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rs r t

rs r - PowerPoint PPT Presentation

Sep 04, 2022 •673 likes •2.1k views

rs r t rt

  1. ❖✉t❧✐♥❡ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❢♊r ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❢♊r ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ❈♊♥❝❧✉s✐♊♥ ✜ ✮ ✺✷

  2. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ ❇❧♊❝❊❝✐♣❀❡rs✮P❡r♠✉t❛t✐♊♥s ❇❧♊❝❊❝✐♣❀❡r ❇❛s❡❞✳ ♣P❡r♠✉t❛t✐♊♥ ❇❛s❡❞✳♣ ï¿œ ï¿œ E E P E ✟ ✮ ✺✷

  3. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ ❇❧♊❝❊❝✐♣❀❡rs • LRW 1 ❛♥❞ LRW 2 ❜② ▲✐s❩♩✈ ❡t ❛❧✳ ❬▲❘❲✵✷❪ ✿ t h ( t ) h ( t ) k k k m c m c E E E • h ✐s ❳❖❘✲✉♥✐✈❡rs❛❧ ❀❛s❀ • ❊✳❣✳✱ h ( t ) = h ⊗ t ❢♊r n ✲❜✐t ✏❊❡②✑ h ✶✵ ✮ ✺✷

  4. ❯s❡❞ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ●❡♥❡r❛❧✐③❡❞ ♠❛s❊✐♥❣✿ ❈❀❛❊r❛❜♩rt② ❛♥❞ ❙❛r❩❛r ❬❈❙✵✻❪ ✿ ❢♊r ▲❋❙❘ ●r❛② ❝♊❞❡s ✭✉s❡❞ ✐♥ ❖❈❇✶ ❛♥❞ ❖❈❇✞✮ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ ❇❧♊❝❊❝✐♣❀❡rs • XE ❛♥❞ XEX ❜② ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ✿ 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) k k m c m c E E • ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ ✶✶ ✮ ✺✷

  5. ●❡♥❡r❛❧✐③❡❞ ♠❛s❊✐♥❣✿ ❈❀❛❊r❛❜♩rt② ❛♥❞ ❙❛r❩❛r ❬❈❙✵✻❪ ✿ ❢♊r ▲❋❙❘ ●r❛② ❝♊❞❡s ✭✉s❡❞ ✐♥ ❖❈❇✶ ❛♥❞ ❖❈❇✞✮ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ ❇❧♊❝❊❝✐♣❀❡rs • XE ❛♥❞ XEX ❜② ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ✿ 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) k k m c m c E E • ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ • ❯s❡❞ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ✶✶ ✮ ✺✷

  6. ❳❚❙✿ ❊①❛♠♣❧❡✿ ❳❊❳ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ❖❈❇✷✿ A 1 A 2 A a ⊕ M i M 1 M 2 M d E N, A 1 ˜ E N, A 2 ˜ E N, A a ˜ E N, M ⊕ ˜ E N, M 1 ˜ E N, M 2 ˜ E N, M d ˜ k k k k k k k C 1 C 2 C d T ✶✷ ✮ ✺✷

  7. ❳❚❙✿ ❊①❛♠♣❧❡✿ ❳❊❳ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ❖❈❇✷✿ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d L 2 · 3 2 L 2 L E k E k E k E k E k E k E k 2 2 L 2 d L 2 L C 1 C 2 C d L = E K ( N ) T ✶✷ ✮ ✺✷

  8. ❳❚❙✿ ❊①❛♠♣❧❡✿ ❳❊❳ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ❖❈❇✷✿ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d L 2 · 3 2 L 2 L E k E k E k E k E k E k E k 2 2 L 2 d L 2 L C 1 C 2 C d L = E K ( N ) T ✶✷ ✮ ✺✷

  9. ❳❚❙✿ ❊①❛♠♣❧❡✿ ❳❊❳ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ❖❈❇✷✿ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d L 2 · 3 2 L 2 L E k E k E k E k E k E k E k 2 2 L 2 d L 2 L C 1 C 2 C d L = E K ( N ) T ✶✷ ✮ ✺✷

  10. ❳❚❙✿ ❊①❛♠♣❧❡✿ ❳❊❳ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ❖❈❇✷✿ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d L 2 · 3 2 L 2 L E k E k E k E k E k E k E k 2 2 L 2 d L 2 L C 1 C 2 C d L = E K ( N ) T ✶✷ ✮ ✺✷

  11. ❳❚❙✿ ❊①❛♠♣❧❡✿ ❳❊❳ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ❖❈❇✷✿ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d L 2 · 3 2 L 2 L E k E k E k E k E k E k E k 2 2 L 2 d L 2 L C 1 C 2 C d L = E K ( N ) T ✶✷ ✮ ✺✷

  12. ❊①❛♠♣❧❡✿ ❳❊❳ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ❖❈❇✷✿ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d L 2 · 3 2 L 2 L E k E k E k E k E k E k E k 2 2 L 2 d L 2 L C 1 C 2 C d L = E K ( N ) T ❳❚❙✿ M 1 M 2 M d E i , 1 E i , 2 ˜ E i , d ˜ ˜ k k k C 1 C 2 C d ✶✷ ✮ ✺✷

  13. ❊①❛♠♣❧❡✿ ❳❊❳ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ❖❈❇✷✿ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d L 2 · 3 2 L 2 L E k E k E k E k E k E k E k 2 2 L 2 d L 2 L C 1 C 2 C d L = E K ( N ) T ❳❚❙✿ M 1 M 2 M d 2 2 L 2 d L 2 L E k E k E k 2 2 L 2 d L 2 L L = E K ( i ) C 1 C 2 C d ✶✷ ✮ ✺✷

  14. ❊①❛♠♣❧❡✿ ❳❊❳ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ❖❈❇✷✿ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d L 2 · 3 2 L 2 L E k E k E k E k E k E k E k 2 2 L 2 d L 2 L C 1 C 2 C d L = E K ( N ) T ❳❚❙✿ M 1 M 2 M d 2 2 L 2 d L 2 L E k E k E k 2 2 L 2 d L 2 L L = E K ( i ) C 1 C 2 C d ✶✷ ✮ ✺✷

  15. ●❡♥❡r❛❧✐③❡❞ ♠❛s❊✐♥❣✿ ❈❀❛❊r❛❜♩rt② ❛♥❞ ❙❛r❩❛r ❬❈❙✵✻❪ ✿ ❢♊r ▲❋❙❘ ●r❛② ❝♊❞❡s ✭✉s❡❞ ✐♥ ❖❈❇✶ ❛♥❞ ❖❈❇✞✮ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ ❇❧♊❝❊❝✐♣❀❡rs • XE ❛♥❞ XEX ❜② ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ✿ 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) k k m c m c E E • ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ • ❯s❡❞ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ ✶✞ ✮ ✺✷

  16. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ ❇❧♊❝❊❝✐♣❀❡rs • XE ❛♥❞ XEX ❜② ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ✿ 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) k k m c m c E E • ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ • ❯s❡❞ ✐♥ ❖❈❇✷ ❛♥❞ ❳❚❙ • ●❡♥❡r❛❧✐③❡❞ ♠❛s❊✐♥❣✿ • ❈❀❛❊r❛❜♩rt② ❛♥❞ ❙❛r❩❛r ❬❈❙✵✻❪ ✿ ϕ α ( E k ( N )) ❢♊r ▲❋❙❘ ϕ • ●r❛② ❝♊❞❡s ✭✉s❡❞ ✐♥ ❖❈❇✶ ❛♥❞ ❖❈❇✞✮ ✶✞ ✮ ✺✷

  17. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ P❡r♠✉t❛t✐♊♥s • ▌✐♥❛❧♣❀❡r✬s TEM ❬❙❚❆✰✶✹❪ ✿ 2 α 3 β 7 γ ( k ï¿œ N ⊕ P ( k ï¿œ N )) P m c • ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ ✶✹ ✮ ✺✷

  18. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ P❡r♠✉t❛t✐♊♥s • PrÞst ❬❑▲▲✰✶✹❪ ✉s❡s ❳❊✭❳✮ ✇✐t❀ ❊✈❡♥✲▌❛♥s♩✉r✿ 2 α 3 β 7 γ E k (0) 2 α 3 β 7 γ E k (0) k m c E ✇✐t❀ E k ( m ) = P ( m ⊕ k ) ⊕ k ✶✺ ✮ ✺✷

  19. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ P❡r♠✉t❛t✐♊♥s • PrÞst ❬❑▲▲✰✶✹❪ ✉s❡s ❳❊✭❳✮ ✇✐t❀ ❊✈❡♥✲▌❛♥s♩✉r✿   2 α 3 β 7 γ E k (0) 2 α 3 β 7 γ E k (0)    (2 α 3 β 7 γ ⊕ 1) k ⊕ 2 α 3 β 7 γ P ( k )  k     m c E    m P c       ✇✐t❀ E k ( m ) = P ( m ⊕ k ) ⊕ k ✶✺ ✮ ✺✷

  20. ❉❡♊①②s ✱ ❆❊❩ ✱ ❈❇❆✱ ❈❖❇❘❆✱ ▌✐♥❛❧♣❀❡r ✱ ❏♩❧t✐❩ ✱ ❈❖P❆ ✱ ❊▲♠❉ ✱ ✐❋❡❡❞✱ PrÞst ❑■❆❙❯✱ ▌❛r❜❧❡✱ ❖❈❇ ✱ ❖▌❉ ✱ ❙❈❘❊❆▌ ❖❚❘ ✱ P❖❊❚ ✱ ❙❍❊▲▲ ♣❧❛✐♥ ❂ ✜rst r♊✉♥❞✱ ❜♩❧❞ ❂ s❡❝♊♥❞ r♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ✐♥ ❈❆❊❙❆❘ 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ ( k ï¿œ N ⊕ P ( k ï¿œ N )) k k ï¿œ m c E m c m P c E t ❉❡❞✐❝❛t❡❞ ❳❊✎❳❊❳✲✐♥s♣✐r❡❞ ❚❊▌✲✐♥s♣✐r❡❞ ✶✻ ✮ ✺✷

  21. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ✐♥ ❈❆❊❙❆❘ 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ ( k ï¿œ N ⊕ P ( k ï¿œ N )) k k ï¿œ m c E m c m P c E t ❉❡❞✐❝❛t❡❞ ❳❊✎❳❊❳✲✐♥s♣✐r❡❞ ❚❊▌✲✐♥s♣✐r❡❞ ❉❡♊①②s ✱ ❆❊❩ ✱ ❈❇❆✱ ❈❖❇❘❆✱ ▌✐♥❛❧♣❀❡r ✱ ❏♩❧t✐❩ ✱ ❈❖P❆ ✱ ❊▲♠❉ ✱ ✐❋❡❡❞✱ PrÞst ❑■❆❙❯✱ ▌❛r❜❧❡✱ ❖❈❇ ✱ ❖▌❉ ✱ ❙❈❘❊❆▌ ❖❚❘ ✱ P❖❊❚ ✱ ❙❍❊▲▲ ♣❧❛✐♥ ❂ ✜rst r♊✉♥❞✱ ❜♩❧❞ ❂ s❡❝♊♥❞ r♊✉♥❞ ✶✻ ✮ ✺✷

  22. ❖✉t❧✐♥❡ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❢♊r ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❢♊r ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ❈♊♥❝❧✉s✐♊♥ ✶✌ ✮ ✺✷

  23. ❬▌❡♥✶✺❜❪ ✱ ❣❡♥❡r❛❧✐③❛t✐♊♥ ♊❢ t❀✐s ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ✐♥ ❈❆❊❙❆❘ k 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ ( k ï¿œ N ⊕ P ( k ï¿œ N )) k ï¿œ m c E P m c m c E t ❉❡❞✐❝❛t❡❞ ❳❊✎❳❊❳✲✐♥s♣✐r❡❞ ❚❊▌✲✐♥s♣✐r❡❞ ❉❡♊①②s ✱ ❆❊❩ ✱ ❈❇❆✱ ❈❖❇❘❆✱ ▌✐♥❛❧♣❀❡r ✱ ❏♩❧t✐❩ ✱ ❈❖P❆ ✱ ❊▲♠❉ ✱ ✐❋❡❡❞✱ PrÞst ❑■❆❙❯✱ ▌❛r❜❧❡✱ ❖❈❇ ✱ ❖▌❉ ✱ ❙❈❘❊❆▌ ❖❚❘ ✱ P❖❊❚ ✱ ❙❍❊▲▲ ♣❧❛✐♥ ❂ ✜rst r♊✉♥❞✱ ❜♩❧❞ ❂ s❡❝♊♥❞ r♊✉♥❞ ✶✜ ✮ ✺✷

  24. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ✐♥ ❈❆❊❙❆❘ k 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ E k ( N ) 2 α 3 β 7 γ ( k ï¿œ N ⊕ P ( k ï¿œ N )) k ï¿œ m c E P m c m c E t ❉❡❞✐❝❛t❡❞ ❳❊✎❳❊❳✲✐♥s♣✐r❡❞ ❚❊▌✲✐♥s♣✐r❡❞ ❉❡♊①②s ✱ ❆❊❩ ✱ ❈❇❆✱ ❈❖❇❘❆✱ ▌✐♥❛❧♣❀❡r ✱ ❏♩❧t✐❩ ✱ ❈❖P❆ ✱ ❊▲♠❉ ✱ ✐❋❡❡❞✱ PrÞst ❑■❆❙❯✱ ▌❛r❜❧❡✱ ❖❈❇ ✱ ❖▌❉ ✱ → ❙❈❘❊❆▌ ❖❚❘ ✱ P❖❊❚ ✱ ❙❍❊▲▲ − − XPX ❬▌❡♥✶✺❜❪ ✱ ❣❡♥❡r❛❧✐③❛t✐♊♥ ♊❢ t❀✐s ♣❧❛✐♥ ❂ ✜rst r♊✉♥❞✱ ❜♩❧❞ ❂ s❡❝♊♥❞ r♊✉♥❞ ✶✜ ✮ ✺✷

  25. ✶ ✏❙t✉♣✐❞✑ ✐♥s❡❝✉r❡ ✷ ✏◆♩r♠❛❧✑ s✐♥❣❧❡✲❊❡② s❡❝✉r❡ ✾ ✏❙tr♊♥❣✑ r❡❧❛t❡❞✲❊❡② s❡❝✉r❡ ❙❡❝✉r✐t② ♊❢ str♊♥❣❧② ❞❡♣❡♥❞s ♊♥ ❝❀♊✐❝❡ ♊❢ ❳P❳ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c ❚✇❡❛❊ ❙❡t • ( t 11 , t 12 , t 21 , t 22 ) ❢r♩♠ s♊♠❡ t✇❡❛❊ s❡t T ⊆ ( { 0 , 1 } n ) 4 • T ❝❛♥ ✭st✐❧❧✮ ❜❡ ❛♥② s❡t ✶✟ ✮ ✺✷

  26. ✶ ✏❙t✉♣✐❞✑ ✐♥s❡❝✉r❡ ✷ ✏◆♩r♠❛❧✑ s✐♥❣❧❡✲❊❡② s❡❝✉r❡ ✾ ✏❙tr♊♥❣✑ r❡❧❛t❡❞✲❊❡② s❡❝✉r❡ ❳P❳ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c ❚✇❡❛❊ ❙❡t • ( t 11 , t 12 , t 21 , t 22 ) ❢r♩♠ s♊♠❡ t✇❡❛❊ s❡t T ⊆ ( { 0 , 1 } n ) 4 • T ❝❛♥ ✭st✐❧❧✮ ❜❡ ❛♥② s❡t • ❙❡❝✉r✐t② ♊❢ XPX str♊♥❣❧② ❞❡♣❡♥❞s ♊♥ ❝❀♊✐❝❡ ♊❢ T ✶✟ ✮ ✺✷

  27. ✷ ✏◆♩r♠❛❧✑ s✐♥❣❧❡✲❊❡② s❡❝✉r❡ ✾ ✏❙tr♊♥❣✑ r❡❧❛t❡❞✲❊❡② s❡❝✉r❡ ❳P❳ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c ❚✇❡❛❊ ❙❡t • ( t 11 , t 12 , t 21 , t 22 ) ❢r♩♠ s♊♠❡ t✇❡❛❊ s❡t T ⊆ ( { 0 , 1 } n ) 4 • T ❝❛♥ ✭st✐❧❧✮ ❜❡ ❛♥② s❡t • ❙❡❝✉r✐t② ♊❢ XPX str♊♥❣❧② ❞❡♣❡♥❞s ♊♥ ❝❀♊✐❝❡ ♊❢ T ✶ ✏❙t✉♣✐❞✑ T − → ✐♥s❡❝✉r❡ ✶✟ ✮ ✺✷

  28. ✾ ✏❙tr♊♥❣✑ r❡❧❛t❡❞✲❊❡② s❡❝✉r❡ ❳P❳ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c ❚✇❡❛❊ ❙❡t • ( t 11 , t 12 , t 21 , t 22 ) ❢r♩♠ s♊♠❡ t✇❡❛❊ s❡t T ⊆ ( { 0 , 1 } n ) 4 • T ❝❛♥ ✭st✐❧❧✮ ❜❡ ❛♥② s❡t • ❙❡❝✉r✐t② ♊❢ XPX str♊♥❣❧② ❞❡♣❡♥❞s ♊♥ ❝❀♊✐❝❡ ♊❢ T ✶ ✏❙t✉♣✐❞✑ T − → ✐♥s❡❝✉r❡ ✷ ✏◆♩r♠❛❧✑ T − → s✐♥❣❧❡✲❊❡② s❡❝✉r❡ ✶✟ ✮ ✺✷

  29. ❳P❳ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c ❚✇❡❛❊ ❙❡t • ( t 11 , t 12 , t 21 , t 22 ) ❢r♩♠ s♊♠❡ t✇❡❛❊ s❡t T ⊆ ( { 0 , 1 } n ) 4 • T ❝❛♥ ✭st✐❧❧✮ ❜❡ ❛♥② s❡t • ❙❡❝✉r✐t② ♊❢ XPX str♊♥❣❧② ❞❡♣❡♥❞s ♊♥ ❝❀♊✐❝❡ ♊❢ T ✶ ✏❙t✉♣✐❞✑ T − → ✐♥s❡❝✉r❡ ✷ ✏◆♩r♠❛❧✑ T − → s✐♥❣❧❡✲❊❡② s❡❝✉r❡ ✾ ✏❙tr♊♥❣✑ T − → r❡❧❛t❡❞✲❊❡② s❡❝✉r❡ ✶✟ ✮ ✺✷

  30. ■❢ ✐s ✐♥✈❛❧✐❞✱ t❀❡♥ ✐s ✐♥s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❊ ❙❡ts ❚❡❝❀♥✐❝❛❧ ❞❡✜♥✐t✐♊♥ t♩ ❡❧✐♠✐♥❛t❡ tr✐✈✐❛❧ ❝❛s❡s ❳P❳✿ ❙t✉♣✐❞ ❚✇❡❛❊s t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c ✷✵ ✮ ✺✷

  31. ■❢ ✐s ✐♥✈❛❧✐❞✱ t❀❡♥ ✐s ✐♥s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❊ ❙❡ts ❚❡❝❀♥✐❝❛❧ ❞❡✜♥✐t✐♊♥ t♩ ❡❧✐♠✐♥❛t❡ tr✐✈✐❛❧ ❝❛s❡s ❳P❳✿ ❙t✉♣✐❞ ❚✇❡❛❊s 0 k ⊕ 0 P ( k ) 0 k ⊕ 0 P ( k ) P m (0 , 0 , 0 , 0) ∈ T ✷✵ ✮ ✺✷

  32. ■❢ ✐s ✐♥✈❛❧✐❞✱ t❀❡♥ ✐s ✐♥s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❊ ❙❡ts ❚❡❝❀♥✐❝❛❧ ❞❡✜♥✐t✐♊♥ t♩ ❡❧✐♠✐♥❛t❡ tr✐✈✐❛❧ ❝❛s❡s ❳P❳✿ ❙t✉♣✐❞ ❚✇❡❛❊s 0 k ⊕ 0 P ( k ) 0 k ⊕ 0 P ( k ) P P ( m ) m (0 , 0 , 0 , 0) ∈ T = ⇒ XPX k ((0 , 0 , 0 , 0) , m ) = P ( m ) ✷✵ ✮ ✺✷

  33. ■❢ ✐s ✐♥✈❛❧✐❞✱ t❀❡♥ ✐s ✐♥s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❊ ❙❡ts ❚❡❝❀♥✐❝❛❧ ❞❡✜♥✐t✐♊♥ t♩ ❡❧✐♠✐♥❛t❡ tr✐✈✐❛❧ ❝❛s❡s ❳P❳✿ ❙t✉♣✐❞ ❚✇❡❛❊s 1 k ⊕ 0 P ( k ) 1 k ⊕ 1 P ( k ) 0 P k (0 , 0 , 0 , 0) ∈ T = ⇒ XPX k ((0 , 0 , 0 , 0) , m ) = P ( m ) (1 , 0 , 1 , 1) ∈ T = ⇒ XPX k ((1 , 0 , 1 , 1) , 0) = k ✷✵ ✮ ✺✷

  34. ■❢ ✐s ✐♥✈❛❧✐❞✱ t❀❡♥ ✐s ✐♥s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❊ ❙❡ts ❚❡❝❀♥✐❝❛❧ ❞❡✜♥✐t✐♊♥ t♩ ❡❧✐♠✐♥❛t❡ tr✐✈✐❛❧ ❝❛s❡s ❳P❳✿ ❙t✉♣✐❞ ❚✇❡❛❊s 1 k ⊕ 0 P ( k ) 0 k ⊕ 2 P ( k ) 0 P 3 P ( k ) (0 , 0 , 0 , 0) ∈ T = ⇒ XPX k ((0 , 0 , 0 , 0) , m ) = P ( m ) (1 , 0 , 1 , 1) ∈ T = ⇒ XPX k ((1 , 0 , 1 , 1) , 0) = k (1 , 0 , 0 , 2) ∈ T = ⇒ XPX k ((1 , 0 , 0 , 2) , 0) = 3 P ( k ) ✷✵ ✮ ✺✷

  35. ■❢ ✐s ✐♥✈❛❧✐❞✱ t❀❡♥ ✐s ✐♥s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❊ ❙❡ts ❚❡❝❀♥✐❝❛❧ ❞❡✜♥✐t✐♊♥ t♩ ❡❧✐♠✐♥❛t❡ tr✐✈✐❛❧ ❝❛s❡s ❳P❳✿ ❙t✉♣✐❞ ❚✇❡❛❊s 1 k ⊕ 0 P ( k ) 0 k ⊕ 2 P ( k ) 0 P 3 P ( k ) (0 , 0 , 0 , 0) ∈ T = ⇒ XPX k ((0 , 0 , 0 , 0) , m ) = P ( m ) (1 , 0 , 1 , 1) ∈ T = ⇒ XPX k ((1 , 0 , 1 , 1) , 0) = k (1 , 0 , 0 , 2) ∈ T = ⇒ XPX k ((1 , 0 , 0 , 2) , 0) = 3 P ( k ) · · · · · · · · · ✷✵ ✮ ✺✷

  36. ■❢ ✐s ✐♥✈❛❧✐❞✱ t❀❡♥ ✐s ✐♥s❡❝✉r❡ ❳P❳✿ ❙t✉♣✐❞ ❚✇❡❛❊s 1 k ⊕ 0 P ( k ) 0 k ⊕ 2 P ( k ) 0 P 3 P ( k ) (0 , 0 , 0 , 0) ∈ T = ⇒ XPX k ((0 , 0 , 0 , 0) , m ) = P ( m ) (1 , 0 , 1 , 1) ∈ T = ⇒ XPX k ((1 , 0 , 1 , 1) , 0) = k (1 , 0 , 0 , 2) ∈ T = ⇒ XPX k ((1 , 0 , 0 , 2) , 0) = 3 P ( k ) · · · · · · · · · ✏❱❛❧✐❞✑ ❚✇❡❛❊ ❙❡ts • ❚❡❝❀♥✐❝❛❧ ❞❡✜♥✐t✐♊♥ t♩ ❡❧✐♠✐♥❛t❡ tr✐✈✐❛❧ ❝❛s❡s ✷✵ ✮ ✺✷

  37. ❳P❳✿ ❙t✉♣✐❞ ❚✇❡❛❊s 1 k ⊕ 0 P ( k ) 0 k ⊕ 2 P ( k ) 0 P 3 P ( k ) (0 , 0 , 0 , 0) ∈ T = ⇒ XPX k ((0 , 0 , 0 , 0) , m ) = P ( m ) (1 , 0 , 1 , 1) ∈ T = ⇒ XPX k ((1 , 0 , 1 , 1) , 0) = k (1 , 0 , 0 , 2) ∈ T = ⇒ XPX k ((1 , 0 , 0 , 2) , 0) = 3 P ( k ) · · · · · · · · · ✏❱❛❧✐❞✑ ❚✇❡❛❊ ❙❡ts • ❚❡❝❀♥✐❝❛❧ ❞❡✜♥✐t✐♊♥ t♩ ❡❧✐♠✐♥❛t❡ tr✐✈✐❛❧ ❝❛s❡s • ■❢ T ✐s ✐♥✈❛❧✐❞✱ t❀❡♥ XPX ✐s ✐♥s❡❝✉r❡ ✷✵ ✮ ✺✷

  38. ✐❢ ✐s ✈❛❧✐❞✱ ❛♥❞ ❢♊r ❛❧❧ t✇❡❛❊s✿ s❡❝✉r✐t② ❛♥❞ ✲r❊✲❙❚P❘P ✲r❊✲❙❚P❘P ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ ❝❛♥ ✐♥✢✉❡♥❝❡ ❊❡②✿ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ ❝❛♥ ✐♥✢✉❡♥❝❡ ❊❡②✿ ♩r ◆♩t❡✿ ♠❛s❊✐♥❣s ✐♥ ❛r❡ ❳P❳✿ ◆♩r♠❛❧ ❛♥❞ ❙tr♊♥❣ ❚✇❡❛❊s ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② • ■❢ T ✐s ✈❛❧✐❞✱ t❀❡♥ XPX ✐s ❙❚P❘P ✷✶ ✮ ✺✷

  39. ✐❢ ✐s ✈❛❧✐❞✱ ❛♥❞ ❢♊r ❛❧❧ t✇❡❛❊s✿ s❡❝✉r✐t② ❛♥❞ ✲r❊✲❙❚P❘P ✲r❊✲❙❚P❘P ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ ❝❛♥ ✐♥✢✉❡♥❝❡ ❊❡②✿ ♩r ◆♩t❡✿ ♠❛s❊✐♥❣s ✐♥ ❛r❡ ❳P❳✿ ◆♩r♠❛❧ ❛♥❞ ❙tr♊♥❣ ❚✇❡❛❊s ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② • ■❢ T ✐s ✈❛❧✐❞✱ t❀❡♥ XPX ✐s ❙❚P❘P Ί ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❊❡②✿ k ￜ→ k ⊕ ÎŽ ✷✶ ✮ ✺✷

  40. ✐❢ ✐s ✈❛❧✐❞✱ ❛♥❞ ❢♊r ❛❧❧ t✇❡❛❊s✿ s❡❝✉r✐t② ❛♥❞ ✲r❊✲❙❚P❘P ✲r❊✲❙❚P❘P ❳P❳✿ ◆♩r♠❛❧ ❛♥❞ ❙tr♊♥❣ ❚✇❡❛❊s ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② • ■❢ T ✐s ✈❛❧✐❞✱ t❀❡♥ XPX ✐s ❙❚P❘P Ί ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❊❡②✿ k ￜ→ k ⊕ ÎŽ Ί P ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❊❡②✿ k ￜ→ k ⊕ ÎŽ ♩r P ( k ) ￜ→ P ( k ) ⊕ Ç« • ◆♩t❡✿ ♠❛s❊✐♥❣s ✐♥ XPX ❛r❡ t i 1 k ⊕ t i 2 P ( k ) ✷✶ ✮ ✺✷

  41. ❳P❳✿ ◆♩r♠❛❧ ❛♥❞ ❙tr♊♥❣ ❚✇❡❛❊s ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② • ■❢ T ✐s ✈❛❧✐❞✱ t❀❡♥ XPX ✐s ❙❚P❘P Ί ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❊❡②✿ k ￜ→ k ⊕ ÎŽ Ί P ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❊❡②✿ k ￜ→ k ⊕ ÎŽ ♩r P ( k ) ￜ→ P ( k ) ⊕ Ç« • ◆♩t❡✿ ♠❛s❊✐♥❣s ✐♥ XPX ❛r❡ t i 1 k ⊕ t i 2 P ( k ) ✐❢ T ✐s ✈❛❧✐❞✱ ❛♥❞ ❢♊r ❛❧❧ t✇❡❛❊s✿ s❡❝✉r✐t② t 12 , t 22 ï¿œ = 0 ❛♥❞ ( t 21 , t 22 ) ï¿œ = (0 , 1) Ί ⊕ ✲r❊✲❙❚P❘P t 11 , t 12 , t 21 , t 22 ï¿œ = 0 Ί P ⊕ ✲r❊✲❙❚P❘P ✷✶ ✮ ✺✷

  42. ❙✐♥❣❧❡✲❊❡② ❙❚P❘P s❡❝✉r❡ ✭s✉r♣r✐s❡❄✮ ●❡♥❡r❛❧❧②✱ ✐❢ ✱ ✐s ❛ ♥♊r♠❛❧ ❜❧♊❝❊❝✐♣❀❡r ❳P❳ ❈♊✈❡rs ❊✈❡♥✲▌❛♥s♩✉r t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) k k − − → P P m c m c ❢♊r T = { (1 , 0 , 1 , 0) } ✷✷ ✮ ✺✷

  43. ●❡♥❡r❛❧❧②✱ ✐❢ ✱ ✐s ❛ ♥♊r♠❛❧ ❜❧♊❝❊❝✐♣❀❡r ❳P❳ ❈♊✈❡rs ❊✈❡♥✲▌❛♥s♩✉r t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) k k − − → P P m c m c ❢♊r T = { (1 , 0 , 1 , 0) } • ❙✐♥❣❧❡✲❊❡② ❙❚P❘P s❡❝✉r❡ ✭s✉r♣r✐s❡❄✮ ✷✷ ✮ ✺✷

  44. ❳P❳ ❈♊✈❡rs ❊✈❡♥✲▌❛♥s♩✉r t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) k k − − → P P m c m c ❢♊r T = { (1 , 0 , 1 , 0) } • ❙✐♥❣❧❡✲❊❡② ❙❚P❘P s❡❝✉r❡ ✭s✉r♣r✐s❡❄✮ • ●❡♥❡r❛❧❧②✱ ✐❢ |T | = 1 ✱ XPX ✐s ❛ ♥♊r♠❛❧ ❜❧♊❝❊❝✐♣❀❡r ✷✷ ✮ ✺✷

  45. ✲r❊ ❙❚P❘P s❡❝✉r❡ ✭✐❢ ✮ ❳P❳ ❈♊✈❡rs ❳❊❳ ❲✐t❀ ❊✈❡♥✲▌❛♥s♩✉r (2 α 3 β 7 γ ⊕ 1) k ⊕ 2 α 3 β 7 γ P ( k ) t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) − − → P P m c m c ï¿œ ( 2 α 3 β 7 γ ⊕ 1 , 2 α 3 β 7 γ , ï¿œ ï¿œ ï¿œ ï¿œ ❢♊r T = ï¿œ ( α, β, γ ) ∈ { XEX ✲t✇❡❛❊s } ( 2 α 3 β 7 γ ⊕ 1 , 2 α 3 β 7 γ ) • ( α, β, γ ) ✐s ✐♥ ❢❛❝t t❀❡ ✏r❡❛❧✑ t✇❡❛❊ ✷✞ ✮ ✺✷

  46. ❳P❳ ❈♊✈❡rs ❳❊❳ ❲✐t❀ ❊✈❡♥✲▌❛♥s♩✉r (2 α 3 β 7 γ ⊕ 1) k ⊕ 2 α 3 β 7 γ P ( k ) t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) − − → P P m c m c ï¿œ ( 2 α 3 β 7 γ ⊕ 1 , 2 α 3 β 7 γ , ï¿œ ï¿œ ï¿œ ï¿œ ❢♊r T = ï¿œ ( α, β, γ ) ∈ { XEX ✲t✇❡❛❊s } ( 2 α 3 β 7 γ ⊕ 1 , 2 α 3 β 7 γ ) • ( α, β, γ ) ✐s ✐♥ ❢❛❝t t❀❡ ✏r❡❛❧✑ t✇❡❛❊ • Ί P ⊕ ✲r❊ ❙❚P❘P s❡❝✉r❡ ✭✐❢ 2 α 3 β 7 γ ï¿œ = 1 ✮ ✷✞ ✮ ✺✷

  47. PrÞst✲❈❖P❆ ❜② ❑❛✈✉♥ ❡t ❛❧✳ ❬❑▲▲✰✶✹❪ ✿ ❈❖P❆ ❜❛s❡❞ ♊♥ ❳❊❳ ❜❛s❡❞ ♊♥ ❊✈❡♥✲▌❛♥s♩✉r ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ ❈❖P❆ A 1 A 2 A a − 1 A a M 1 M 2 M d M 1 ⊕···⊕ M d 3 3 L 2 · 3 3 L 2 a -2 3 3 L 2 a -1 3 4 L 2 d -1 3 L 2 d -1 3 2 L 3 L 2 · 3 L E k E k E k E k E k E k E k L E k E k E k E k E k L = E K (0) 2 2 L 2 d L 2 d -1 7 L 2 L C 1 C 2 C d T • ❇② ❆♥❞r❡❡✈❛ ❡t ❛❧✳ ❬❆❇▲✰✶✹❪ • ■♠♣❧✐❝✐t❧② ❜❛s❡❞ ♊♥ XEX ❜❛s❡❞ ♊♥ ❆❊❙ ✷✹ ✮ ✺✷

  48. ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ ❈❖P❆ A 1 A 2 A a − 1 A a M 1 M 2 M d M 1 ⊕···⊕ M d 3 3 L 2 · 3 3 L 2 a -2 3 3 L 2 a -1 3 4 L 2 d -1 3 L 2 d -1 3 2 L 3 L 2 · 3 L E k E k E k E k E k E k E k L E k E k E k E k E k L = E K (0) 2 2 L 2 d L 2 d -1 7 L 2 L C 1 C 2 C d T • ❇② ❆♥❞r❡❡✈❛ ❡t ❛❧✳ ❬❆❇▲✰✶✹❪ • ■♠♣❧✐❝✐t❧② ❜❛s❡❞ ♊♥ XEX ❜❛s❡❞ ♊♥ ❆❊❙ • PrÞst✲❈❖P❆ ❜② ❑❛✈✉♥ ❡t ❛❧✳ ❬❑▲▲✰✶✹❪ ✿ ❈❖P❆ ❜❛s❡❞ ♊♥ ❳❊❳ ❜❛s❡❞ ♊♥ ❊✈❡♥✲▌❛♥s♩✉r ✷✹ ✮ ✺✷

  49. ✳ ✳ ✲r❊ ✳ ✳ s❊ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♊❢ ❈❖P❆ ❆♣♣r♊❛❝❀ ❣❡♥❡r❛❧✐③❡s ❢♊r ❛♥② ✭♣r♊♊❢ ✐♥ ❬▌❡♥✶✺❜❪ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ❈❖P❆ ✲r❊ ✲r❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ ❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♊❢ ❈❖P❆ σ 2 σ 2 ï¿œ ï¿œ ï¿œ ï¿œ O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❖P❆ XEX E s❊ s❊ ✷✺ ✮ ✺✷

  50. ✳ ✳ s❊ ✳ ✳ ✲r❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ ❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♊❢ ❈❖P❆ σ 2 σ 2 ï¿œ ï¿œ ï¿œ ï¿œ O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❖P❆ XEX E s❊ s❊ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♊❢ ❈❖P❆ • ❆♣♣r♊❛❝❀ ❣❡♥❡r❛❧✐③❡s ❢♊r ❛♥② Ί ✭♣r♊♊❢ ✐♥ ❬▌❡♥✶✺❜❪ ✮ ï¿œ σ 2 ï¿œ ï¿œ σ 2 ï¿œ O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❖P❆ XEX E Ί ✲r❊ Ί ✲r❊ ✷✺ ✮ ✺✷

  51. ❆♣♣r♊❛❝❀ ❣❡♥❡r❛❧✐③❡s ✭♣r♊♊❢ ✐♥ ❬▌❡♥✶✺❜❪ ✮ ✲r❊ s❊ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❖P❆ ✲r❊ ✲r❊ ✲r❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ PrÞst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ σ 2 σ 2 ï¿œ ï¿œ ï¿œ ï¿œ O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❖P❆ XEX E P s❊ s❊ ✷✻ ✮ ✺✷

  52. ❆♣♣r♊❛❝❀ ❣❡♥❡r❛❧✐③❡s ✭♣r♊♊❢ ✐♥ ❬▌❡♥✶✺❜❪ ✮ ✲r❊ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❖P❆ ✲r❊ ✲r❊ ✲r❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ PrÞst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ σ 2 σ 2 σ 2 ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ O O O 2 n 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P s❊ s❊ s❊ ✷✻ ✮ ✺✷

  53. ❆♣♣r♊❛❝❀ ❣❡♥❡r❛❧✐③❡s ✭♣r♊♊❢ ✐♥ ❬▌❡♥✶✺❜❪ ✮ ✲r❊ ✲r❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ PrÞst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ σ 2 σ 2 σ 2 ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ O O O 2 n 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P s❊ s❊ s❊ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ ï¿œ σ 2 ï¿œ ï¿œ σ 2 ï¿œ O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❖P❆ XEX E P Ί ✲r❊ Ί ✲r❊ ✷✻ ✮ ✺✷

  54. ❆♣♣r♊❛❝❀ ❣❡♥❡r❛❧✐③❡s ✭♣r♊♊❢ ✐♥ ❬▌❡♥✶✺❜❪ ✮ ✲r❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ PrÞst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ σ 2 σ 2 σ 2 ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ O O O 2 n 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P s❊ s❊ s❊ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ ï¿œ σ 2 ï¿œ ï¿œ σ 2 ï¿œ ï¿œ ï¿œ O O ℩ 1 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P Ί ✲r❊ Ί ✲r❊ Ί ✲r❊ ✷✻ ✮ ✺✷

  55. ❆♣♣r♊❛❝❀ ❣❡♥❡r❛❧✐③❡s ✭♣r♊♊❢ ✐♥ ❬▌❡♥✶✺❜❪ ✮ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ PrÞst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ σ 2 σ 2 σ 2 ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ ï¿œ O O O 2 n 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P s❊ s❊ s❊ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♊❢ PrÞst✲❈❖P❆ ï¿œ σ 2 ï¿œ ï¿œ σ 2 ï¿œ ï¿œ ï¿œ O O ℩ 1 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P Ί ✲r❊ Ί ✲r❊ Ί ✲r❊ ï¿œ ï¿œ σ 2 O 2 n Ί P ⊕ ✲r❊ ✷✻ ✮ ✺✷

  56. ✲r❊ ❇❛s❡❞ ♊♥ ✇✐t❀ ✳ ✳ ✳ ✳ ✳ ✳ ▌✐♥❛❧♣❀✳ ✲r❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ ▌✐♥❛❧♣❀❡r A 1 A 2 A a − 1 A a M 1 M 2 M d − 1 M d 2 L ′ 2 2 L ′ 2 a -1 L ′ 2 L 2 3 L 2 2 d -3 L 2 2 d -1 L P P P P P P P 2 2 L ′ 2 a -1 L ′ 2 3 L 2 2 d -3 L 2 2 d -1 L 2 L ′ 2 L C 1 C 2 C d − 1 C d 2 a -1 3 L ′ 2 2 L 2 4 L 2 2 d -2 L L ′ = k ï¿œ flag ï¿œ 0 ⊕ P ( k ï¿œ flag ï¿œ 0) P P P P 2 a -1 3 L ′ 2 2 d -2 L 2 2 L 2 4 L L = k ï¿œ flag ï¿œ N ⊕ P ( k ï¿œ flag ï¿œ N ) 2 2 d -1 3 L • ❇② ❙❛s❛❩✐ ❡t ❛❧✳ ❬❙❚❆✰✶✹❪ P • ❊①tr❛ ♥♊♥❝❡ N ❝♊♥❝❛t❡♥❛t❡❞ t♩ k 2 2 d -1 3 L T ✷✌ ✮ ✺✷

  57. ✲r❊ ✳ ✳ ✳ ✳ ✳ ✳ ▌✐♥❛❧♣❀✳ ✲r❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ ▌✐♥❛❧♣❀❡r A 1 A 2 A a − 1 A a M 1 M 2 M d − 1 M d 2 L ′ 2 2 L ′ 2 a -1 L ′ 2 L 2 3 L 2 2 d -3 L 2 2 d -1 L P P P P P P P 2 2 L ′ 2 a -1 L ′ 2 3 L 2 2 d -3 L 2 2 d -1 L 2 L ′ 2 L C 1 C 2 C d − 1 C d 2 a -1 3 L ′ 2 2 L 2 4 L 2 2 d -2 L L ′ = k ï¿œ flag ï¿œ 0 ⊕ P ( k ï¿œ flag ï¿œ 0) P P P P 2 a -1 3 L ′ 2 2 d -2 L 2 2 L 2 4 L L = k ï¿œ flag ï¿œ N ⊕ P ( k ï¿œ flag ï¿œ N ) 2 2 d -1 3 L • ❇② ❙❛s❛❩✐ ❡t ❛❧✳ ❬❙❚❆✰✶✹❪ P • ❊①tr❛ ♥♊♥❝❡ N ❝♊♥❝❛t❡♥❛t❡❞ t♩ k 2 2 d -1 3 L T • ❇❛s❡❞ ♊♥ XPX ✇✐t❀ T = { (2 α 3 β , 2 α 3 β , 2 α 3 β , 2 α 3 β ) } ✷✌ ✮ ✺✷

  58. ✲r❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ ▌✐♥❛❧♣❀❡r A 1 A 2 A a − 1 A a M 1 M 2 M d − 1 M d 2 L ′ 2 2 L ′ 2 a -1 L ′ 2 L 2 3 L 2 2 d -3 L 2 2 d -1 L P P P P P P P 2 2 L ′ 2 a -1 L ′ 2 3 L 2 2 d -3 L 2 2 d -1 L 2 L ′ 2 L C 1 C 2 C d − 1 C d 2 a -1 3 L ′ 2 2 L 2 4 L 2 2 d -2 L L ′ = k ï¿œ flag ï¿œ 0 ⊕ P ( k ï¿œ flag ï¿œ 0) P P P P 2 a -1 3 L ′ 2 2 d -2 L 2 2 L 2 4 L L = k ï¿œ flag ï¿œ N ⊕ P ( k ï¿œ flag ï¿œ N ) 2 2 d -1 3 L • ❇② ❙❛s❛❩✐ ❡t ❛❧✳ ❬❙❚❆✰✶✹❪ P • ❊①tr❛ ♥♊♥❝❡ N ❝♊♥❝❛t❡♥❛t❡❞ t♩ k 2 2 d -1 3 L T • ❇❛s❡❞ ♊♥ XPX ✇✐t❀ T = { (2 α 3 β , 2 α 3 β , 2 α 3 β , 2 α 3 β ) } σ 2 ï¿œ ï¿œ O 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → ▌✐♥❛❧♣❀✳ XPX P Ί ✲r❊ ✷✌ ✮ ✺✷

  59. ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ❆❊✿ ▌✐♥❛❧♣❀❡r A 1 A 2 A a − 1 A a M 1 M 2 M d − 1 M d 2 L ′ 2 2 L ′ 2 a -1 L ′ 2 L 2 3 L 2 2 d -3 L 2 2 d -1 L P P P P P P P 2 2 L ′ 2 a -1 L ′ 2 3 L 2 2 d -3 L 2 2 d -1 L 2 L ′ 2 L C 1 C 2 C d − 1 C d 2 a -1 3 L ′ 2 2 L 2 4 L 2 2 d -2 L L ′ = k ï¿œ flag ï¿œ 0 ⊕ P ( k ï¿œ flag ï¿œ 0) P P P P 2 a -1 3 L ′ 2 2 d -2 L 2 2 L 2 4 L L = k ï¿œ flag ï¿œ N ⊕ P ( k ï¿œ flag ï¿œ N ) 2 2 d -1 3 L • ❇② ❙❛s❛❩✐ ❡t ❛❧✳ ❬❙❚❆✰✶✹❪ P • ❊①tr❛ ♥♊♥❝❡ N ❝♊♥❝❛t❡♥❛t❡❞ t♩ k 2 2 d -1 3 L T • ❇❛s❡❞ ♊♥ XPX ✇✐t❀ T = { (2 α 3 β , 2 α 3 β , 2 α 3 β , 2 α 3 β ) } σ 2 σ 2 ï¿œ ï¿œ ï¿œ ï¿œ O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ▌✐♥❛❧♣❀✳ XPX P Ί ✲r❊ Ί P ⊕ ✲r❊ ✷✌ ✮ ✺✷

  60. ❇❛s❡❞ ♊♥ ✇✐t❀ ✳ ✳ ✳ ✳ ✳ ✳ ❈❀❛s❊❡② s❊ s❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ▌❆❈✿ ❈❀❛s❊❡② k M 1 M 2 M d 2 k 2 k 0 P P P T k M 1 M 2 M d 10 ∗ 4 k 4 k T 0 P P P • ❇② ▌♊✉❀❛ ❡t ❛❧✳ ❬▌▌❱✰✶✹❪ ✷✜ ✮ ✺✷

  61. ✳ ✳ ✳ ✳ ✳ ✳ ❈❀❛s❊❡② s❊ s❊ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ▌❆❈✿ ❈❀❛s❊❡② k M 1 M 2 M d 2 k 2 k 0 P P P T k M 1 M 2 M d 10 ∗ 4 k 4 k T 0 P P P • ❇② ▌♊✉❀❛ ❡t ❛❧✳ ❬▌▌❱✰✶✹❪ • ❇❛s❡❞ ♊♥ XPX ✇✐t❀ T = { (1 , 0 , 1 , 0) , (3 , 0 , 2 , 0) , (5 , 0 , 4 , 0) } ✷✜ ✮ ✺✷

  62. ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ❳P❳ t♩ ▌❆❈✿ ❈❀❛s❊❡② k M 1 M 2 M d 2 k 2 k 0 P P P T k M 1 M 2 M d 10 ∗ 4 k 4 k T 0 P P P • ❇② ▌♊✉❀❛ ❡t ❛❧✳ ❬▌▌❱✰✶✹❪ • ❇❛s❡❞ ♊♥ XPX ✇✐t❀ T = { (1 , 0 , 1 , 0) , (3 , 0 , 2 , 0) , (5 , 0 , 4 , 0) } ï¿œ σ 2 ï¿œ ï¿œ σ 2 ï¿œ O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❀❛s❊❡② P XPX s❊ s❊ ✷✜ ✮ ✺✷

  63. ✳ ✳ ✳ ✳ ✳ ✳ ❈❀❛s❊❡② ✲r❊ ✲r❊ ❆♣♣r♊❛❝❀ ❛❧s♩ ❛♣♣❧✐❡s t♩ ❑❡②❡❞ ❙♣♊♥❣❡s ❆♣♣❧✐❝❛t✐♊♥ t♩ ▌❆❈✿ ❆❞❥✉st❡❞ ❈❀❛s❊❡② k M 1 M 2 M d 2 k 2 k 0 P P P P T k M 1 M 2 M d 10 ∗ 4 k 4 k T 0 P P P P • ❊①tr❛ P ✲❝❛❧❧ • ❇❛s❡❞ ♊♥ XPX ✇✐t❀ T ′ = { (0 , 1 , 0 , 1) , (2 , 1 , 2 , 0) , (4 , 1 , 4 , 0) } ✷✟ ✮ ✺✷

  64. ❆♣♣r♊❛❝❀ ❛❧s♩ ❛♣♣❧✐❡s t♩ ❑❡②❡❞ ❙♣♊♥❣❡s ❆♣♣❧✐❝❛t✐♊♥ t♩ ▌❆❈✿ ❆❞❥✉st❡❞ ❈❀❛s❊❡② k M 1 M 2 M d 2 k 2 k 0 P P P P T k M 1 M 2 M d 10 ∗ 4 k 4 k T 0 P P P P • ❊①tr❛ P ✲❝❛❧❧ • ❇❛s❡❞ ♊♥ XPX ✇✐t❀ T ′ = { (0 , 1 , 0 , 1) , (2 , 1 , 2 , 0) , (4 , 1 , 4 , 0) } ï¿œ σ 2 ï¿œ ï¿œ σ 2 ï¿œ O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❀❛s❊❡② P XPX Ί ✲r❊ Ί ⊕ ✲r❊ ✷✟ ✮ ✺✷

  65. ❆♣♣❧✐❝❛t✐♊♥ t♩ ▌❆❈✿ ❆❞❥✉st❡❞ ❈❀❛s❊❡② k M 1 M 2 M d 2 k 2 k 0 P P P P T k M 1 M 2 M d 10 ∗ 4 k 4 k T 0 P P P P • ❊①tr❛ P ✲❝❛❧❧ • ❇❛s❡❞ ♊♥ XPX ✇✐t❀ T ′ = { (0 , 1 , 0 , 1) , (2 , 1 , 2 , 0) , (4 , 1 , 4 , 0) } ï¿œ σ 2 ï¿œ ï¿œ σ 2 ï¿œ O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❀❛s❊❡② P XPX Ί ✲r❊ Ί ⊕ ✲r❊ • ❆♣♣r♊❛❝❀ ❛❧s♩ ❛♣♣❧✐❡s t♩ ❑❡②❡❞ ❙♣♊♥❣❡s ✷✟ ✮ ✺✷

  66. ❖✉t❧✐♥❡ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❢♊r ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❢♊r ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ❈♊♥❝❧✉s✐♊♥ ✞✵ ✮ ✺✷

  67. ▌❛s❊✐♥❣ ❝♊♠❜✐♥❡s ❛❞✈❛♥t❛❣❡s ♊❢✿ P♊✇❡r✐♥❣✲✉♣ ♠❛s❊✐♥❣ ▲❋❙❘ ♠❛s❊✐♥❣ ◆❡✇ ♠❛s❊✐♥❣ ✐s s✐♠♣❧❡r✱ ❝♊♥st❛♥t✲t✐♠❡ ✭❜② ❞❡❢❛✉❧t✮✱ ♠♩r❡ ❡✣❝✐❡♥t ▌❊▌ • MEM ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▌◆✶✺❪ ✿ ϕ γ 2 ◩ ϕ β 1 ◩ ϕ α 0 ◩ P ( N ï¿œ k ) P m c • ϕ i ❛r❡ ✜①❡❞ ▲❋❙❘s✱ ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ ✞✶ ✮ ✺✷

  68. ◆❡✇ ♠❛s❊✐♥❣ ✐s s✐♠♣❧❡r✱ ❝♊♥st❛♥t✲t✐♠❡ ✭❜② ❞❡❢❛✉❧t✮✱ ♠♩r❡ ❡✣❝✐❡♥t ▌❊▌ • MEM ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▌◆✶✺❪ ✿ ϕ γ 2 ◩ ϕ β 1 ◩ ϕ α 0 ◩ P ( N ï¿œ k ) P m c • ϕ i ❛r❡ ✜①❡❞ ▲❋❙❘s✱ ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ • ▌❛s❊✐♥❣ ❝♊♠❜✐♥❡s ❛❞✈❛♥t❛❣❡s ♊❢✿ • P♊✇❡r✐♥❣✲✉♣ ♠❛s❊✐♥❣ • ▲❋❙❘ ♠❛s❊✐♥❣ ✞✶ ✮ ✺✷

  69. ▌❊▌ • MEM ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▌◆✶✺❪ ✿ ϕ γ 2 ◩ ϕ β 1 ◩ ϕ α 0 ◩ P ( N ï¿œ k ) P m c • ϕ i ❛r❡ ✜①❡❞ ▲❋❙❘s✱ ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ • ▌❛s❊✐♥❣ ❝♊♠❜✐♥❡s ❛❞✈❛♥t❛❣❡s ♊❢✿ • P♊✇❡r✐♥❣✲✉♣ ♠❛s❊✐♥❣ • ▲❋❙❘ ♠❛s❊✐♥❣ ◆❡✇ ♠❛s❊✐♥❣ ✐s s✐♠♣❧❡r✱ ❝♊♥st❛♥t✲t✐♠❡ ✭❜② ❞❡❢❛✉❧t✮✱ ♠♩r❡ ❡✣❝✐❡♥t ✞✶ ✮ ✺✷

  70. ✵✳✺✺ ❝♣❜ ✇✐t❀ r❡❞✉❝❡❞✲r♊✉♥❞ ❇▲❆❑❊✷❜ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ▌❊▌ t♩ ❆❊✿ ❖PP A 0 A 1 A a –1 ⊕ M i M 0 M 1 M d –1 ϕ 0 ( L ) ϕ 1 ( L ) ϕ a –1 ( L ) ϕ 2 1 ◩ ϕ a –1 ( L ) ϕ 2 ◩ ϕ 0 ( L ) ϕ 2 ◩ ϕ 1 ( L ) ϕ 2 ◩ ϕ d –1 ( L ) P P P P P P P ϕ 0 ( L ) ϕ 1 ( L ) ϕ a –1 ( L ) ϕ 2 1 ◩ ϕ a –1 ( L ) ϕ 2 ◩ ϕ 0 ( L ) ϕ 2 ◩ ϕ 1 ( L ) ϕ 2 ◩ ϕ d –1 ( L ) C 1 C 2 C d T L = P ( N ï¿œ k ) ϕ 1 = ϕ ⊕ id , ϕ 2 = ϕ 2 ⊕ ϕ ⊕ id • ❖✛s❡t P✉❜❧✐❝ P❡r♠✉t❛t✐♊♥ ✭❖PP✮ ❬●❏▌◆✶✺❪ • ●❡♥❡r❛❧✐③❛t✐♊♥ ♊❢ ❖❈❇✞✿ • P❡r♠✉t❛t✐♊♥✲❜❛s❡❞ • ▌♊r❡ ❡✣❝✐❡♥t ▌❊▌✲♠❛s❊✐♥❣ • ❙❡❝✉r✐t② ❛❣❛✐♥st ♥♊♥❝❡✲r❡s♣❡❝t✐♥❣ ❛❞✈❡rs❛r✐❡s ✞✷ ✮ ✺✷

  71. ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ▌❊▌ t♩ ❆❊✿ ❖PP A 0 A 1 A a –1 ⊕ M i M 0 M 1 M d –1 ϕ 0 ( L ) ϕ 1 ( L ) ϕ a –1 ( L ) ϕ 2 1 ◩ ϕ a –1 ( L ) ϕ 2 ◩ ϕ 0 ( L ) ϕ 2 ◩ ϕ 1 ( L ) ϕ 2 ◩ ϕ d –1 ( L ) P P P P P P P ϕ 0 ( L ) ϕ 1 ( L ) ϕ a –1 ( L ) ϕ 2 1 ◩ ϕ a –1 ( L ) ϕ 2 ◩ ϕ 0 ( L ) ϕ 2 ◩ ϕ 1 ( L ) ϕ 2 ◩ ϕ d –1 ( L ) C 1 C 2 C d T L = P ( N ï¿œ k ) ϕ 1 = ϕ ⊕ id , ϕ 2 = ϕ 2 ⊕ ϕ ⊕ id • ❖✛s❡t P✉❜❧✐❝ P❡r♠✉t❛t✐♊♥ ✭❖PP✮ ❬●❏▌◆✶✺❪ • ●❡♥❡r❛❧✐③❛t✐♊♥ ♊❢ ❖❈❇✞✿ • P❡r♠✉t❛t✐♊♥✲❜❛s❡❞ • ▌♊r❡ ❡✣❝✐❡♥t ▌❊▌✲♠❛s❊✐♥❣ • ❙❡❝✉r✐t② ❛❣❛✐♥st ♥♊♥❝❡✲r❡s♣❡❝t✐♥❣ ❛❞✈❡rs❛r✐❡s ✵✳✺✺ ❝♣❜ ✇✐t❀ r❡❞✉❝❡❞✲r♊✉♥❞ ❇▲❆❑❊✷❜ ✞✷ ✮ ✺✷

  72. ✶✳✵✻ ❝♣❜ ✇✐t❀ r❡❞✉❝❡❞✲r♊✉♥❞ ❇▲❆❑❊✷❜ ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ▌❊▌ t♩ ❆❊✿ ▌❘❖ T ï¿œ 0 T ï¿œ d –1 A 0 A a –1 M 0 M d –1 | A |ï¿œ| M | ϕ 0 ( L ) ϕ a –1 ( L ) ϕ 1 ◩ ϕ 0 ( L ) ϕ 1 ◩ ϕ d –1 ( L ) ϕ 2 ( L ) ϕ 2 ( L ) P P P P P P ϕ a –1 ( L ) ϕ 1 ◩ ϕ d –1 ( L ) ϕ 0 ( L ) ϕ 1 ◩ ϕ 0 ( L ) ϕ 2 ( L ) ⊕ M 0 ϕ 2 ( L ) ⊕ M d –1 ϕ 2 1 ( L ) C 1 C d P L = P ( N ï¿œ k ) ϕ 1 = ϕ ⊕ id , ϕ 2 = ϕ 2 ⊕ ϕ ⊕ id ϕ 2 1 ( L ) T • ▌✐s✉s❡✲❘❡s✐st❛♥t ❖PP ✭▌❘❖✮ ❬●❏▌◆✶✺❪ • ❋✉❧❧② ♥♊♥❝❡✲♠✐s✉s❡ r❡s✐st❛♥t ✈❡rs✐♊♥ ♊❢ ❖PP ✾✾ ✮ ✺✷

  73. ❆♣♣❧✐❝❛t✐♊♥ ♊❢ ▌❊▌ t♩ ❆❊✿ ▌❘❖ T ï¿œ 0 T ï¿œ d –1 A 0 A a –1 M 0 M d –1 | A |ï¿œ| M | ϕ 0 ( L ) ϕ a –1 ( L ) ϕ 1 ◩ ϕ 0 ( L ) ϕ 1 ◩ ϕ d –1 ( L ) ϕ 2 ( L ) ϕ 2 ( L ) P P P P P P ϕ a –1 ( L ) ϕ 1 ◩ ϕ d –1 ( L ) ϕ 0 ( L ) ϕ 1 ◩ ϕ 0 ( L ) ϕ 2 ( L ) ⊕ M 0 ϕ 2 ( L ) ⊕ M d –1 ϕ 2 1 ( L ) C 1 C d P L = P ( N ï¿œ k ) ϕ 1 = ϕ ⊕ id , ϕ 2 = ϕ 2 ⊕ ϕ ⊕ id ϕ 2 1 ( L ) T • ▌✐s✉s❡✲❘❡s✐st❛♥t ❖PP ✭▌❘❖✮ ❬●❏▌◆✶✺❪ • ❋✉❧❧② ♥♊♥❝❡✲♠✐s✉s❡ r❡s✐st❛♥t ✈❡rs✐♊♥ ♊❢ ❖PP ✶✳✵✻ ❝♣❜ ✇✐t❀ r❡❞✉❝❡❞✲r♊✉♥❞ ❇▲❆❑❊✷❜ ✾✾ ✮ ✺✷

  74. ❖✉t❧✐♥❡ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❢♊r ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❢♊r ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚❇❈s ❈♊♥❝❧✉s✐♊♥ ✞✹ ✮ ✺✷

  75. ❙❡❝✉r✐t② ♊❢ ❆❊✬s ✐s ♠♩st❧② ❞♊♠✐♥❛t❡❞ ❜② s❡❝✉r✐t② ♊❢ ❋♩r s♊♠❡ ❆❊✬s ✭❡✳❣✳✱ ❖❈❇✱ ♣❖▌❉✱ ❖PP✱ ✳ ✳ ✳ ✮✿ ✳ ✳ ✳ ✳ ✳ ✳ ❆❊ ♩r ❈❛♥ ✇❡ ✐♠♣r♊✈❡ t❀✐s❄ ❙❡❝✉r✐t② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞❄ • ❆❧❧ r❡s✉❧ts s♩ ❢❛r✿ ✉♣ t♩ ❜✐rt❀❞❛② ❜♊✉♥❞ ✞✺ ✮ ✺✷

  76. ❋♩r s♊♠❡ ❆❊✬s ✭❡✳❣✳✱ ❖❈❇✱ ♣❖▌❉✱ ❖PP✱ ✳ ✳ ✳ ✮✿ ✳ ✳ ✳ ✳ ✳ ✳ ❆❊ ♩r ❈❛♥ ✇❡ ✐♠♣r♊✈❡ t❀✐s❄ ❙❡❝✉r✐t② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞❄ • ❆❧❧ r❡s✉❧ts s♩ ❢❛r✿ ✉♣ t♩ ❜✐rt❀❞❛② ❜♊✉♥❞ • ❙❡❝✉r✐t② ♊❢ ❆❊✬s ✐s ♠♩st❧② ❞♊♠✐♥❛t❡❞ ❜② s❡❝✉r✐t② ♊❢ ï¿œ E ✞✺ ✮ ✺✷

  77. ❈❛♥ ✇❡ ✐♠♣r♊✈❡ t❀✐s❄ ❙❡❝✉r✐t② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞❄ • ❆❧❧ r❡s✉❧ts s♩ ❢❛r✿ ✉♣ t♩ ❜✐rt❀❞❛② ❜♊✉♥❞ • ❙❡❝✉r✐t② ♊❢ ❆❊✬s ✐s ♠♩st❧② ❞♊♠✐♥❛t❡❞ ❜② s❡❝✉r✐t② ♊❢ ï¿œ E • ❋♩r s♊♠❡ ❆❊✬s ✭❡✳❣✳✱ ❖❈❇✱ ♣❖▌❉✱ ❖PP✱ ✳ ✳ ✳ ✮✿ ï¿œ ï¿œ ï¿œ σ 2 ï¿œ σ O O 2 n 2 n → ✳ ✳ ✳ ✳ → ✳ ✳ ï¿œ ❆❊ − − − − − − − − E ♩r P E ✞✺ ✮ ✺✷

  78. ❙❡❝✉r✐t② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞❄ • ❆❧❧ r❡s✉❧ts s♩ ❢❛r✿ ✉♣ t♩ ❜✐rt❀❞❛② ❜♊✉♥❞ • ❙❡❝✉r✐t② ♊❢ ❆❊✬s ✐s ♠♩st❧② ❞♊♠✐♥❛t❡❞ ❜② s❡❝✉r✐t② ♊❢ ï¿œ E • ❋♩r s♊♠❡ ❆❊✬s ✭❡✳❣✳✱ ❖❈❇✱ ♣❖▌❉✱ ❖PP✱ ✳ ✳ ✳ ✮✿ ï¿œ ï¿œ ï¿œ σ 2 ï¿œ σ O O 2 n 2 n → ✳ ✳ ✳ ✳ → ✳ ✳ ï¿œ ❆❊ − − − − − − − − E ♩r P E → − − ❈❛♥ ✇❡ ✐♠♣r♊✈❡ t❀✐s❄ ✞✺ ✮ ✺✷

  79. ✿ s❡❝✉r❡ ✉♣ t♩ q✉❡r✐❡s ❬▲❙❚✶✷✱Pr♊✶✹❪ ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♩ q✉❡r✐❡s ❬▲❙✶✞❪ ❈♊♥❥❡❝t✉r❡✿ ♊♣t✐♠❛❧ s❡❝✉r✐t② ❇❇❇ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ ❇❧♊❝❊❝✐♣❀❡rs h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h ρ − 1 ( t ) ⊕ h ρ ( t ) h ρ ( t ) k 1 k 2 k ρ m · · · · · · c E E E • LRW 2 [ ρ ] ✿ ❝♊♥❝❛t❡♥❛t✐♊♥ ♊❢ ρ LRW 2 ✬s • k 1 , . . . , k ρ ❛♥❞ h 1 , . . . , h ρ ✐♥❞❡♣❡♥❞❡♥t ✞✻ ✮ ✺✷

  80. ❇❇❇ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ ❇❧♊❝❊❝✐♣❀❡rs h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h ρ − 1 ( t ) ⊕ h ρ ( t ) h ρ ( t ) k 1 k 2 k ρ m · · · · · · c E E E • LRW 2 [ ρ ] ✿ ❝♊♥❝❛t❡♥❛t✐♊♥ ♊❢ ρ LRW 2 ✬s • k 1 , . . . , k ρ ❛♥❞ h 1 , . . . , h ρ ✐♥❞❡♣❡♥❞❡♥t • ρ = 2 ✿ s❡❝✉r❡ ✉♣ t♩ 2 2 n/ 3 q✉❡r✐❡s ❬▲❙❚✶✷✱Pr♊✶✹❪ • ρ ≥ 2 ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♩ 2 ρn/ ( ρ +2) q✉❡r✐❡s ❬▲❙✶✞❪ • ❈♊♥❥❡❝t✉r❡✿ ♊♣t✐♠❛❧ 2 ρn/ ( ρ +1) s❡❝✉r✐t② ✞✻ ✮ ✺✷

  81. ✿ s❡❝✉r❡ ✉♣ t♩ q✉❡r✐❡s ❬❈▲❙✶✺❪ ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♩ q✉❡r✐❡s ❬❈▲❙✶✺❪ ❈♊♥❥❡❝t✉r❡✿ ♊♣t✐♠❛❧ s❡❝✉r✐t② ❇❇❇ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ P❡r♠✉t❛t✐♊♥s h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h ρ − 1 ( t ) ⊕ h ρ ( t ) h ρ ( t ) P 1 P 2 P ρ m · · · · · · c • TEM [ ρ ] ✿ ❝♊♥❝❛t❡♥❛t✐♊♥ ♊❢ ρ TEM ✲❧✐❊❡✬s • P 1 , . . . , P ρ ❛♥❞ h 1 , . . . , h ρ ✐♥❞❡♣❡♥❞❡♥t ✞✌ ✮ ✺✷

  82. ❇❇❇ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❢r♩♠ P❡r♠✉t❛t✐♊♥s h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h ρ − 1 ( t ) ⊕ h ρ ( t ) h ρ ( t ) P 1 P 2 P ρ m · · · · · · c • TEM [ ρ ] ✿ ❝♊♥❝❛t❡♥❛t✐♊♥ ♊❢ ρ TEM ✲❧✐❊❡✬s • P 1 , . . . , P ρ ❛♥❞ h 1 , . . . , h ρ ✐♥❞❡♣❡♥❞❡♥t • ρ = 2 ✿ s❡❝✉r❡ ✉♣ t♩ 2 2 n/ 3 q✉❡r✐❡s ❬❈▲❙✶✺❪ • ρ ≥ 2 ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♩ 2 ρn/ ( ρ +2) q✉❡r✐❡s ❬❈▲❙✶✺❪ • ❈♊♥❥❡❝t✉r❡✿ ♊♣t✐♠❛❧ 2 ρn/ ( ρ +1) s❡❝✉r✐t② ✞✌ ✮ ✺✷

  83. ❙t❛t❡ ♊❢ t❀❡ ❆rt ✭❇❧♊❝❊❝✐♣❀❡r ❇❛s❡❞✮ ❝♩st s❡❝✉r✐t② ❊❡② s❝❀❡♠❡ ✭ log 2 ✮ ❧❡♥❣t❀ E ⊗ /h n/ 2 n ✷ ✵ LRW 1 n/ 2 2 n ✶ ✶ LRW 2 n/ 2 n ✷ ✵ XEX LRW 2 [2] 2 n/ 3 4 n ✷ ✷ LRW 2 [ ρ ] ρn/ ( ρ +2) 2 ρn ρ ρ max { n/ 2 , n −| t |} ❖♣t✐♠❛❧ 2 n s❡❝✉r✐t② ♊♥❧② ✐❢ ❊❡② ❧❡♥❣t❀ ❛♥❞ ❝♩st → ∞ ❄ ✞✜ ✮ ✺✷

  84. ✳ ✳ ❙❡❝✉r✐t② t✇❡❛❊ s❝❀❡❞✉❧❡ str♊♥❣❡r t❀❛♥ ❊❡② s❝❀❡❞✉❧❡ ❚✇❡❛❊ ❛♥❞ ❊❡② ❝❀❛♥❣❡ ❛♣♣r♩①✐♠❛t❡❧② ❡q✉❛❧❧② ❡①♣❡♥s✐✈❡ ❚❲❊❆❑❊❚ ❬❏◆P✶✹❪ ❊❡② s❝❀❡❞✉❧✐♥❣ ❜❧❡♥❞s ❊❡② ❛♥❞ t✇❡❛❊ ❚✇❡❛❊✲❉❡♣❡♥❞❡♥t ❑❡②s ✳ ✳ ❊✣❝✐❡♥❝② t✇❡❛❊ s❝❀❡❞✉❧❡ ❧✐❣❀t❡r t❀❛♥ ❊❡② s❝❀❡❞✉❧❡ ✞✟ ✮ ✺✷

  85. ❚✇❡❛❊ ❛♥❞ ❊❡② ❝❀❛♥❣❡ ❛♣♣r♩①✐♠❛t❡❧② ❡q✉❛❧❧② ❡①♣❡♥s✐✈❡ ❚❲❊❆❑❊❚ ❬❏◆P✶✹❪ ❊❡② s❝❀❡❞✉❧✐♥❣ ❜❧❡♥❞s ❊❡② ❛♥❞ t✇❡❛❊ ❚✇❡❛❊✲❉❡♣❡♥❞❡♥t ❑❡②s ✳ ✳ ✳ ✳ ❊✣❝✐❡♥❝② ❙❡❝✉r✐t② t✇❡❛❊ s❝❀❡❞✉❧❡ ❧✐❣❀t❡r t✇❡❛❊ s❝❀❡❞✉❧❡ str♊♥❣❡r t❀❛♥ ❊❡② s❝❀❡❞✉❧❡ t❀❛♥ ❊❡② s❝❀❡❞✉❧❡ ✞✟ ✮ ✺✷

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