ts rtss - PowerPoint PPT Presentation

❋♦✉♥❞❛t✐♦♥s ♦❢ ❝r②♣t❛♥❛❧②s✐s✿ ♦♥ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❆♥♥❡ ❈❛♥t❡❛✉t ❆♥♥❡✳❈❛♥t❡❛✉t❅✐♥r✐❛✳❢r ❤tt♣✿✴✴✇✇✇✲r♦❝q✳✐♥r✐❛✳❢r✴s❡❝r❡t✴❆♥♥❡✳❈❛♥t❡❛✉t✴ ■❝❡ ❇r❡❛❦ ✷✵✶✸

❖✉t❧✐♥❡ • ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s • ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❛♥❞ ❲❛❧s❤ tr❛♥s❢♦r♠ • ❘❡s✐st❛♥❝❡ t♦ ❞✐✛❡r❡♥t✐❛❧ ❛tt❛❝❦s • ❋✐♥❞✐♥❣ ❣♦♦❞ ❙❜♦①❡s ✶

❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✷

❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❆ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s ✐s ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ F n ❉❡✜♥✐t✐♦♥✳ 2 ✐♥t♦ F 2 ✳ ❚r✉t❤ t❛❜❧❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✳ x 1 ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶ x 2 ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ x 3 ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ f ( x 1 , x 2 , x 3 ) ✵ ✶ ✵ ✵ ✵ ✶ ✶ ✶ ❱❛❧✉❡ ✈❡❝t♦r ♦❢ f ✿ ✇♦r❞ ♦❢ 2 n ❜✐ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛❧❧ f ( x ) , x ∈ F n 2 ✳ ✸

❱❡❝t♦r✐❛❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❉❡✜♥✐t✐♦♥✳ ❆ ✈❡❝t♦r✐❛❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ✐s ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ F n 2 ✐♥t♦ F m 2 ✿ F n F m − → S : 2 2 ( x 1 , . . . , x n ) �− → ( y 1 , . . . , y m ) ❊❛❝❤ ❢✉♥❝t✐♦♥ S i : ( x 1 , . . . , x n ) �− → y i ✐s ❝❛❧❧❡❞ ❛ ❝♦♦r❞✐♥❛t❡ ♦❢ S ✳ ❊①❛♠♣❧❡✳ x ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ S ( x ) ❢ ❡ ❜ ❝ ✻ ❞ ✼ ✽ ✵ ✸ ✾ ❛ ✹ ✷ ✶ ✺ S 1 ( x ) ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✶ ✶ S 2 ( x ) ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✵ S 3 ( x ) ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ S 4 ( x ) ✶ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✵ ✹

❍❛♠♠✐♥❣ ✇❡✐❣❤t ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❍❛♠♠✐♥❣ ✇❡✐❣❤t ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❍❛♠♠✐♥❣ ✇❡✐❣❤t ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f ✱ wt ( f ) ✱ ✐s t❤❡ ❍❛♠♠✐♥❣ ✇❡✐❣❤t ♦❢ ✐ts ✈❛❧✉❡ ✈❡❝t♦r✳ ❆ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s ✐s ❜❛❧❛♥❝❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ wt ( f ) = 2 n − 1 ✳ Pr♦♣♦s✐t✐♦♥✳ ❆ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ S ✇✐t❤ n ✐♥♣✉ts ❛♥❞ n ♦✉t♣✉ts ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛♥② ♥♦♥③❡r♦ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✐ts ❝♦♦r❞✐♥❛t❡s n � x �− → λ = ( λ 1 , . . . , λ n ) � = 0 λ i S i ( x ) , i =1 ✐s ❛ ❜❛❧❛♥❝❡❞ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✳ ✺

❆❧❣❡❜r❛✐❝ ♥♦r♠❛❧ ❢♦r♠ ✭❆◆❋✮ ▼♦♥♦♠✐❛❧s ✐♥ x 1 , . . . , x n ✿ n ✇❤❡r❡ x u = x u i x u , u ∈ F n � � � i . 2 i =1 ❊①❛♠♣❧❡✿ x 1 1 x 0 2 x 1 3 x 1 4 = x 1 x 3 x 4 = x 1011 ✳ Pr♦♣♦s✐t✐♦♥✳ ❆♥② ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s ❤❛s ❛ ✉♥✐q✉❡ ♣♦❧②♥♦♠✐❛❧ r❡♣r❡s❡♥t❛t✐♦♥✿ a u x u , � a u ∈ F 2 . f ( x 1 , . . . , x n ) = u ∈ F n 2 ▼♦r❡♦✈❡r✱ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❆◆❋ ❛♥❞ t❤❡ ✈❛❧✉❡s ♦❢ f s❛t✐s❢②✿ � � a u = f ( x ) ❛♥❞ f ( u ) = a x , x � u x � u ✇❤❡r❡ x � u ✐❢ ❛♥❞ ♦♥❧② ✐❢ x i ≤ u i ❢♦r ❛❧❧ 1 ≤ i ≤ n ✳ ✻

❊①❛♠♣❧❡ x 1 ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶ x 2 ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ x 3 ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ f ( x 1 , x 2 , x 3 ) ✵ ✶ ✵ ✵ ✵ ✶ ✶ ✶ a 000 = f (000) = 0 a 100 = f (100) ⊕ f (000) = 1 a 010 = f (010) ⊕ f (000) = 0 a 110 = f (110) ⊕ f (010) ⊕ f (100) ⊕ f (000) = 1 a 001 = f (001) ⊕ f (000) = 0 a 101 = f (101) ⊕ f (001) ⊕ f (100) ⊕ f (000) = 0 a 011 = f (011) ⊕ f (001) ⊕ f (010) ⊕ f (000) = 1 a 111 = � 2 f ( x ) = wt ( f ) mod 2 = 0 x ∈ F 3 f = x 1 ⊕ x 1 x 2 ⊕ x 2 x 3 . ✼

❈♦♠♣✉t✐♥❣ t❤❡ ❆◆❋ n = 3 ✿ ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ f (0) f (1) f (2) f (3) f (4) f (5) f (6) f (7) f (0) ⊕ f (1) f (2) ⊕ f (3) f (4) ⊕ f (5) f (6) ⊕ f (7) f (0) f (2) f (4) f (6) f (0) ⊕ f (1) f (0) ⊕ f (2) f (0) ⊕ f (1) f (4) ⊕ f (5) f (4) ⊕ f (6) f (4) ⊕ f (5) f (0) f (4) ⊕ f (2) ⊕ f (3) ⊕ f (6) ⊕ f (7) f (0) ⊕ f (1) f (0) ⊕ f (2) f (0) ⊕ f (1) f (0) ⊕ f (4) f (0) ⊕ f (1) f (0) ⊕ f (2) f (0) ⊕ f (1) f (0) ⊕ f (2) ⊕ f (3) f (4) ⊕ f (5) ⊕ f (4) ⊕ f (6) ⊕ f (2) ⊕ f (3) ⊕ f (4) ⊕ f (5) ⊕ f (6) ⊕ f (7) ✜rst st❡♣✿ f (2 i + 1) ← f (2 i + 1) ⊕ f (2 i ) s❡❝♦♥❞ st❡♣✿ f (4 i + j + 2) ← f (4 i + j + 2) ⊕ f (4 i + j ) , ∀ 0 ≤ j < 2 t❤✐r❞ st❡♣✿ f (8 i + j + 4) ← f (8 i + j + 4) ⊕ f (8 i + j ) , ∀ 0 ≤ j < 4 ✽

❈♦♠♣✉t✐♥❣ t❤❡ ❆◆❋ ❲❤❡♥ t❤❡ ✈❛❧✉❡ ✈❡❝t♦r ✐s st♦r❡❞ ❛s ❛ 32 ✲❜✐t ✐♥t❡❣❡r ① ✿ ① ❫❂ ✭① ✫ ✵①✺✺✺✺✺✺✺✺✮ ❁❁ ✶❀ ① ❫❂ ✭① ✫ ✵①✸✸✸✸✸✸✸✸✮ ❁❁ ✷❀ ① ❫❂ ✭① ✫ ✵①✵❢✵❢✵❢✵❢✮ ❁❁ ✹❀ ① ❫❂ ✭① ✫ ✵①✵✵❢❢✵✵❢❢✮ ❁❁ ✽❀ ① ❫❂ ① ❁❁ ✶✻❀ ✾

❉❡❣r❡❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❉❡✜♥✐t✐♦♥✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❧❛r❣❡st ♠♦♥♦✲ ♠✐❛❧ ✐♥ ✐ts ❆◆❋✳ Pr♦♣♦s✐t✐♦♥✳ ❚❤❡ ✇❡✐❣❤t ♦❢ ❛♥ n ✲✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥ f ✐s ♦❞❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ deg f = n ✳ ❉❡✜♥✐t✐♦♥✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ S ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ✐s t❤❡ ♠❛①✐♠❛❧ ❞❡❣r❡❡ ♦❢ ✐ts ❝♦♦r❞✐♥❛t❡s✳ ✶✵

❊①❛♠♣❧❡ x ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ S ( x ) ❢ ❡ ❜ ❝ ✻ ❞ ✼ ✽ ✵ ✸ ✾ ❛ ✹ ✷ ✶ ✺ S 1 ( x ) ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✶ ✶ S 2 ( x ) ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✵ S 3 ( x ) ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ S 4 ( x ) ✶ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✵ = 1 + x 1 + x 3 + x 2 x 3 + x 4 + x 2 x 4 + x 3 x 4 + x 1 x 3 x 4 + x 2 x 3 x 4 S 1 = 1 + x 1 x 2 + x 1 x 3 + x 1 x 2 x 3 + x 4 + x 1 x 4 + x 1 x 2 x 4 + x 1 x 3 x 4 S 2 = 1 + x 2 + x 1 x 2 + x 2 x 3 + x 4 + x 2 x 4 + x 1 x 2 x 4 + x 3 x 4 + x 1 x 3 x 4 S 3 = 1 + x 3 + x 1 x 3 + x 4 + x 2 x 4 + x 3 x 4 + x 1 x 3 x 4 + x 2 x 3 x 4 S 4 ✶✶

■❞❡♥t✐❢②✐♥❣ F n 2 ✇✐t❤ ❛ ✜♥✐t❡ ✜❡❧❞ 2 ✐s ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ✜♥✐t❡ ✜❡❧❞ ✇✐t❤ 2 n ❡❧❡♠❡♥ts✳ F n F 2 n = { 0 } ∪ { α i , 0 ≤ i ≤ 2 n − 2 } ✇❤❡r❡ α ✐s ❛ r♦♦t ♦❢ ❛ ♣r✐♠✐t✐✈❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ n ✳ n − 1 α i = λ j α j � ⇒ ❢♦r ❛♥② i, j =0 ❊①❛♠♣❧❡ ❢♦r n = 4 ✿ ♣r✐♠✐t✐✈❡ ♣♦❧②♥♦♠✐❛❧✿ 1 + x + x 4 ✱ α ❛ r♦♦t ♦❢ t❤✐s ♣♦❧②♥♦♠✐❛❧✳ α 2 α 3 α 4 α 5 α 6 α 7 F 2 4 0 1 α α 2 + α α 3 + α 2 α 3 + α + 1 α 2 α 3 0 1 α + 1 α F 4 ✵✵✵✵ ✵✵✵✶ ✵✵✶✵ ✵✶✵✵ ✶✵✵✵ ✵✵✶✶ ✵✶✶✵ ✶✶✵✵ ✶✵✶✶ 2 α 8 α 9 α 10 α 11 α 12 α 13 α 14 α 2 + 1 α 3 + α α 2 + α + 1 α 3 + α 2 + α α 3 + α 2 + α + 1 α 3 + α 2 + 1 α 3 + 1 ✵✶✵✶ ✶✵✶✵ ✵✶✶✶ ✶✶✶✵ ✶✶✶✶ ✶✶✵✶ ✶✵✵✶ ✶✷