Pseudorandomness and complexity

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Randomness has become an indispensable tool in many areas of computer science: algorithm design, networking, error-correcting codes, cryptography, etc. While the use of randomness is unavoidable for certain applications (e.g., cryptographic constructions), other applications can be made completely deterministic. One of the main challenges of theoretical computer science is to understand the exact power of randomness in computation. In particular, it is a big open question whether every randomized polynomial-time algorithm has an equivalent deterministic polynomial-time algorithm (i.e., can be efficiently derandomized). Answering this question is important for algorithm design, as fast deterministic algorithms are usually preferable to randomized ones. On the other hand, there is a deep connection between trying to derandomize probabilistic algorithms and trying to prove lower bounds on the computational power of Boolean circuits. The latter problem has been actively researched for several decades with the goal of resolving the "P vs. NP" question. However, no strong circuit lower bo unds are known for any explicit Boolean function.