Deterministic Extractors for Affine Sources over Large Fields

  • Authors:

  • Ariel Gabizon;Ran Raz

  • Affiliations:

  • Weizmann Institute;Weizmann Institute

  • Venue:
  • FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
  • Year:
  • 2005

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Abstract

An (n, k)-affine source over a finite field F is a random variable X = (X_1 ,...,X_n ) \in F^n, which is uniformly distributed over an (unknown) k-dimensional affine subspace of F^n. We show how to (deterministically) extract practically all the randomness from affine sources, for any field of size larger than nc (where c is a large enough constant). Our main results are as follows: 1.(For arbitrary k): For any n, k and any F of size larger than n^20, we give an explicit construction for a function D:F^n \to F^{k - 1} , such that for any (n,k)- affine source X over F, the distribution of D(X) is \in -close to uniform, where \in is polynomially small in |F|. 2. (For k = 1): For any n and any F of size larger than nc, we give an explicit construction for a function D : D:F^n\to \{ 0,1\} ^{(1 - 8)\log _2 |F|} |, such that for any (n, 1)- affine source X over F, the distribution of D(X) is \in -close to uniform, where\in is polynomially small in |F|. Here, \delta 0 is an arbitrary small constant, and c is a constant depending on \delta.