Deterministic Extractors - Lecture Notes

Randomness is used in many places in our daily lives. Some examples are gambling, statistics, algorithms, cryptography etc. For such applications one typically assumes a supply of completely unbiased and independent randim bits. This raises the problem of where to get these assumed random bits from. We could try and use natural sources of randomness such as sun spots, the stock market or the weather, which seem unpredictable at least to some degree. The problem is that these natural sources are not completely random, only somewhat random, in the sense that samples from them are often correlated and biased. The goal of deterministic extractors is to extract completely random bits from such “weak random sources”. One hopes to obtain extractors which would achieve this task for as large and general families of weak sources. We will concentrate on the problem of extracting only a single random bit, which turns out to demonstrate the variety of problems, results and techniques in this area (in the general case we would like to get many uncorrelated bits). We will not discuss here seeded extractors (e.g. see survey [Sha02] and the recent [DW08]).