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TR07-132 | 8th December 2007 00:00
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#### The sum of d small-bias generators fools polynomials of degree d

**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.