TY - JOUR
T1 - Deterministic extractors for affine sources over large fields
AU - Gabizon, Ariel
AU - Raz, Ran
N1 - Funding Information:
* Research supported by Israel Science Foundation (ISF) grant. 1 A line is a 1-dimensional affine subspace of Fn.
PY - 2008/7
Y1 - 2008/7
N2 - An (n,k)-affine source over a finite field F is a random variable X = (X 1,..., X n ) ε Fn which is uniformly distributed over an (unknown) k-dimensional affine subspace of Fn . 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. (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 Fn → Fk - 1, such that for any (n,k)-affine source X over F, the distribution of D(X) is aŞ-close to uniform, where aŞ is polynomially small in | F |. 2. (For k=1): For any n and any F of size larger than n c , we give an explicit construction for a function D: Fn \to \{ 0,1\} {(1 - \delta )log-2 |F| , such that for any (n, 1)-affine source X over F , the distribution of D(X) is aŞ-close to uniform, where aŞ is polynomially small in | F |. Here, δ>0 is an arbitrary small constant, and c is a constant depending on δ.
AB - An (n,k)-affine source over a finite field F is a random variable X = (X 1,..., X n ) ε Fn which is uniformly distributed over an (unknown) k-dimensional affine subspace of Fn . 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. (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 Fn → Fk - 1, such that for any (n,k)-affine source X over F, the distribution of D(X) is aŞ-close to uniform, where aŞ is polynomially small in | F |. 2. (For k=1): For any n and any F of size larger than n c , we give an explicit construction for a function D: Fn \to \{ 0,1\} {(1 - \delta )log-2 |F| , such that for any (n, 1)-affine source X over F , the distribution of D(X) is aŞ-close to uniform, where aŞ is polynomially small in | F |. Here, δ>0 is an arbitrary small constant, and c is a constant depending on δ.
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U2 - 10.1007/s00493-008-2259-3
DO - 10.1007/s00493-008-2259-3
M3 - Article
AN - SCOPUS:58549103289
SN - 0209-9683
VL - 28
SP - 415
EP - 440
JO - Combinatorica
JF - Combinatorica
IS - 4
ER -