Some applications of pairwise independence Research Seminar in Cryptography

This report provides a brief review of lectures given by Michael Luby and Avi Wigderson in 1995 on the topic of pairwise independence and derandomization. We show that BPP ⊂ ∆2 with a proof given by Sipser in 1983 and then exhibit different ways of recycling random bits given by Chor and Goldreich (1989), Nisan (1992), Impagliazzo and Zuckerman (1990), Karp, Pippenger and Sipser (1986) and by Ajtai, Komloś and Szemerédi (1987). The article is mainly intended to show the power and applications of pairwise independent distributions in both complexity theory and in generating and recycling pseudorandom bits. 1 Pairwise independence and tools of the trade We look at a group of random variables Z = {Zu : u ∈ U} where U is an index set for the variables with |U | = N . Each Zi takes values