Randomness extractors for independent sources and applications

The use of randomized algorithms and protocols is ubiquitous in computer science. Randomized solutions are typically faster and simpler than deterministic ones for the same problem. In addition, many computational problems (for example in cryptography and distributed computing) are impossible to solve without access to randomness. In computer science, access to randomness is usually modeled as access to a string of uncorrelated uniformly random bits. Although it is widely believed that many physical phenomena are inherently unpredictable, there is a gap between the computer science model of randomness and what is actually available. It is not clear where one could find such a source of uniformly distributed bits. In practice, computers generate random bits in ad-hoc ways, with no guarantees on the quality of their distribution. One aim of this thesis is to close this gap and identify the weakest assumption on the source of randomness that would still permit the use of randomized algorithms and protocols. This is achieved by building randomness extractors. A randomness extractor is an algorithm that computes a function Ext : {0, 1}n → {0, 1} m, with the property that for any defective source of randomness X satisfying minimal assumptions, Ext(X) is close to uniformly distributed. Such an algorithm would allow us to use a compromised source of randomness to obtain truly random bits, which we could then use in our original application. Randomness extractors are interesting in their own right as combinatorial objects that look random in strong ways. They fall into the class of objects whose existence is easy to check using the probabilistic method (i.e., almost all functions are good randomness extractors), yet finding explicit examples of a single such object is non-trivial. Expander graphs, error correcting codes, hard functions, epsilon biased sets and Ramsey graphs are just a few examples of other such objects. Finding explicit examples of extractors is part of the bigger project in the area of derandomization of constructing such objects which can be used to reduce the dependence of computer science solutions on randomness. These objects are often used as basic building blocks to solve problems in computer science. The main results of this thesis are: Extractors for Independent Sources. The central model that we study is the model of independent sources. Here the only assumption we make (beyond the necessary one that the source of randomness has some entropy/unpredictability), is that the source can be broken up into two or more independent parts. We show how to deterministically extract true randomness from such sources as long as a constant (as small as 3) number of sources is available with a small amount of entropy. Extractors for Small Space Sources. In this model we assume that the source is generated by a computationally bounded processes—a bounded width branching program or an algorithm that uses small memory. This seems like a plausible model for sources of randomness produced by a defective physical device. We build on our work on extractors for independent sources to obtain extractors for such sources. Extractors for Low Weight Affine Sources. In this model, we assume that the source gives a random point from some unknown low dimensional affine subspace with a low-weight basis. This model generalizes the well studied model of bit-fixing sources. We give new extractors for this model that have exponentially small error, a parameter that is important for an application in cryptography. The techniques that go into solving this problem are inspired by the techniques that give our extractors for independent sources. Ramsey Graphs. A Ramsey graph is a graph thathas no large clique or independent set. We show how to use our extractors and many other ideas to construct new explicit Ramsey graphs that avoid cliques and independent sets of the smallest size to date. Distributed Computing with Weak Randomness. Finally, we give an application of extractors for independent sources to distributed computing. We give new protocols for Byzantine Agreement and Leader Election that work when the players involved only have access to defective sources of randomness, even in the presence of completely adversarial behavior at many players and limited adversarial behavior at every player. In fact, we show how to simulate any distributed computing protocol that assumes that each player has access to private truly random bits, with the aid of defective sources of randomness.