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Paper:

TR07-132 | 8th December 2007 00:00

The sum of d small-bias generators fools polynomials of degree d

TR07-132
Authors: Emanuele Viola
Publication: 16th December 2007 05:01
Keywords:

Abstract:

We prove that the sum of $d$ small-bias generators $L : \F^s \to \F^n$ fools degree-$d$ polynomials in $n$
variables over a prime field $\F$, for any fixed
degree $d$ and field $\F$, including $\F = \F_2 = {0,1}$.

Our result improves on both the work by Bogdanov and
Viola (FOCS '07) and the beautiful follow-up by Lovett
(ECCC '07). The first relies on a conjecture that
turned out to be true only for some degrees and
fields, while the latter considers the sum of $2^d$
small-bias generators (as opposed to $d$ in our
result).

Our proof builds on and somewhat simplifies the
arguments by Bogdanov and Viola (FOCS '07) and by
Lovett (ECCC '07). Its core is a case analysis based
on the \emph{bias} of the polynomial to be fooled.

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