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r t t t t r t s

rt ttt rt - PowerPoint PPT Presentation

Feb 01, 2024 •171 likes •1.98k views

rt ttt rt s rt rst trs

  1. ●❡♥❡r✐❝❛❧❧② ✐♥s❡❝✉r❡ ▌✐❧❞❧② ✐♥s❡❝✉r❡ ▌♊st s❡❝✉r❡ ✈❛r✐❛♥t ▌❆❈ P❛❞❞✐♥❣ ♩r❛❝❧❡ ❈✐♣❀❡rt❡①t ✐♥t❡❣r✐t② ❛tt❛❝❩ ●❡♥❡r✐❝ ❈♊♠♣♊s✐t✐♊♥ • ●❡♥❡r✐❝ ❝♊♥str✉❝t✐♊♥s ❢♊r ❆❊✿ • ❊♥❝ ✰ ▌❆❈ ❂ ❆❊ • ❇❡❧❧❛r❡ ❛♥❞ ◆❛♠♣r❡♠♣r❡ ✭✷✵✵✵✮✿ ✾ ❜❛s✐❝ ❛♣♣r♊❛❝❀❡s ❊✫▌ â–Œt❊ ❊tâ–Œ m m m Enc k MAC l MAC l Enc k Enc k MAC l c c c t t t • ❯s❡❞ ✐♥ ❙❙❍ • ❯s❡❞ ✐♥ ❚▲❙ • ❯s❡❞ ✐♥ ■P❙❡❝ ✶✶ ✮ ✺✌

  2. ▌✐❧❞❧② ✐♥s❡❝✉r❡ ▌♊st s❡❝✉r❡ ✈❛r✐❛♥t P❛❞❞✐♥❣ ♩r❛❝❧❡ ❈✐♣❀❡rt❡①t ✐♥t❡❣r✐t② ❛tt❛❝❩ ●❡♥❡r✐❝ ❈♊♠♣♊s✐t✐♊♥ • ●❡♥❡r✐❝ ❝♊♥str✉❝t✐♊♥s ❢♊r ❆❊✿ • ❊♥❝ ✰ ▌❆❈ ❂ ❆❊ • ❇❡❧❧❛r❡ ❛♥❞ ◆❛♠♣r❡♠♣r❡ ✭✷✵✵✵✮✿ ✾ ❜❛s✐❝ ❛♣♣r♊❛❝❀❡s ❊✫▌ â–Œt❊ ❊tâ–Œ m m m Enc k MAC l MAC l Enc k Enc k MAC l c c c t t t • ❯s❡❞ ✐♥ ❙❙❍ • ❯s❡❞ ✐♥ ❚▲❙ • ❯s❡❞ ✐♥ ■P❙❡❝ • ●❡♥❡r✐❝❛❧❧② ✐♥s❡❝✉r❡ • ▌❆❈ L ( m ) = m ï¿œ t ✶✶ ✮ ✺✌

  3. ▌♊st s❡❝✉r❡ ✈❛r✐❛♥t ❈✐♣❀❡rt❡①t ✐♥t❡❣r✐t② ●❡♥❡r✐❝ ❈♊♠♣♊s✐t✐♊♥ • ●❡♥❡r✐❝ ❝♊♥str✉❝t✐♊♥s ❢♊r ❆❊✿ • ❊♥❝ ✰ ▌❆❈ ❂ ❆❊ • ❇❡❧❧❛r❡ ❛♥❞ ◆❛♠♣r❡♠♣r❡ ✭✷✵✵✵✮✿ ✾ ❜❛s✐❝ ❛♣♣r♊❛❝❀❡s ❊✫▌ â–Œt❊ ❊tâ–Œ m m m Enc k MAC l MAC l Enc k Enc k MAC l c c c t t t • ❯s❡❞ ✐♥ ❙❙❍ • ❯s❡❞ ✐♥ ❚▲❙ • ❯s❡❞ ✐♥ ■P❙❡❝ • ●❡♥❡r✐❝❛❧❧② ✐♥s❡❝✉r❡ • ▌✐❧❞❧② ✐♥s❡❝✉r❡ • ▌❆❈ L ( m ) = m ï¿œ t • P❛❞❞✐♥❣ ♩r❛❝❧❡ ❛tt❛❝❩ ✶✶ ✮ ✺✌

  4. ●❡♥❡r✐❝ ❈♊♠♣♊s✐t✐♊♥ • ●❡♥❡r✐❝ ❝♊♥str✉❝t✐♊♥s ❢♊r ❆❊✿ • ❊♥❝ ✰ ▌❆❈ ❂ ❆❊ • ❇❡❧❧❛r❡ ❛♥❞ ◆❛♠♣r❡♠♣r❡ ✭✷✵✵✵✮✿ ✾ ❜❛s✐❝ ❛♣♣r♊❛❝❀❡s ❊✫▌ â–Œt❊ ❊tâ–Œ m m m Enc k MAC l MAC l Enc k Enc k MAC l c c c t t t • ❯s❡❞ ✐♥ ❙❙❍ • ❯s❡❞ ✐♥ ❚▲❙ • ❯s❡❞ ✐♥ ■P❙❡❝ • ●❡♥❡r✐❝❛❧❧② ✐♥s❡❝✉r❡ • ▌✐❧❞❧② ✐♥s❡❝✉r❡ • ▌♊st s❡❝✉r❡ ✈❛r✐❛♥t • ▌❆❈ L ( m ) = m ï¿œ t • P❛❞❞✐♥❣ ♩r❛❝❧❡ • ❈✐♣❀❡rt❡①t ✐♥t❡❣r✐t② ❛tt❛❝❩ ✶✶ ✮ ✺✌

  5. P❛r❛❧❧❡❧✐③❛❜❧❡ ❊✈❛❧✉❛t❡s ♊♥❧② ✭♥♊ ✮ Pr♊✈❛❜❧② s❡❝✉r❡ ✭✐❢ ✐s P❘P✮ ❱❡r② ❡✣❝✐❡♥t ✐♥ ❍❲ ❘❡❛s♊♥❛❜❧② ❡✣❝✐❡♥t ✐♥ ❙❲ ❲❀❛t ❀❛♣♣❡♥s ✐❢ ♥♊♥❝❡ ✐s r❡✲✉s❡❞❄ ●❈▌ ❢♊r 96 ✲❜✐t ♥♊♥❝❡ N • ▌❝●r❡✇ ❛♥❞ ❱✐❡❣❛ ✭✷✵✵✹✮ N ï¿œ 1 N ï¿œ 2 N ï¿œ 3 N ï¿œ ( m + 1) • ❊tâ–Œ ❞❡s✐❣♥ • ❲✐❞❡❧② ✉s❡❞ ✭❚▲❙✊✮ E K E K E K E K • P❛t❡♥t✲❢r❡❡ M 1 M 2 M m ENC C 1 C 2 C m A GHASH L MAC T ✶✷ ✮ ✺✌

  6. ❲❀❛t ❀❛♣♣❡♥s ✐❢ ♥♊♥❝❡ ✐s r❡✲✉s❡❞❄ ●❈▌ ❢♊r 96 ✲❜✐t ♥♊♥❝❡ N • ▌❝●r❡✇ ❛♥❞ ❱✐❡❣❛ ✭✷✵✵✹✮ N ï¿œ 1 N ï¿œ 2 N ï¿œ 3 N ï¿œ ( m + 1) • ❊tâ–Œ ❞❡s✐❣♥ • ❲✐❞❡❧② ✉s❡❞ ✭❚▲❙✊✮ E K E K E K E K • P❛t❡♥t✲❢r❡❡ M 1 M 2 M m ENC • P❛r❛❧❧❡❧✐③❛❜❧❡ C 1 C 2 C m • ❊✈❛❧✉❛t❡s E ♊♥❧② ✭♥♊ E − 1 ✮ • Pr♊✈❛❜❧② s❡❝✉r❡ A GHASH L ✭✐❢ E ✐s P❘P✮ MAC • ❱❡r② ❡✣❝✐❡♥t ✐♥ ❍❲ T • ❘❡❛s♊♥❛❜❧② ❡✣❝✐❡♥t ✐♥ ❙❲ ✶✷ ✮ ✺✌

  7. ●❈▌ ❢♊r 96 ✲❜✐t ♥♊♥❝❡ N • ▌❝●r❡✇ ❛♥❞ ❱✐❡❣❛ ✭✷✵✵✹✮ N ï¿œ 1 N ï¿œ 2 N ï¿œ 3 N ï¿œ ( m + 1) • ❊tâ–Œ ❞❡s✐❣♥ • ❲✐❞❡❧② ✉s❡❞ ✭❚▲❙✊✮ E K E K E K E K • P❛t❡♥t✲❢r❡❡ M 1 M 2 M m ENC • P❛r❛❧❧❡❧✐③❛❜❧❡ C 1 C 2 C m • ❊✈❛❧✉❛t❡s E ♊♥❧② ✭♥♊ E − 1 ✮ • Pr♊✈❛❜❧② s❡❝✉r❡ A GHASH L ✭✐❢ E ✐s P❘P✮ MAC • ❱❡r② ❡✣❝✐❡♥t ✐♥ ❍❲ T • ❘❡❛s♊♥❛❜❧② ❡✣❝✐❡♥t ✐♥ ❙❲ ❲❀❛t ❀❛♣♣❡♥s ✐❢ ♥♊♥❝❡ ✐s r❡✲✉s❡❞❄ ✶✷ ✮ ✺✌

  8. ■♥❀❡r✐ts ●❈▌ ❢❡❛t✉r❡s ❙❡❝✉r❡ ❛❣❛✐♥st ♥♊♥❝❡✲r❡✉s❡ Pr♊♊❢✿ ■✇❛t❛ ❛♥❞ ❙❡✉r✐♥ ✭✷✵✶✌✮ ●❈▌✲❙■❱ T +0 T +1 T +( m − 1) N • ●✉❡r♊♥ ❛♥❞ ▲✐♥❞❡❧❧ ✭✷✵✶✺✮ KeyGen E k E K E K E K • â–Œt❊ ❞❡s✐❣♥ • ❖♥❣♊✐♥❣ st❛♥❞❛r❞✐③❛t✐♊♥ C 1 C 2 C m ENC ( K, L ) ✭■❊❚❋ ❘❋❈✮ KEY M 1 M 2 M m • P❛t❡♥t✲❢r❡❡ A GHASH L N E K MAC T ✶✞ ✮ ✺✌

  9. ●❈▌✲❙■❱ T +0 T +1 T +( m − 1) N • ●✉❡r♊♥ ❛♥❞ ▲✐♥❞❡❧❧ ✭✷✵✶✺✮ KeyGen E k E K E K E K • â–Œt❊ ❞❡s✐❣♥ • ❖♥❣♊✐♥❣ st❛♥❞❛r❞✐③❛t✐♊♥ C 1 C 2 C m ENC ( K, L ) ✭■❊❚❋ ❘❋❈✮ KEY M 1 M 2 M m • P❛t❡♥t✲❢r❡❡ A GHASH L • ■♥❀❡r✐ts ●❈▌ ❢❡❛t✉r❡s • ❙❡❝✉r❡ ❛❣❛✐♥st ♥♊♥❝❡✲r❡✉s❡ N • Pr♊♊❢✿ ■✇❛t❛ ❛♥❞ ❙❡✉r✐♥ E K MAC ✭✷✵✶✌✮ T ✶✞ ✮ ✺✌

  10. ❖✉t❧✐♥❡ ●❡♥❡r✐❝ ❈♊♠♣♊s✐t✐♊♥ ▲✐♥❊ ❲✐t❀ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ ◆♊♥❝❡✲❘❡✉s❡ ❈♊♥❝❧✉s✐♊♥ ✶✹ ✮ ✺✌

  11. ❚✇❡❛❊✿ ✢❡①✐❜✐❧✐t② t♩ t❀❡ ❝✐♣❀❡r ❊❛❝❀ t✇❡❛❊ ❣✐✈❡s ❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♊♥ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs k m c E ✶✺ ✮ ✺✌

  12. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs k ï¿œ m c E t • ❚✇❡❛❊✿ ✢❡①✐❜✐❧✐t② t♩ t❀❡ ❝✐♣❀❡r • ❊❛❝❀ t✇❡❛❊ ❣✐✈❡s ❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♊♥ ✶✺ ✮ ✺✌

  13. tr✐❡s t♩ ❞❡t❡r♠✐♥❡ ✇❀✐❝❀ ♩r❛❝❧❡ ✐t ❝♊♠♠✉♥✐❝❛t❡s ✇✐t❀ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡r ❙❡❝✉r✐t② ï¿œ E k ï¿œ π IC tweakable blockcipher random tweakable permutation distinguisher D • ï¿œ E k s❀♊✉❧❞ ❧♩♩❩ ❧✐❊❡ r❛♥❞♊♠ ♣❡r♠✉t❛t✐♊♥ ❢♊r ❡✈❡r② t • ❉✐✛❡r❡♥t t✇❡❛❊s − → ♣s❡✉❞♊✲✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♊♥s ✶✻ ✮ ✺✌

  14. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡r ❙❡❝✉r✐t② ï¿œ E k π ï¿œ IC tweakable blockcipher random tweakable permutation distinguisher D • ï¿œ E k s❀♊✉❧❞ ❧♩♩❩ ❧✐❊❡ r❛♥❞♊♠ ♣❡r♠✉t❛t✐♊♥ ❢♊r ❡✈❡r② t • ❉✐✛❡r❡♥t t✇❡❛❊s − → ♣s❡✉❞♊✲✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♊♥s • D tr✐❡s t♩ ❞❡t❡r♠✐♥❡ ✇❀✐❝❀ ♩r❛❝❧❡ ✐t ❝♊♠♠✉♥✐❝❛t❡s ✇✐t❀ ï¿œ ￜￜ ï¿œ ï¿œ ï¿œ π − 1 = 1 ï¿œ ï¿œ E k , ï¿œ ï¿œ E − 1 Adv stprp D ï¿œ π, ï¿œ ( D ) = ï¿œ Pr D = 1 − Pr ï¿œ k ï¿œ E ✶✻ ✮ ✺✌

  15. ✐♥ ❈❆❊❙❆❘ ❑■❆❙❯✱ ❈❇❆✱ ❈❖❇❘❆✱ ✐❋❡❡❞✱ PrÞst✱ ❏♩❧t✐❩ ✱ ▌❛r❜❧❡✱ ❖▌❉ ✱ P❖❊❚ ✱ ▌✐♥❛❧♣❀❡r ❙❈❘❊❆▌ ✱ ❙❍❊▲▲ ✱ ❆❊❩ ✱ ❈❖P❆ ✮ ❉❡♊①②s ❊▲♠❉ ✱ ❖❈❇ ✱ ❖❚❘ ✜rst r♊✉♥❞✱ s❡❝♊♥❞ r♊✉♥❞ ✱ t❀✐r❞ r♊✉♥❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡r ❉❡s✐❣♥s ï¿œ ï¿œ E E ï¿œ E P E t ❉❡❞✐❝❛t❡❞ ❇❧♊❝❊❝✐♣❀❡r✲❇❛s❡❞ P❡r♠✉t❛t✐♊♥✲❇❛s❡❞ ✶✌ ✮ ✺✌

  16. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡r ❉❡s✐❣♥s ✐♥ ❈❆❊❙❆❘ ï¿œ ï¿œ E E ï¿œ E P E t ❉❡❞✐❝❛t❡❞ ❇❧♊❝❊❝✐♣❀❡r✲❇❛s❡❞ P❡r♠✉t❛t✐♊♥✲❇❛s❡❞ ❑■❆❙❯✱ ❈❇❆✱ ❈❖❇❘❆✱ ✐❋❡❡❞✱ PrÞst✱ ❏♩❧t✐❩ ✱ ▌❛r❜❧❡✱ ❖▌❉ ✱ P❖❊❚ ✱ ▌✐♥❛❧♣❀❡r ❙❈❘❊❆▌ ✱ ❙❍❊▲▲ ✱ ❆❊❩ ✱ ❈❖P❆ ✮ ❉❡♊①②s ❊▲♠❉ ✱ ❖❈❇ ✱ ❖❚❘ ✜rst r♊✉♥❞✱ s❡❝♊♥❞ r♊✉♥❞ ✱ t❀✐r❞ r♊✉♥❞ ✶✌ ✮ ✺✌

  17. ■♥t❡r♥❛❧❧② ❜❛s❡❞ ♊♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ❚✇❡❛❊ ✐s ✉♥✐q✉❡ ❢♊r ❡✈❡r② ❡✈❛❧✉❛t✐♊♥ ❉✐✛❡r❡♥t ❜❧♩❝❩s ❛❧✇❛②s tr❛♥s❢♊r♠❡❞ ✉♥❞❡r ❞✐✛❡r❡♥t t✇❡❛❊ ❚r✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②✿ ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✶✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ 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 T • ●❡♥❡r❛❧✐③❡❞ ❖❈❇ ❜② ❘♊❣❛✇❛② ❡t ❛❧✳ ❬❘❇❇❑✵✶✱❘♊❣✵✹✱❑❘✶✶❪ ✶✜ ✮ ✺✌

  18. ❚r✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②✿ ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✶✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ 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 T • ●❡♥❡r❛❧✐③❡❞ ❖❈❇ ❜② ❘♊❣❛✇❛② ❡t ❛❧✳ ❬❘❇❇❑✵✶✱❘♊❣✵✹✱❑❘✶✶❪ • ■♥t❡r♥❛❧❧② ❜❛s❡❞ ♊♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ï¿œ E • ❚✇❡❛❊ ( N, tweak ) ✐s ✉♥✐q✉❡ ❢♊r ❡✈❡r② ❡✈❛❧✉❛t✐♊♥ • ❉✐✛❡r❡♥t ❜❧♩❝❩s ❛❧✇❛②s tr❛♥s❢♊r♠❡❞ ✉♥❞❡r ❞✐✛❡r❡♥t t✇❡❛❊ ✶✜ ✮ ✺✌

  19. ❚r✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②✿ ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✶✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ 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 T • ●❡♥❡r❛❧✐③❡❞ ❖❈❇ ❜② ❘♊❣❛✇❛② ❡t ❛❧✳ ❬❘❇❇❑✵✶✱❘♊❣✵✹✱❑❘✶✶❪ • ■♥t❡r♥❛❧❧② ❜❛s❡❞ ♊♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ï¿œ E • ❚✇❡❛❊ ( N, tweak ) ✐s ✉♥✐q✉❡ ❢♊r ❡✈❡r② ❡✈❛❧✉❛t✐♊♥ • ❉✐✛❡r❡♥t ❜❧♩❝❩s ❛❧✇❛②s tr❛♥s❢♊r♠❡❞ ✉♥❞❡r ❞✐✛❡r❡♥t t✇❡❛❊ Adv ae E k ] ( σ ) AE [ ï¿œ ✶✜ ✮ ✺✌

  20. ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✶✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ N, t M1 N, t M2 N, t Md N, t Aa π ˜ π ˜ π ˜ π ˜ ˜ π π ˜ ˜ π C 1 C 2 C d T • ●❡♥❡r❛❧✐③❡❞ ❖❈❇ ❜② ❘♊❣❛✇❛② ❡t ❛❧✳ ❬❘❇❇❑✵✶✱❘♊❣✵✹✱❑❘✶✶❪ • ■♥t❡r♥❛❧❧② ❜❛s❡❞ ♊♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ï¿œ E • ❚✇❡❛❊ ( N, tweak ) ✐s ✉♥✐q✉❡ ❢♊r ❡✈❡r② ❡✈❛❧✉❛t✐♊♥ • ❉✐✛❡r❡♥t ❜❧♩❝❩s ❛❧✇❛②s tr❛♥s❢♊r♠❡❞ ✉♥❞❡r ❞✐✛❡r❡♥t t✇❡❛❊ • ❚r✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②✿ Adv ae E k ] ( σ ) ≀ Adv ae π ] ( σ ) AE [ ï¿œ AE [ ï¿œ ✶✜ ✮ ✺✌

  21. ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✶✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ N, t M1 N, t M2 N, t Md N, t Aa π ˜ π ˜ π ˜ ˜ π π ˜ π ˜ ˜ π C 1 C 2 C d T • ●❡♥❡r❛❧✐③❡❞ ❖❈❇ ❜② ❘♊❣❛✇❛② ❡t ❛❧✳ ❬❘❇❇❑✵✶✱❘♊❣✵✹✱❑❘✶✶❪ • ■♥t❡r♥❛❧❧② ❜❛s❡❞ ♊♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ï¿œ E • ❚✇❡❛❊ ( N, tweak ) ✐s ✉♥✐q✉❡ ❢♊r ❡✈❡r② ❡✈❛❧✉❛t✐♊♥ • ❉✐✛❡r❡♥t ❜❧♩❝❩s ❛❧✇❛②s tr❛♥s❢♊r♠❡❞ ✉♥❞❡r ❞✐✛❡r❡♥t t✇❡❛❊ • ❚r✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②✿ π ] ( σ ) + Adv stprp Adv ae E k ] ( σ ) ≀ Adv ae ( σ ) AE [ ï¿œ AE [ ï¿œ ï¿œ E ✶✜ ✮ ✺✌

  22. ❚❛❣ ❢♊r❣❡❞ ✇✐t❀ ♣r♩❜❛❜✐❧✐t② ❛t ♠♩st ❞❡s✐❣♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ❚♩ ❞♊✿ ◆♊♥❝❡ ✉♥✐q✉❡♥❡ss t✇❡❛❊ ✉♥✐q✉❡♥❡ss ❊♥❝r②♣t✐♊♥ ❝❛❧❧s ❜❡❀❛✈❡ ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥s✿ ❆✉t❀❡♥t✐❝❛t✐♊♥ ❜❡❀❛✈❡s ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥ ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✷✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ N, t M1 N, t M2 N, t Md N, t Aa π ˜ π ˜ π ˜ π ˜ ˜ π π ˜ π ˜ C 1 C 2 C d T ✶✟ ✮ ✺✌

  23. ❚❛❣ ❢♊r❣❡❞ ✇✐t❀ ♣r♩❜❛❜✐❧✐t② ❛t ♠♩st ❞❡s✐❣♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ❚♩ ❞♊✿ ❊♥❝r②♣t✐♊♥ ❝❛❧❧s ❜❡❀❛✈❡ ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥s✿ ❆✉t❀❡♥t✐❝❛t✐♊♥ ❜❡❀❛✈❡s ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥ ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✷✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ N, t M1 N, t M2 N, t Md N, t Aa π ˜ π ˜ π ˜ π ˜ ˜ π π ˜ ˜ π C 1 C 2 C d T • ◆♊♥❝❡ ✉♥✐q✉❡♥❡ss ⇒ t✇❡❛❊ ✉♥✐q✉❡♥❡ss ✶✟ ✮ ✺✌

  24. ❚❛❣ ❢♊r❣❡❞ ✇✐t❀ ♣r♩❜❛❜✐❧✐t② ❛t ♠♩st ❞❡s✐❣♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ❚♩ ❞♊✿ ❆✉t❀❡♥t✐❝❛t✐♊♥ ❜❡❀❛✈❡s ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥ ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✷✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ N, t M1 N, t M2 N, t Md N, t Aa π ˜ π ˜ π ˜ π ˜ π ˜ π ˜ ˜ π C 1 C 2 C d T • ◆♊♥❝❡ ✉♥✐q✉❡♥❡ss ⇒ t✇❡❛❊ ✉♥✐q✉❡♥❡ss • ❊♥❝r②♣t✐♊♥ ❝❛❧❧s ❜❡❀❛✈❡ ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥s✿ AE [ ï¿œ π ] = $ ✶✟ ✮ ✺✌

  25. ❞❡s✐❣♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ❚♩ ❞♊✿ ❚❛❣ ❢♊r❣❡❞ ✇✐t❀ ♣r♩❜❛❜✐❧✐t② ❛t ♠♩st ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✷✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ N, t M1 N, t M2 N, t Md N, t Aa π ˜ π ˜ π ˜ π ˜ π ˜ π ˜ ˜ π C 1 C 2 C d T • ◆♊♥❝❡ ✉♥✐q✉❡♥❡ss ⇒ t✇❡❛❊ ✉♥✐q✉❡♥❡ss • ❊♥❝r②♣t✐♊♥ ❝❛❧❧s ❜❡❀❛✈❡ ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥s✿ AE [ ï¿œ π ] = $ • ❆✉t❀❡♥t✐❝❛t✐♊♥ ❜❡❀❛✈❡s ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥ ✶✟ ✮ ✺✌

  26. ❞❡s✐❣♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ❚♩ ❞♊✿ ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✷✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ N, t M1 N, t M2 N, t Md N, t Aa π ˜ π ˜ π ˜ π ˜ ˜ π π ˜ ˜ π C 1 C 2 C d T • ◆♊♥❝❡ ✉♥✐q✉❡♥❡ss ⇒ t✇❡❛❊ ✉♥✐q✉❡♥❡ss • ❊♥❝r②♣t✐♊♥ ❝❛❧❧s ❜❡❀❛✈❡ ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥s✿ AE [ ï¿œ π ] = $ • ❆✉t❀❡♥t✐❝❛t✐♊♥ ❜❡❀❛✈❡s ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥ • ❚❛❣ ❢♊r❣❡❞ ✇✐t❀ ♣r♩❜❛❜✐❧✐t② ❛t ♠♩st 1 / (2 n − 1) ✶✟ ✮ ✺✌

  27. ❞❡s✐❣♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ❚♩ ❞♊✿ ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✷✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ N, t M1 N, t M2 N, t Md N, t Aa π ˜ π ˜ π ˜ π ˜ ˜ π π ˜ ˜ π C 1 C 2 C d T • ◆♊♥❝❡ ✉♥✐q✉❡♥❡ss ⇒ t✇❡❛❊ ✉♥✐q✉❡♥❡ss • ❊♥❝r②♣t✐♊♥ ❝❛❧❧s ❜❡❀❛✈❡ ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥s✿ AE [ ï¿œ π ] = $ • ❆✉t❀❡♥t✐❝❛t✐♊♥ ❜❡❀❛✈❡s ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥ • ❚❛❣ ❢♊r❣❡❞ ✇✐t❀ ♣r♩❜❛❜✐❧✐t② ❛t ♠♩st 1 / (2 n − 1) π ] ( σ ) ≀ 1 / (2 n − 1) Adv ae AE [ ï¿œ ✶✟ ✮ ✺✌

  28. ❊①❛♠♣❧❡ ❯s❡ ✐♥ ❖❈❇① ✭✷✎✷✮ A 1 A 2 A a ⊕ M i M 1 M 2 M d N, t A1 N, t A2 N, t M ⊕ N, t M1 N, t M2 N, t Md N, t Aa π ˜ π ˜ π ˜ π ˜ ˜ π π ˜ ˜ π C 1 C 2 C d ❞❡s✐❣♥ t✇❡❛❊❛❜❧❡ ❜❧♊❝❊❝✐♣❀❡r ❚♩ ❞♊✿ T • ◆♊♥❝❡ ✉♥✐q✉❡♥❡ss ⇒ t✇❡❛❊ ✉♥✐q✉❡♥❡ss • ❊♥❝r②♣t✐♊♥ ❝❛❧❧s ❜❡❀❛✈❡ ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥s✿ AE [ ï¿œ π ] = $ • ❆✉t❀❡♥t✐❝❛t✐♊♥ ❜❡❀❛✈❡s ❧✐❊❡ r❛♥❞♊♠ ❢✉♥❝t✐♊♥ • ❚❛❣ ❢♊r❣❡❞ ✇✐t❀ ♣r♩❜❛❜✐❧✐t② ❛t ♠♩st 1 / (2 n − 1) π ] ( σ ) ≀ 1 / (2 n − 1) Adv ae AE [ ï¿œ ✶✟ ✮ ✺✌

  29. ❉❡❞✐❝❛t❡❞ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs • ❍❛st② P✉❞❞✐♥❣ ❈✐♣❀❡r ❬❙❝❀✟✜❪ • ❆❊❙ s✉❜♠✐ss✐♊♥✱ ✏✜rst t✇❡❛❊❛❜❧❡ ❝✐♣❀❡r✑ • ▌❡r❝② ❬❈r♊✵✶❪ • ❉✐s❊ ❡♥❝r②♣t✐♊♥ • ❚❀r❡❡✜s❀ ❬❋▲❙✰✵✌❪ • ❙❍❆✲✞ s✉❜♠✐ss✐♊♥ ❙❊❡✐♥ • ❚❲❊❆❑❊❚ ❢r❛♠❡✇♊r❊ ❬❏◆P✶✹❪ • ❋♩✉r ❈❆❊❙❆❘ s✉❜♠✐ss✐♊♥s • ❙❑■◆◆❹ ✫ ▌❆◆❚■❙ ✷✵ ✮ ✺✌

  30. ❙❡❝✉r✐t② ♠❡❛s✉r❡❞ t❀r♊✉❣❀ ❝r②♣t❛♥❛❧②s✐s ❖✉r ❢♊❝✉s✿ ♠♩❞✉❧❛r ❞❡s✐❣♥ ❚❲❊❆❑❊❚ ❋r❛♠❡✇♊r❊ • ❚❲❊❆❑❊❚ ❜② ❏❡❛♥ ❡t ❛❧✳ ❬❏◆P✶✹❪ ✿ ( k, t ) h h h · · · · · · g g g g f f f m c · · · · · · • f ✿ r♊✉♥❞ ❢✉♥❝t✐♊♥ • g ✿ s✉❜❊❡② ❝♊♠♣✉t❛t✐♊♥ • h ✿ tr❛♥s❢♊r♠❛t✐♊♥ ♊❢ ( k, t ) ✷✶ ✮ ✺✌

  31. ❚❲❊❆❑❊❚ ❋r❛♠❡✇♊r❊ • ❚❲❊❆❑❊❚ ❜② ❏❡❛♥ ❡t ❛❧✳ ❬❏◆P✶✹❪ ✿ ( k, t ) h h h · · · · · · g g g g f f f m c · · · · · · • f ✿ r♊✉♥❞ ❢✉♥❝t✐♊♥ • g ✿ s✉❜❊❡② ❝♊♠♣✉t❛t✐♊♥ • h ✿ tr❛♥s❢♊r♠❛t✐♊♥ ♊❢ ( k, t ) • ❙❡❝✉r✐t② ♠❡❛s✉r❡❞ t❀r♊✉❣❀ ❝r②♣t❛♥❛❧②s✐s • ❖✉r ❢♊❝✉s✿ ♠♩❞✉❧❛r ❞❡s✐❣♥ ✷✶ ✮ ✺✌

  32. ❖✉t❧✐♥❡ ●❡♥❡r✐❝ ❈♊♠♣♊s✐t✐♊♥ ▲✐♥❊ ❲✐t❀ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ • ■♥t✉✐t✐♊♥ • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② • ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ◆♊♥❝❡✲❘❡✉s❡ ❈♊♥❝❧✉s✐♊♥ ✷✷ ✮ ✺✌

  33. ❜❧❡♥❞ ✐t ✇✐t❀ t❀❡ ❊❡② ❜❧❡♥❞ ✐t ✇✐t❀ t❀❡ st❛t❡ ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k t ? m c E • ❈♊♥s✐❞❡r ❛ ❜❧♊❝❊❝✐♣❀❡r E ✇✐t❀ κ ✲❜✐t ❊❡② ❛♥❞ n ✲❜✐t st❛t❡ ❍♩✇ t♩ ♠✐♥❣❧❡ t❀❡ t✇❡❛❊ ✐♥t♩ t❀❡ ❡✈❛❧✉❛t✐♊♥❄ ✷✞ ✮ ✺✌

  34. ■♥t✉✐t✐♊♥✿ ❉❡s✐❣♥ k t ? m c E • ❈♊♥s✐❞❡r ❛ ❜❧♊❝❊❝✐♣❀❡r E ✇✐t❀ κ ✲❜✐t ❊❡② ❛♥❞ n ✲❜✐t st❛t❡ ❍♩✇ t♩ ♠✐♥❣❧❡ t❀❡ t✇❡❛❊ ✐♥t♩ t❀❡ ❡✈❛❧✉❛t✐♊♥❄ − − − − − − ← ← ❜❧❡♥❞ ✐t ✇✐t❀ t❀❡ ❊❡② ❜❧❡♥❞ ✐t ✇✐t❀ t❀❡ st❛t❡ ✷✞ ✮ ✺✌

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

  36. ❚❲❊❆❑❊❚ ❜❧❡♥❞✐♥❣ ❬❏◆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 ✷✹ ✮ ✺✌

  37. ■♥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❡ ❛❞✈❛♥❝❡❞ ✷✹ ✮ ✺✌

  38. ❚✇♊✲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❊ ✷✺ ✮ ✺✌

  39. ❚✇♊✲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 ) ✷✺ ✮ ✺✌

  40. ❚✇♊✲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 ✷✺ ✮ ✺✌

  41. ❚✇♊✲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 ✷✺ ✮ ✺✌

  42. ❚✇♊✲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 k ( t, c ) ⊕ ï¿œ E − 1 E − 1 k ( t ⊕ C, c ) = h ⊗ C ✷✺ ✮ ✺✌

  43. ■♥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 k ( t, c ) ⊕ ï¿œ E − 1 E − 1 k ( t ⊕ C, c ) = h ⊗ C • ❚✇♊✲s✐❞❡❞ ♠❛s❊✐♥❣ ♥❡❝❡ss❛r② ✷✺ ✮ ✺✌

  44. ▌❛❥♊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❡❄ ✷✻ ✮ ✺✌

  45. ▌❛❥♊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 ✷✻ ✮ ✺✌

  46. ▌❛❥♊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 ✷✻ ✮ ✺✌

  47. ▌❛❥♊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♊❜❧❡♠ ✷✻ ✮ ✺✌

  48. ■♥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♊❜❧❡♠ ✷✻ ✮ ✺✌

  49. ❙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 ✷✌ ✮ ✺✌

  50. ❙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 ✷✌ ✮ ✺✌

  51. ■♥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 ✷✌ ✮ ✺✌

  52. ❋♩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 ✷✜ ✮ ✺✌

  53. ❯♥❧✐❊❡❧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 ′ ) ✷✜ ✮ ✺✌

  54. ■♠♣❧✐❝❛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 ✷✜ ✮ ✺✌

  55. ❙❝❀❡♠❡ ❝❛♥ ❜❡ ❜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 ′ ✷✜ ✮ ✺✌

  56. ■♥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 ✷✜ ✮ ✺✌

  57. ❚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 ♣r❡s❡♥t❛t✐♊♥✿ 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❡ ✐♥✈♊❧✈❡❞ ✷✟ ✮ ✺✌

  58. ❆❧❧ ❝♊♥str✉❝t✐♊♥s ✐♥ t❀✐s ♣r❡s❡♥t❛t✐♊♥✿ 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 ✷✟ ✮ ✺✌

  59. ■♥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 ♣r❡s❡♥t❛t✐♊♥✿ s❡❝✉r❡ ✉♣ t♩ ≈ 2 n/ 2 ❡✈❛❧✉❛t✐♊♥s ✷✟ ✮ ✺✌

  60. ❖✉t❧✐♥❡ ●❡♥❡r✐❝ ❈♊♠♣♊s✐t✐♊♥ ▲✐♥❊ ❲✐t❀ ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡rs ❇❛s❡❞ ♊♥ ▌❛s❊✐♥❣ • ■♥t✉✐t✐♊♥ • ❙t❛t❡ ♊❢ t❀❡ ❆rt • ■♠♣r♊✈❡❞ ❊✣❝✐❡♥❝② • ■♠♣r♊✈❡❞ ❙❡❝✉r✐t② ◆♊♥❝❡✲❘❡✉s❡ ❈♊♥❝❧✉s✐♊♥ ✞✵ ✮ ✺✌

  61. 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 ✞✶ ✮ ✺✌

  62. ❚✇❡❛❊❛❜❧❡ ❇❧♊❝❊❝✐♣❀❡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 ✞✶ ✮ ✺✌

  63. ❖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 ✞✷ ✮ ✺✌

  64. ❯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✐♠♣❧✐✜❡❞✮ ✾✾ ✮ ✺✌

  65. 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 ✾✾ ✮ ✺✌

  66. 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 ❛❧✳ ❬❈▲❙✶✺❪ ✮ ✾✾ ✮ ✺✌

  67. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷ 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 T L = E k ( N ) ✞✹ ✮ ✺✌

  68. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷ 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 T L = E k ( N ) ✞✹ ✮ ✺✌

  69. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷ 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 T L = E k ( N ) ✞✹ ✮ ✺✌

  70. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷ 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 T L = E k ( N ) ✞✹ ✮ ✺✌

  71. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷ 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 T L = E k ( N ) ✞✹ ✮ ✺✌

  72. ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷ 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 T L = E k ( N ) ✞✹ ✮ ✺✌

  73. P♊✇❡r✐♥❣✲❯♣ ▌❛s❊✐♥❣ ✐♥ ❖❈❇✷ 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 T L = E k ( N ) • ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ • ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ • ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ • ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s ✞✹ ✮ ✺✌

  74. ■♥t❡r♠❡③③♊✿ ❲❀② ❙t❛rt ❛t 2 · E k ( N ) ❄ 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 T L = E k ( N ) • ❯♣❞❛t❡ ♊❢ ♠❛s❊✿ • ❙❀✐❢t ❛♥❞ ❝♊♥❞✐t✐♊♥❛❧ ❳❖❘ • ❱❛r✐❛❜❧❡ t✐♠❡ ❝♊♠♣✉t❛t✐♊♥ • ❊①♣❡♥s✐✈❡ ♊♥ ❝❡rt❛✐♥ ♣❧❛t❢♊r♠s ✞✺ ✮ ✺✌

  75. ❉✐st✐♥❣✉✐s❀❡r ❝❛♥ ♠❛❊❡ ✐♥✈❡rs❡ q✉❡r✐❡s P✉tt✐♥❣ ❣✐✈❡s ❉✐st✐♥❣✉✐s❀❡r ❊♥♊✇s s♩ ❧❡❛r♥s ✏s✉❜❊❡②✑ ■♥t❡r♠❡③③♊✿ ❲❀② ❙t❛rt ❛t 2 · E k ( N ) ❄ • ❙✉♣♣♊s❡ ✇❡ ✇♩✉❧❞ ♠❛s❊ ✇✐t❀ E k ( N ) ✿ E k ( N ) m E k c ✞✻ ✮ ✺✌

  76. P✉tt✐♥❣ ❣✐✈❡s ❉✐st✐♥❣✉✐s❀❡r ❊♥♊✇s s♩ ❧❡❛r♥s ✏s✉❜❊❡②✑ ■♥t❡r♠❡③③♊✿ ❲❀② ❙t❛rt ❛t 2 · E k ( N ) ❄ • ❙✉♣♣♊s❡ ✇❡ ✇♩✉❧❞ ♠❛s❊ ✇✐t❀ E k ( N ) ✿ E k ( N ) E − 1 m c k • ❉✐st✐♥❣✉✐s❀❡r ❝❛♥ ♠❛❊❡ ✐♥✈❡rs❡ q✉❡r✐❡s ✞✻ ✮ ✺✌

  77. ❉✐st✐♥❣✉✐s❀❡r ❊♥♊✇s s♩ ❧❡❛r♥s ✏s✉❜❊❡②✑ ■♥t❡r♠❡③③♊✿ ❲❀② ❙t❛rt ❛t 2 · E k ( N ) ❄ • ❙✉♣♣♊s❡ ✇❡ ✇♩✉❧❞ ♠❛s❊ ✇✐t❀ E k ( N ) ✿ E k ( N ) E − 1 N ⊕ E k ( N ) 0 k • ❉✐st✐♥❣✉✐s❀❡r ❝❛♥ ♠❛❊❡ ✐♥✈❡rs❡ q✉❡r✐❡s • P✉tt✐♥❣ c = 0 ❣✐✈❡s m = N ⊕ E k ( N ) ✞✻ ✮ ✺✌

  78. ■♥t❡r♠❡③③♊✿ ❲❀② ❙t❛rt ❛t 2 · E k ( N ) ❄ • ❙✉♣♣♊s❡ ✇❡ ✇♩✉❧❞ ♠❛s❊ ✇✐t❀ E k ( N ) ✿ E k ( N ) E − 1 N ⊕ E k ( N ) 0 k • ❉✐st✐♥❣✉✐s❀❡r ❝❛♥ ♠❛❊❡ ✐♥✈❡rs❡ q✉❡r✐❡s • P✉tt✐♥❣ c = 0 ❣✐✈❡s m = N ⊕ E k ( N ) • ❉✐st✐♥❣✉✐s❀❡r ❊♥♊✇s N s♩ ❧❡❛r♥s ✏s✉❜❊❡②✑ E k ( N ) ✞✻ ✮ ✺✌

  79. ❙✐♥❣❧❡ ❳❖❘ ▲♊❣❛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 ) ✞✌ ✮ ✺✌

  80. ●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✐♥❣✲✉♣ ❬❑❘✶✶❪ ✞✌ ✮ ✺✌

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