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[]\OT1/cmr/m/n/12 We need to show that for ev-ery $\OML/cmm/m/it/12 y \OMS/cmsy
/m/n/12 6\OT1/cmr/m/n/12 = \OML/cmm/m/it/12 z$\OT1/cmr/m/n/12 , the ran-dom var
i-able $(\OML/cmm/m/it/12 f\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT1/cmr/m/n/12 )
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[]\OT1/cmr/m/n/12 We use the linear-algebraic method. See the book by Babai--Fr
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\OT1/cmr/bx/n/12 Theorem 13.\OT1/cmr/m/n/12 [[[]]] Let $\OML/cmm/m/it/12 X[]; X
[]; [] ; X[]$ \OT1/cmr/m/n/12 be ran-dom vari-ables over $\OMS/cmsy/m/n/12 f\OT
1/cmr/m/n/12 0\OML/cmm/m/it/12 ; \OT1/cmr/m/n/12 1\OMS/cmsy/m/n/12 g$\OT1/cmr/m
/n/12 ,
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[]\OT1/cmr/m/n/12 This fun-da-men-tal the-o-rem ap-peared in the con-fer-ence v
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\OML/cmm/m/it/12 T[] \OT1/cmr/m/n/12 = []  \U/msb/m/n/12 E[] [] \OMS/cmsy/m/n/1
2  [] [] [][]  [][] [][] \OT1/cmr/m/n/12 = [][] \OML/cmm/m/it/12 :
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\OT1/cmr/m/n/12 By the choice of $\OML/cmm/m/it/12 n[]$ \OT1/cmr/m/n/12 and the
 pre-vi-ous ar-gu-ment, we also know that $\OMS/cmsy/m/n/12 j [][]\OT1/cmr/m/n/
12 [[][] \OML/cmm/m/it/12 X[] \OT1/cmr/m/n/12 =
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[]\OT1/cmr/m/n/12 Consider a sin-gle OR gate with in-put gates $\OML/cmm/m/it/1
2 g[]; [] ; g[]$\OT1/cmr/m/n/12 .
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\U/msb/m/n/12 E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 f\OT1/cmr/m/n/12 (\OML/cmm/m/i
t/12 x\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 f\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x\OT
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[]\OT1/cmr/m/n/12 We can now show our re-sult, which is known as \OT1/cmr/bx/n/
12 Vazi-rani's XOR Lemma\OT1/cmr/m/n/12 : 
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[] [] \OT1/cmr/m/n/12 = []\OMS/cmsy/m/n/12 f \OML/cmm/m/it/12  \OMS/cmsy/m/n/1
2 j 9\OML/cmm/m/it/12 g \OMS/cmsy/m/n/12 2 \U/msb/m/n/12 R[]\OT1/cmr/m/n/12 [\O
ML/cmm/m/it/12 X\OT1/cmr/m/n/12 ] : \OMS/cmsy/m/n/12 8\OML/cmm/m/it/12 x; [] \O
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[]\OT1/cmr/m/n/12 More gen-er-ally $\OML/cmm/m/it/12 p[]\OT1/cmr/m/n/12 (\OML/c
mm/m/it/12 X[]; [] ; X[]\OT1/cmr/m/n/12 )$ is a sym-met-ric poly-no-mial. As $\
OMS/cmsy/m/n/12 f\OT1/cmr/m/n/12 ([][] \OML/cmm/m/it/12 X[]\OT1/cmr/m/n/12 )[]\
OMS/cmsy/m/n/12 g[]$
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[]\OT1/cmr/m/n/12 where the rows of $\OML/cmm/m/it/12 A$ \OT1/cmr/m/n/12 ex-cep
t the first two cor-re-spond to some $\OML/cmm/m/it/12 S \OMS/cmsy/m/n/12  f\O
T1/cmr/m/n/12 1\OML/cmm/m/it/12 ; [] ; n\OMS/cmsy/m/n/12 g$
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[]$\OML/cmm/m/it/12 d[]\OT1/cmr/m/n/12 ($AND-OR$([]))$ un-der the promise that 
$[][] \OML/cmm/m/it/12 z[] \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 N$\OT1/cmr/m/n/12
 , is at least $\OML/cmm/m/it/12 d[]\OT1/cmr/m/n/12 ($AND-
 []


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[]\OT1/cmr/m/n/12 Let $\OML/cmm/m/it/12 P\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x[];
 [] ; x[]\OT1/cmr/m/n/12 )$ be a poly-no-mial for SURJ, where $\OML/cmm/m/it/12
 x[] \OT1/cmr/m/n/12 = (\OML/cmm/m/it/12 x[]\OT1/cmr/m/n/12 )[]\OML/cmm/m/it/12
 ; [] ; \OT1/cmr/m/n/12 (\OML/cmm/m/it/12 x[]\OT1/cmr/m/n/12 )[]$.
 []


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\U/msb/m/n/12 E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 y[] [] y[]\OT1/cmr/m/n/12 ] = 
\U/msb/m/n/12 E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 y[]\OT1/cmr/m/n/12 ]\U/msb/m/n
/12 E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 y[]\OMS/cmsy/m/n/12 j\OML/cmm/m/it/12 y[
] \OT1/cmr/m/n/12 = 1] [] \U/msb/m/n/12 E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 y[]\
OMS/cmsy/m/n/12 j\OML/cmm/m/it/12 y[] \OT1/cmr/m/n/12 = [] = \OML/cmm/m/it/12 y
[] \OT1/cmr/m/n/12 = 1]\OML/cmm/m/it/12 :
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[]\OT1/cmr/m/n/12 We now study the com-mu-ni-ca-tion com-plex-ity of some prob-
lems on groups.
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\OT1/cmr/m/n/12 The prod-uct $\OML/cmm/m/it/12 a \OMS/cmsy/m/n/12 
 \OML/cmm/m/
it/12 x$ \OT1/cmr/m/n/12 for a uni-formly ran-dom $\OML/cmm/m/it/12 a$\OT1/cmr/
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[]\OT1/cmr/m/n/12 For the group $\OML/cmm/m/it/12 A[]$ \OT1/cmr/m/n/12 the know
n bounds on $\OML/cmm/m/it/12 d$ \OT1/cmr/m/n/12 yield com-mu-ni-ca-tion $
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.
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[]\OT1/cmr/m/n/12 Ben Green.  An ar-gu-ment of Shkre-dov in the
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\OT1/cmr/m/n/12 fi-nite field set-ting, 2005.  Avail-able at peo-
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[]\OT1/cmr/m/n/12 Nathan Linial and Noam Nisan.  Ap-prox-i-mate inclusion-
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