"This project is concerned with the necessity and availability of randomness in computation. This research area (often referred to as the ""Theory of Derandomization'') is one of the most active and exciting areas in theoretical computer science. We intend to study some of the main questions of this area: In what setups can randomized algorithms be efficiently simulated by deterministic ones? In cases that randomness is essential, how can computers obtain random bits? More specifically, how can computers generate secret random keys for running secure cryptographic protocols? Furthermore, once random keys are generated, how can computers maintain the secrecy of their keys in the presence of side-channel attacks? These are important real-world problems and our research is intended to lay the theoretical foundation for achieving actual solutions.Our main approach for these problems is to design ``pseudorandom generators'' and ``randomness extractors'' that are efficient algorithms that manipulate randomness in various ways. The PI is one of the leading figures in this research area and many of the concrete research directions suggested in this proposal are related to and build on past work of the PI.The goals outlined in this proposal are important open problems in this area. Some of them (such as derandomizing bounded memory randomized algorithms and constructing 2-source extractors and dispersers for low min-entropy) are famous longstanding open problems and solving either of them will be a dramatic breakthrough in theoretical computer science. We suggest concrete (and we believe novel) approaches to attack these problems. Along the way we identify important and accessible intermediate goals.In addition to the development of the theory of derandomization, past work in this area (including work of the PI) had big impact on other areas of Computer Science and Mathematics such as Combinatorics, Cryptography, Coding Theory and Ramsey Theory."