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rs rt r t

rs - PowerPoint PPT Presentation

Dec 22, 2022 •123 likes •1.89k views

rs rt rt rt rst

  1. ❚❲❊❆❑❊❚ ❜❧❡♥❞✐♥❣ ❬❏◆P✶✹❪ ✐s ♠♩r❡ ❛❞✈❛♥❝❡❞ ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k t m c E • ❇❧❡♥❞✐♥❣ t✇❡❛❊ ❛♥❞ ❊❡② ✇♩r❊s✳ ✳ ✳ • ✳ ✳ ✳ ❜✉t✿ ❝❛r❡❢✉❧ ✇✐t❀ r❡❧❛t❡❞✲❊❡② ❛tt❛❝❩s✩ • ❋♩r ⊕ ✲♠✐①✐♥❣✱ ❊❡② ❝❛♥ ❜❡ r❡❝♊✈❡r❡❞ ✐♥ 2 κ/ 2 ❡✈❛❧✉❛t✐♊♥s • ❙❝❀❡♠❡ ✐s ✐♥s❡❝✉r❡ ✐❢ E ✐s ❊✈❡♥✲▌❛♥s♩✉r ✶✵ ✮ ✹✜

  2. ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k t m c E • ❇❧❡♥❞✐♥❣ t✇❡❛❊ ❛♥❞ ❊❡② ✇♩r❊s✳ ✳ ✳ • ✳ ✳ ✳ ❜✉t✿ ❝❛r❡❢✉❧ ✇✐t❀ r❡❧❛t❡❞✲❊❡② ❛tt❛❝❩s✩ • ❋♩r ⊕ ✲♠✐①✐♥❣✱ ❊❡② ❝❛♥ ❜❡ r❡❝♊✈❡r❡❞ ✐♥ 2 κ/ 2 ❡✈❛❧✉❛t✐♊♥s • ❙❝❀❡♠❡ ✐s ✐♥s❡❝✉r❡ ✐❢ E ✐s ❊✈❡♥✲▌❛♥s♩✉r • ❚❲❊❆❑❊❚ ❜❧❡♥❞✐♥❣ ❬❏◆P✶✹❪ ✐s ♠♩r❡ ❛❞✈❛♥❝❡❞ ✶✵ ✮ ✹✜

  3. ❚✇♊✲s✐❞❡❞ ♠❛s❊✐♥❣ ♥❡❝❡ss❛r② ❙♊♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ ❙t✐❧❧ ❞♊❡s ♥♊t ✇♩r❊ ✐❢ ❛❞✈❡rs❛r② ❀❛s ❛❝❝❡ss t♩ ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♊❢ t✇❡❛❊ ❛♥❞ st❛t❡ ❞♊❡s ♥♊t ✇♩r❊ ✶✶ ✮ ✹✜

  4. ❚✇♊✲s✐❞❡❞ ♠❛s❊✐♥❣ ♥❡❝❡ss❛r② ❙♊♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ ❙t✐❧❧ ❞♊❡s ♥♊t ✇♩r❊ ✐❢ ❛❞✈❡rs❛r② ❀❛s ❛❝❝❡ss t♩ ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♊❢ t✇❡❛❊ ❛♥❞ st❛t❡ ❞♊❡s ♥♊t ✇♩r❊ • ï¿œ E k ( t, m ) = ï¿œ E k ( t ⊕ C, m ⊕ C ) ✶✶ ✮ ✹✜

  5. ❚✇♊✲s✐❞❡❞ ♠❛s❊✐♥❣ ♥❡❝❡ss❛r② ❙t✐❧❧ ❞♊❡s ♥♊t ✇♩r❊ ✐❢ ❛❞✈❡rs❛r② ❀❛s ❛❝❝❡ss t♩ ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k h ⊗ t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♊❢ t✇❡❛❊ ❛♥❞ st❛t❡ ❞♊❡s ♥♊t ✇♩r❊ • ï¿œ E k ( t, m ) = ï¿œ E k ( t ⊕ C, m ⊕ C ) • ❙♊♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ h ✶✶ ✮ ✹✜

  6. ❚✇♊✲s✐❞❡❞ ♠❛s❊✐♥❣ ♥❡❝❡ss❛r② ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k h ⊗ t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♊❢ t✇❡❛❊ ❛♥❞ st❛t❡ ❞♊❡s ♥♊t ✇♩r❊ • ï¿œ E k ( t, m ) = ï¿œ E k ( t ⊕ C, m ⊕ C ) • ❙♊♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ h • ❙t✐❧❧ ❞♊❡s ♥♊t ✇♩r❊ ✐❢ ❛❞✈❡rs❛r② ❀❛s ❛❝❝❡ss t♩ ï¿œ E − 1 k ✶✶ ✮ ✹✜

  7. ❚✇♊✲s✐❞❡❞ ♠❛s❊✐♥❣ ♥❡❝❡ss❛r② ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k h ⊗ t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♊❢ t✇❡❛❊ ❛♥❞ st❛t❡ ❞♊❡s ♥♊t ✇♩r❊ • ï¿œ E k ( t, m ) = ï¿œ E k ( t ⊕ C, m ⊕ C ) • ❙♊♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ h • ❙t✐❧❧ ❞♊❡s ♥♊t ✇♩r❊ ✐❢ ❛❞✈❡rs❛r② ❀❛s ❛❝❝❡ss t♩ ï¿œ E − 1 k • ï¿œ E − 1 k ( t, c ) ⊕ ï¿œ E − 1 k ( t ⊕ C, c ) = h ⊗ C ✶✶ ✮ ✹✜

  8. ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k h ⊗ t h ⊗ t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♊❢ t✇❡❛❊ ❛♥❞ st❛t❡ ❞♊❡s ♥♊t ✇♩r❊ • ï¿œ E k ( t, m ) = ï¿œ E k ( t ⊕ C, m ⊕ C ) • ❙♊♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ h • ❙t✐❧❧ ❞♊❡s ♥♊t ✇♩r❊ ✐❢ ❛❞✈❡rs❛r② ❀❛s ❛❝❝❡ss t♩ ï¿œ E − 1 k • ï¿œ E − 1 k ( t, c ) ⊕ ï¿œ E − 1 k ( t ⊕ C, c ) = h ⊗ C • ❚✇♊✲s✐❞❡❞ ♠❛s❊✐♥❣ ♥❡❝❡ss❛r② ✶✶ ✮ ✹✜

  9. ▌❛❥♊r✐t② ♊❢ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡rs ❢♊❧❧♊✇ ♠❛s❊✲ ✲♠❛s❊ ♣r✐♥❝✐♣❧❡ ●❡♥❡r❛❧✐③✐♥❣ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥ ❱❛r✐❛t✐♊♥ ✐♥ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥s ❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ ❄ ❯s✉❛❧❧② ♥♊ ♣r♊❜❧❡♠ ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k h ⊗ t h ⊗ t m c E • ❚✇♊✲s✐❞❡❞ s❡❝r❡t ♠❛s❊✐♥❣ s❡❡♠s t♩ ✇♩r❊ • ❈❛♥ ✇❡ ❣❡♥❡r❛❧✐③❡❄ ✶✷ ✮ ✹✜

  10. ▌❛❥♊r✐t② ♊❢ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡rs ❢♊❧❧♊✇ ♠❛s❊✲ ✲♠❛s❊ ♣r✐♥❝✐♣❧❡ ❱❛r✐❛t✐♊♥ ✐♥ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥s ❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ ❄ ❯s✉❛❧❧② ♥♊ ♣r♊❜❧❡♠ ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k f ( t ) f ( t ) m c E • ❚✇♊✲s✐❞❡❞ s❡❝r❡t ♠❛s❊✐♥❣ s❡❡♠s t♩ ✇♩r❊ • ❈❛♥ ✇❡ ❣❡♥❡r❛❧✐③❡❄ • ●❡♥❡r❛❧✐③✐♥❣ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥ f ✶✷ ✮ ✹✜

  11. ▌❛❥♊r✐t② ♊❢ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡rs ❢♊❧❧♊✇ ♠❛s❊✲ ✲♠❛s❊ ♣r✐♥❝✐♣❧❡ ❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ ❄ ❯s✉❛❧❧② ♥♊ ♣r♊❜❧❡♠ ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k f 1 ( t ) f 2 ( t ) m c E • ❚✇♊✲s✐❞❡❞ s❡❝r❡t ♠❛s❊✐♥❣ s❡❡♠s t♩ ✇♩r❊ • ❈❛♥ ✇❡ ❣❡♥❡r❛❧✐③❡❄ • ●❡♥❡r❛❧✐③✐♥❣ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥ f • ❱❛r✐❛t✐♊♥ ✐♥ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥s f 1 , f 2 ✶✷ ✮ ✹✜

  12. ▌❛❥♊r✐t② ♊❢ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡rs ❢♊❧❧♊✇ ♠❛s❊✲ ✲♠❛s❊ ♣r✐♥❝✐♣❧❡ ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ f 1 ( t ) f 2 ( t ) P m c • ❚✇♊✲s✐❞❡❞ s❡❝r❡t ♠❛s❊✐♥❣ s❡❡♠s t♩ ✇♩r❊ • ❈❛♥ ✇❡ ❣❡♥❡r❛❧✐③❡❄ • ●❡♥❡r❛❧✐③✐♥❣ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥ f • ❱❛r✐❛t✐♊♥ ✐♥ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥s f 1 , f 2 • ❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ E ❄ ❯s✉❛❧❧② ♥♊ ♣r♊❜❧❡♠ ✶✷ ✮ ✹✜

  13. ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ f 1 ( t ) f 2 ( t ) ▌❛❥♊r✐t② ♊❢ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡rs P m c ❢♊❧❧♊✇ ♠❛s❊✲ E k /P ✲♠❛s❊ ♣r✐♥❝✐♣❧❡ • ❚✇♊✲s✐❞❡❞ s❡❝r❡t ♠❛s❊✐♥❣ s❡❡♠s t♩ ✇♩r❊ • ❈❛♥ ✇❡ ❣❡♥❡r❛❧✐③❡❄ • ●❡♥❡r❛❧✐③✐♥❣ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥ f • ❱❛r✐❛t✐♊♥ ✐♥ ♠❛s❊✐♥❣❄ ❉❡♣❡♥❞s ♊♥ ❢✉♥❝t✐♊♥s f 1 , f 2 • ❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ E ❄ ❯s✉❛❧❧② ♥♊ ♣r♊❜❧❡♠ ✶✷ ✮ ✹✜

  14. ❙t❡♣ ✶✿ ❍♩✇ ♠❛♥② ❡✈❛❧✉❛t✐♊♥s ❞♊❡s ♥❡❡❞ ❛t ♠♩st❄ ❙t❡♣ ✶✿ ❇♩✐❧s ❞♊✇♥ t♩ ✜♥❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❩s ❙t❡♣ ✷✿ ❍♩✇ ♠❛♥② ❡✈❛❧✉❛t✐♊♥s ❞♊❡s ♥❡❡❞ ❛t ❧❡❛st❄ ❙t❡♣ ✷✿ ❇♩✐❧s ❞♊✇♥ t♩ ♣r♊✈❛❜❧❡ s❡❝✉r✐t② ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ï¿œ E k s❀♊✉❧❞ ✏❧♩♩❩ ❧✐❊❡✑ r❛♥❞♊♠ ♣❡r♠✉t❛t✐♊♥ ❢♊r ❡✈❡r② t • ❈♊♥s✐❞❡r ❛❞✈❡rs❛r② D t❀❛t ♠❛❊❡s q ❡✈❛❧✉❛t✐♊♥s ♊❢ ï¿œ E k ✶✞ ✮ ✹✜

  15. ❙t❡♣ ✷✿ ❍♩✇ ♠❛♥② ❡✈❛❧✉❛t✐♊♥s ❞♊❡s ♥❡❡❞ ❛t ❧❡❛st❄ ❙t❡♣ ✷✿ ❇♩✐❧s ❞♊✇♥ t♩ ♣r♊✈❛❜❧❡ s❡❝✉r✐t② ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ï¿œ E k s❀♊✉❧❞ ✏❧♩♩❩ ❧✐❊❡✑ r❛♥❞♊♠ ♣❡r♠✉t❛t✐♊♥ ❢♊r ❡✈❡r② t • ❈♊♥s✐❞❡r ❛❞✈❡rs❛r② D t❀❛t ♠❛❊❡s q ❡✈❛❧✉❛t✐♊♥s ♊❢ ï¿œ E k • ❙t❡♣ ✶✿ • ❍♩✇ ♠❛♥② ❡✈❛❧✉❛t✐♊♥s ❞♊❡s D ♥❡❡❞ ❛t ♠♩st❄ ❙t❡♣ ✶✿ • ❇♩✐❧s ❞♊✇♥ t♩ ✜♥❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❩s ✶✞ ✮ ✹✜

  16. ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ï¿œ E k s❀♊✉❧❞ ✏❧♩♩❩ ❧✐❊❡✑ r❛♥❞♊♠ ♣❡r♠✉t❛t✐♊♥ ❢♊r ❡✈❡r② t • ❈♊♥s✐❞❡r ❛❞✈❡rs❛r② D t❀❛t ♠❛❊❡s q ❡✈❛❧✉❛t✐♊♥s ♊❢ ï¿œ E k • ❙t❡♣ ✶✿ • ❍♩✇ ♠❛♥② ❡✈❛❧✉❛t✐♊♥s ❞♊❡s D ♥❡❡❞ ❛t ♠♩st❄ ❙t❡♣ ✶✿ • ❇♩✐❧s ❞♊✇♥ t♩ ✜♥❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❩s • ❙t❡♣ ✷✿ • ❍♩✇ ♠❛♥② ❡✈❛❧✉❛t✐♊♥s ❞♊❡s D ♥❡❡❞ ❛t ❧❡❛st❄ ❙t❡♣ ✷✿ • ❇♩✐❧s ❞♊✇♥ t♩ ♣r♊✈❛❜❧❡ s❡❝✉r✐t② ✶✞ ✮ ✹✜

  17. ❋♩r ❛♥② t✇♩ q✉❡r✐❡s ✱ ✿ ❯♥❧✐❊❡❧② t♩ ❀❛♣♣❡♥ ❢♊r r❛♥❞♊♠ ❢❛♠✐❧② ♊❢ ♣❡r♠✉t❛t✐♊♥s ■♠♣❧✐❝❛t✐♊♥ st✐❧❧ ❀♊❧❞s ✇✐t❀ ❞✐✛❡r❡♥❝❡ ①♩r❡❞ t♩ ❙❝❀❡♠❡ ❝❛♥ ❜❡ ❜r♊❊❡♥ ✐♥ ❡✈❛❧✉❛t✐♊♥s ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c ✶✹ ✮ ✹✜

  18. ❯♥❧✐❊❡❧② t♩ ❀❛♣♣❡♥ ❢♊r r❛♥❞♊♠ ❢❛♠✐❧② ♊❢ ♣❡r♠✉t❛t✐♊♥s ■♠♣❧✐❝❛t✐♊♥ st✐❧❧ ❀♊❧❞s ✇✐t❀ ❞✐✛❡r❡♥❝❡ ①♩r❡❞ t♩ ❙❝❀❡♠❡ ❝❛♥ ❜❡ ❜r♊❊❡♥ ✐♥ ❡✈❛❧✉❛t✐♊♥s ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ❋♩r ❛♥② t✇♩ q✉❡r✐❡s ( t, m, c ) ✱ ( t ′ , m ′ , c ′ ) ✿ m ⊕ f 1 ( t ) = m ′ ⊕ f 1 ( t ′ ) = ⇒ c ⊕ f 2 ( t ) = c ′ ⊕ f 2 ( t ′ ) ✶✹ ✮ ✹✜

  19. ■♠♣❧✐❝❛t✐♊♥ st✐❧❧ ❀♊❧❞s ✇✐t❀ ❞✐✛❡r❡♥❝❡ ①♩r❡❞ t♩ ❙❝❀❡♠❡ ❝❛♥ ❜❡ ❜r♊❊❡♥ ✐♥ ❡✈❛❧✉❛t✐♊♥s ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ❋♩r ❛♥② t✇♩ q✉❡r✐❡s ( t, m, c ) ✱ ( t ′ , m ′ , c ′ ) ✿ m ⊕ f 1 ( t ) = m ′ ⊕ f 1 ( t ′ ) = ⇒ c ⊕ f 2 ( t ) = c ′ ⊕ f 2 ( t ′ ) • ❯♥❧✐❊❡❧② t♩ ❀❛♣♣❡♥ ❢♊r r❛♥❞♊♠ ❢❛♠✐❧② ♊❢ ♣❡r♠✉t❛t✐♊♥s ✶✹ ✮ ✹✜

  20. ❙❝❀❡♠❡ ❝❛♥ ❜❡ ❜r♊❊❡♥ ✐♥ ❡✈❛❧✉❛t✐♊♥s ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ❋♩r ❛♥② t✇♩ q✉❡r✐❡s ( t, m, c ) ✱ ( t ′ , m ′ , c ′ ) ✿ m ⊕ f 1 ( t ) = m ′ ⊕ f 1 ( t ′ ) = ⇒ c ⊕ f 2 ( t ) = c ′ ⊕ f 2 ( t ′ ) • ❯♥❧✐❊❡❧② t♩ ❀❛♣♣❡♥ ❢♊r r❛♥❞♊♠ ❢❛♠✐❧② ♊❢ ♣❡r♠✉t❛t✐♊♥s • ■♠♣❧✐❝❛t✐♊♥ st✐❧❧ ❀♊❧❞s ✇✐t❀ ❞✐✛❡r❡♥❝❡ C ①♩r❡❞ t♩ m, m ′ ✶✹ ✮ ✹✜

  21. ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ❋♩r ❛♥② t✇♩ q✉❡r✐❡s ( t, m, c ) ✱ ( t ′ , m ′ , c ′ ) ✿ m ⊕ f 1 ( t ) = m ′ ⊕ f 1 ( t ′ ) = ⇒ c ⊕ f 2 ( t ) = c ′ ⊕ f 2 ( t ′ ) • ❯♥❧✐❊❡❧② t♩ ❀❛♣♣❡♥ ❢♊r r❛♥❞♊♠ ❢❛♠✐❧② ♊❢ ♣❡r♠✉t❛t✐♊♥s • ■♠♣❧✐❝❛t✐♊♥ st✐❧❧ ❀♊❧❞s ✇✐t❀ ❞✐✛❡r❡♥❝❡ C ①♩r❡❞ t♩ m, m ′ ❙❝❀❡♠❡ ❝❛♥ ❜❡ ❜r♊❊❡♥ ✐♥ ≈ 2 n/ 2 ❡✈❛❧✉❛t✐♊♥s ✶✹ ✮ ✹✜

  22. ❚②♣✐❝❛❧ ❛♣♣r♊❛❝❀✿ ❈♊♥s✐❞❡r ❛♥② tr❛♥s❝r✐♣t ❛♥ ❛❞✈❡rs❛r② ♠❛② s❡❡ ▌♊st ✬s s❀♊✉❧❞ ❜❡ ❡q✉❛❧❧② ❧✐❊❡❧② ✐♥ ❜♩t❀ ✇♩r❧❞s ❖❞❞ ♊♥❡s s❀♊✉❧❞ ❀❛♣♣❡♥ ✇✐t❀ ✈❡r② s♠❛❧❧ ♣r♩❜❛❜✐❧✐t② ❆❧❧ ❝♊♥str✉❝t✐♊♥s ♊❢ t❀✐s ❊✐♥❞✿ s❡❝✉r❡ ✉♣ t♩ ❡✈❛❧✉❛t✐♊♥s ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ❚❀❡ ❢✉♥ st❛rts ❀❡r❡✊ • ▌♊r❡ t❡❝❀♥✐❝❛❧ ❛♥❞ ♊❢t❡♥ ♠♩r❡ ✐♥✈♊❧✈❡❞ ✶✺ ✮ ✹✜

  23. ❆❧❧ ❝♊♥str✉❝t✐♊♥s ♊❢ t❀✐s ❊✐♥❞✿ s❡❝✉r❡ ✉♣ t♩ ❡✈❛❧✉❛t✐♊♥s ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ❚❀❡ ❢✉♥ st❛rts ❀❡r❡✊ • ▌♊r❡ t❡❝❀♥✐❝❛❧ ❛♥❞ ♊❢t❡♥ ♠♩r❡ ✐♥✈♊❧✈❡❞ • ❚②♣✐❝❛❧ ❛♣♣r♊❛❝❀✿ • ❈♊♥s✐❞❡r ❛♥② tr❛♥s❝r✐♣t τ ❛♥ ❛❞✈❡rs❛r② ♠❛② s❡❡ • ▌♊st τ ✬s s❀♊✉❧❞ ❜❡ ❡q✉❛❧❧② ❧✐❊❡❧② ✐♥ ❜♩t❀ ✇♩r❧❞s • ❖❞❞ ♊♥❡s s❀♊✉❧❞ ❀❛♣♣❡♥ ✇✐t❀ ✈❡r② s♠❛❧❧ ♣r♩❜❛❜✐❧✐t② ✶✺ ✮ ✹✜

  24. ■♥t✉✐t✐♊♥✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • ❚❀❡ ❢✉♥ st❛rts ❀❡r❡✊ • ▌♊r❡ t❡❝❀♥✐❝❛❧ ❛♥❞ ♊❢t❡♥ ♠♩r❡ ✐♥✈♊❧✈❡❞ • ❚②♣✐❝❛❧ ❛♣♣r♊❛❝❀✿ • ❈♊♥s✐❞❡r ❛♥② tr❛♥s❝r✐♣t τ ❛♥ ❛❞✈❡rs❛r② ♠❛② s❡❡ • ▌♊st τ ✬s s❀♊✉❧❞ ❜❡ ❡q✉❛❧❧② ❧✐❊❡❧② ✐♥ ❜♩t❀ ✇♩r❧❞s • ❖❞❞ ♊♥❡s s❀♊✉❧❞ ❀❛♣♣❡♥ ✇✐t❀ ✈❡r② s♠❛❧❧ ♣r♩❜❛❜✐❧✐t② ❆❧❧ ❝♊♥str✉❝t✐♊♥s ♊❢ t❀✐s ❊✐♥❞✿ s❡❝✉r❡ ✉♣ t♩ ≈ 2 n/ 2 ❡✈❛❧✉❛t✐♊♥s ✶✺ ✮ ✹✜

  25. ❖✉t❧✐♥❡ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ • ■♥t✉✐t✐♊♥ • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ • ■♠♣r♊✈❡❞ ❆tt❛❝❩ • ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❇♊✉♥❞ ❈♊♥❝❧✉s✐♊♥ ✶✻ ✮ ✹✜

  26. t②♣✐❝❛❧❧② ✶✷✜ ❜✐ts ♠✉❝❀ ❧❛r❣❡r✿ ✷✺✻✲✶✻✵✵ ❜✐ts ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ ❇❧♊❝❊❝✐♣❀❡r✲❇❛s❡❞✳ ♣P❡r♠✉t❛t✐♊♥✲❇❛s❡❞✳♣ tweak-based mask tweak-based mask m E k c m P c ✶✌ ✮ ✹✜

  27. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ ❇❧♊❝❊❝✐♣❀❡r✲❇❛s❡❞✳ ♣P❡r♠✉t❛t✐♊♥✲❇❛s❡❞✳♣ tweak-based mask tweak-based mask m E k c m P c t②♣✐❝❛❧❧② ✶✷✜ ❜✐ts ♠✉❝❀ ❧❛r❣❡r✿ ✷✺✻✲✶✻✵✵ ❜✐ts ✶✌ ✮ ✹✜

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

  29. ❯s❡❞ ✐♥ ❖❈❇✷ ❛♥❞ ✶✹ ❈❆❊❙❆❘ ❝❛♥❞✐❞❛t❡s P❡r♠✉t❛t✐♊♥✲❜❛s❡❞ ✈❛r✐❛♥ts ✐♥ ▌✐♥❛❧♣❀❡r ❛♥❞ PrÞst ✭❣❡♥❡r❛❧✐③❡❞ ❜② ❈♊❣❧✐❛t✐ ❡t ❛❧✳ ❬❈▲❙✶✺❪ ✮ P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✭❳❊❳✮ • XEX ❜② ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ✿ 2 α 3 β 7 γ · E k ( N ) E k m c • ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ ✶✟ ✮ ✹✜

  30. P❡r♠✉t❛t✐♊♥✲❜❛s❡❞ ✈❛r✐❛♥ts ✐♥ ▌✐♥❛❧♣❀❡r ❛♥❞ PrÞst ✭❣❡♥❡r❛❧✐③❡❞ ❜② ❈♊❣❧✐❛t✐ ❡t ❛❧✳ ❬❈▲❙✶✺❪ ✮ P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✭❳❊❳✮ • XEX ❜② ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ✿ 2 α 3 β 7 γ · E k ( N ) E k m c • ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ • ❯s❡❞ ✐♥ ❖❈❇✷ ❛♥❞ ± ✶✹ ❈❆❊❙❆❘ ❝❛♥❞✐❞❛t❡s ✶✟ ✮ ✹✜

  31. P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✭❳❊❳✮ • XEX ❜② ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ✿ 2 α 3 β 7 γ · ( k ï¿œ N ⊕ P ( k ï¿œ N )) 2 α 3 β 7 γ · E k ( N ) E k P m c m c • ( α, β, γ, N ) ✐s t✇❡❛❊ ✭s✐♠♣❧✐✜❡❞✮ • ❯s❡❞ ✐♥ ❖❈❇✷ ❛♥❞ ± ✶✹ ❈❆❊❙❆❘ ❝❛♥❞✐❞❛t❡s • P❡r♠✉t❛t✐♊♥✲❜❛s❡❞ ✈❛r✐❛♥ts ✐♥ ▌✐♥❛❧♣❀❡r ❛♥❞ PrÞst ✭❣❡♥❡r❛❧✐③❡❞ ❜② ❈♊❣❧✐❛t✐ ❡t ❛❧✳ ❬❈▲❙✶✺❪ ✮ ✶✟ ✮ ✹✜

  32. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷✲▲✐❊❡ ❈♊♥str✉❝t✐♊♥ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t M ⊕ N, t A1 N, t A2 N, t M1 N, t M2 N, t Md N, t Aa ˜ ˜ ˜ ˜ ˜ ˜ ˜ E E E E E E E k k k k k k k C 1 C 2 C d L = E k ( N ) T ✷✵ ✮ ✹✜

  33. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷✲▲✐❊❡ ❈♊♥str✉❝t✐♊♥ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 · 3 2 L 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d 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 ✷✵ ✮ ✹✜

  34. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷✲▲✐❊❡ ❈♊♥str✉❝t✐♊♥ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 · 3 2 L 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d 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 ✷✵ ✮ ✹✜

  35. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷✲▲✐❊❡ ❈♊♥str✉❝t✐♊♥ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 · 3 2 L 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d 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 ✷✵ ✮ ✹✜

  36. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷✲▲✐❊❡ ❈♊♥str✉❝t✐♊♥ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 · 3 2 L 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d 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 ✷✵ ✮ ✹✜

  37. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷✲▲✐❊❡ ❈♊♥str✉❝t✐♊♥ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 · 3 2 L 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d 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 ✷✵ ✮ ✹✜

  38. P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷✲▲✐❊❡ ❈♊♥str✉❝t✐♊♥ A 1 A 2 A a ⊕ M i M 1 M 2 M d 2 · 3 2 L 2 2 3 2 L 2 a 3 2 L 2 d 3 L 2 2 L 2 d 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 • ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ • ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ • ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ • ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s ✷✵ ✮ ✹✜

  39. ❙✐♥❣❧❡ ❳❖❘ ▲♊❣❛r✐t❀♠✐❝ ❛♠♊✉♥t ♊❢ ✜❡❧❞ ❞♊✉❜❧✐♥❣s ✭♣r❡❝♊♠♣✉t❡❞✮ ▌♊r❡ ❡✣❝✐❡♥t t❀❛♥ ♣♊✇❡r✐♥❣✲✉♣ ❬❑❘✶✶❪ ●r❛② ❈♊❞❡ ▌❛s❊✐♥❣ • ❖❈❇✶ ❛♥❞ ❖❈❇✾ ✉s❡ ●r❛② ❈♊❞❡s✿ ï¿œ ï¿œ α ⊕ ( α ≫ 1) · E k ( N ) E k m c • ( α, N ) ✐s t✇❡❛❊ • ❯♣❞❛t✐♥❣✿ G ( α ) = G ( α − 1) ⊕ 2 ntz ( α ) ✷✶ ✮ ✹✜

  40. ●r❛② ❈♊❞❡ ▌❛s❊✐♥❣ • ❖❈❇✶ ❛♥❞ ❖❈❇✾ ✉s❡ ●r❛② ❈♊❞❡s✿ ï¿œ ï¿œ α ⊕ ( α ≫ 1) · E k ( N ) E k m c • ( α, N ) ✐s t✇❡❛❊ • ❯♣❞❛t✐♥❣✿ G ( α ) = G ( α − 1) ⊕ 2 ntz ( α ) • ❙✐♥❣❧❡ ❳❖❘ • ▲♊❣❛r✐t❀♠✐❝ ❛♠♊✉♥t ♊❢ ✜❡❧❞ ❞♊✉❜❧✐♥❣s ✭♣r❡❝♊♠♣✉t❡❞✮ • ▌♊r❡ ❡✣❝✐❡♥t t❀❛♥ ♣♊✇❡r✐♥❣✲✉♣ ❬❑❘✶✶❪ ✷✶ ✮ ✹✜

  41. ❖✉t❧✐♥❡ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ • ■♥t✉✐t✐♊♥ • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ • ■♠♣r♊✈❡❞ ❆tt❛❝❩ • ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❇♊✉♥❞ ❈♊♥❝❧✉s✐♊♥ ✷✷ ✮ ✹✜

  42. ❈♊♠❜✐♥❡s ❛❞✈❛♥t❛❣❡s ♊❢✿ P♊✇❡r✐♥❣✲✉♣ ♠❛s❊✐♥❣ ❲♊r❞✲❜❛s❡❞ ▲❋❙❘s ❙✐♠♣❧❡r✱ ❝♊♥st❛♥t✲t✐♠❡ ✭❜② ❞❡❢❛✉❧t✮✱ ♠♩r❡ ❡✣❝✐❡♥t ▌❛s❊❡❞ ❊✈❡♥✲▌❛♥s♩✉r ✭ MEM ✮ • 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✐♠♣❧✐✜❡❞✮ ✷✞ ✮ ✹✜

  43. ❙✐♠♣❧❡r✱ ❝♊♥st❛♥t✲t✐♠❡ ✭❜② ❞❡❢❛✉❧t✮✱ ♠♩r❡ ❡✣❝✐❡♥t ▌❛s❊❡❞ ❊✈❡♥✲▌❛♥s♩✉r ✭ MEM ✮ • 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 ❛❞✈❛♥t❛❣❡s ♊❢✿ • P♊✇❡r✐♥❣✲✉♣ ♠❛s❊✐♥❣ • ❲♊r❞✲❜❛s❡❞ ▲❋❙❘s ✷✞ ✮ ✹✜

  44. ▌❛s❊❡❞ ❊✈❡♥✲▌❛♥s♩✉r ✭ MEM ✮ • 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 ❛❞✈❛♥t❛❣❡s ♊❢✿ • P♊✇❡r✐♥❣✲✉♣ ♠❛s❊✐♥❣ • ❲♊r❞✲❜❛s❡❞ ▲❋❙❘s • ❙✐♠♣❧❡r✱ ❝♊♥st❛♥t✲t✐♠❡ ✭❜② ❞❡❢❛✉❧t✮✱ ♠♩r❡ ❡✣❝✐❡♥t ✷✞ ✮ ✹✜

  45. ❙❛♠♣❧❡ ▲❋❙❘s ✭st❛t❡ s✐③❡ ❛s ✇♩r❞s ♊❢ ❜✐ts✮✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❲♊r❊ ❡①❝❡♣t✐♊♥❛❧❧② ✇❡❧❧ ❢♊r ❆❘❳ ♣r✐♠✐t✐✈❡s MEM ✿ ❉❡s✐❣♥ ❈♊♥s✐❞❡r❛t✐♊♥s • P❛rt✐❝✉❧❛r❧② s✉✐t❡❞ ❢♊r ❧❛r❣❡ st❛t❡s ✭♣❡r♠✉t❛t✐♊♥s✮ • ▲♊✇ ♊♣❡r❛t✐♊♥ ❝♊✉♥ts ❜② ❝❧❡✈❡r ❝❀♊✐❝❡ ♊❢ ▲❋❙❘ ✷✹ ✮ ✹✜

  46. ❲♊r❊ ❡①❝❡♣t✐♊♥❛❧❧② ✇❡❧❧ ❢♊r ❆❘❳ ♣r✐♠✐t✐✈❡s MEM ✿ ❉❡s✐❣♥ ❈♊♥s✐❞❡r❛t✐♊♥s • P❛rt✐❝✉❧❛r❧② s✉✐t❡❞ ❢♊r ❧❛r❣❡ st❛t❡s ✭♣❡r♠✉t❛t✐♊♥s✮ • ▲♊✇ ♊♣❡r❛t✐♊♥ ❝♊✉♥ts ❜② ❝❧❡✈❡r ❝❀♊✐❝❡ ♊❢ ▲❋❙❘ • ❙❛♠♣❧❡ ▲❋❙❘s ✭st❛t❡ s✐③❡ b ❛s n ✇♩r❞s ♊❢ w ❜✐ts✮✿ b w n ϕ 128 8 16 ( x 1 , . . . , x 15 , ( x 0 ≪ 1) ⊕ ( x 9 ≫ 1) ⊕ ( x 10 ≪ 1)) 128 32 4 ( x 1 , . . . , x 3 , ( x 0 ≪ 5) ⊕ x 1 ⊕ ( x 1 ≪ 13)) 128 64 2 ( x 1 , ( x 0 ≪ 11) ⊕ x 1 ⊕ ( x 1 ≪ 13)) 256 64 4 ( x 1 , . . . , x 3 , ( x 0 ≪ 3) ⊕ ( x 3 ≫ 5)) 512 32 16 ( x 1 , . . . , x 15 , ( x 0 ≪ 5) ⊕ ( x 3 ≫ 7)) 512 64 8 ( x 1 , . . . , x 7 , ( x 0 ≪ 29) ⊕ ( x 1 ≪ 9)) 1024 64 16 ( x 1 , . . . , x 15 , ( x 0 ≪ 53) ⊕ ( x 5 ≪ 13)) 1600 32 50 ( x 1 , . . . , x 49 , ( x 0 ≪ 3) ⊕ ( x 23 ≫ 3)) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✮ ✹✜

  47. MEM ✿ ❉❡s✐❣♥ ❈♊♥s✐❞❡r❛t✐♊♥s • P❛rt✐❝✉❧❛r❧② s✉✐t❡❞ ❢♊r ❧❛r❣❡ st❛t❡s ✭♣❡r♠✉t❛t✐♊♥s✮ • ▲♊✇ ♊♣❡r❛t✐♊♥ ❝♊✉♥ts ❜② ❝❧❡✈❡r ❝❀♊✐❝❡ ♊❢ ▲❋❙❘ • ❙❛♠♣❧❡ ▲❋❙❘s ✭st❛t❡ s✐③❡ b ❛s n ✇♩r❞s ♊❢ w ❜✐ts✮✿ b w n ϕ 128 8 16 ( x 1 , . . . , x 15 , ( x 0 ≪ 1) ⊕ ( x 9 ≫ 1) ⊕ ( x 10 ≪ 1)) 128 32 4 ( x 1 , . . . , x 3 , ( x 0 ≪ 5) ⊕ x 1 ⊕ ( x 1 ≪ 13)) 128 64 2 ( x 1 , ( x 0 ≪ 11) ⊕ x 1 ⊕ ( x 1 ≪ 13)) 256 64 4 ( x 1 , . . . , x 3 , ( x 0 ≪ 3) ⊕ ( x 3 ≫ 5)) 512 32 16 ( x 1 , . . . , x 15 , ( x 0 ≪ 5) ⊕ ( x 3 ≫ 7)) 512 64 8 ( x 1 , . . . , x 7 , ( x 0 ≪ 29) ⊕ ( x 1 ≪ 9)) 1024 64 16 ( x 1 , . . . , x 15 , ( x 0 ≪ 53) ⊕ ( x 5 ≪ 13)) 1600 32 50 ( x 1 , . . . , x 49 , ( x 0 ≪ 3) ⊕ ( x 23 ≫ 3)) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ • ❲♊r❊ ❡①❝❡♣t✐♊♥❛❧❧② ✇❡❧❧ ❢♊r ❆❘❳ ♣r✐♠✐t✐✈❡s ✷✹ ✮ ✹✜

  48. ✻✹ ✶✷✜ ✷✺✻ ✺✶✷ ✶✵✷✹ s♊❧✈❡❞ ❜② r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ❡✳❣✳✱ ❜② PrÞst ✭✷✵✶✹✮ s♊❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▌◆✶✻❪ MEM ✿ ❯♥✐q✉❡♥❡ss ♊❢ ▌❛s❊✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❊✐♥❣ ❣♊❡s ✇❡❧❧ ❛s ❧♊♥❣ ❛s ϕ γ 2 ◩ ϕ β 0 ï¿œ = ϕ γ ′ 2 ◩ ϕ β ′ 1 ◩ ϕ α ′ 1 ◩ ϕ α 0 ❢♊r ❛♥② ( α, β, γ ) ï¿œ = ( α ′ , β ′ , γ ′ ) • ❈❀❛❧❧❡♥❣❡✿ s❡t ♣r♊♣❡r ❞♊♠❛✐♥ ❢♊r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♊♠♣✉t❛t✐♊♥ ♊❢ ❞✐s❝r❡t❡ ❧♊❣❛r✐t❀♠s ✷✺ ✮ ✹✜

  49. s♊❧✈❡❞ ❜② r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ❡✳❣✳✱ ❜② PrÞst ✭✷✵✶✹✮ s♊❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▌◆✶✻❪ MEM ✿ ❯♥✐q✉❡♥❡ss ♊❢ ▌❛s❊✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❊✐♥❣ ❣♊❡s ✇❡❧❧ ❛s ❧♊♥❣ ❛s ϕ γ 2 ◩ ϕ β 0 ï¿œ = ϕ γ ′ 2 ◩ ϕ β ′ 1 ◩ ϕ α ′ 1 ◩ ϕ α 0 ❢♊r ❛♥② ( α, β, γ ) ï¿œ = ( α ′ , β ′ , γ ′ ) • ❈❀❛❧❧❡♥❣❡✿ s❡t ♣r♊♣❡r ❞♊♠❛✐♥ ❢♊r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♊♠♣✉t❛t✐♊♥ ♊❢ ❞✐s❝r❡t❡ ❧♊❣❛r✐t❀♠s ✻✹ ✶✷✜ ✷✺✻ ✺✶✷ ✶✵✷✹ ✷✺ ✮ ✹✜

  50. r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❡✳❣✳✱ ❜② PrÞst ✭✷✵✶✹✮ s♊❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▌◆✶✻❪ MEM ✿ ❯♥✐q✉❡♥❡ss ♊❢ ▌❛s❊✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❊✐♥❣ ❣♊❡s ✇❡❧❧ ❛s ❧♊♥❣ ❛s ϕ γ 2 ◩ ϕ β 0 ï¿œ = ϕ γ ′ 2 ◩ ϕ β ′ 1 ◩ ϕ α ′ 1 ◩ ϕ α 0 ❢♊r ❛♥② ( α, β, γ ) ï¿œ = ( α ′ , β ′ , γ ′ ) • ❈❀❛❧❧❡♥❣❡✿ s❡t ♣r♊♣❡r ❞♊♠❛✐♥ ❢♊r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♊♠♣✉t❛t✐♊♥ ♊❢ ❞✐s❝r❡t❡ ❧♊❣❛r✐t❀♠s ✻✹ ✶✷✜ ✷✺✻ ✺✶✷ ✶✵✷✹ ï¿œ ￜￜ ï¿œ s♊❧✈❡❞ ❜② ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ✷✺ ✮ ✹✜

  51. s♊❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▌◆✶✻❪ MEM ✿ ❯♥✐q✉❡♥❡ss ♊❢ ▌❛s❊✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❊✐♥❣ ❣♊❡s ✇❡❧❧ ❛s ❧♊♥❣ ❛s ϕ γ 2 ◩ ϕ β 0 ï¿œ = ϕ γ ′ 2 ◩ ϕ β ′ 1 ◩ ϕ α ′ 1 ◩ ϕ α 0 ❢♊r ❛♥② ( α, β, γ ) ï¿œ = ( α ′ , β ′ , γ ′ ) • ❈❀❛❧❧❡♥❣❡✿ s❡t ♣r♊♣❡r ❞♊♠❛✐♥ ❢♊r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♊♠♣✉t❛t✐♊♥ ♊❢ ❞✐s❝r❡t❡ ❧♊❣❛r✐t❀♠s ✻✹ ✶✷✜ ✷✺✻ ✺✶✷ ✶✵✷✹ ï¿œ ￜￜ ï¿œ ï¿œ ￜￜ ï¿œ s♊❧✈❡❞ ❜② r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ❡✳❣✳✱ ❜② PrÞst ✭✷✵✶✹✮ ✷✺ ✮ ✹✜

  52. MEM ✿ ❯♥✐q✉❡♥❡ss ♊❢ ▌❛s❊✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❊✐♥❣ ❣♊❡s ✇❡❧❧ ❛s ❧♊♥❣ ❛s ϕ γ 2 ◩ ϕ β 0 ï¿œ = ϕ γ ′ 2 ◩ ϕ β ′ 1 ◩ ϕ α ′ 1 ◩ ϕ α 0 ❢♊r ❛♥② ( α, β, γ ) ï¿œ = ( α ′ , β ′ , γ ′ ) • ❈❀❛❧❧❡♥❣❡✿ s❡t ♣r♊♣❡r ❞♊♠❛✐♥ ❢♊r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♊♠♣✉t❛t✐♊♥ ♊❢ ❞✐s❝r❡t❡ ❧♊❣❛r✐t❀♠s ✻✹ ✶✷✜ ✷✺✻ ✺✶✷ ✶✵✷✹ ï¿œ ￜￜ ï¿œ ï¿œ ￜￜ ï¿œ s♊❧✈❡❞ ❜② r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❘♊❣❛✇❛② ❬❘♊❣✵✹❪ ❡✳❣✳✱ ❜② PrÞst ✭✷✵✶✹✮ ï¿œ ￜￜ ï¿œ s♊❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▌◆✶✻❪ ✷✺ ✮ ✹✜

  53. ❆♣♣❧✐❝❛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 ◩ ϕ 2 1 ◩ ϕ d –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 ◩ ϕ 2 1 ◩ ϕ d –1 ( L ) ϕ 2 ◩ ϕ 0 ( L ) ϕ 2 ◩ ϕ 1 ( L ) ϕ 2 ◩ ϕ d –1 ( L ) C 1 C 2 C d L = P ( N ï¿œ k ) T 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♊✉♥❞ ❇▲❆❑❊✷❜ ✷✻ ✮ ✹✜

  54. ❆♣♣❧✐❝❛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 0 ( L ) ϕ a –1 ( L ) ϕ 1 ◩ ϕ 0 ( L ) ϕ 1 ◩ ϕ d –1 ( L ) ϕ 2 ( L ) ⊕ M 0 ϕ 2 ( L ) ⊕ M d –1 ϕ 2 1 ( L ) C 1 C d L = P ( N ï¿œ k ) P 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♊✉♥❞ ❇▲❆❑❊✷❜ ✷✌ ✮ ✹✜

  55. ❖✉t❧✐♥❡ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ • ■♥t✉✐t✐♊♥ • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ • ■♠♣r♊✈❡❞ ❆tt❛❝❩ • ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❇♊✉♥❞ ❈♊♥❝❧✉s✐♊♥ ✷✜ ✮ ✹✜

  56. ■❢ ✐s ❧❛r❣❡ ❡♥♊✉❣❀ ♥♊ ♣r♊❜❧❡♠ ■❢ ✐s s♠❛❧❧ ✏❜❡②♊♥❞ ❜✐rt❀❞❛② ❜♊✉♥❞✑ s♩❧✉t✐♊♥s ❚✇❡❛❊✲r❡❊❡②✐♥❣ ❬▌✐♥✵✟✱▌❡♥✶✺✱❲●❩✰✶✻✱❏▲▌✰✶✌✱▲▲✶✜❪ ❈❛s❝❛❞✐♥❣ ✭♥♊✇✮ ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs f 1 ( t ) f 2 ( t ) E k /P m c • ✏❇✐rt❀❞❛② ❜♊✉♥❞✑ 2 n/ 2 s❡❝✉r✐t② ❛t ❜❡st • ❖✈❡r❧②✐♥❣ ♠♊❞❡s ✐♥❀❡r✐t s❡❝✉r✐t② ❜♊✉♥❞ ✷✟ ✮ ✹✜

  57. ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs f 1 ( t ) f 2 ( t ) E k /P m c • ✏❇✐rt❀❞❛② ❜♊✉♥❞✑ 2 n/ 2 s❡❝✉r✐t② ❛t ❜❡st • ❖✈❡r❧②✐♥❣ ♠♊❞❡s ✐♥❀❡r✐t s❡❝✉r✐t② ❜♊✉♥❞ • ■❢ n ✐s ❧❛r❣❡ ❡♥♊✉❣❀ − → ♥♊ ♣r♊❜❧❡♠ • ■❢ n ✐s s♠❛❧❧ − → ✏❜❡②♊♥❞ ❜✐rt❀❞❛② ❜♊✉♥❞✑ s♩❧✉t✐♊♥s • ❚✇❡❛❊✲r❡❊❡②✐♥❣ ❬▌✐♥✵✟✱▌❡♥✶✺✱❲●❩✰✶✻✱❏▲▌✰✶✌✱▲▲✶✜❪ • ❈❛s❝❛❞✐♥❣ ✭♥♊✇✮ ✷✟ ✮ ✹✜

  58. ✿ s❡❝✉r❡ ✉♣ t♩ q✉❡r✐❡s ❬▲❙❚✶✷✱Pr♊✶✹❪ ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♩ q✉❡r✐❡s ❬▲❙✶✞❪ ❇❡st ❛tt❛❝❊✿ q✉❡r✐❡s ❈❛s❝❛❞✐♥❣ LRW 2 ✬s h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h ρ − 1 ( t ) ⊕ h ρ ( t ) h ρ ( t ) m E k 1 E k 2 E k ρ c · · · · · · • LRW 2 [ ρ ] ✿ ❝♊♥❝❛t❡♥❛t✐♊♥ ♊❢ ρ LRW 2 ✬s ✏❈❛s❝❛❞❡❞ LRW 2 ✑ • k 1 , . . . , k ρ ❛♥❞ h 1 , . . . , h ρ ✐♥❞❡♣❡♥❞❡♥t ❂ LRW 2 [2] ✞✵ ✮ ✹✜

  59. ❈❛s❝❛❞✐♥❣ LRW 2 ✬s h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h ρ − 1 ( t ) ⊕ h ρ ( t ) h ρ ( t ) m E k 1 E k 2 E k ρ c · · · · · · • LRW 2 [ ρ ] ✿ ❝♊♥❝❛t❡♥❛t✐♊♥ ♊❢ ρ LRW 2 ✬s ✏❈❛s❝❛❞❡❞ LRW 2 ✑ • k 1 , . . . , k ρ ❛♥❞ h 1 , . . . , h ρ ✐♥❞❡♣❡♥❞❡♥t ❂ LRW 2 [2] • 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 ❬▲❙✶✞❪ • ❇❡st ❛tt❛❝❊✿ 2 n q✉❡r✐❡s ✞✵ ✮ ✹✜

  60. ✿ s❡❝✉r❡ ✉♣ t♩ q✉❡r✐❡s ❬❈▲❙✶✺❪ ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♩ q✉❡r✐❡s ❬❈▲❙✶✺❪ ❇❡st ❛tt❛❝❊✿ q✉❡r✐❡s ❬❇❑▲✰✶✷❪ ❈❛s❝❛❞✐♥❣ ❚❊▌✬s h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h ρ − 1 ( t ) ⊕ h ρ ( t ) h ρ ( t ) m P 1 P 2 P ρ c · · · · · · • TEM [ ρ ] ✿ ❝♊♥❝❛t❡♥❛t✐♊♥ ♊❢ ρ TEM ✬s • P 1 , . . . , P ρ ❛♥❞ h 1 , . . . , h ρ ✐♥❞❡♣❡♥❞❡♥t ✞✶ ✮ ✹✜

  61. ❈❛s❝❛❞✐♥❣ ❚❊▌✬s h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h ρ − 1 ( t ) ⊕ h ρ ( t ) h ρ ( t ) m P 1 P 2 P ρ 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 ❬❈▲❙✶✺❪ • ❇❡st ❛tt❛❝❊✿ 2 ρn/ ( ρ +1) q✉❡r✐❡s ❬❇❑▲✰✶✷❪ ✞✶ ✮ ✹✜

  62. ■♠♣r♊✈❡❞ ✐♥ ❬▌❡♥✶✜❪ n/ 2 2 n/ 3 3 n/ 4 5 n/ 6 n ❙t❛t❡ ♊❢ t❀❡ ❆rt LRW 2 [1] LRW 2 [2] ❣❛♣ LRW 2 [3] ❣❛♣ LRW 2 [4] ❣❛♣ LRW 2 [5] ❣❛♣ LRW 2 [6] ❣❛♣ LRW 2 [7] ❣❛♣ LRW 2 [8] ❣❛♣ LRW 2 [9] ❣❛♣ ❣❛♣ LRW 2 [10] ❣❛♣ LRW 2 [11] n/ 2 2 n/ 3 3 n/ 4 5 n/ 6 n TEM [1] TEM [2] TEM [3] ❣❛♣ TEM [4] ❣❛♣ TEM [5] ❣❛♣ ❣❛♣ TEM [6] TEM [7] ❣❛♣ TEM [8] ❣❛♣ TEM [9] ❣❛♣ TEM [10] ❣❛♣ ❣❛♣ TEM [11] ✞✷ ✮ ✹✜

  63. n/ 2 2 n/ 3 3 n/ 4 5 n/ 6 n ❙t❛t❡ ♊❢ t❀❡ ❆rt LRW 2 [1] LRW 2 [2] ❣❛♣ → LRW 2 [3] ❣❛♣ − LRW 2 [4] ❣❛♣ − LRW 2 [5] ❣❛♣ LRW 2 [6] ❣❛♣ ■♠♣r♊✈❡❞ LRW 2 [7] ❣❛♣ ✐♥ ❬▌❡♥✶✜❪ LRW 2 [8] ❣❛♣ LRW 2 [9] ❣❛♣ ❣❛♣ LRW 2 [10] ❣❛♣ LRW 2 [11] n/ 2 2 n/ 3 3 n/ 4 5 n/ 6 n TEM [1] TEM [2] TEM [3] ❣❛♣ TEM [4] ❣❛♣ TEM [5] ❣❛♣ ❣❛♣ TEM [6] TEM [7] ❣❛♣ TEM [8] ❣❛♣ TEM [9] ❣❛♣ TEM [10] ❣❛♣ ❣❛♣ TEM [11] ✞✷ ✮ ✹✜

  64. ❖✉t❧✐♥❡ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ • ■♥t✉✐t✐♊♥ • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ • ■♠♣r♊✈❡❞ ❆tt❛❝❩ • ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❇♊✉♥❞ ❈♊♥❝❧✉s✐♊♥ ✾✾ ✮ ✹✜

  65. ✐♠♣r♊✈❡❞ ❜♊✉♥❞ ✭❝♊♥❞✐t✐♊♥❛❧❧②✮ ✐♠♣r♊✈❡❞ ❛tt❛❝❩ ✭❣❡♥❡r❛❧✐③❡❞ ❝♊♥str✉❝t✐♊♥✮ ❝❛rr✐❡s ♊✈❡r t♩ ✕ ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h 2 ( t ) m E k 1 E k 2 c n/ 2 2 n/ 3 3 n/ 4 n ❣❛♣ ✞✹ ✮ ✹✜

  66. ✐♠♣r♊✈❡❞ ❜♊✉♥❞ ✭❝♊♥❞✐t✐♊♥❛❧❧②✮ ❝❛rr✐❡s ♊✈❡r t♩ ✕ ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h 2 ( t ) m E k 1 E k 2 c n/ 2 2 n/ 3 3 n/ 4 n ❣❛♣ ✐♠♣r♊✈❡❞ ❛tt❛❝❩ ✭❣❡♥❡r❛❧✐③❡❞ ❝♊♥str✉❝t✐♊♥✮ ✞✹ ✮ ✹✜

  67. ❝❛rr✐❡s ♊✈❡r t♩ ✕ ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h 2 ( t ) m E k 1 E k 2 c n/ 2 2 n/ 3 3 n/ 4 n ✐♠♣r♊✈❡❞ ❜♊✉♥❞ ✭❝♊♥❞✐t✐♊♥❛❧❧②✮ ✐♠♣r♊✈❡❞ ❛tt❛❝❩ ✭❣❡♥❡r❛❧✐③❡❞ ❝♊♥str✉❝t✐♊♥✮ ✞✹ ✮ ✹✜

  68. ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h 2 ( t ) m E k 1 E k 2 c n/ 2 2 n/ 3 3 n/ 4 n ✐♠♣r♊✈❡❞ ❜♊✉♥❞ ✭❝♊♥❞✐t✐♊♥❛❧❧②✮ ✐♠♣r♊✈❡❞ ❛tt❛❝❩ ✭❣❡♥❡r❛❧✐③❡❞ ❝♊♥str✉❝t✐♊♥✮ ❝❛rr✐❡s ♊✈❡r t♩ LRW 2 [3] ✕ LRW 2 [5] ✞✹ ✮ ✹✜

  69. ❖✉t❧✐♥❡ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ • ■♥t✉✐t✐♊♥ • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ • ■♠♣r♊✈❡❞ ❆tt❛❝❩ • ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❇♊✉♥❞ ❈♊♥❝❧✉s✐♊♥ ✞✺ ✮ ✹✜

  70. ●❡♥❡r✐❝ ❞✐st✐♥❣✉✐s❀✐♥❣ ❛tt❛❝❩ ✐♥ ❡✈❛❧✉❛t✐♊♥s ■♠♣r♊✈❡❞ ❆tt❛❝❩ • GCL ✭●❡♥❡r❛❧✐③❡❞ ❈❛s❝❛❞❡❞ LRW 2 ✮✿ f 1 ( t ) f 2 ( t ) f 3 ( t ) m E k 1 E k 2 c • f i ❛r❡ ❛r❜✐tr❛r② ❢✉♥❝t✐♊♥s • p i := E k i ❛r❡ r❛♥❞♊♠ ♣❡r♠✉t❛t✐♊♥s ✞✻ ✮ ✹✜

  71. ■♠♣r♊✈❡❞ ❆tt❛❝❩ • GCL ✭●❡♥❡r❛❧✐③❡❞ ❈❛s❝❛❞❡❞ LRW 2 ✮✿ f 1 ( t ) f 2 ( t ) f 3 ( t ) m E k 1 E k 2 c • f i ❛r❡ ❛r❜✐tr❛r② ❢✉♥❝t✐♊♥s • p i := E k i ❛r❡ r❛♥❞♊♠ ♣❡r♠✉t❛t✐♊♥s ●❡♥❡r✐❝ ❞✐st✐♥❣✉✐s❀✐♥❣ ❛tt❛❝❩ ✐♥ 2 n 1 / 2 2 3 n/ 4 ❡✈❛❧✉❛t✐♊♥s ✞✻ ✮ ✹✜

  72. ❙✉♣♣♊s❡ ✐t ♠❛❊❡s q✉❡r✐❡s s✉❝❀ t❀❛t ◆❡❝❡ss❛r✐❧②✱ ❙t❛t❡❞ ❞✐✛❡r❡♥t❧②✿ ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❘❛t✐♊♥❛❧❡ • ❉✐st✐♥❣✉✐s❀❡r D ♠❛❊❡s ✈❛r✐♩✉s q✉❡r✐❡s ❢♊r t✇♩ ❞✐✛❡r❡♥t t✇❡❛❊s✿ t ❛♥❞ t ′ m 1 p 1 p 2 c 1 f 1 ( t ) f 2 ( t ) f 3 ( t ) m 3 p 1 p 2 c 3 m ′ p 1 p 2 c ′ 2 2 f 1 ( t ′ ) f 2 ( t ′ ) f 3 ( t ′ ) m ′ p 1 p 2 c ′ 4 4 ✞✌ ✮ ✹✜

  73. ◆❡❝❡ss❛r✐❧②✱ ❙t❛t❡❞ ❞✐✛❡r❡♥t❧②✿ ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❘❛t✐♊♥❛❧❡ • ❉✐st✐♥❣✉✐s❀❡r D ♠❛❊❡s ✈❛r✐♩✉s q✉❡r✐❡s ❢♊r t✇♩ ❞✐✛❡r❡♥t t✇❡❛❊s✿ t ❛♥❞ t ′ • ❙✉♣♣♊s❡ ✐t ♠❛❊❡s 4 q✉❡r✐❡s s✉❝❀ t❀❛t m 1 p 1 p 2 c 1 m 1 ⊕ f 1 ( t ) = m ′ 2 ⊕ f 1 ( t ′ ) f 1 ( t ) f 2 ( t ) f 3 ( t ) c ′ 2 ⊕ f 3 ( t ′ ) = c 3 ⊕ f 3 ( t ) m 3 p 1 p 2 c 3 m 3 ⊕ f 1 ( t ) = m ′ 4 ⊕ f 1 ( t ′ ) m ′ p 1 p 2 c ′ 2 2 f 1 ( t ′ ) f 2 ( t ′ ) f 3 ( t ′ ) m ′ p 1 p 2 c ′ 4 4 ✞✌ ✮ ✹✜

  74. ❙t❛t❡❞ ❞✐✛❡r❡♥t❧②✿ ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❘❛t✐♊♥❛❧❡ • ❉✐st✐♥❣✉✐s❀❡r D ♠❛❊❡s ✈❛r✐♩✉s q✉❡r✐❡s ❢♊r t✇♩ ❞✐✛❡r❡♥t t✇❡❛❊s✿ t ❛♥❞ t ′ • ❙✉♣♣♊s❡ ✐t ♠❛❊❡s 4 q✉❡r✐❡s s✉❝❀ t❀❛t m 1 p 1 p 2 c 1 m 1 ⊕ f 1 ( t ) = m ′ 2 ⊕ f 1 ( t ′ ) f 1 ( t ) f 2 ( t ) f 3 ( t ) c ′ 2 ⊕ f 3 ( t ′ ) = c 3 ⊕ f 3 ( t ) m 3 p 1 p 2 c 3 m 3 ⊕ f 1 ( t ) = m ′ 4 ⊕ f 1 ( t ′ ) • ◆❡❝❡ss❛r✐❧②✱ m ′ p 1 p 2 c ′ 2 2 c 1 ⊕ f 3 ( t ) = c ′ 4 ⊕ f 3 ( t ′ ) f 1 ( t ′ ) f 2 ( t ′ ) f 3 ( t ′ ) m ′ p 1 p 2 c ′ 4 4 ✞✌ ✮ ✹✜

  75. ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❘❛t✐♊♥❛❧❡ • ❉✐st✐♥❣✉✐s❀❡r D ♠❛❊❡s ✈❛r✐♩✉s q✉❡r✐❡s ❢♊r t✇♩ ❞✐✛❡r❡♥t t✇❡❛❊s✿ t ❛♥❞ t ′ • ❙✉♣♣♊s❡ ✐t ♠❛❊❡s 4 q✉❡r✐❡s s✉❝❀ t❀❛t m 1 p 1 p 2 c 1 m 1 ⊕ f 1 ( t ) = m ′ 2 ⊕ f 1 ( t ′ ) f 1 ( t ) f 2 ( t ) f 3 ( t ) c ′ 2 ⊕ f 3 ( t ′ ) = c 3 ⊕ f 3 ( t ) m 3 p 1 p 2 c 3 m 3 ⊕ f 1 ( t ) = m ′ 4 ⊕ f 1 ( t ′ ) • ◆❡❝❡ss❛r✐❧②✱ m ′ p 1 p 2 c ′ 2 2 c 1 ⊕ f 3 ( t ) = c ′ 4 ⊕ f 3 ( t ′ ) f 1 ( t ′ ) f 2 ( t ′ ) f 3 ( t ′ ) m ′ p 1 p 2 c ′ • ❙t❛t❡❞ ❞✐✛❡r❡♥t❧②✿ 4 4 m 1 ⊕ m ′ 2 = m 3 ⊕ m ′ 4 = f 1 ( t ) ⊕ f 1 ( t ′ ) c ′ 2 ⊕ c 3 = c 1 ⊕ c ′ 4 = f 3 ( t ) ⊕ f 3 ( t ′ ) ✞✌ ✮ ✹✜

  76. ❇✉t ❞♊❡s ♥♊t ❊♥♊✇ ❈❀♊♊s❡ t❀❡ ✬s ❛♥❞ ✬s s✉❝❀ t❀❛t ❢♊r ❛♥② ✱ t❀❡r❡ ❛r❡ q✉❛❞r✉♣❧❡s s✉❝❀ t❀❛t ✭❝♩sts q✉❡r✐❡s ❢♊r ❜♩t❀ ❛♥❞ ✮ s♩❧✉t✐♊♥s t♩ ❄ ✐❢ ✱ ♩t❀❡r✇✐s❡ ❊①t❡♥❞ t❀❡ ♥✉♠❜❡r ♊❢ q✉❡r✐❡s ❜② ❢❛❝t♩r t♩ ❡❧✐♠✐♥❛t❡ ❢❛❧s❡ ♣♊s✐t✐✈❡s ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❘❛t✐♊♥❛❧❡ • ❙t❛t❡❞ ❞✐✛❡r❡♥t❧②✿ m 1 ⊕ m ′ 2 = m 3 ⊕ m ′ 4 = f 1 ( t ) ⊕ f 1 ( t ′ ) c ′ 2 ⊕ c 3 = c 1 ⊕ c ′ 4 = f 3 ( t ) ⊕ f 3 ( t ′ ) m 1 p 1 p 2 c 1 f 1 ( t ) f 2 ( t ) f 3 ( t ) m 3 p 1 p 2 c 3 m ′ p 1 p 2 c ′ 2 2 f 1 ( t ′ ) f 2 ( t ′ ) f 3 ( t ′ ) m ′ p 1 p 2 c ′ 4 4 ✞✜ ✮ ✹✜

  77. ❈❀♊♊s❡ t❀❡ ✬s ❛♥❞ ✬s s✉❝❀ t❀❛t ❢♊r ❛♥② ✱ t❀❡r❡ ❛r❡ q✉❛❞r✉♣❧❡s s✉❝❀ t❀❛t ✭❝♩sts q✉❡r✐❡s ❢♊r ❜♩t❀ ❛♥❞ ✮ s♩❧✉t✐♊♥s t♩ ❄ ✐❢ ✱ ♩t❀❡r✇✐s❡ ❊①t❡♥❞ t❀❡ ♥✉♠❜❡r ♊❢ q✉❡r✐❡s ❜② ❢❛❝t♩r t♩ ❡❧✐♠✐♥❛t❡ ❢❛❧s❡ ♣♊s✐t✐✈❡s ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❘❛t✐♊♥❛❧❡ • ❙t❛t❡❞ ❞✐✛❡r❡♥t❧②✿ m 1 ⊕ m ′ 2 = m 3 ⊕ m ′ 4 = f 1 ( t ) ⊕ f 1 ( t ′ ) c ′ 2 ⊕ c 3 = c 1 ⊕ c ′ 4 = f 3 ( t ) ⊕ f 3 ( t ′ ) m 1 p 1 p 2 c 1 f 1 ( t ) f 2 ( t ) f 3 ( t ) • ❇✉t D ❞♊❡s ♥♊t ❊♥♊✇ f 1 ( t ) ⊕ f 1 ( t ′ ) m 3 p 1 p 2 c 3 m ′ p 1 p 2 c ′ 2 2 f 1 ( t ′ ) f 2 ( t ′ ) f 3 ( t ′ ) m ′ p 1 p 2 c ′ 4 4 ✞✜ ✮ ✹✜

  78. s♩❧✉t✐♊♥s t♩ ❄ ✐❢ ✱ ♩t❀❡r✇✐s❡ ❊①t❡♥❞ t❀❡ ♥✉♠❜❡r ♊❢ q✉❡r✐❡s ❜② ❢❛❝t♩r t♩ ❡❧✐♠✐♥❛t❡ ❢❛❧s❡ ♣♊s✐t✐✈❡s ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❘❛t✐♊♥❛❧❡ • ❙t❛t❡❞ ❞✐✛❡r❡♥t❧②✿ m 1 ⊕ m ′ 2 = m 3 ⊕ m ′ 4 = f 1 ( t ) ⊕ f 1 ( t ′ ) c ′ 2 ⊕ c 3 = c 1 ⊕ c ′ 4 = f 3 ( t ) ⊕ f 3 ( t ′ ) m 1 p 1 p 2 c 1 f 1 ( t ) f 2 ( t ) f 3 ( t ) • ❇✉t D ❞♊❡s ♥♊t ❊♥♊✇ f 1 ( t ) ⊕ f 1 ( t ′ ) m 3 p 1 p 2 c 3 • ❈❀♊♊s❡ t❀❡ m i ✬s ❛♥❞ m ′ i ✬s s✉❝❀ t❀❛t ❢♊r ❛♥② d ✱ t❀❡r❡ ❛r❡ 2 n q✉❛❞r✉♣❧❡s s✉❝❀ t❀❛t m 1 ⊕ m ′ 2 = m 3 ⊕ m ′ 4 = d m ′ p 1 p 2 c ′ 2 2 ✭❝♩sts 2 3 n/ 4 q✉❡r✐❡s ❢♊r ❜♩t❀ t ❛♥❞ t ′ ✮ f 1 ( t ′ ) f 2 ( t ′ ) f 3 ( t ′ ) m ′ p 1 p 2 c ′ 4 4 ✞✜ ✮ ✹✜

  79. ❊①t❡♥❞ t❀❡ ♥✉♠❜❡r ♊❢ q✉❡r✐❡s ❜② ❢❛❝t♩r t♩ ❡❧✐♠✐♥❛t❡ ❢❛❧s❡ ♣♊s✐t✐✈❡s ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❘❛t✐♊♥❛❧❡ • ❙t❛t❡❞ ❞✐✛❡r❡♥t❧②✿ m 1 ⊕ m ′ 2 = m 3 ⊕ m ′ 4 = f 1 ( t ) ⊕ f 1 ( t ′ ) c ′ 2 ⊕ c 3 = c 1 ⊕ c ′ 4 = f 3 ( t ) ⊕ f 3 ( t ′ ) m 1 p 1 p 2 c 1 f 1 ( t ) f 2 ( t ) f 3 ( t ) • ❇✉t D ❞♊❡s ♥♊t ❊♥♊✇ f 1 ( t ) ⊕ f 1 ( t ′ ) m 3 p 1 p 2 c 3 • ❈❀♊♊s❡ t❀❡ m i ✬s ❛♥❞ m ′ i ✬s s✉❝❀ t❀❛t ❢♊r ❛♥② d ✱ t❀❡r❡ ❛r❡ 2 n q✉❛❞r✉♣❧❡s s✉❝❀ t❀❛t m 1 ⊕ m ′ 2 = m 3 ⊕ m ′ 4 = d m ′ p 1 p 2 c ′ 2 2 ✭❝♩sts 2 3 n/ 4 q✉❡r✐❡s ❢♊r ❜♩t❀ t ❛♥❞ t ′ ✮ f 1 ( t ′ ) f 2 ( t ′ ) f 3 ( t ′ ) • E [ s♩❧✉t✐♊♥s t♩ c ′ 2 ⊕ c 3 = c 1 ⊕ c ′ 4 ] ❄ m ′ p 1 p 2 c ′ 2 ✐❢ d = f 1 ( t ) ⊕ f 1 ( t ′ ) ✱ 1 ♩t❀❡r✇✐s❡ 4 4 ✞✜ ✮ ✹✜

  80. ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❘❛t✐♊♥❛❧❡ • ❙t❛t❡❞ ❞✐✛❡r❡♥t❧②✿ m 1 ⊕ m ′ 2 = m 3 ⊕ m ′ 4 = f 1 ( t ) ⊕ f 1 ( t ′ ) c ′ 2 ⊕ c 3 = c 1 ⊕ c ′ 4 = f 3 ( t ) ⊕ f 3 ( t ′ ) m 1 p 1 p 2 c 1 f 1 ( t ) f 2 ( t ) f 3 ( t ) • ❇✉t D ❞♊❡s ♥♊t ❊♥♊✇ f 1 ( t ) ⊕ f 1 ( t ′ ) m 3 p 1 p 2 c 3 • ❈❀♊♊s❡ t❀❡ m i ✬s ❛♥❞ m ′ i ✬s s✉❝❀ t❀❛t ❢♊r ❛♥② d ✱ t❀❡r❡ ❛r❡ 2 n q✉❛❞r✉♣❧❡s s✉❝❀ t❀❛t m 1 ⊕ m ′ 2 = m 3 ⊕ m ′ 4 = d m ′ p 1 p 2 c ′ 2 2 ✭❝♩sts 2 3 n/ 4 q✉❡r✐❡s ❢♊r ❜♩t❀ t ❛♥❞ t ′ ✮ f 1 ( t ′ ) f 2 ( t ′ ) f 3 ( t ′ ) • E [ s♩❧✉t✐♊♥s t♩ c ′ 2 ⊕ c 3 = c 1 ⊕ c ′ 4 ] ❄ m ′ p 1 p 2 c ′ 2 ✐❢ d = f 1 ( t ) ⊕ f 1 ( t ′ ) ✱ 1 ♩t❀❡r✇✐s❡ 4 4 • ❊①t❡♥❞ t❀❡ ♥✉♠❜❡r ♊❢ q✉❡r✐❡s ❜② ❢❛❝t♩r n 1 / 2 t♩ ❡❧✐♠✐♥❛t❡ ❢❛❧s❡ ♣♊s✐t✐✈❡s ✞✜ ✮ ✹✜

  81. ❊①♣❡r✐♠❡♥t❛❧ ❱❡r✐✜❝❛t✐♊♥ ❙♠❛❧❧✲s❝❛❧❡ ✐♠♣❧❡♠❡♥t❛t✐♊♥ ❢♊r ✐s t❀❡ ♥✉♠❜❡r ♊❢ ❀✐ts ✐♥ r❡❛❧ ✇♩r❧❞ ❢♊r ✐♥ ✐❞❡❛❧ ✇♩r❧❞ ❢♊r r❛♥❞♊♠ r❛♥❞♊♠ ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❱❡r✐✜❝❛t✐♊♥ ❚❀❡♊r❡t✐❝❛❧ ❱❡r✐✜❝❛t✐♊♥ • ❆ss✉♠✐♥❣ n ≥ 27 ✱ t❀❡ s✉❝❝❡ss ♣r♩❜❛❜✐❧✐t② ♊❢ D ✐s ❛t ❧❡❛st 1 / 2 ï¿œ ï¿œ ï¿œ ï¿œ π = 1 D ï¿œ E k = 1 D ï¿œ • ❆♥❛❧②s✐s ❝♊♥s✐sts ♊❢ ♣r♊♣❡r❧② ❜♊✉♥❞✐♥❣ Pr ❛♥❞ Pr ✞✟ ✮ ✹✜

  82. ■♠♣r♊✈❡❞ ❆tt❛❝❊✿ ❱❡r✐✜❝❛t✐♊♥ ❚❀❡♊r❡t✐❝❛❧ ❱❡r✐✜❝❛t✐♊♥ • ❆ss✉♠✐♥❣ n ≥ 27 ✱ t❀❡ s✉❝❝❡ss ♣r♩❜❛❜✐❧✐t② ♊❢ D ✐s ❛t ❧❡❛st 1 / 2 ï¿œ ï¿œ ï¿œ ï¿œ π = 1 D ï¿œ E k = 1 D ï¿œ • ❆♥❛❧②s✐s ❝♊♥s✐sts ♊❢ ♣r♊♣❡r❧② ❜♊✉♥❞✐♥❣ Pr ❛♥❞ Pr ❊①♣❡r✐♠❡♥t❛❧ ❱❡r✐✜❝❛t✐♊♥ • ❙♠❛❧❧✲s❝❛❧❡ ✐♠♣❧❡♠❡♥t❛t✐♊♥ ❢♊r n = 16 , 20 , 24 • N d ✐s t❀❡ ♥✉♠❜❡r ♊❢ ❀✐ts c ′ 2 ⊕ c 3 = c 1 ⊕ c ′ 4 N d ✐♥ r❡❛❧ ✇♩r❧❞ ❢♊r d = N d ✐♥ ✐❞❡❛❧ ✇♩r❧❞ ❢♊r d = n 1 / 2 ≈ f 1 ( t ) ⊕ f 1 ( t ′ ) f 1 ( t ) ⊕ f 1 ( t ′ ) n q r❛♥❞♊♠ r❛♥❞♊♠ 4 · 2 12 16 2 256 . 593750 129 . 781250 127 . 093750 127 . 375000 4 · 2 15 20 2 265 . 531250 133 . 312500 125 . 625000 128 . 750000 4 · 2 18 24 2 246 . 750000 131 . 375000 120 . 625000 129 . 875000 ✞✟ ✮ ✹✜

  83. ❖✉t❧✐♥❡ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ • ■♥t✉✐t✐♊♥ • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② ❇❡②♊♥❞ ❇✐rt❀❞❛② ❇♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ❚✐❣❀t ❙❡❝✉r✐t② ♊❢ ❈❛s❝❛❞❡❞ LRW 2 ❄ • ■♠♣r♊✈❡❞ ❆tt❛❝❩ • ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❇♊✉♥❞ ❈♊♥❝❧✉s✐♊♥ ✹✵ ✮ ✹✜

  84. ❈❛s❝❛❞❡❞ ✐s s❡❝✉r❡ ✉♣ t♩ ❡✈❛❧✉❛t✐♊♥s ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ❇♊✉♥❞ • ❈❛s❝❛❞❡❞ LRW 2 ✿ h 1 ( t ) h 1 ( t ) ⊕ h 2 ( t ) h 2 ( t ) m E k 1 E k 2 c • E k i ❛r❡ ❙P❘P✲s❡❝✉r❡ • h i ❛r❡ 4 ✲✇✐s❡ ✐♥❞❡♣❡♥❞❡♥t ❳❖❘✲✉♥✐✈❡rs❛❧ ❀❛s❀ • ◆♩ t✇❡❛❊ ✐s q✉❡r✐❡❞ ♠♩r❡ t❀❛♥ 2 n/ 4 t✐♠❡s ✹✶ ✮ ✹✜

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