present applications to streaming algorithms, and to implicit probe search [8]. We also refine our method to handle product distributions, where the i’th sample comes from one of several arbitrary unknown distributions. This requires creating a new set of tools, which may also be of independent interest. 1

Kuznetsov and Tsybakov [11] considered the problem of storing information in a memory where some of the cells are ‘stuck ’ at certain values. More precisely, For 0 < r, p < 1 we want to store a string z ∈ {0, 1} rn in an n-bit memory x = (x1,..., xn) in which a subset S ⊆ [n] of size pn are stuck at certain values u1,..., upn and cannot be modified. The encoding procedure receives S, u1,..., upn and z and can modify the cells outside of S. The decoding procedure should be able to recover z given x (without having to know S or u1,..., upn). This problem is related to, and harder than, the Write-Once-Memory (WOM) problem (in which once we raise a cell xi from zero to one, it is stuck at this value). We give explicit schemes with rate r ≥ 1 − p − o(1) (note that r ≤ 1 − p is a trivial lower bound). This is the first explicit scheme with asymptotically optimal rate. We are able to guarantee the same rate even if following the encoding, the memory x is corrupted in o ( √ n) adversarially chosen positions. This more general setup was first considered by Tsybakov [26] (see also [10, 8]) and our scheme improves upon previous results. Our approach utilizes a recent connection observed by Shpilka [23] between the WOM problem