1. No category

Finding Vector Orthogonal to Roughly half a Collections of Vectors

 1 !! "#$ %"&$2 '($ )*" 2 +, $-" .$2 "&
+ /0" 1* ! ''/ 1
234567 89:;<=>:?@ 5ABCD< E<=9B=D F 2GH9 I
JKLMNNM OPQRNSLT UVTMWQRX OYQZ[R\\M]^_XL`abcZXdOeN
2f47 ghHA< iH=jBA< 6Ck@=:<C=< D< 2GH9
Jl[[RX_Mb OmMRXnMb UKR\oRb OpZLNRX UVcb`RLX OKMNLeMb]^MX\abcZX OeN
1
2
qrstuvwtx y :j:?=: z=:{H=:<; |B> >|H}9 ?|B? ~H= B9G ~Bj:AG H~ N ;<h
?H=> :9 ?|< dD:j<9>:H9BA A:9<B= >kBh< E = (F ) 7 ?|<=< <€:>?> B ;<h?H=
:9
}|:h|
:>
H=?|H{H9BA
?H
B?
A<B>?
B9D
B?
jH>?
;<h?H=>
H~
?|<
E
N/3
2N/3
~Bj:AG ‚< >|H} ?|B? ?|<
=B9{<
hB9
ƒ<
=<kABh<D
ƒG ?|< jCh|
[N/3,
2N/3]
>jBAA<= =B9{< [N/2 − √N /2, N/2 + √N /2] B9D }< {:;< B9 <„h:<9?7 D<
?<=j:9:>?:h kB=BAA<A BA{H=:?|j }|:h| …9D> B ;<h?H= Bh|:<;:9{ ?|:> ƒHC9D
† |< Hk?:jBA:?G H~ ?|< ƒHC9D :> BA>H :9;<>?:{B?<D
‡ ˆ‰Š‹ŒŽ BA{<ƒ=B:h hHjkA<€:?G7 D<h:>:H9 ?=<<>7 kB=BAA<A BA{H=:?|j>7 D<
=B9DHj:B?:H9
d
2
‘ ’“”•–—˜™”š–“
› ! œ*- žŸ ' '*¡" 0*" $*(*" 0*$ !¢ " " ! "'
*. m $( , 0'#.(' *. & D " Rn (" '*$-& , *0*$*($
&('*" ' *. &0 O(n log(mD)) £ ¤ " *0*($ &('*" ' ž¥¦ ¥§ "*&'
$$$& , ¨©ª«¬©¨©­ 0*$, "* ! $' £ £ (*' *. -$#*" ' "*& £
1 ® , "&" " ' "*"(*"'#(- 0**. ' .*$$*¡" (* ! "*$
$ !! £ ¯ F ¡*°$ ! " ±$& £ ²* " , . ! $, *. N -(*' " d°
2
&! "'*"$ $"
'0( (F )d ³'' -(* " E ¡( ' ***"$
2
* $' N/3 "& ! E*'=2N/3
-(*' *. . ! $, **"$ , ' &±"&
¡ '0( * F2 °-$#& µ"" 0*&#(¶ u.v = Pd£ ´uivi '($ , '0®" ' ' *. (*#' "* µ*"'¶ "" 0*&#( '"( .*i=1"'"(· -(* (" ***"$ * '$.¸ £
" *& * ³0$* (*"'#(- '0(' *. œ*-¹' 0*" $*(*" °
* !¤ ' #'.#$ * - (*"'#(- -'*" *. ' (* ! "*$ $ !! £ º *" ! " *$ ' * *" "¡ "'. * ! ' .* $( -'*"' *. -' £ » 0*$ ! £ ¤ ' ¡$$ ®"*¡" 0*" $*(*" 0*$ ! " "°
! "' *. , 00$"' (" '*$-& ¼("$, , $" &('*" ' ž½ ¾ ¥¿ £
1' ¡' ! " ("($ **$ " 0**. -' £ » 0*$ ! .* $ "# ! ' ¡ &&*" "& *& ' À#-$" * ($''($
0*$ ! žÁ § £
 f? :> j:>A<BD:9{AG hBAA<D Ã=B9{< ><B=h|:9{ k=HƒA<jÄ :9 ÅÆÇ B9D ÅÈǁ
3C?|H=> 6Ckk=<>><D y C< ?H €h<>>:;< 2<9{?|
' '#'& " žÁ "& ž #"&'"&" *. 0*" $*(*" " "°
! "' *. , 0'#.(' ¡$$ ! ® 0*''$ * *" "'. * ! ' .* ( ! *&$ *. (* ! 0#*" " ¡( ! #$0$(*" ' $$*¡& ·0(' '°
! "' "& 0**.' ¡$$ 0*-&& " '0 00¸ £ 1 *$' *. 0'"
00 * ! 0*- œ*-¹' $ !! "& * - (*"'#(- -'*" *.
£ » ! $ , ¡ '*¡ " [N/3, 2N/3] (" 0$(& , ! #(
' ! $$ " [N/2 − √N /2, N/2 + √N /2] "& ¡ - " ¼(" & ! "'(
0$$$ $*
! ¡( ±"&' -(* (-" ' *#"& # $* ! '
$*'0( #".* ! » £ £ (" ! 0$ ! "& , . ! $ , *.£ ´$*'0( #".* !
**$" ((#' *. 0*$, "* ! $ ' "& 0*$ , $* ! ( &0 £
œ*-¹' $ !! ' '& " žŸ "& """ *. ' "*&#(*" "
$"# *. $" $ £ 1 ' " À#-$" .* ! #$*" " 0#$,
'°*( $"# ! $ , ¡ -" ' *. &'"( '#'' *. ±" ' X £ 1 *$ '£ »* ±"& '#' F *. X '#(F N*#$ , |F |/2 $ ! "'
*. F - " "'(*" ¡ F *. -" (&"$ , £ 1' '°*( 0*" *.
-¡ ' &-$*0& " '(*" " '(*" ¥ ¡ - 0*$'( 0**. *.
(* ! "*$ $ !! ¡(£ ,¤ $&' ! £0*-& " [N/2 − √N/2, N/2 +
√
**- ¡ '*¡ "&* ! '#' F ⊆ X ¡$$ .$$ " '$$,
N /2] £ 
" [N/2 − √N, N/2 + √N ] ¡ 0*$
, $' ¦ Á '* ' À# ' ! 0$ "&* ! & $* ! .* *# 0*$ ! £ " '*¡ & ! "'( $* ! (" *"& , &"&* ! " 0*$'( 0**.
*. (* ! "*$ $ !! £ ¤ " '(*" £ ¡ - "* ·"*"°0*$'(¸
0**. *. $ !! ¡( (-' ' ! *#"& ' 0*$'( 0**. £ 1
*0! $ , *. ' *#"& ' &'(#''& " '(*" £¦ "& "* & ! "'(
'À#"$ $* ! '& *" *# '(*"& 0**. ' -" " '(*" Á #"
* $"# *. $" $ " '(*" ¦ * &'( *# 0$$$£ £ $*
!
* ' $* ! $' *" $ ! " , .(' *# ³"'*"' *. ±"£
»±$&'
' ! * *. " ""'($$, $( "# '* ² $& ³"'*"'
$"£ $(
0*" *. -¡ ' ! ' ! *' 00*0 * ' "& 0*- '#$' *. '(*" £
¡*#$& "'" * ±"& *# ¡ 0*$'( 0**. *. '(°
*"¤ £¥ (" &"&* ! & * , $& "* *"$, " ¼(" 'À#"$ $* ! # $'* " ¼(" 0$$$ $* ! ·! * *" ' "& *. '(*" £ ¥¸ £
(*"($#& ' "*&#(*" ¡ $*" À#* .* ! ž¥¥ µ "#$ 00*(
*¡&'
&°"&* ! " $* ! ' ' * ±"& ! *& .* '(" ''*(°
& ' ! 0$ Ω .* **& 0*" w ¡ '0( * -" "0# "'"( I £ œ-"
'#( 0*" $* ! A(I, w) ' "*¡ & ! "'( $* ! "& '
#"& *w±"& (*( '*$#*"
! .(& " '(" ' ! 0$
£ 1"0*$
'0( ' ' "$$ , ³0*""$
' £ 1 '#$ *. &$ ! " '*¡"
RP ⊆ P/poly ! 0$' ' ! 0$ '0( Ω ''*(& ¡ "&* ! &
$* ! $¡ , ' (*""' 0*$, "* ! $°' & '#'0( ¡( ' **& 0*"
.* ( 0*''$ "0# "'"( £ º *¡- ' '#$ ' $ , "*"°(*"'#(-
"& 00' (""* #'& * (#$$, &°"&* ! $* ! ' £¶ ´ #
:9D:9{ B <h?H= =?|H{H9BA ?H HC{|AG BA~ B 5HAA<h?:H9 H~ <h?H=>
00 -' " ³ ! 0$ *. 0*$ ! .* ¡( ' µ &$ ! ""¶ 00*( *
&"&* ! *" ' (#$$ , .'$ £ ¤ "&& *# 0$$$ $* ! (*"'#(' 0*$, "* ! $°' $' *. µ("&& -(*'¶
¡( .* " , ' *. N "0# -(*'
' #"& * (*"" -(* ***"$ * *#$, N/2 "0#
-(*' £ 1'
$' ' ! & #0 *. $$ -(*' " 0*$, "* ! $°' . ! $, *. µ("&& '#'0('¶
*. ' ! $$ $* ! (¸ &! "'*" £ ´ "( $' ' (*"'#(& ¡ *"$ , - *
·
'*$- " ³#'-
'( 0*$ ! "& ' (" &*" À# '$ , " 0$$$ £
” ” –• ”š™ –š“” – š ¡ '#& , ' *( .* ! #$*" *. *# 0*$ ! ' ¤ " '''(*"
±"
"& F ' *. N "*" ! 0 , &'"( '#'' *. X £ 1 *$ ' X* ±"&
'#' F *. X '#( "# ! *. $ ! "' *. F ¡( - " *&&
"'(*"
¡ F ' ' ($*' ' 0*''$ * |F | £
2
Y =
N
X
1 ±' "#$ & .* ' 0*$ ! ' * ® .* F "&* ! '#' *. X £
¬
ª ¨ «¬ ¬ ¨ F ª ¨ ¬ N !" ¬­ # ª ¬ X $
%& © « ¨ # ª ¬ X
# '& ¬& ¨¬
F ⊆X
√
√
¥¸
N
N
N
( −
}| −
≤ |{Fi ∈ F : |F ∩ Fi |
≤
.
·
2
2
2
$
"'
*.
) © $$
(**'
"&*
!
! '#' *.
F£ $
1 , . . . , FN
*"&
, F'$("
* "* - , $ ! "
*. X ¡ 0*$ , 1/2 £ F X
¯ Yi "&* ! -$ &±"& , .
' -" "& Y = −1 *¡' £
Yi = 1 |F ∩ Fi |
i
1.* ¡ "'& " "&* ! -$
i=1
Yi = |{i : |F ∩Fi |
-" }|−|{i : |F ∩F | *&& }| = 2|{i : |F ∩F | -" }|−N.
i
i
¡" * '*¡ ³'' " F .* ¡( |Y | ≤ √N £ £ Y 2 ≤ N £
' $ #' 0*- P (Yi = 1) = 1/2 £ 1' .*$$*¡' !! &$ , .* ! .('² - , '#' F *((#'
¡ ' ! 0*$ , "& '
"
*&&
'
-"
'#''
"
(
1#' ) = 0 £
! ,
1" ¡ 0*- -"' F{Yi £ = 1} E(Y
i" ¨«© * «+ "&0"&" £ ²* '
i
$ #' (*"'& ¡* $ ! "' F "& F *. F £ - * 0*- 1
2
P (Y1 = 1 ∩ Y2 = 1) = P (Y1 = 1)P (Y2 = 1) = 1/4.
· ¸
1 ('' , f? hB9 ƒ< >|H}9 ?|B? ?|<>< <;<9?> B=< 9H? BA}BG> }:>< :9D<k<9D<9?
3C?|H=> 6Ckk=<>><D y C< ?H €h<>>:;< 2<9{?|
"& F &' *" £ ¤ " ' (' ' ($ -"' "&0"&" £
F1
2
1' (' (" &#(& * 0-*#' *" .* F "& F \ F
F1 ⊆ F2 £
1
2
1
"& ¡ '$$ - · ¸ £
1 '' A = F \ F B = F ∩ F C = F \ F "*" ! 0 , £ 1"
2
"& Y = 1 ' 1À#-$"
*1|A ∩2 F | ≡ |B ∩2F | ≡1 |C ∩ F | ! *& £ #
Y
1 =1
2
'"( ' '' &' *" ¡ - 0*$ , * " ('
-"°-"°-" "& 1/8 * " (' *&&°*&&°*&& £1/8-"#$$ , ¡ $'*
- · ¸ £
+ "( -"' 0¡' "&0"&" ¡ - i Yj ) = E(Yi )E(Yj ) = 0
. i 6= j £ ²# ! * E(Y 2) = 1 '* , $" , *. E(Y³0(*"
¡ -
Æ
i
N
X
X
Yi Yj ) = N.
E(Y 2 ) = E(
Yi2 +
i=1
i6=j
º "( ³'' F .* ¡( Y 2 ≤ N ' ' &'& ' £
u
t
! ¨© " *- 0**. ®" "* ((*#" .( 2
.*
$ ¤
Y = N2
¡
*"
.*
¡( 1#'
³''
'
2
£
F
=∅
E(Y
|F 6= ∅) <" N
F'
"À#$
'
'(
*
¡*&'
³''
''. , "
,
2
F
'*" "À#$£ ,£ Y < N £ ¤
√
N
< |{Fi ∈ F : |F ∩ Fi |
−
2
√
-" }| − N < N .
2
2
! ¨© 1 0¡' "&0"&"( *. "$'
#'
*
-$# -°
i
"( *. Y $ V ar(Y ) = PN V ar(Y ) = N £ , Y1(
(
¹'
"À#$
,
, ¡ - i
i=1
√
√
√
P (|Y − E(Y )| > 2 N ) = P (|Y | > 2 N ) < V ar(Y )/(2 N )2 = 1/4.
1' "'#' $' 3/4 *. '#'' F .$$ ¡" " [N/2 −
"& , $&' -$ "&* ! & $* ! .* ±"&" '#( ' £ 1 & ! "'( $* ! ' *. *0*'*" ¥ '(*" £Á "& '(*" ¦
(- *¡- " [N/2 − √N/2, N/2 + √N /2] *"& " * ! £
"*¡ '*¡ *¡ * &"&* ! 0**. *. 1* ! ¥ , ! *& *.
(*"&*"$ ³0(*"' " *& * *" & ! "'( $* ! £ » * ' ! 0$ & ! "'( $* ! ¡$$ 0'"& " '(*" £Á £
%& " © %& © ! '¨ ª ©¨ ! « # « ¬& ! ¬& ' «¬« ¨ " ' ¬¨¬« $ %& « ­« ¨ " ­ ! «¨ ¬« ! ¬ © ! « « ¬« ' ¨ ©«¬& ! © √ « ¨ ¬ ( « ¬ © ' ¬« * «¬& ¨¬ ¨ ¬ N/2 − √N /2 ¨ ¨¬
&
! ¬
N/2 + N /2 ¬ Fi $
) © *$$*¡" 0**. *. 1* ! ¥ ' ! *#"' * ±"&" ' .*
¡($ Y² 2 ≤ N £ #$& '#( ' , "# ! " $ ! "' *. XF "&
&(&" " #" .* ( x ∈ X ¡ ! #' $*" * F £ $*" ¡ ,
√
√
N , N/2 + N ]
:9D:9{ B <h?H= =?|H{H9BA ?H HC{|AG BA~ B 5HAA<h?:H9 H~ <h?H=>
¡ ®0 E(Y 2) *#"&& *- , N #' -" #" ±"$ '
¡$$ - ³0(& 0*0 , £
F """ ¡ ®"*¡ .* ! 0**. *. 1* ! ¥ 2) ≤ N £ ( '#'À#" '0 ¡ - $& , & ! "& .* '* ! $E(Y
! "' ¡
, $*" * F $ #' ($$ C ' (*"&*" ·.* ³ ! 0$ C ≡ (x ∈ F ) ∧
¸ "&#(*" , 0*'' ¡ - E(Y 2 |C) ≤ N £ 1 "³ 1'0 ' *
(x2 6∈ F ) £ ,
& ! " ¡ " $ ! " x ∈ X ' " F £ - E(Y 2 |C) = 1/2(E(Y 2 |C ∧ (x ∈ F )) + E(Y 2 |C ∧ (x 6∈ F ))).
1.* ³'' (*( c · c ≡ (x ∈ F ) * c ≡ (x 6∈ F )¸ .* ¡(
" ! *- *" * "³ '0 ((*&"
E(Y 2 |C ∧ c) ≤ E(Y 2 |C) ≤ N £ * ' (*( ' ¡$$ "'# "&#(*" , 0*'' ' ''±& "³ '0 £ "& *. $* ! £ £ ¡" - , $ ! " x ∈ X ' "
(*"'&& ' *"& ''±' E(Y 2 |F ) ≤ N "& "( ' ! "
*. 1* ! ¥ £ F
1 *"$ , ! "" 0*" * '$ ' *¡ * (* ! 0# E(Y 2|C ∧ (x ∈ F ))
"& E(Y 2|C ∧ (x 6∈ F )) ¡ "& ' -$#' " *& * ! ®
*# (*( £ *
"$$, ¡ ¡" * $ * (* ! 0# E(Y 2|C) .* " , (*"&*"
C
C≡
^
(x ∈ F ) ∧
x∈A
^
x∈B
(x 6∈ F ).
¯ Ti "&* ! -$ &±"& ,
.
' -" "& T = −1 *¡' £
Ti = 1 |(Fi \ (A ∪ B)) ∩ F |
i
- E(Y |C) = (−1)|F ∩A|E(T ).
i
(" À#$ ·-" . , ''# ! 0*" * i '* ! '' F \ (A ∪ B)
»
i
¹'
&
"¸ "& (" -" ! 0 , #' -$#"
! *#"' *
2
Fi
(* ! 0#" ³0(*" *. Z 2 ¡ Z = P α Y .* E(Y
' |C)
*.
{F1 , . .. , Fk }
i i i
0*''$
0
¸
'#''
*.
*
¡
¡'
'
"
,
!
,
X
α1 , . . . , αk ∈ Z £
· *. 1* ! ¥ -"'
0**.
0¡' "&0"&"
{Yi = 1}
²# ! *
. F = ∅ " *. (*#' E(Y ) = 1 *¡' E(Y ) = 0 £ ² "$$, £E(Y
.*
2
i ) = 1
" , i i £ 1#' (* ! 0#" E(Z 2i ) ' ' , (#' i
i
X
X
X
X
E(Z 2 ) = E(
α2i Yi2 +
αi αj Yi Yj ) =
α2i E(Yi2 ) +
αi αj E(Yi )E(Yj ).
i
i6=j
i
i6=j
1' ! 0$' " 0*$ , "* ! $ ! *" (" (* ! 0# 2|C ∧ (x ∈ F ))
"& E(Y 2 |C ∧ (x 6∈ F )) "& &(& ¡ x '*#$& E(Y
®" " * "*
1 (*"'#(*" *. F #' À#' |X| '0' ( (* ! 0#$ " 0*$F , "* ! $£
! *-$$ & ! "'( $* ! ±"&' ' ¡ ³0(& 0*0 ,
" 0*$ , "* ! $ ! £
u
t
' ³0$"& " "*&#(*" ! " *$ *. '(*" ¦ ' * *" &°
! "'( 0$$$ $* ! .* *# 0*$ ! £ ¤ ¡*#$& "'" * *"
'#( " $* ! .* ! & " &"&* ! *" *. 1* ! ¥ £ 1 ! "
È
3C?|H=> 6Ckk=<>><D y C< ?H €h<>>:;< 2<9{?|
&"&* ! *" ! *& , $&' ¼(" 0$$$ $* ! ' ' ! *&
*. *#"&& "&0"&"( ' &'(& .* "'"( " '(*" ¥§ £ *. ž¥ £ ±'
' $**®' $® ' ! *& !  00$($ '"( 0**. *. 1* ! ¥
' '& *" 0¡' "&0"&"( *. "&* ! -$' Y £ ".*#"$ ,
i
! *& ' "* 00$($ &($, (#' Y ' &±"& *"$ , "&($
, *#
i
.* ! #$
.
' -" "& Y = −1 *¡' £
Yi = 1 |F ∩ Fi |
i
"
#'
.*
(*"'#(
'
$$
'
0$
'0( "* .* Y # .* ´ ! ' 1' ' (-& " !'(*"!¦ *#
i
"&*
" &°*( ! *&
! F£
£
¡" * ±"& '#' ! " ! ' " ¡" |{i : |F ∩F | -" }|
"& |{i : |F ∩ F | *&& }| £F # ' ! "' ³($ , ±"&" F ! ³!i ' i
"# ! *. 0' {F , F } ¡ |F ∩ F | 6≡ |F ∩ F | ! *& 2 £ ¤ "&& . t &"*'
*&& }|−i N j "# ! *.i'#( 0' 'j ³($, (N/2 − t)(N/2 + t) =
|{i : |F ∩Fi |
2
N 2 /4 − t2 £
1 (#($ .( ' . F ⊆ X "& F F ¡* $ ! "' *. F i
j
*&
*& 2.
|F ∩ Fi | 6≡ |F ∩ Fj | !
2 ⇐⇒ |F ∩ (Fi 4 Fj )| ≡ 1 !
1#' ±"&" F ! "! ' " ¡" |{i : |F ∩ F | -" }| "&
*&& }| ' ³($, ±"&" F ! ³ ! ' |{(i, j) i: |F ∩ (F 4
|{i : |F ∩ Fi |
i
*&& }| £
Fj )| (*"'&
.*$$*¡" 0 0 (V, E) ¡
"& V = P(X) V = V1 ∪ V2 V1 = {(i, j) : 1*&&
≤ i < j ≤ N }
2
((i, j), F ) ∈ E
|F ∩ (Fi 4 Fj )|
£
¡ $**®" .* ' -³ *. V *. ! ³ ! # ! & £ ¯ N (x)
2
&"*
' *. "*#' *. x £ ¡$$ *"$ , "&
* 00$, .*$$*¡" $ !! .* A = V2 ' " ¯ !! £ º *¡- #"' *# ¡ (" (( "
!! ¥ $$ '#'' A ⊆ V2 .* ¡( 0**. '$$ *$&' ·' $'* ! ® ¦
"¯ '(*" ¦¸ £
¬
# '& ¬& ¨¬
⊆ V2 ª
∅ ∈ A ¨ ∀F, F 0 ∈ A, (F 4 F 0 ) ∈ A $
#! ! © (A
&
© ¬ ¨¬ ∀x ∈ V1 N (x) ∩ A 6= ∅ $ %& ∀x ∈ V1 , |N (x) ∩ A| =
|A|
.
2
) © , 0*'' ³'' F ∈ A '#( (x, F ) ' " &
¯ x ∈V"&1 £ , *. $ 0
, * , 0*'' .*$$*¡" ! 0 ' ¡$$°&±"& £
φ : A −→ A
F 0 7−→ (F 4 F 0 )
"& ' (*" *. N (x) ∩ A *"* A \ N (x) ¡( 0*-' '#$ £
u
t
:9D:9{ B <h?H= =?|H{H9BA ?H HC{|AG BA~ B 5HAA<h?:H9 H~ <h?H=>
%& © « ¬ ¨ # ª ¬ A ⊆ V2 ¨¬« ­« ¬& & ­" ¬& « ! !
¨ $
¤ '#¼(' * ® A = V2 £
u
t
%& © « ¬ # '& ¬& ¨¬
F ∈ V2
|N (F )| ≥ |V2 |
) © - , x ∈ V ' |N (x)| = |V |/2 "*#' £ , &*#$
$ , ¯ !!
1
2
(*#""
³'' " F ∈ V ''.
*. $ !! £ ut
, " , 0*''
2
√
%& © « ¬ # '& ¬& ¨¬
( F ⊂X
||{i : |F ∩ Fi |
}| − N2 | ≤ 2N
) © -" , **$$ , ¥ £ ›±" t = |{i : |F ∩ F | *&& }| − N £ 1"
$ ¯ F
2
"& , , 0*'' *" F i
|N (F )| = ( N2 + t)( N2 − t) = N4 − t2
©
) $
1
2
|V1 |
N (N − 1)
N2 − N
N2
− t2 ≥
=
=
4
2
4
4
¡( ! 0$' |t| ≤ √N £
u
t
2
$0 *. 1* ! ¥ ¡ ®"*¡ ' 0*''$ * ( ³0(&
-$# ¡" " *. *& √N £ ´ " (" ¡*"& ¡ ' 0*''$ *
"'# (*"'" " £ 1 .*$$*¡" ³ ! 0$' 0*- ' ' ! 0*''$ £
' X ¡ n = 4k2 + 1 $ ! "' "& F ' *. $$
¯ #'*. (*"'&
'#''
*.
'
= |F | = n(n − 1)/2 £ ¤ " ' (*"³ 0*$ !
X
2 £ ¯¡* N0'
' * 0*"
"*
"& (*#" "# ! *. &' *# (# ¡( X0('$ , '' *. F ¡ *&& "'(*" £ ¡" * ±"&
'#( a(n − a) ' ' ($*' ' 0*''$ * N/2 = k2(4k2 + 1) £ # 0 ≤ a ≤ n/2
"&
(2k 2 − k)(2k 2 + k + 1) = 4k 4 + k 2 − k
(2k 2 − k + 1)(2k 2 + k) = 4k 4 + k 2 + k.
1 .#"(*" a 7−→ a(n−a) " "('" *" [0, n/2] ' 0*-' ' ¡* ' -$#'
"& * ' $' ¡( ' *. *& *. N 1/4 £
¤ ' 0*''$ * ±" ' # ! " .# £ ²*k "'"( (*"'&*"
*.
'#'' ¡ ¦ $ ! "' "'& *. , $&' .*$$*¡" '#$ £
¬
ª ¬& ¨ ! « ­ # ª ¬ ¬& © ! ¬ {1, . . . , n} $
n
%& © « ¬ ¨ ' ¬¨ F
¬ c > 0 # '& ¬ & ¨¬ © « «¬ ­ ! ¨ ­ n © ¨ ­ # ª ¬ G
{1, . . . , n} |{F ∈ Fn : |F ∩ G| ( }| − |Fn | ≥ c|Fn |1/3 .
2 3C?|H=> 6Ckk=<>><D y C< ?H €h<>>:;< 2<9{?|
) © '#' *. (&"$ a £ 1 "# ! *. $ ! "'
$ ¯ F ⊆ {1, . . . , n}
*. F ¡*' "'(*" ¡ F ' *. *&& (&"$ , ' " a n − a + a £
n
3
2
1.* $
f (a) =
a(n − a)(n − a − 1) a(a − 1)(a − 2) n(n − 1)(n − 2)
+
−
2
6
12
& "( ¡ |F |/2 £ ! '*¡" f ' . .* ! * *"
n
" -$#' ¡" n ' ¡$$
(*'"
1 *' *. f n/2 "& n/2 ± √£ 3n − 2/2 £ ²* ! -*"' *. f ¡ '
"' i '* |f (i)| ' ! " ! $ ! *" '³ "' *#"&
*' £ ¤ "#-$ , ' -$#' '*#$& ! ³! & . *' . .* !
"' · ' . , " $.°"'¸ £ 1' À#' n * *&& "&
√
* " " " · £ £ 3n − 2 ' 4k2 .* '* ! k ¸ £
3n − 2/2
1' (*"'&*"' $& * (*( n = 4k2/3 + 1 ¡ k ≡ 0 ! *& ¦ £
1 " n ' " *&& '*
f (bn/2c) = f (n/2 − 1/2) = n/4 − 1/4
f (dn/2e) = f (n/2 + 1/2) = −n/4 + 1/4
# ! * . k ≥ 2 " √3n − 2 = √4k2 + 1 ' ! *' 1/8 ¡ , .* ! 2k '*² f (bn/2 +
√
3n − 2/2c) = f (n/2 − 1/2 + k)
√
√
= f (n/2 − 1/2 + 3n − 3/2) = −n/2 + O( n).
' !  £ + "(
+ ! $$ , * "' *#"& *' -
*$ "# ! N *. '#'' *. $ ! "' ! *" n Ω(n)
' O(n3 ) *
'
$' Ω(N 1/3) £
u
t
1 ' ! ®"& *. ($(#$*"' .* '#'' ¡ § $ ! "' , $&' "
$*¡ *#"& £ 1 ' $*¡ *#"& ¡ - *"& '
2/5
Ω(|F
pn | )
' '*¡" $*¡ ' $ ! *' *0 ! $ $*¡ *#"& '
Ω( |Fn |/(log |Fn |)1/4 ) £
(-& , ®" .* Fn ' *. $$ '#'' *. ' (n − 1)/2 £
¬
& ¨ ! « ­ # ª ¬ ! ¬ Fn ª ¬
(n−1)/2 {1, . . . , n} * & © n « ¨ « ¬ © $ %& © « ¬ ¨ ' ¬¨ ¬ c > 0 # '& ¬& ¨¬ © « «¬ ­
! ¨­
# n © ¨ ­ ª ¬ G {1, . . . , n} ©
) $
p
|{F ∈ Fn : |F ∩ G| ( }| − |Fn | ≥ c |Fn |/(log |Fn |)1/4 .
2
($$ &±"*" *. "* ! $ (*¼(" .* x ∈ R "& k ∈ N Qk−1
(x − i)
x
.
= i=0
k
k!
:9D:9{ B <h?H= =?|H{H9BA ?H HC{|AG BA~ B 5HAA<h?:H9 H~ <h?H=>
1 '0($ (' ¡" x ' $. " " ¡$$ #'.#$ £ » ! $ , .* n < k−1
¡ -
·¦¸
*¡ $ #' (*"'& ' ¡ = 4k + 1 $ ! "' "& $ F '
*. $$» '#'' *. X *. ' 2k £ X1 "# !n
*. '' " F ' Y *. (&"$
"'('
"
-"
"#
*.
'
'
!
!
j
Qk−1
(n + 1/2 − i)
n + 1/2
(−1)k−n+1 (2n + 1)!(2k − 2n − 3)!
= i=0
=
.
k
k!
22k−2 n!(k − n − 2)!k!
X j n − j f (j) =
.
p 2k − p
p even
1 *$ "# ! *. '' '
|F | =
n
2k
X j n − j =
,
p 2k − p
p
'* ¡ "'& " À#" ,
g(j) = f (j) −
1X
|F |
j
n−j
=
(−1)p
.
2
2 p
p 2k − p
´ # !! & *$ ' * 0*- g(j) = −42k
·Á¸
(j − 1)/2
j/2
−
.
2k + 1
2k + 1
' .* ! .*$$*¡" &" , · ž¥Á &" , ¦ £Á * ž¥ ¸ X
p
(−1)p
j
2m − j
(j − 1)/2
= (−4)m
.
p
m−p
m
¤ ' "* &¼(#$ * ((® j
4k + 1 − j
j
4k + 1 − j
j
4k + 2 − j
j + 1 4k + 2 − (j + 1)
−
=
−
.
p
2k − p
p − 1 2k − (p − 1)
p 2k + 1 − p
p
2k + 1 − p
' (*"'À#"( 2
p j
p (−1) p
P
4k+1−j
2k−p
=
=
=
h j
i
4k+1−j
j
4k+1−j
− p−1
2k−p
2k−(p−1) i
h
4k+2−(j+1)
P
j 4k+2−j
p
− j+1
p (−1)
p 2k+1−p
p
2k+1−p
j/2
(−4)2k+1 (j−1)/2
−
,
2k+1
2k+1
P
p
p (−1)
p
¡( 0*-' ·Á¸ £ " j ' -" g(j) &#(' *
g(j) = −(−4)2k
(j − 1)/2
.
2k + 1
3C?|H=> 6Ckk=<>><D y C< ?H €h<>>:;< 2<9{?|
1' ' 0*&#( *. $. "' 1' 0*&#( ' .* ! "! $ " '*°
$# -$# ¡" ' ("& *#"&£ 0 ' ¡" j = 2k * j = 2k + 2 £ ¤ "
* ('' ¡ -
I
k − 1/2 .
|g(j)| = 42k 2k + 1 " j ' *&& |g(j)| &#(' *
|g(j)| = 42k
j/2
2k + 1
¡( ' ! " ! $ ¡" j = 2k − 1 * j = 2k + 1 £ 1 ! " ! # ! ' ' ! '
" -" (' £ , ·¦¸ ! "! $ '*$# -$# g ®' ' .*
2k−1
k − 1/2 = 2k − 1 ∼ 2√
µ = 42k 2k + 1
k
πk
¡'
º "( µ = Ω(
p
|F | =
24k+1
4k + 1
.
∼ √
2k
2πk
p
|F |/ 4 log |F |).
u
t
"*¡ 0'" - , '! 0$ 0*$ , "* ! $ $* ! ¡( ±"&' '#'
(-" "À#$ , · ¥¸ .* ! 1* ! ¥ £ ¡*® .* ! 0*" *. -¡ &°F
'(& " '#'(*" -" '#'' ¡ "& * ±"& '#' ' " *&& "'(*"£ ¡ ! * " $.Fi *. F 4 F ·(*"'&&F ' ! #$'¸ £ » * ' ' ,!! ( & "(' $$i "*"j! 0 , '"( Fi
&'"( £ 1 $* ! *' ' ¡ , £
¥ £ (*"'#( $$ '' F 4 F "& &"* , G ! #$' *"& £
!i#$'j *. $$ $ ! "' *. G "* (*""" x £
X £ ¯ G0
£ ¯00$x,∈(#'-$
, $* ! * X \ {x} "& G 0 £ 1#' ¡  '#' F 0
*. X \ {x} ' " *&& "'(*"
¡ ! * " $. *. $ ! "'
*. G 0 £ » *¡ ¡* ('' ' " *&& "'(*" ¡ ! * " $. *. $ ! "' *. G \ G 0 £
F"0 ' ('
' '*$#*" * *# 0*$ ! £
¤ ¡' '"(
F = F0
$*"'
* $$ $ ! "' *. G \ G 0 ®" F = F 0 ∪ {x}
´ '*$#*" x
-'
£
š“ •
• š™ –š“” – š " ' '(*" ¡ (*"("& ¡ 0$$$ $* ! .* *# 0*$ ! £ *
¤0('$
, ¡ '$$ #$& " ¨ ' # « © ! . ! $ , *. ((#' *. 0*$, $* ! (
:9D:9{ B <h?H= =?|H{H9BA ?H HC{|AG BA~ B 5HAA<h?:H9 H~ <h?H=>
&0 .* *# 0*$ ! " ! "! ¡ $& * ³ "* 0*$ , "* ! $°
! 'À#"$ $*£ ¤ ! ¡( ' ±' '0 *¡&' 0$$$ *" £
#' ("À#' *. $" $ &$" "*¡ ¡ ¿°¥ -(*' "'&
*. ''
£ ¯ #' ±' .* ! #$ 1* ! ¥ " ' ! ' £
¬
'
(' ¬ © %& © $
u1 , . . . , u&N ∈ E = (F2 )d ª « ¬« ¬ © « ¬ ¨ (' ¬ ©
# '& ¬ ¨¬
v∈E
√
√
N
N
N
−
≤ |{1 ≤ i ≤ N : ui .v = 0}| −
≤
.
2
2
2
-(* v ∈ E ' " (**$$ , ' ($$& µ**&¶ .*
¤ " ¡ .*$$*¡'
"*¡
#" * ¡* $* ! ' .* ±"&" **& -(* £ ' "0#
u1 , . . . , uN £ ¡ - N &'"( "*" * -(*' u , . . . , u *. E -" , (**&"'
N
¸ £ 1
*#0# ¡$$ **& -(* .*
*. "0# ' *. *& 1
N
d
·"( '1
0"(0$
*.
$*
'
'
*
'( '( * ' ! $$
!
u1 , . . . , uN £
' V ¡ '#$ -(* v ' #"& * ³' £ ¤ . ' µ' ! 0$ '0(¶ '
' ! $$ "*# ¡ ¡$$ " $ * ±"& -(* , ³#'- '( £
# " ' * & '& & & V
ui − uj $ $
&
&
1 ≤ i < j ≤ N v∈V v.(ui − uj ) = 1 &
%& & (' ui $
v ∈ V u1 , . . . , uN $
) $
v1 , . . . , vk
V
V
Pk
ui − uj
u0i
u0i = l=1 (ui .vl )vl
i 6= j
l
ui .vl 6= uj .vl V
ui
u0i
u0i
V
V
: VP× V → FP
2
P
(v1 , . . . , vk )
( l λl vl ) ( lP
µl vl ) = l λl µl
w = l λl vl √
√
0
N/2
−
N
/2
u
N/2
+
N
/2
w u0i =
i
P
P
0
w
E
l λl vl ui =
l λl (ui .vl ) = w.ui
E
u
t
! $
«
ª ¨ ª ¨ « « ©¬
¨ ¬
¬
¬ © «
¬ ¨¬
¨ ¬
¬
¬ © «¬ ¨
©
©
'' *. 1 (*"&*" ' ***"$ * "*"
©
*. ¯
! 0$' "¡£ -(*' &±"& ,
0¡' &'"( £ 1' ' (#' .* $$
³'' '#( **-
(*"&*"
'
***"$
* "*" *. ! 0$' £"*" *. ' À#$ * * £ ¤ " * ! ( ! ' ! , *# *. ' 0* (*" *"* £
* "*¡ &±" *" "¡ 0*&#(
·''*(&
''
¸ , .* ! #$
*
'
"¡
0*&#( **$$ , ¦ ''' ³'"( *.
¡( ' £ ²°***"$
* $'
-(*' "& ! *'
£ # ' ¡*$
"&
#'
'
$'*
'#$
.*
¡
#'#$
0*&#(
*"
¸
£
·
1
*-
$
("
$'* &-& .* ! ' *( 0*"
¨©
!!
*. -¡ ' (*"'À#"( *. ¯ !! ¥ £ » * " 0(#$ " ¯ !! ¥ , 0*'' *" A *. '$ , #"& ' ,!! ( & "(' '! 0$ , ! "' .* ! $" $ -¡0*" A ' $" '#'0( £
"*¡ '*¡ '#'0( *. !! ¦ (" - ' ! $$ & ! "'*" £ ($$
E ' -(* '0( *- F *. &!¯ "'*" d £
2
¬ ª ¨ # ª ¬ ¬ ' ¬¨« « %& ¬& © « ¬ ¨ # ª " ¨ '
U
E
& '& $ ! '
W E « « ≥ d − log(|U | + 1) * « ¬ « ¬ © ¬ U $
¬
©¨
II
3C?|H=> 6Ckk=<>><D y C< ?H €h<>>:;< 2<9{?|
) © , "&#(*" *" & ! "'*" d *. E £ ²* d = 0 |U | = 0 "& '#$
-$$$ ,.*$$*¡' £
''# ! . | = 2d − 1 £ £ U = E \ {0} ¡ (" (**' W = {0} £
d >''#
0 £ ¤ |U
"(
¡
'$$
! ³'' "*" * -(* w0 " E \ U £ ¯ W0
º '#'0( ¡
¡* $ ! "'¸ "& , w £ ¤ . |U | ≥ 2d−1 − 1 "
· ¡'
0
'#'
*#
"&'
*. & ! "'*" − 1 "&
E/W0 ' -(* '0(
£
´
W0
¡ (" 00$, "&#(*" , 0*'' * ' Ū *. ($''' *.d$
! "'
*. U ¡( $$ & " .* ! * £ 1' ' ''±' |Ū | ≤ |U | "( ³'' '#'0( W̄ *. E/W *. &! "'*" ≥ d − 1 − log(|U | + 1) ¡( &*'
0
"* "'( Ū £ 1
$$ W '#'0( *. E *. & ! "'*" 1+dim(W̄ ) (*"''" *. $$ $ ! "'
1
*. $$ ($'''
*. W̄ £ , &±"*" *. E/W W &*'1 "* "'( U "& ' *.
0
1
1
&! "'*" ≥ d − log(|U
| + 1) £
u
t
"*¡
00$
*
Á
, ¯ !!
U = {ui −uj : 1 ≤ i < j ≤ N }∪{ui : 1 ≤ i ≤ N }
¡ -
"(
³''
'#'0(
*.
*.
&
"'*"
!
+ 1)/2.
$' d|U−| 2=logN N(N
&*'º"* (*"" " , *. u − uW"& E*. u £ 1
i
***"$ V *. W ' " *. & ! "'*" ≤ 2 log N "&
'j ***"$ *i "*"
*. "& * "*" *. ·(#' V ⊥ = W ⊥⊥ = W ' ' '$,
uj
i
$ ! "' £
-±&¸ui£ »−*
V (*""' !u*'
N2
1' -' 0*$, "* ! $ 'À#"$ $*
! .* ±"&" **& -(* ¡
*"$ , '®( '"( ¡ - $& , &'(& '! 0$ 'À#"$ $* !· "
'(*" £Á¸ ¥ £ ² "& '' e , . . . , e *. '#'0( W *. & ! "'*" ≥ d − 2 log N ¡(
1
&*' "* (*""
" , b*. u − u "& *. u ·.* 1 ≤ i < j ≤ N ¸ £ 1'
i
j
' &*" , "&#(*" ®" À#*"
'0( i ( '0 ' " 0**.
*. ¯ !! Á £
£ ² "& ***"$ '0( V *. W £ 1' ' &*" , '*$-" $" ' , ' !
i .x = 0)1≤i≤b £
¦ £ (e² "&
**& -(* v " V £ 1' ' &*" , ³#'- '( £
I
' " 'À#"$ $* ! '®(& *- ¡ 0$" * 0.* ! " ³#'-
'( " ' ! $$ ' ! 0$ '0( £ 1 #' *. ¯ !! Á .* ±"&" ' ! 0$ '0(
' #".*#"$, ""'($$ , 'À#"$ '"( 0**. ¡*®' "&#(-$ , " À#*" '0( £
' "* '*" * '( '( * *"$ , *" '#'0( "
¤ " .( '(
³#'-
(" $'* 0.* ! & " 0*$, "* ! $$ , ! " , '#'0(' *.
' ! $$ & ! "'*" " 0$$$ £ " & * *-(* ! &¼(#$ , *. #'" ¯ !! Á " (*"''' " .*$$*¡" """ *. $* ! ¡ #$& . ! $ , *. '#'0(' *. $ & !£ "'*" W = {W1, . . . , Wk } ' µ"(¶ "
† |:> ~HAAH}> ~=Hj ?|< :9< CBA:?G log(N (N + 1)/2 + 1) ≤ 2 log N 7 }|:h| |HAD> ?=C<
~H= N ≥ 2  † |<=< :> 9H AH>> H~ {<9<=BA:?G :9 B>>Cj:9{ ?|B? N ≥ 2 >:9h< B9G ;<h?H=
v ∈ E }:AA >B?:>~G 5H=HAAB=G ~H= N = 1 
:9D:9{ B <h?H= =?|H{H9BA ?H HC{|AG BA~ B 5HAA<h?:H9 H~ <h?H=>
'"' .* $$ '#'' U ⊆ E \ {0} *. (&"$ N (N + 1)/2 ³''
.* ¡( U ∩ W = ∅ £
Wl ∈ W
*
0(#$ (*(l U = {u −u : 1 ≤ i < j ≤ N }∪{u : 1 ≤ i ≤ N } ¡
²
i
' $' *" W (*""' "*"
*.j u − u * *. ui '* , ¯ !! ¦
l
i
#' (*"" **& -(* £ ¤ . W ¹' *.j '#¼("$, i $ &! "'*" Wl⊥ !
l
' *"$, 0*$, "* ! $$ , ! " , $ ! "' "& (" '(& ¼("$ , £ 1'
Wl⊥
$&'
, .*$$*¡" * ! ¡( ' 0*-& " 'À#$ £
© « ¨ ¨©¨ ¨ ©«¬
« « ¬
«¬« « ¬ ©
ª « « ¬«
¨ ¨ «­
¬
¬ ¨¬ ¬¨« © ¨ ­ « ¬« ¬
© ¬©
¨ ¬©
¬ ¨¬
%& "
& ! * & '& ( * " ( N
&
#
! d *
! O(log N + log d log log(dN ))
N ≤ 2d F &
!
'
'
(F )d
(' d2 N 2 (N + 1)2 (' 2 # '& &
u1 , . . . , uN ∈ (F2 )d v
√
√
N/2 − N /2 ≤ |{1 ≤ i ≤ N : ui .v = 0}| ≤ N/2 + N /2.
¨
"
«¬
& ¨ # ¬« ( ¨© '& « ¬& « ¨ ! « ­ '¨ ¬& © © ª " © © ! « O(log(dN ))
¨©¨ ¬« ! ¨ª « # ¬ ¨ (' ¬ © v « " # ¬ u1, . . . , uN « " ­ &! ' "
!
¨©«¬ « ¨©¨ ¬« O(log N + log d log log(dN )) $
" '(*" ¦ £¦ ¡ '*¡ "( . ! $ , W = {W , . . . , W } .* '' U
1
*. '¤ N (N + 1)/2
"&& ³'' "& (" #$ ¼("$
, £ ´ #k . ! $, ' *.
(&"$ , k ≤ 2d|U | "& ( '#'0( W *. &! "'*" $' d−1−log(d|U |) £
1 W ¹' ¡$$
-" ' "'(*" *. ,l 00$"' '* '0""" . ! $,
l
*. W ⊥ ' !! &$ , .*#"& £ ¤ " '(*" ¦ £¦ U &"*' " , '#' *.
l
' ³0$"& *- , 0($ (*( .*
¡$$ {u − u : 1 ≤ i <
E \ {0} £
U
i
j
j ≤ N } ∪ {ui : 1 ≤ i ≤ N } £
1* $$*¡ ! * ** ! ¡ ±' ¡*® " ±$& ³"'*" *. £ * 0('$ , ¡
±³ " ³"'*" K *. & e > log((d − 1)|U |) '* F2
! * "
$
"'
"
*
.*
'#$
(*( '
!
(d − 1)|U |
K£ »
|U | = N (N + 1)/2
1'
'
(*(
¡(
¡$$
&
"
'(*"
¦ £Á £
!
e = blog(dN (N + 1))c £
$**®
'
¡
'
"
&#($
0*$
"*
,
! $ *.
K
F2 [X]/(P (X))
P (X)
*.
&
¡$$
-¡&
'
($'''
*.
0*$
"*
1#'
$
"'
*.
!
,
! $'
F2*&#$*
! [X] P £ ´ "(e £ 0*$ , "* ! $ P 'K.*#"& ' ' , * ($(#$ " K ,
! "0#$" 0*$ , "* ! $' *. & $'' " e ¡ (*¼("' " F2 ·&$'
¡$$ -" " '(*" ¦ £§¸ £
" d ¡ $ * ±"& , 00$"' '* - , ' *. (&"$
' ¤ "K ! 0 , "'(*" ¡|K| $' *" *. ! £ ²* - , θ ∈ K $ |U#'|
"&& (*"'& , 00$" H *. K d &±"& , À#*" x + θx +
1
2
1 θ |K| > (d − 1)|U | & " , 00$"'
"
2
£
θ'
x3 .+ · $· · + θd−1 xd =
0
! , (Hθ )θ∈K "& 0*" a ∈ K d \ {0} $*"' * ! *' d − 1 &'"(
, 00$"' ' ' &# * .( ! *' d − 1 &'"( **' *.
0*$ , "* ! $ P (θ) = a + a θ + · · · + a θd−1 £ 1#' ! *"
' , 00$"' 1
2
d
$' *" &*' "* "'(
U£
I
3C?|H=> 6Ckk=<>><D y C< ?H €h<>>:;< 2<9{?|
1* *" *# . ! $, *- F ·"'& *. K ¸ ¡ "*¡ (*"'& ( *. H
2
*" Fd £ ²* (x , . . . , x ) ∈ Fd À#*" *. , 00$" H (" ¡"θ
1
d
θ
2
((*&"
* 0*¡' *.2X IÆ
·! *& ¸
i=0
¡ µ F °$" (* ! "*"' *. x · (*¼(" *. x " µ '
2
j
i
À#$ * i X i °(**&"
*. θj−1 " F °''
*. jK ¸ £ 1
1, X, . . . , X e−1
2
"'(*" W = H ∩ Fd ' " &±"& , ' , ' ! *. À#*"' µ (x) = 0
θ
2
¡ i "'θ *- {0,
' .* '#'0( *. E = i(F )d *.
1, . . . , e − 1} £ ¤
2
(*& ! "'*" ! *' e £
1' (*"'#(*" , $&' . ! $ ,
*. k ≤ 2e '#'0('
1 , . . . , Wk }
¡ ³0(& "( , 0*0 ,W .*= $${W'#''
*. (&"$
U ⊆ E \ {0}
.*
¡(
("
®"
³''
"(
N (N + 1)/2 W
U ∩ Wl *.
= ∅&£ + "'*"
e $'
l ∈ W
¡

*'
'#'0('
!
!
≤ 1 + log(d|U |)) 2d|U |
( £ ' 0* ! '& '
'#'0(' -" ' "'(*"'d −*.
1 − log(d|U |)
, 00$"' £
x1 + θx2 + · · · + θd−1 xd ≡
e−1
X
µi (x1 , . . . , xd )X i
#' '# ! #0 ! " '0' *. ' 0$$$ $* ! £ ¤ ' ! 0$ ! "*" "&
¯"$
, '' &'(#''& " "³ '(*" 1 "0# ' ' {u1, . . . , uN } *. N
***"$ *
&'"( "*" * -(*' *. E = (F )d "&£ *#0# ' -(*
$' N/2 − √N /2 "& ! *' 2N/2 + √N /2 *. ! £
¥ £ ¯ e = blog(dN (N + 1))c £ , "# ! " " 0$$$ $$ 0*$ , "* ! $' *.
*. & e ±"& " &#($ 0*$, "* ! $ P £ ¯ K = F [X]/(P (X)) £
F2 [X]
*"'&
*. , 00$"' " K d (*"''" *. 2|K| = 2e , °
F
£ 00$"' . ! $ , &'(&
" '(*" ¦ ¦ ¡ À#*" *. (
, 00$"(H*.θ )Fθ∈K' ' , ' ! *. e À#*"'£ "£ F £ 1' ' *"$ , "°
! " *. ! ' £ *" *" '#'0( Wθ *. (F22 )d *. (*& ! "'*" ! *'
.* ( , 00$" Hθ £ ' ¡*$ ' "( . ! $ , #' (*""' e *'
! e '#'0(' *. 2 )d £
¦ £ + (2 " 0$$$ " (F
.* $$ W " "( . ! $, £ **& -(*
W⊥
#'
³'
"
$'
*"
*.
! "* ' *"$, ' & '0 ¡(
!
(#$$ , &0"&' *" "0#¸ £ ·
' ³0$"& " "³ '(*" ³(#*" ! *. ' $* ! '
0*$, $* ! ( " ' dN *. "0# £
"& "*¡ ³0$" *¡ * 0.* ! ' 0*(&# À#(®$ , " 0$$$ ' " *& * ±"& " &#($ 0*$, "* ! $ P ∈ F [X] *. & e ¡ £!²$,
"# ! " 0$$$ $$ 0*$, "* ! $' A ∈ F [X]2 *. & e "& ' 2
:9D:9{ B <h?H= =?|H{H9BA ?H HC{|AG BA~ B 5HAA<h?:H9 H~ <h?H=>
&#($ , £ 1 2e ≤ dN (N + 1) '#( 0*$, "* ! $' £ 1 0*$ , "* ! $ A '
&#($ . "& *"$, . ' "* &-'$ , "* "*"°(*"'" 0*$, "* ! $
*. & ≤ e/2 £ 1' , $&' '.*¡& &#($ , ' (* ! 0# "
0$$$ &-'*" ¡ ! "& *. A , $$ "*"°(*"'" 0*$ , "* ! $' B
*. & ≤ e/2 "& ' ¡ *" *. ! "&' ' * £ ² "&" P
.* ®'
0$$$ ! ¡ T (e) ' (*' *. &-'*"
" F [X] £ º "( ¡ *"$ , "&O(e)
* #'+ T (e)
&-'*" $*
! *. 0$$$ (* ! 0$³ ,
2
"
"*#'
*#"&
¡
-"
!,
, " 0$$$ $$ 0*''$
£ "& ((® ¡!
O(e)
À#*"'
*
0$$$
&-'*" $* ! ' A = BQ £ + !
*. (*#' Q! #( .' ·# *-(*
0$(&
.*
0*$
!
! "&¸ ' .*
"'"( ž¦ £
"*¡ 0*(& * '(*"& '0 *. $* ! ¡( ¡ " ¡ 0$! " , (* ! 0#*" £ ¯ P &#($ 0*$ , "* ! $ .*#"& ±'
'0 "& $ K = F [X]/(P (X)) ±$& ¡ 2e $ ! "' £ ±' (* ! 0#
.* $$ i ∈2 [e, 2(e − 1)] £ 1 ±' $ ! " *. ' 'À#"( ' *"&
X i mod P
!! &$ , .* ! "& X i+1 mod P (" *"& " (*"'" 0$$$ ! .* ! X e mod P "&P X i mod
'($$ , , '. *. (*¼("' .*$$*¡& , P ·
*'
*"
&&*"
"
¸
1
¡*$ 'À#"( (" .* (*"'#(&
!
F2 [X] £
" ! O(e) £
'0 *# ! " '® ' * (* ! 0# i .* $$
"& $$ θ ∈ K £
θ .*
i = 0, . . . , d−1#$0$(*"'
.'
³0*""*"
("
*"&
"
! θ , O(log d) !
,
θi
(
*.
(*'
"&&
*
0.*
'#(
#$0$(*"
¡
ª ¨ !
!
K
O(log e) £ ¤
! #$0$ , ¡* 0*$ , "* ! $' *. & ≤ e − 1 ¡ (*¼("' " F2 "& ® ! "& ! *&#$* P (X) £ 1 (*' *. ! #$0$(*" " F [X] ' O(log e) 2
"& , $&' 0*$, "* ! $ *. & ! *' 2(e − 1) £ """
*. '0 ¡ - 0(* ! 0#& 0'"*" ! *&#$* P (X) *. $$ ! *"* ! $' ¡(
(" 0*''$, *((# " ' 0*$ , "* ! $ £ º "( ' ! 0$ , ! "' * && #0 ! *' e 0*$ , "* ! $' *. & ≤ e − 1 £ 1' (" &*" " 0$$$ ! O(log e) £
1 0$$$ (*' *. "" *# "( . ! $, *. 2e '#'0(' ' .*
¡( ' O(log d log log dN ) 1 ***"$ *. ( '#'0( W
O(log d log e) (*""' ! *' 2e 0*"'
'"( ' *. &! "'*" ! *' e £ $* ¡ -θ
! *' (2e )2 0*"' " #"*" *. $$ ***"$' £ + "( 2e ≤ dN (N + 1) '
, $&' *#"& d2N 2(N + 1)2 *. 1* ! ¦ £ 1 &&*"$ (*' *. ³0$(
"# ! *" *. $$ *' 0*"' ' O(log e) '"( ( 0*" ' '# ! *. ! *'
'0""" -(*' *. '* ! ***"$ £
e
"$$ , ¡ (" ±"& **& -(* ! *" 2 2(N + 1)2 ("&&' "
" 0$$$ ""
! ² O(log(dN )) , ³#'- '( £ ² ' ¡ (*d !N0#
0*&#(' u .v .* $$ "0#' u "& $$ ("&& -(*' v £ 1' ' &*" " &0
i
i
- * '# ! *- $$ * *" "# ! 1"
.* ±³& v ¡
£
O(log
d)
*. i '#( u .v = 1 £ ¤ ' ¡$$ ®"*¡" '#( u"i & &&*" ("
0.* ! & " i&0 O(log N ) ' .* "'"( ž 0**. *. 1* ! ¥ £ £ ¸ £
· ® '*$# -$# '* .* - ,
1* '# ! ¡ '#'( N/2 "&
("&& ¡ - (* ! 0#& ≤ i ≤ N : u .v = 1}| − N/2| £ "*¡ -
i
* ±"& v! " ! # ! ! *" ||{1
-$#'
' (" &*" " &0
d2 N 2 (N + 1)2
" # ' "
'"(
(*
0#"
2 2
2
!
! ! !
O(log(d N (N + 1) )) = O(log(dN ))
AC 0
I
IÈ
3C?|H=> 6Ckk=<>><D y C< ?H €h<>>:;< 2<9{?|
0*$ ! ·' .* "'"( ž ³ ! 0$ Ÿ £ £ ¸ £ 1#' ³#'- '( À#'
0$$$ ! O(log(dN )) ' ($ ! & " 1* ! ¦ £
1 *-$$ 0$$$ ³(#*" ! *. *# $* ! ' .* O(log N +
¡( 0*-' * ! £
log d log log(dN )) ! ¨© 1' 0$$$ $* ! (" ! 0$ ! "& , . ! $, *. $*'0(
$
#".* ! **$"
((#' *. 0*$, "* ! $ ' "& 0*$ , $* ! ( &0 '"( (
*. '0' *. $* ! (" "* ' '* ! &#"&"( , " '
·
' ! " '"( $*'0( *#"&& 1#"
! (" (" *"$, (*"'#( ((#
. ! $' *. 0*$ , "* ! $ ' ¸ £
! ¨© 1 ((# &0 *"& " 1* ! ¦ ' , "* ! "' *0 ! $
£ - (*'"$ * &'( (*"'#(*" *. $' *. $$ d2 N 2(N +1)2 ("&&
-(*' ³0$($, ' 0 *. *# 0$$$ $* ! # . ¡ ¡*® ¡ $*'0(
#".* ! ((#' " , 0(* ! 0#*" À#" *"$ , $* ! ( '0( ' $$*¡& £
' "* &¼(#$ * ((® ¡*$ $' *. ("&& -(*' (" "&& ¤(*"'#(&
" $* ! ( '0( , -*" *" *# 0$$$ $* ! ¸ £ .
· ³#'- '( ¡( (" $ & "
*" ' ! 0$, ' * 0.* ! "
&0
' ³0$"& *- £ 1' '*¡' *# 0*$ ! ' " $*'0(
dN )
#".* !O(log
(" #& *¡- $*'0( #".* ! , ' "* NC1 ·
 #".*
(*"&*"
.* 1 ' .* "'"( ž¥Ÿ (0 Á¸ £ ²* ! ! ,
'0( (* ! 0$³ , 0*" *. -¡ NC
*# 0*$ ! ' " '"( ³#'- '(
(" 0.* ! & " $* ! ( '0( ' ¡$$ £ ¯
• “™ I i 3AH9 B9D  6k<9h<= ˆ Œ‹ Ž ˆ ‹ Žˆ‹  ˆ ‹  ‚:A<G f9?<=
>h:<9h< 6<=:<> :9 y :>h=<?< B?|<jB?:h> B9D k?:j:B?:H9 ‚ :A<G7 
 4 5AH?< B9D  =B9B :> ‹‹ ˆ ‹ Ž  ‹ ‹ ‹ˆ Ž  †<€?> :9
† |<H=<?:hBA 5HjkC?<= 6h:<9h< B9 3† 56 6<=:<>  6k=:9{<=7 
 ‚  ƒ<=AG <=G ~B>? kB=BAA<A kHAG9Hj:BA B=:?|j<?:h ‹ Œ ‹ ‹ 7
I ! F È7 I 
Ɓ  HC=9:<= B9D 4 H:=B9 3=< AH}<= ƒHC9D> <B>:<= H;<= ?|< =<BA> " f9 Œ‹# $ %
‰ ‹Ž ‹ ˆ‹Œ‰ ‹& ‹ 7 kB{<>
F I 7 I 
  HC=9:<= B9D 4 H:=B9 2H}<= ƒHC9D> B=< 9H? <B>:<= H;<= ?|< =<BA> ! f9>:D< 4  f9
Œ‹ # ' ( ˆŒ ‹ ‹ ‹ ) ‹ ‹ * + ˆŽ  Œ‹ Œ 7
;HACj< I H~ + ˆ Œˆ , ‹ ˆŽ ‹ ˆŒ ˆ ˆ7 kB{<> F Æ  6k=:9{<=7

ȁ y  z=:{H=:<; †HkHAH{:hBA hHjkA<€:?G H~ ?|< =B9{< ><B=h|:9{ ‹ Œ ‹& ‹ ˆ- ‰ 7
IÈ ! F 7 
 4 H:=B9 5:=hC:?> ;<=>C> ?=<<> :9 BA{<ƒ=B:h hHjkA<€:?G f9 Œ‹# ' %%% 7
;HACj< I H~ + ˆ Œˆ , ‹ ˆŽ ‹ ˆŒ ˆ ˆ7 kB{<> F  6k=:9{<= <=AB{7

 6 <:><= 4H:9? AHhB?:H9 :9 B==B9{<j<9?> H~ |Gk<=kAB9<> & ‹Œ ‹  ‹ .
‹ 7 I È ! ÈF 7 I 
  <G<= BC~ D<= <:D< 3 kHAG9Hj:BA A:9<B= ><B=h| BA{H=:?|j ~H= ?|< nD:j<9>:H9BA
9Bk>Bh k=HƒA<j  ‹ Œ ‹& ˆ 7 I !ÈÈ FÈ È7 I Ɓ
I   <G<= BC~ D<= <:D< B>? BA{H=:?|j> ~H= nD:j<9>:H9BA =<>?=:h?:H9> H~ |B=D
k=HƒA<j> ‹ Œ ‹& ˆ 7 ! Æ F Æ 7 I 
II
I
I 
IƁ
I
Iȁ
:9D:9{ B <h?H= =?|H{H9BA ?H HC{|AG BA~ B 5HAA<h?:H9 H~ <h?H=>
 H?}B9:7  iBH=7 B9D  iBH= † |< k=HƒBƒ:A:>?:h j<?|HD G:<AD> D<?<=j:9:>?:h
kB=BAA<A BA{H=:?|j> ‹ Œ ‹& ‹ ˆŒ  ‰Ž ˆ ˆ ˆŽ 7 Æ !Æ F IÈ7 I Ɓ
H|9 :H=DB9 ‹ ‹Œ ˆ ˆŽ  ‚:A<G7 i<} H= 7 I

6 6jBA< 9 ?|< ?HkHAH{G H~ BA{H=:?|j> f ‹ Œ ‹& ‹ ˆ- ‰ 7 ! IF 7 I 
<9H 6k=C{9HA: 5Hjƒ:9B?H=:BA fD<9?:?:<> †<h|9:hBA =<kH=?7 89:;<=>:? D: :=<9<7
Ɓ
 3 B>>:A:<; 9 D<h:>:H9 ?=<<> ~H= H=?|B9?> & ‹Œ ‹ Œ‹ ˆŽŽ + ˆ ˆŒŽ 7
È ! È F È 7 I 
 HAAj<= Œ‹ ‹ ‹ Œ ‹ ˆ- ‰ & ‹Œ Œ‹  † <€?> :9
† |<H=<?:hBA 5HjkC?<= 6h:<9h< B9 3† 56 6<=:<>  6k=:9{<=7 I 
I

 Download  Report

Paperzz.com

Your Paperzz

© Copyright 2026 Paperzz