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2008 IEEE 23rd Annual Conference on Computational Complexity
The Sum of d Small-Bias Generators Fools Polynomials of Degree d
June 22-June 26
ISBN: 978-0-7695-3169-4
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 $\F = \F_2 =\zo$. Our result improves on both the work by Bogdanov and Viola (FOCS '07) and the beautiful follow-up by Lovett (STOC '08). 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 (STOC '08). Its core is a case analysis based on the \emph{bias} of the polynomial to be fooled.
Citation:
Emanuele Viola, "The Sum of d Small-Bias Generators Fools Polynomials of Degree d," ccc, pp.124-127, 2008 IEEE 23rd Annual Conference on Computational Complexity, 2008
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