Variable-Length Extractors: Efficiently Extracting Randomness from Weak Stochastic Processes

The topic of generating random bits from imperfect random sources was studied extensively. Existing work provides solutions that are based on the concept of a fixed-length extractor; namely, an algorithm that gets a fixed number of random bits from a source and generates random bits. Typically, the number of random bits extracted using fixed-length extractors is upperbounded by the min-entropy. However, based on Shannon’s theory, the information theoretic limit for randomness extraction is the source’s entropy that is typically strictly larger than the min-entropy. In this paper, we consider a scenario where the source is a weak stochastic process and the output sequence is required to be ε-close to the uniform distribution on {0, 1}, where m is a prescribed value. We introduce a new class of extractors with a variable input-length and fixed output-length. Our variable-length extractors have two important properties: (i) they approach the information theoretic upper bound on efficiency (entropy of the source) while the extracted bits are random in the sense of statistical difference; and (ii) they minimize the expected number of symbols read from the source in order to reach a prescribed number of random bits.