rtr s tt - PowerPoint PPT Presentation

❈r②♣t♦❣r❛♣❤✐❝ ❙❜♦①❡s ❆♥♥❡ ❈❛♥t❡❛✉t ❆♥♥❡✳❈❛♥t❡❛✉t❅✐♥r✐❛✳❢r ❤tt♣✿✴✴✇✇✇✲r♦❝q✳✐♥r✐❛✳❢r✴s❡❝r❡t✴❆♥♥❡✳❈❛♥t❡❛✉t✴ ❙✉♠♠❡r ❙❝❤♦♦❧✱ ➆✐❜❡♥✐❦✱ ❏✉♥❡ ✷✵✶✹

❱❡❝t♦r✐❛❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❆ ✈❡❝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 ) ❊①❛♠♣❧❡✳ x ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ S ( x ) ❢ ❡ ❜ ❝ ✻ ❞ ✼ ✽ ✵ ✸ ✾ ❛ ✹ ✷ ✶ ✺ ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✶ ✶ S 1 ( x ) ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✵ S 2 ( x ) ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ S 3 ( x ) ✶ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✵ S 4 ( x ) ✶

❘♦✉♥❞ ❢✉♥❝t✐♦♥ ✐♥ ❛ s✉❜st✐t✉t✐♦♥✲♣❡r♠✉t❛t✐♦♥ ♥❡t✇♦r❦ x ( i ) ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ S S S S S S S S S S ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❧✐♥❡❛r ❞✐✛✉s✐♦♥ ✗✔ ❄ k i + ✲ ✖✕ ❄ x ( i +1) ✷

❖✉t❧✐♥❡ • ❆❧❣❡❜r❛✐❝ ❞❡❣r❡❡ • ❉✐✛❡r❡♥t✐❛❧ ✉♥✐❢♦r♠✐t② • ◆♦♥❧✐♥❡❛r✐t② • ❋✐♥❞✐♥❣ ❣♦♦❞ ❙❜♦①❡s ✸

❉❡❣r❡❡ ♦❢ ❛♥ ❙❜♦① n x u i � � f ( x 1 , . . . , x n ) = a u ∈ F 2 . a u i , u ∈ F n i =1 2 ❉❡✜♥✐t✐♦♥✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❧❛r❣❡st ♠♦♥♦♠✐❛❧ ✐♥ ✐ts ❛❧❣❡❜r❛✐❝ ♥♦r♠❛❧ ❢♦r♠✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ S ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ✐s t❤❡ ♠❛①✐♠❛❧ ❞❡❣r❡❡ ♦❢ ✐ts ❝♦♦r❞✐♥❛t❡s✳ Pr♦♣♦s✐t✐♦♥✳ ■❢ S ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦❢ F n 2 ✱ t❤❡♥ deg S ≤ n − 1 ✳ ✹

❊①❛♠♣❧❡ 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 ✺

❘❡s✐st❛♥❝❡ t♦ ❞✐✛❡r❡♥t✐❛❧ ❛tt❛❝❦s ✻

❉✐✛❡r❡♥❝❡ t❛❜❧❡ ♦❢ ❛♥ ❙❜♦① a \ b ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ ✷ ✵ ✹ ✷ ✵ ✷ ✷ ✵ ✵ ✵ ✷ ✵ ✵ ✵ ✷ ✶ ✷ ✷ ✵ ✷ ✹ ✵ ✷ ✵ ✹ ✵ ✵ ✵ ✵ ✵ ✵ ✷ ✷ ✵ ✹ ✵ ✷ ✵ ✵ ✵ ✵ ✻ ✵ ✵ ✵ ✷ ✵ ✸ ✷ ✵ ✷ ✹ ✵ ✵ ✵ ✷ ✷ ✵ ✵ ✷ ✵ ✵ ✷ ✹ ✵ ✹ ✷ ✵ ✵ ✵ ✷ ✷ ✵ ✵ ✹ ✷ ✵ ✵ ✵ ✺ ✹ ✵ ✵ ✵ ✵ ✹ ✵ ✹ ✵ ✵ ✵ ✵ ✹ ✵ ✵ ✻ ✵ ✷ ✵ ✵ ✷ ✷ ✷ ✵ ✷ ✷ ✷ ✵ ✵ ✷ ✵ ✼ ✵ ✹ ✵ ✵ ✵ ✹ ✵ ✵ ✵ ✵ ✵ ✵ ✹ ✵ ✹ ✽ ✷ ✷ ✵ ✷ ✷ ✵ ✵ ✵ ✹ ✵ ✵ ✷ ✵ ✷ ✵ ✾ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✷ ✵ ✷ ✷ ✵ ✵ ✵ ✷ ❛ ✵ ✵ ✷ ✵ ✹ ✵ ✷ ✷ ✵ ✵ ✵ ✻ ✵ ✵ ✵ ❜ ✵ ✷ ✵ ✵ ✵ ✷ ✵ ✵ ✷ ✷ ✷ ✷ ✵ ✹ ✵ ❝ ✷ ✵ ✵ ✵ ✷ ✵ ✵ ✵ ✵ ✷ ✵ ✵ ✽ ✷ ✵ ❞ ✵ ✵ ✵ ✵ ✵ ✵ ✹ ✵ ✵ ✵ ✹ ✵ ✵ ✹ ✹ ❡ ✵ ✵ ✵ ✹ ✵ ✵ ✵ ✹ ✷ ✷ ✵ ✷ ✵ ✵ ✷ ❢ δ S ( a, b ) = # { X ∈ F n 2 , S ( X ⊕ a ) ⊕ S ( X ) = b } ✼

❘❡s✐st❛♥❝❡ t♦ ❞✐✛❡r❡♥t✐❛❧ ❛tt❛❝❦s ❬◆②❜❡r❣ ❑♥✉❞s❡♥ ✾✷❪✱❬◆②❜❡r❣ ✾✸❪ ❈r✐t❡r✐♦♥ ♦♥ t❤❡ ❙❜♦①✳ ❆❧❧ ❡♥tr✐❡s ✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ t❛❜❧❡ ♦❢ S s❤♦✉❧❞ ❜❡ s♠❛❧❧✳ a,b � =0 # { X ∈ F n δ ( S ) = max 2 , S ( X ⊕ a ) ⊕ S ( X ) = b } ♠✉st ❜❡ ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡✳ δ ( S ) ✐s ❝❛❧❧❡❞ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ✉♥✐❢♦r♠✐t② ♦❢ S ✭❛❧✇❛②s ❡✈❡♥✮✳ ❋♦r ❛♥② ❙❜♦① S ✇✐t❤ n ✐♥♣✉ts ❛♥❞ n ♦✉t♣✉ts✱ ❚❤❡♦r❡♠✳ δ ( S ) ≥ 2 . ❚❤❡ ❢✉♥❝t✐♦♥s ❛❝❤✐❡✈✐♥❣ t❤✐s ❜♦✉♥❞ ❛r❡ ❝❛❧❧❡❞ ❛❧♠♦st ♣❡r❢❡❝t ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥s ✭❆P◆✮✳ ✽

❋♦r ❙P◆ ✉s✐♥❣ S ❊①♣❡❝t❡❞ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ 2 ✲r♦✉♥❞ ❝❤❛r❛❝t❡r✐st✐❝ � d � δ ( S ) ≤ 2 n ✇❤❡r❡ d ✐s t❤❡ ❜r❛♥❝❤ ♥✉♠❜❡r ♦❢ t❤❡ ❧✐♥❡❛r ❧❛②❡r✳ ❊①♣❡❝t❡❞ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ 2 ✲r♦✉♥❞ ❞✐✛❡r❡♥t✐❛❧ ❬❉❛❡♠❡♥ ❘✐❥♠❡♥ ✵✷❪ � d − 1 � δ ( S ) MEDP 2 ≤ 2 n ❊✳❣✳✱ ❢♦r t❤❡ 4 ✲r♦✉♥❞ ❆❊❙✱ 2 − 6 � 16 � MEDP 4 ≤ ❘❡✜♥❡♠❡♥ts ✐♥✈♦❧✈✐♥❣ t❤❡ ✇❤♦❧❡ ❞✐✛❡r❡♥❝❡ t❛❜❧❡ ❬P❛r❦ ❡t ❛❧✳ ✵✸❪✳ ✾

❘❡s✐st❛♥❝❡ t♦ ❧✐♥❡❛r ❛tt❛❝❦s ✶✵

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❛♥ ❙❜♦① a \ b ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ ✲✹ ✳ ✹ ✳ ✲✹ ✽ ✲✹ ✹ ✽ ✹ ✳ ✲✹ ✳ ✹ ✳ ✶ ✹ ✲✹ ✳ ✲✹ ✳ ✳ ✹ ✹ ✽ ✳ ✹ ✽ ✲✹ ✲✹ ✳ ✷ ✽ ✹ ✹ ✲✹ ✹ ✳ ✳ ✳ ✳ ✹ ✲✹ ✲✹ ✲✹ ✳ ✽ ✸ ✳ ✲✹ ✹ ✹ ✲✹ ✳ ✳ ✲✽ ✳ ✹ ✹ ✹ ✹ ✳ ✽ ✹ ✲✹ ✹ ✳ ✹ ✽ ✳ ✹ ✲✹ ✽ ✳ ✲✹ ✳ ✹ ✲✹ ✳ ✺ ✲✹ ✳ ✹ ✳ ✹ ✽ ✹ ✹ ✲✽ ✹ ✳ ✹ ✳ ✲✹ ✳ ✻ ✳ ✳ ✳ ✽ ✳ ✲✽ ✳ ✳ ✳ ✳ ✽ ✳ ✽ ✳ ✳ ✼ ✳ ✲✹ ✹ ✲✽ ✳ ✹ ✹ ✲✽ ✳ ✲✹ ✲✹ ✳ ✳ ✹ ✲✹ ✽ ✲✹ ✲✶✷ ✳ ✳ ✹ ✲✹ ✳ ✹ ✳ ✳ ✲✹ ✲✹ ✳ ✳ ✹ ✾ ✲✹ ✳ ✲✶✷ ✲✹ ✳ ✹ ✳ ✲✹ ✳ ✹ ✳ ✳ ✲✹ ✳ ✹ ❛ ✳ ✳ ✳ ✹ ✲✹ ✹ ✲✹ ✳ ✳ ✲✽ ✲✽ ✹ ✲✹ ✲✹ ✹ ❜ ✳ ✳ ✳ ✲✹ ✲✹ ✲✹ ✲✹ ✳ ✳ ✽ ✲✽ ✹ ✹ ✲✹ ✲✹ ❝ ✲✹ ✳ ✹ ✹ ✳ ✲✹ ✳ ✲✹ ✳ ✹ ✳ ✳ ✲✶✷ ✳ ✲✹ ❞ ✹ ✲✹ ✳ ✳ ✹ ✹ ✲✽ ✲✹ ✳ ✳ ✹ ✲✹ ✳ ✲✽ ✲✹ ❡ ✲✽ ✹ ✹ ✲✽ ✳ ✲✹ ✲✹ ✳ ✳ ✲✹ ✹ ✳ ✳ ✲✹ ✹ ❢ Pr[ a · x + b · S ( x ) = 0] = 1 1 + W [ a, b ] � � 2 n 2 ❋♦r ✐♥st❛♥❝❡✱ ❢♦r a = 0x9 ❛♥❞ b = 0x2 ✱ ✇❡ ❤❛✈❡ p = 1 2 (1 − 12 16 ) = 1 8 ✳ ✶✶

❲❛❧s❤ tr❛♥s❢♦r♠ ♦❢ ❛♥ ❙❜♦① ❲❛❧s❤ tr❛♥s❢♦r♠ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f ♦❢ n ✈❛r✐❛❜❧❡s F n − → Z 2 2 ( − 1) f ( x )+ a · x �− → W f ( a ) = � a x ∈ F n ❲❛❧s❤ tr❛♥s❢♦r♠ ♦❢ ❛♥ ❙❜♦① S ✿ F n 2 × F m − → Z 2 2 ( − 1) b · S ( x )+ a · x = W b · S ( a ) → W S ( a, b ) = � ( a, b ) �− x ∈ F n ✶✷

▲✐♥❡❛r✐t② ♦❢ ❛♥ ❙❜♦① ❈r✐t❡r✐♦♥ ♦♥ t❤❡ ❙❜♦①✳ ❆❧❧ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ S s❤♦✉❧❞ ❤❛✈❡ ❛ s♠❛❧❧ ❜✐❛s✱ ✐✳❡✳ ✱ L ( S ) = max 2 ,b � =0 |W S ( a, b ) | a ∈ F n 2 , b ∈ F n ♠✉st ❜❡ ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡✳ P❛rs❡✈❛❧✬s ❡q✉❛❧✐t②✿ ❢♦r ❛♥② ♦✉t♣✉t ♠❛s❦ b ✱ S ( a, b ) = 2 2 n . W 2 � a ∈ F n 2 ✶✸