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  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❡❞✳♣ � � 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 � N ⊕ P ( k � N )) P m c • ( α, β, γ, N ) ✐s t✇❡❛❦ ✭s✐♠♣❧✐✜❡❞✮ ✶✹ ✴ ✺✷

  18. ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❢r♦♠ P❡r♠✉t❛t✐♦♥s • Prøst ❬❑▲▲✰✶✹❪ ✉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 • Prøst ❬❑▲▲✰✶✹❪ ✉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❆ ✱ ❊▲♠❉ ✱ ✐❋❡❡❞✱ Prøst ❑■❆❙❯✱ ▼❛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 � N ⊕ P ( k � N )) k k � 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 � N ⊕ P ( k � N )) k k � m c E m c m P c E t ❉❡❞✐❝❛t❡❞ ❳❊✴❳❊❳✲✐♥s♣✐r❡❞ ❚❊▼✲✐♥s♣✐r❡❞ ❉❡♦①②s ✱ ❆❊❩ ✱ ❈❇❆✱ ❈❖❇❘❆✱ ▼✐♥❛❧♣❤❡r ✱ ❏♦❧t✐❦ ✱ ❈❖P❆ ✱ ❊▲♠❉ ✱ ✐❋❡❡❞✱ Prøst ❑■❆❙❯✱ ▼❛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 � N ⊕ P ( k � N )) k � m c E P m c m c E t ❉❡❞✐❝❛t❡❞ ❳❊✴❳❊❳✲✐♥s♣✐r❡❞ ❚❊▼✲✐♥s♣✐r❡❞ ❉❡♦①②s ✱ ❆❊❩ ✱ ❈❇❆✱ ❈❖❇❘❆✱ ▼✐♥❛❧♣❤❡r ✱ ❏♦❧t✐❦ ✱ ❈❖P❆ ✱ ❊▲♠❉ ✱ ✐❋❡❡❞✱ Prøst ❑■❆❙❯✱ ▼❛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 � N ⊕ P ( k � N )) k � m c E P m c m c E t ❉❡❞✐❝❛t❡❞ ❳❊✴❳❊❳✲✐♥s♣✐r❡❞ ❚❊▼✲✐♥s♣✐r❡❞ ❉❡♦①②s ✱ ❆❊❩ ✱ ❈❇❆✱ ❈❖❇❘❆✱ ▼✐♥❛❧♣❤❡r ✱ ❏♦❧t✐❦ ✱ ❈❖P❆ ✱ ❊▲♠❉ ✱ ✐❋❡❡❞✱ Prøst ❑■❆❙❯✱ ▼❛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 Φ ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❦❡②✿ k �→ k ⊕ δ ✷✶ ✴ ✺✷

  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 Φ ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❦❡②✿ k �→ k ⊕ δ Φ P ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❦❡②✿ k �→ k ⊕ δ ♦r P ( k ) �→ P ( k ) ⊕ ǫ • ◆♦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 Φ ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❦❡②✿ k �→ k ⊕ δ Φ P ⊕ ✲❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ✭❙✐♠♣❧✐✜❡❞✮ • D ❝❛♥ ✐♥✢✉❡♥❝❡ ❦❡②✿ k �→ k ⊕ δ ♦r P ( k ) �→ P ( k ) ⊕ ǫ • ◆♦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 � = 0 ❛♥❞ ( t 21 , t 22 ) � = (0 , 1) Φ ⊕ ✲r❦✲❙❚P❘P t 11 , t 12 , t 21 , t 22 � = 0 Φ 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 � ( 2 α 3 β 7 γ ⊕ 1 , 2 α 3 β 7 γ , � � � � ❢♦r T = � ( 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 � ( 2 α 3 β 7 γ ⊕ 1 , 2 α 3 β 7 γ , � � � � ❢♦r T = � ( 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✇❡❛❦ • Φ P ⊕ ✲r❦ ❙❚P❘P s❡❝✉r❡ ✭✐❢ 2 α 3 β 7 γ � = 1 ✮ ✷✸ ✴ ✺✷

  47. Prøst✲❈❖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❡❞ ♦♥ ❆❊❙ • Prøst✲❈❖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 � � � � 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 � � � � O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❖P❆ XEX E s❦ s❦ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ ❈❖P❆ • ❆♣♣r♦❛❝❤ ❣❡♥❡r❛❧✐③❡s ❢♦r ❛♥② Φ ✭♣r♦♦❢ ✐♥ ❬▼❡♥✶✺❜❪ ✮ � σ 2 � � σ 2 � O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❖P❆ XEX E Φ ✲r❦ Φ ✲r❦ ✷✺ ✴ ✺✷

  51. ❆♣♣r♦❛❝❤ ❣❡♥❡r❛❧✐③❡s ✭♣r♦♦❢ ✐♥ ❬▼❡♥✶✺❜❪ ✮ ✲r❦ s❦ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❖P❆ ✲r❦ ✲r❦ ✲r❦ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❳P❳ t♦ ❆❊✿ Prøst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ σ 2 σ 2 � � � � O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❖P❆ XEX E P s❦ s❦ ✷✻ ✴ ✺✷

  52. ❆♣♣r♦❛❝❤ ❣❡♥❡r❛❧✐③❡s ✭♣r♦♦❢ ✐♥ ❬▼❡♥✶✺❜❪ ✮ ✲r❦ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❖P❆ ✲r❦ ✲r❦ ✲r❦ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❳P❳ t♦ ❆❊✿ Prøst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ σ 2 σ 2 σ 2 � � � � � � 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♦ ❆❊✿ Prøst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ σ 2 σ 2 σ 2 � � � � � � O O O 2 n 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P s❦ s❦ s❦ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ � σ 2 � � σ 2 � O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❖P❆ XEX E P Φ ✲r❦ Φ ✲r❦ ✷✻ ✴ ✺✷

  54. ❆♣♣r♦❛❝❤ ❣❡♥❡r❛❧✐③❡s ✭♣r♦♦❢ ✐♥ ❬▼❡♥✶✺❜❪ ✮ ✲r❦ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❳P❳ t♦ ❆❊✿ Prøst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ σ 2 σ 2 σ 2 � � � � � � O O O 2 n 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P s❦ s❦ s❦ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ � σ 2 � � σ 2 � � � O O Ω 1 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P Φ ✲r❦ Φ ✲r❦ Φ ✲r❦ ✷✻ ✴ ✺✷

  55. ❆♣♣r♦❛❝❤ ❣❡♥❡r❛❧✐③❡s ✭♣r♦♦❢ ✐♥ ❬▼❡♥✶✺❜❪ ✮ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❳P❳ t♦ ❆❊✿ Prøst✲❈❖P❆ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ σ 2 σ 2 σ 2 � � � � � � O O O 2 n 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P s❦ s❦ s❦ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ � σ 2 � � σ 2 � � � O O Ω 1 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → − − − − → ❈❖P❆ XEX E P Φ ✲r❦ Φ ✲r❦ Φ ✲r❦ � � σ 2 O 2 n Φ 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 � flag � 0 ⊕ P ( k � flag � 0) P P P P 2 a -1 3 L ′ 2 2 d -2 L 2 2 L 2 4 L L = k � flag � N ⊕ P ( k � flag � 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 � flag � 0 ⊕ P ( k � flag � 0) P P P P 2 a -1 3 L ′ 2 2 d -2 L 2 2 L 2 4 L L = k � flag � N ⊕ P ( k � flag � 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 � flag � 0 ⊕ P ( k � flag � 0) P P P P 2 a -1 3 L ′ 2 2 d -2 L 2 2 L 2 4 L L = k � flag � N ⊕ P ( k � flag � 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 � � O 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → ▼✐♥❛❧♣❤✳ XPX P Φ ✲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 � flag � 0 ⊕ P ( k � flag � 0) P P P P 2 a -1 3 L ′ 2 2 d -2 L 2 2 L 2 4 L L = k � flag � N ⊕ P ( k � flag � 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 � � � � O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ▼✐♥❛❧♣❤✳ XPX P Φ ✲r❦ Φ 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) } � σ 2 � � σ 2 � 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) } � σ 2 � � σ 2 � O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❤❛s❦❡② P XPX Φ ✲r❦ Φ ⊕ ✲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) } � σ 2 � � σ 2 � O O 2 n 2 n ✳ ✳ ✳ ✳ ✳ ✳ − − − − → − − − − → ❈❤❛s❦❡② P XPX Φ ✲r❦ Φ ⊕ ✲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 � 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 � 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 � 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 � 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 � 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 � 0 T � d –1 A 0 A a –1 M 0 M d –1 | A |�| 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 � 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 � 0 T � d –1 A 0 A a –1 M 0 M d –1 | A |�| 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 � 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 ♦❢ � 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 ♦❢ � E • ❋♦r s♦♠❡ ❆❊✬s ✭❡✳❣✳✱ ❖❈❇✱ ♣❖▼❉✱ ❖PP✱ ✳ ✳ ✳ ✮✿ � � � σ 2 � σ O O 2 n 2 n → ✳ ✳ ✳ ✳ → ✳ ✳ � ❆❊ − − − − − − − − 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 ♦❢ � E • ❋♦r s♦♠❡ ❆❊✬s ✭❡✳❣✳✱ ❖❈❇✱ ♣❖▼❉✱ ❖PP✱ ✳ ✳ ✳ ✮✿ � � � σ 2 � σ O O 2 n 2 n → ✳ ✳ ✳ ✳ → ✳ ✳ � ❆❊ − − − − − − − − 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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