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❙✐♠✐❧❛r✐t✐❡s ❜❡t✇❡❡♥ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥✿ ❤♦✇ ❢❛r ❝❛♥ ✇❡ ❣♦❄ ❆♥♥❡ ❈❛♥t❡❛✉t ■♥r✐❛✱ ❋r❛♥❝❡ ❛♥❞ ❉❚❯✱ ❉❡♥♠❛r❦ ❆♥♥❡✳❈❛♥t❡❛✉t❅✐♥r✐❛✳❢r ❤tt♣✿✴✴✇✇✇✲r♦❝q✳✐♥r✐❛✳❢r✴s❡❝r❡t✴❆♥♥❡✳❈❛♥t❡❛✉t✴ ❙❆❈ ✷✵✶✸ ❜❛s❡❞ ♦♥ ❛ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ▲❛rs ❑♥✉❞s❡♥ ❛♥❞ ●r❡❣♦r ▲❡❛♥❞❡r

❖✉t❧✐♥❡ • ▲♦✇✲❧❛t❡♥❝② ❛♥❞ ❧✐❣❤t✇❡✐❣❤t ❝✐♣❤❡rs • ▼✐♥✐♠✐③✐♥❣ t❤❡ ♦✈❡r❤❡❛❞ ♦❢ ❞❡❝r②♣t✐♦♥✿ ✐♥✈♦❧✉t✐♦♥❛❧ ❝✐♣❤❡rs ❛♥❞ ✐♥✈♦❧✉t✐♦♥❛❧ ❜✉✐❧❞✐♥❣✲❜❧♦❝❦s • ▼✐♥✐♠✐③✐♥❣ t❤❡ ♦✈❡r❤❡❛❞ ♦❢ ❞❡❝r②♣t✐♦♥✿ r❡✢❡❝t✐♦♥ ❝✐♣❤❡rs • P❘■◆❈❊ ✶

■t❡r❛t❡❞ ❜❧♦❝❦ ❝✐♣❤❡rs K ♠❛st❡r ❦❡② ❄ ❦❡② s❝❤❡❞✉❧❡ ❄ ❄ ❄ k 1 k 2 k r ❡ ❡ ❡ y ❝✐♣❤❡rt❡①t ♣❧❛✐♥t❡①t x ✲ F (2) ✳✳✳ ✲ F (1) ✲ ✲ F ( r ) ✲ ✇❤❡r❡ ❡❛❝❤ F ( i ) ✐s ❛ ❦❡②❡❞ ♣❡r♠✉t❛t✐♦♥ ♦❢ F n 2 ✳ ✷

▲✐❣❤t✇❡✐❣❤t ❜❧♦❝❦ ❝✐♣❤❡rs ❆❊❙ ❬❉❛❡♠❡♥✲❘✐❥♠❡♥ ✾✽❪❬❋■P❙ P❯❇ ✶✾✼❪ • ❜❧♦❝❦s✐③❡✿ 128 ❜✐ts • ❙❜♦① ♦♣❡r❛t❡s ♦♥ 8 ❜✐ts � 4 � • ❧✐♥❡❛r ❞✐✛✉s✐♦♥ ❧❛②❡r ✐s ❛ ❧✐♥❡❛r ♣❡r♠✉t❛t✐♦♥ ♦❢ F 2 8 ❚♦ ♠❛❦❡ ✐t s♠❛❧❧❡r ✐♥ ❤❛r❞✇❛r❡✿ • ❜❧♦❝❦s✐③❡✿ 64 ❜✐ts • s♠❛❧❧❡r ❙❜♦①✱ ♦♥ 3 ♦r 4 ❜✐ts • ❧✐♥❡❛r ❞✐✛✉s✐♦♥ ❧❛②❡r ♦✈❡r ❛ s♠❛❧❧❡r ❛❧♣❤❛❜❡t • s✐♠♣❧✐✜❡❞ ❦❡②✲s❝❤❡❞✉❧❡ ✸

❚❤❡ ✉s✉❛❧ ❞❡s✐❣♥ str❛t❡❣②✿ P❘❊❙❊◆❚ ❬❇♦❣❞❛♥♦✈ ❡t ❛❧✳ ✵✼❪ 64 bits sk i S S S S S S S S S S S S S S S S 31 r♦✉♥❞s ✭✰ ❛ ❦❡② ❛❞❞✐t✐♦♥✮ ✹

▲✐❣❤t✇❡✐❣❤t ❜✉t s❡❝✉r❡✳✳✳ ■♥❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ r♦✉♥❞s✦ • P❘❊❙❊◆❚ ❬❇♦❣❞❛♥♦✈ ❡t ❛❧✳ ✵✼❪✳ ✸✶ r♦✉♥❞s • ▲❊❉ ❬●✉♦ ❡t ❛❧✳ ✶✶❪✿ ▲❊❉✲✻✹✿ ✸✷ r♦✉♥❞s✱ ▲❊❉✲✶✷✽✿ ✹✽ r♦✉♥❞s • ❙P❊❈❑ ❬❇❡❛✉❧✐❡✉ ❡t ❛❧✳ ✶✸❪✿ ❙P❊❈❑✻✹✴✶✷✽✿ ✷✼ r♦✉♥❞s✱ ❙P❊❈❑✶✷✽✴✷✺✻✿ ✸✹ r♦✉♥❞s • ❙■▼❖◆ ❬❇❡❛✉❧✐❡✉ ❡t ❛❧✳ ✶✸❪✿ ❙■▼❖◆✻✹✴✶✷✽✿ ✹✹ r♦✉♥❞s✱ ❙■▼❖◆✶✷✽✴✷✺✻✿ ✼✷ r♦✉♥❞s ✺

❉♦❡s ❧✐❣❤t✇❡✐❣❤t ♠❡❛♥ ✏❧✐❣❤t ✰ ✇❛✐t✑❄ ❬❑♥❡➸❡✈✐➣ ❡t ❛❧✳ ✶✷❪ ✻

❉♦❡s ❧✐❣❤t✇❡✐❣❤t ♠❡❛♥ ✏❧✐❣❤t ✰ ✇❛✐t✑❄ ❬❑♥❡➸❡✈✐➣ ❡t ❛❧✳ ✶✷❪ ❙❊❈❯❘■❚❨ ✓ ✼ ♦ ❙ ✓ ❙ ✓ ❙ ✓ ❙ ✓ ❙ ✓ ❙ ✓ ❙ ✓ ❙ ✓ ❙ ✴ ✓ ❙ ✇ ✛ ✲ ❙P❊❊❉ ❆❘❊❆ ▲♦✇✲❧❛t❡♥❝② ❡♥❝r②♣t✐♦♥✳ • ▼❡♠♦r② ❡♥❝r②♣t✐♦♥ • ❱❆◆❊❚ ✭❱❡❤✐❝✉❧❛r ❛❞✲❤♦❝ ♥❡t✇♦r❦✮ • ❡♥❝r②♣t✐♦♥ ❢♦r ❤✐❣❤✲s♣❡❡❞ ♥❡t✇♦r❦✐♥❣✳✳✳ ✼

❍♦✇ ❝❛♥ ✇❡ ❞❡s✐❣♥ ❛ ❢❛st ❛♥❞ ❧✐❣❤t✇❡✐❣❤t ❝✐♣❤❡r❄ ❯♥r♦❧❧❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥✳ • s♠❛❧❧ ♥✉♠❜❡r ♦❢ r♦✉♥❞s❀ • ❡❛❝❤ r♦✉♥❞ ♦❢ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ s❤♦✉❧❞ ❤❛✈❡ ❛ ❧♦✇ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❝♦st❀ • t❤❡ r♦✉♥❞s ❞♦ ♥♦t ♥❡❡❞ t♦ ❜❡ s✐♠✐❧❛r✳ ❘❡❧❛t❡❞ ♦♣❡♥ ♣r♦❜❧❡♠✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦✈✐❞❡ s❡❝✉r✐t② ❛r❣✉♠❡♥ts ❢♦r ❛ ❝✐♣❤❡r ✐t❡r❛t✐♥❣ ✈❡r② ❞✐✛❡r❡♥t r♦✉♥❞s❄ ✽

▼✐♥✐♠✐③✐♥❣ t❤❡ ♦✈❡r❤❡❛❞ ♦❢ ❞❡❝r②♣t✐♦♥✿ ✐♥✈♦❧✉t✐♦♥❛❧ ❜✉✐❧❞✐♥❣✲❜❧♦❝❦s ✾

❲❤❡♥ ❧✐❣❤t✇❡✐❣❤t ❡♥❝r②♣t✐♦♥ ✇❛s r❡❛❧❧② ❛♥ ✐ss✉❡✳✳✳ ❤tt♣✿✴✴✇✇✇✳♥s❛✳❣♦✈✴♠✉s❡✉♠✴❡♥✐❣♠❛✳❤t♠❧ ✶✵

❙❝❤❡r❜✐✉s✬ s♦❧✉t✐♦♥✿ ❛❞❞ ❛ r❡✢❡❝t♦r ☛ ✟ ☛ ✟ ☛ ✟ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ✡ ✠ ✡ ✠ ✡ ✠ ❦❡②❜♦❛r❞ ❧❛♠♣❜♦❛r❞ r❡✢❡❝t♦r E K = F − 1 ◦ M ◦ F K ✇❤❡r❡ M = M − 1 K ✶✶

❈❛♥ E K ❜❡ ❛♥ ✐♥✈♦❧✉t✐♦♥❄ ❋✐①❡❞ ♣♦✐♥ts✳ ❬❨♦✉ss❡❢✲❚❛✈❛r❡s✲❍❡②s ✾✻❪ • ❆ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ♦❢ F n 2 ❤❛s 1 ✜①❡❞ ♣♦✐♥t ♦♥ ❛✈❡r❛❣❡❀ n • ❆ r❛♥❞♦♠ ✐♥✈♦❧✉t✐♦♥ ♦❢ F n 2 + O (1) ✜①❡❞ ♣♦✐♥ts✳ 2 ❤❛s 2 ■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r E K = F − 1 ◦ M ◦ F K K E K ❤❛s t❤❡ s❛♠❡ ❝②❝❧❡ str✉❝t✉r❡ ✭❛♥❞ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ ✜①❡❞ ♣♦✐♥ts✮ ❛s M ✳ • ❊♥✐❣♠❛✿ t❤❡ r❡✢❡❝t♦r ❤❛s ♥♦ ✜①❡❞ ♣♦✐♥ts❀ • ❉❊❙ ✇✐t❤ ❛ ✇❡❛❦ ❦❡②✿ M ✐s t❤❡ s✇❛♣♣✐♥❣ ♦❢ t❤❡ ✷ ❤❛❧✈❡s → ■t ❤❛s 2 32 ✜①❡❞ ♣♦✐♥ts ❬❈♦♣♣❡rs♠✐t❤ ✽✺❪✳ ✶✷

❆❞❞ s♦♠❡ ✇❤✐t❡♥✐♥❣ ❦❡②s ❬❘✐✈❡st ✽✹❪ k 1 F X ❝♦♥str✉❝t✐♦♥ k 0 k 2 ❄ ✎☞ ✎☞ ✲ ❄ ✲ ❄ m c ✰ ✲ ✰ ✲ F ✍✌ ✍✌ n +1 ❙❧✐❞❡ ❛tt❛❝❦ ✇✐t❤ ❝♦♠♣❧❡①✐t② 2 2 ❬❨♦✉ss❡❢✲❚❛✈❛r❡s✲❍❡②s ✾✻❪❬❉✉♥❦❡❧♠❛♥ ❡t ❛❧✳ ✶✷❪ k 1 k 0 k 2 ❄ y x ✎☞ ✎☞ ❄ ❄ c ✲ ✲ ✲ ✲ ✰ ✰ m F ✍✌ ✍✌ k 1 k 0 k 2 ❄ y x ✎☞ ❄ ✎☞ ❄ m ′ c ′ ✲ ✰ ✲ ✲ ✰ ✲ F ✍✌ ✍✌ ■❢ ( m, c ) ❛♥❞ ( m ′ , c ′ ) s❛t✐s❢② m ⊕ c = m ′ ⊕ c ′ ✱ t❤❡♥ ❝❤❡❝❦ ✇❤❡t❤❡r k 0 ⊕ k 2 = m ′ ⊕ c ✳ ✶✸

❯s✐♥❣ ✐♥✈♦❧✉t✐♦♥❛❧ ❜✉✐❧❞✐♥❣✲❜❧♦❝❦s ❊①❛♠♣❧❡s✿ • ❋❡✐st❡❧ ❝✐♣❤❡rs • ✐♥✈♦❧✉t✐♦♥❛❧ ❙P◆s ❬❨♦✉ss❡❢✲❚❛✈❛r❡s✲❍❡②s ✾✻❪ • ❑❤❛③❛❞ ❬❇❛rr❡t♦✲❘✐❥♠❡♥ ✵✵❪ • ❆◆❯❇■❙ ❬❇❛rr❡t♦✲❘✐❥♠❡♥ ✵✵❪ • ◆❖❊❑❊❖◆ ❬❉❛❡♠❡♥ ❡t ❛❧✳ ✵✵❪ • ■❈❊❇❊❘● ❬❙t❛♥❞❛❡rt ❡t ❛❧✳ ✵✹❪✳✳✳ ✶✹

❆❊❙ s✉♣❡r❜♦① ❄ ❄ ❄ ❄ S S S S ❄ ❄ ❄ ❄ L ❄ ❄ ❄ ❄ K ✲ ✲ ✲ ✲ ❦ ❦ ❦ ❦ ✰ ✰ ✰ ✰ ❄ ❄ ❄ ❄ S S S S ❄ ❄ ❄ ❄ S ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦✈❡r F m 2 ❚❤❡ ❞✐✛✉s✐♦♥ ❧❛②❡r ✐s ❧✐♥❡❛r ♦✈❡r F 2 m ❛♥❞ ❤❛s ♠❛①✐♠❛❧ ❜r❛♥❝❤ ♥✉♠❜❡r✳ ✶✺

■♥✈♦❧✉t✐♦♥❛❧ ❙❜♦①❡s ✇✐t❤ ❛♥ ❙P◆ ▼❛①✐♠❛❧ ❡①♣❡❝t❡❞ ♣r♦❜❛❜✐❧✐t② ❢♦r ❛ t✇♦✲r♦✉♥❞ ❞✐✛❡r❡♥t✐❛❧✿ MEDP 2 = max a � =0 ,b Pr x,K [∆ E K ( x ) = b | ∆ x = a ] ❋♦r t❤❡ ❆❊❙ ❙❜♦① S ( x ) = ℓ ( x 254 ) ✿ MEDP 2 = 53 × 2 − 34 ❬❑❡❧✐❤❡r✲❙✉✐ ✵✼❪ ❋♦r t❤❡ ♥❛✐✈❡ ❙❜♦① S ( x ) = x 254 ✿ MEDP 2 = 79 × 2 − 34 ❬❉❛❡♠❡♥✲❘✐❥♠❡♥ ✵✻❪ → ❍✐❣❤❡st ♣♦ss✐❜❧❡ ✈❛❧✉❡ ❢♦r ❛ ❢✉♥❝t✐♦♥ ❤❛✈✐♥❣ s✐♠✐❧❛r ✈❛❧✉❡s ✐♥ ✐ts ❞✐✛❡r❡♥❝❡ t❛❜❧❡ ❬P❛r❦ ❡t ❛❧✳ ✵✸❪ ✶✻