Abstract
A polynomial source of randomness over F n is a random variable X = f(Z) where f is a polynomial map and Z is a random variable distributed uniformly over F r for some integer r. The three main parameters of interest associated with a polynomial source are the order q of the field, the (total) degree D of the map f , and the base-q logarithm of the size of the range of f over inputs in F r , denoted by k. For simplicity we call X a (q; D; k)-source.
Highlights
This paper is part of a long and active line of research devoted to the problem of “randomness extraction”: Given a family of distributions all guaranteed to have a certain structure, devise a method that can convert a sample from any distribution in this family to a sequence of uniformly distributed bits—or at least a sequence statistically close to the uniform distribution
It is easy to prove that a random function is, with high probability, a good extractor for the given family, and the challenge is to give an explicit construction of such an extractor
In this paper we construct extractors for polynomial sources, which are distributions that are sampled by applying low-degree polynomials to uniform inputs as defined
Summary
This paper is part of a long and active line of research devoted to the problem of “randomness extraction”: Given a family of distributions all guaranteed to have a certain structure, devise a method that can convert a sample from any distribution in this family to a sequence of uniformly distributed bits—or at least a sequence statistically close to the uniform distribution. It is easy to prove that a random function is, with high probability, a good extractor for the given family, and the challenge is to give an explicit construction of such an extractor. Let C be a class of random variables taking values in Γ. We say a random variable P taking values in Ω is ε-close to uniform if for every A ⊆ Ω,. A function E : Γ → Ω is an ε-extractor for C if for every X ∈ C, the random variable E(X) is ε-close to uniform. A function D : Γ → Ω is a disperser for C if for every X ∈ C, the random variable D(X) takes more than one value in Ω with nonzero probability
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