Summary
An \((n,k)\)-affine source over a finite field \({\mathbb F}\) is a random variable \(X=(X_1,...,X_n) \in {\mathbb F}^n\), 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). This chapter is based on [25].
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Notes
- 1.
A line is a 1-dimensional affine subspace of \({\mathbb F}^n\).
- 2.
Our extractors will sometimes output bits and sometimes output field elements. Therefore, the definition here uses a general output domain.
- 3.
Actually, we can construct a \({deterministic\, (k,\epsilon)-affine\,source\,extractor}\) that outputs \(k-1\) random elements in \({\mathbb F_q}\) and \(\lfloor(1-\delta) \cdot \log q\rfloor\) random bits for any constant \(0<\delta<1\).
- 4.
See Lemma 3.10 for an exact formulation of such an instantiation.
- 5.
- 6.
We use a slightly different expression than the one given here to ensure that f will not be of a certain restricted form on which Weil’s theorems don’t apply.
- 7.
The Reed-Solomon encoding of \(x=(x_1,\ldots,x_n) \in {\mathbb F_q}^n\) at location \(u\in {\mathbb F_q}\) is defined as \(\sum_{i=1}^n x_i\cdot u^i\).
- 8.
A character χ of \({\mathbb F_q}^{*}\) is extended to 0 by \(\chi(0)=0\).
- 9.
It is known that \(Tr(a)\in\mathbb F_2\) for every \(a\in {\mathbb F_q}\).
- 10.
We interpret the field elements 0 and 1 as the corresponding integers.
- 11.
Characters of higher order are also extractors, but with larger error.
- 12.
In [26] the authors assume all distributions are over binary strings, but it is easy to see that the proof follows in the case stated here.
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Gabizon, A. (2011). Deterministic Extractors for Affine Sources over Large Fields. In: Deterministic Extraction from Weak Random Sources. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14903-0_3
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