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� !"#$ %& '()*$+( ,--. L G4 555D GFD GF-HIEIG),-(*-555 � !"#$ %& '()*$+( ,--. 0(78-8",(!' &-9.,!"!'9)() / :(;!%/<;!%(- =>>5 • &!"*-+ :(.)+$-;$ .$$.&1< :#":"*-/ ,"# >("&1 &):!-#* • /.$. x &"0>)+-/ 6)$! 1-7 k 4 x ⊗ k • /-M+- /)N-#-+&- ", /.$. x 1 .+/ x 2 .* 0(78-8",(!' &-9.,!"!'9)() ∆( x 1 , x 2 ) = x 1 ⊗ x − 1 2 • /)N-#-+&- )+9.#).+$ .,$-# &"0>)+.$)"+ ", 1-7 ∆( x 1 ⊗ k, x 2 ⊗ k ) x 1 ⊗ k ⊗ k − 1 ⊗ x − 1 = = ∆( x 1 , x 2 ) 2 • =-M+)$)"+ ", !"#$#%&# #-(.$)9- $" &):!-# D",$-+ -;"#F � !"#$ %& '()*$+( ,--. � !"#$ %& '()*$+( ,--. ?@!%.'8A F : { 0 , 1 } 4 → { 0 , 1 } 4 0(78-8",(!' &-9.,!"!'9)() ' 0 ' 1 x ↓ ↓ O E G P @ Q R S T U EO EE EG EP E@ EQ F ( x ) m 1 − → ⊕ − → u 1 − → F − → v 1 − → ⊕ − → c 1 R @ EG Q O S G E@ E EQ P EP T EO U EE ' 0 ' 1 • &"+*)/-# )+:'$* x .+/ y 6!-#- y )* $!- >)$6)*- &"0:(-0-+$ ", x ↓ ↓ m 2 − → ⊕ − → u 2 − → F − → v 2 − → ⊕ − → c 2 • *'&! )+:'$* !.9- /)N-#-+&- x ⊕ y = 15 • m 1 ⊕ m 2 = α )0:()-* u 1 ⊕ u 2 = α • G )+:'$* ", /)N-#-+&- 15 ❀ G "'$:'$* ", /)N-#-+&- EP )+ EO ", ER &.*-* • .**'0- u 1 ⊕ u 2 = α )0:()-* v 1 ⊕ v 2 = β 6)$! :#">.>)()$7 p F • 6- *.7 $!.$ ∆ = 15 → ∆ = 13 6)$! :#">.>)()$7 10 / 16 • $!-+ m 1 ⊕ m 2 = α )0:()-* c 1 ⊕ c 2 = β 6)$! :#">.>)()$7 p • :#">.>)()$7 &"0:'$-/ "9-# .(( )+:'$* D1-7*F � !"#$ %& '()*$+( ,--. � !"#$ %& '()*$+( ,--.

  3. !"#$#%&!'() x y F ( x ) F ( y ) F ( x ) ⊕ F ( y ) 0 1 2 3 ↓ ↓ ↓ ↓ m 1 − → ⊕ − → F − → ⊕ − → F − → ⊕ − → F − → ⊕ − → c 1 !" # !! !$ ! !% % & !$ ' !$ !' ! # 0 1 2 3 ↓ ↓ ↓ ↓ $ !' " ( !$ m 2 − → ⊕ − → F − → ⊕ − → F − → ⊕ − → F − → ⊕ − → c 2 % !! !$ !$ " ! ) $ % • *+, ,-./0/+1/2 3-45 5-65 7089:9-;-4-/2 4508<65 358;/ 1-75/0 # & ' !" !$ ) ( !% ! !" F F F F ( ) ! !% !" • ∆ m = α 0 − → α 1 − → α 2 − → · · · − → α r = ∆ c & # !" ' !$ F r F ! " $ ) % → α r 3-45 7089 p = � r • α i − 1 − → α i 3-45 7089 p i = α 0 − i =1 p i !! % !$ !$ !' $ ( " !$ • 7089:9-;-4> 8? ,-./0/+4-:; 4:@/+ :2 :A/0:6/ 8A/0 :;; @/>2 !$ ' ! !' # � !"#$ %& '()*$+( ,--. !% ! & % !$ !" !! # !$ !"#$#%&!'( *$+,&'%'(+)!) -.$ /')/ -0%*&!.%) ! ' $ % " # ) ( & ! !! !' !$ !% !" !# B B B B B B B B B B B B B B B CD:E7;/F 9;81@ 1-75/0 9:2/, 5:25 ?<+14-8+ ! B B # B B B B ' B ' B B ' B % B ' B # # B B B B B B ' ' B B B B B $ B B B # B ' B B ' B B B % B ' B • G:4>:2BG/>/0BH2/:2 h i = e h i − 1 ( m i ) ⊕ m i % B B B ' B ' % B B ' ' ' B B ' B " B ' ' B % B B % ' B B ' B B B B e • *+, 5-65B7089:9-;-4> ,-./0/+4-:; ?80 e 2<15 45:4 α → α # B B ' B % B B ' ' B ' ' ' B B B ) B B B B B % % B ' ' ' ' B B B B • :22<E/ m :+, m ′ = m ⊕ α :0/ 2<15 45:4 ( B B B B B ' B ' % B B % B ' B ' & B ' B B B ' ' ' B % ' B B B B ' e h i − 1 ( m ) = e h i − 1 ( m ′ ) ⊕ α, ! B B B B ' ' B B B % % B ' ' B B ?80 28E/ A:;</ 8? h i − 1 !! B B B ' ' B ' ' ' B B % B B ' B !' B % B ' B ' B B ' B B B B B # B • 9<4 45/+ h i = h ′ !$ B B B B B B ' ' B B B B # ' B % i = :+, 45/0/ -2 : 18;;-2-8+I !% B ' B % ' B B B B B ' B B B B # � !"#$ %& '()*$+( ,--. !" B B B B ' B ' B B B B B B ! B ' !"#$#%&!'( *$+,&'%'(+)!) 1 #%*$+,&!.% 2) /')/!%3 !"#$%&'(! )*+,'!- @/> 48 e *D/,= +8 18+408; +84 *D/,= <+,/0 J28E/K 8? :44:1@/0 18+408; 8? :44:1@/0 4&&'*5) 1 #6'7,(#) *+, 7:-02 2:4-2?>-+6 15/1@ 1-75/04/D42 15/1@ :?4/0 /:15 08<+, ,-./0/+4-:; /:0;> :9804 240:4/6> +:4<0/ 8? ,-./0/+4-:; :+> 27/1-:; ?80E 380@;8:, 48 *+, < 2 n/ 2 < 2 n ,-./0/+4-:; 258<;, 9/ � !"#$ %& '()*$+( ,--. � !"#$ %& '()*$+( ,--.

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