Deterministic extractors for affine sources over large fields

Abstract

An (n,k)-affine source over a finite field \( \mathbb{F} \) is a random variable X = (X 1,..., X n ) ∈ \( \mathbb{F}^n \), which is uniformly distributed over an (unknown) k-dimensional affine subspace of \( \mathbb{F}^n \). We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than n c (where c is a large enough constant). Our main results are as follows:

  1. 1.

    (For arbitrary k): For any n,k and any \( \mathbb{F} \) of size larger than n 20, we give an explicit construction for a function D : \( \mathbb{F}^n \)\( \mathbb{F}^{k - 1} \), such that for any (n,k)-affine source X over \( \mathbb{F} \), the distribution of D(X) is -close to uniform, where is polynomially small in |\( \mathbb{F} \)|.

  2. 2.

    (For k=1): For any n and any \( \mathbb{F} \) of size larger than n c, we give an explicit construction for a function D: \( \mathbb{F}^n \to \{ 0,1\} ^{(1 - \delta )log_2 |\mathbb{F}|} \), such that for any (n, 1)-affine source X over \( \mathbb{F} \), the distribution of D(X) is -close to uniform, where is polynomially small in |\( \mathbb{F} \)|. Here, δ>0 is an arbitrary small constant, and c is a constant depending on δ.

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Correspondence to Ariel Gabizon.

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Research supported by Israel Science Foundation (ISF) grant.

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Gabizon, A., Raz, R. Deterministic extractors for affine sources over large fields. Combinatorica 28, 415–440 (2008). https://doi.org/10.1007/s00493-008-2259-3

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Mathematics Subject Classification (2000)

  • 11T24
  • 68Q99
  • 68R05
  • 05D10