trt t rs - PowerPoint PPT Presentation

✐♥ ❈❆❊❙❆❘ ❑■❆❙❯✱ ❈❇❆✱ ❈❖❇❘❆✱ ✐❋❡❡❞✱ Prøst✱ ❏♦❧t✐❦ ✱ ▼❛r❜❧❡✱ ❖▼❉ ✱ P❖❊❚ ✱ ▼✐♥❛❧♣❤❡r ❙❈❘❊❆▼ ✱ ❙❍❊▲▲ ✱ ❆❊❩ ✱ ❈❖P❆ ✴ ❉❡♦①②s ❊▲♠❉ ✱ ❖❈❇ ✱ ❖❚❘ ✜rst r♦✉♥❞✱ s❡❝♦♥❞ r♦✉♥❞ ✱ t❤✐r❞ r♦✉♥❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡r ❉❡s✐❣♥s � � E E � E P E t ❉❡❞✐❝❛t❡❞ ❇❧♦❝❦❝✐♣❤❡r✲❇❛s❡❞ P❡r♠✉t❛t✐♦♥✲❇❛s❡❞ ✶✵ ✴ ✺✸

❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡r ❉❡s✐❣♥s ✐♥ ❈❆❊❙❆❘ � � E E � E P E t ❉❡❞✐❝❛t❡❞ ❇❧♦❝❦❝✐♣❤❡r✲❇❛s❡❞ P❡r♠✉t❛t✐♦♥✲❇❛s❡❞ ❑■❆❙❯✱ ❈❇❆✱ ❈❖❇❘❆✱ ✐❋❡❡❞✱ Prøst✱ ❏♦❧t✐❦ ✱ ▼❛r❜❧❡✱ ❖▼❉ ✱ P❖❊❚ ✱ ▼✐♥❛❧♣❤❡r ❙❈❘❊❆▼ ✱ ❙❍❊▲▲ ✱ ❆❊❩ ✱ ❈❖P❆ ✴ ❉❡♦①②s ❊▲♠❉ ✱ ❖❈❇ ✱ ❖❚❘ ✜rst r♦✉♥❞✱ s❡❝♦♥❞ r♦✉♥❞ ✱ t❤✐r❞ r♦✉♥❞ ✶✵ ✴ ✺✸

❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥ ✶✶ ✴ ✺✸

❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥ ✶✷ ✴ ✺✸

❉❡❞✐❝❛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 • ❙❑■◆◆❨ ✫ ▼❆◆❚■❙ ✶✸ ✴ ✺✸

❙❡❝✉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 ) ✶✹ ✴ ✺✸

❚❲❊❆❑❊❨ ❋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✐❣♥ ✶✹ ✴ ✺✸

❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥ ✶✺ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ✶ ❉❡t❡r♠✐♥❡ ❛♣♣r♦♣r✐❛t❡ s❡❝✉r✐t② ♠♦❞❡❧ ✷ ❉❡s✐❣♥ t❤❡ s❝❤❡♠❡ ✸ P❡r❢♦r♠ s❡❝✉r✐t② ❛♥❛❧②s✐s ✶✻ ✴ ✺✸

❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t② s❤♦✉❧❞ ❧♦♦❦ ❧✐❦❡ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② ❉✐✛❡r❡♥t t✇❡❛❦s ♣s❡✉❞♦✲✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❙tr♦♥❣ ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t② ❆❞✈❡rs❛r② ♠❛② ❤❛✈❡ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ❛❝❝❡ss t♦ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✶✿ ❙❡❝✉r✐t② ▼♦❞❡❧ ✳ k ✳ ← − − − − → � m c E t ✶✼ ✴ ✺✸

❙tr♦♥❣ ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t② ❆❞✈❡rs❛r② ♠❛② ❤❛✈❡ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ❛❝❝❡ss t♦ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✶✿ ❙❡❝✉r✐t② ▼♦❞❡❧ ✳ k ✳ ← − − − − → � m c E t ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t② • � E k s❤♦✉❧❞ ❧♦♦❦ ❧✐❦❡ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t • ❉✐✛❡r❡♥t t✇❡❛❦s − → ♣s❡✉❞♦✲✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ✶✼ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✶✿ ❙❡❝✉r✐t② ▼♦❞❡❧ ✳ k ✳ ← − − − − → � m c E t ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t② • � E k s❤♦✉❧❞ ❧♦♦❦ ❧✐❦❡ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t • ❉✐✛❡r❡♥t t✇❡❛❦s − → ♣s❡✉❞♦✲✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❙tr♦♥❣ ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t② • ❆❞✈❡rs❛r② ♠❛② ❤❛✈❡ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ❛❝❝❡ss t♦ � E ✶✼ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✶✿ ❙❡❝✉r✐t② ▼♦❞❡❧ ❊①❛♠♣❧❡ 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 • ❚❛❣ ❣❡♥❡r❛t✐♦♥✿ � E k ❡✈❛❧✉❛t❡❞ ✐♥ ❢♦r✇❛r❞ ❞✐r❡❝t✐♦♥ ♦♥❧② • ❊♥❝r②♣t✐♦♥✴❞❡❝r②♣t✐♦♥✿ � E k ❡✈❛❧✉❛t❡❞ ✐♥ ❜♦t❤ ❞✐r❡❝t✐♦♥s ✶✽ ✴ ✺✸

❜❧❡♥❞ ✐t ✇✐t❤ t❤❡ ❦❡② ❜❧❡♥❞ ✐t ✇✐t❤ t❤❡ st❛t❡ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ k t ? m c E • ❈♦♥s✐❞❡r ❛ ❜❧♦❝❦❝✐♣❤❡r E ✇✐t❤ κ ✲❜✐t ❦❡② ❛♥❞ n ✲❜✐t st❛t❡ ❍♦✇ t♦ ♠✐♥❣❧❡ t❤❡ t✇❡❛❦ ✐♥t♦ t❤❡ ❡✈❛❧✉❛t✐♦♥❄ ✶✾ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ 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❡ ✶✾ ✴ ✺✸

❋♦r ✲♠✐①✐♥❣✱ ❦❡② ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ✐♥ ❡✈❛❧✉❛t✐♦♥s ❙❝❤❡♠❡ ✐s ✐♥s❡❝✉r❡ ✐❢ ✐s ❊✈❡♥✲▼❛♥s♦✉r ❚❲❊❆❑❊❨ ❜❧❡♥❞✐♥❣ ✐s ♠♦r❡ ❛❞✈❛♥❝❡❞ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ k t m c E • ❇❧❡♥❞✐♥❣ t✇❡❛❦ ❛♥❞ ❦❡② ✇♦r❦s✳ ✳ ✳ • ✳ ✳ ✳ ❜✉t✿ ❝❛r❡❢✉❧ ✇✐t❤ r❡❧❛t❡❞✲❦❡② ❛tt❛❝❦s✦ ✷✵ ✴ ✺✸

❚❲❊❆❑❊❨ ❜❧❡♥❞✐♥❣ ✐s ♠♦r❡ ❛❞✈❛♥❝❡❞ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ 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 ✷✵ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ 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 • ❚❲❊❆❑❊❨ ❜❧❡♥❞✐♥❣ ✐s ♠♦r❡ ❛❞✈❛♥❝❡❞ ✷✵ ✴ ✺✸

❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r② ❙♦♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ ❙t✐❧❧ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ❛❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ k t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♦❢ t✇❡❛❦ ❛♥❞ st❛t❡ ❞♦❡s ♥♦t ✇♦r❦ ✷✶ ✴ ✺✸

❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r② ❙♦♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ ❙t✐❧❧ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ❛❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ k t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♦❢ t✇❡❛❦ ❛♥❞ st❛t❡ ❞♦❡s ♥♦t ✇♦r❦ • � E k ( t, m ) = � E k ( t ⊕ C, m ⊕ C ) ✷✶ ✴ ✺✸

❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r② ❙t✐❧❧ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ❛❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ k h ⊗ t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♦❢ t✇❡❛❦ ❛♥❞ st❛t❡ ❞♦❡s ♥♦t ✇♦r❦ • � E k ( t, m ) = � E k ( t ⊕ C, m ⊕ C ) • ❙♦♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ h ✷✶ ✴ ✺✸

❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r② ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ k h ⊗ t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♦❢ t✇❡❛❦ ❛♥❞ st❛t❡ ❞♦❡s ♥♦t ✇♦r❦ • � E k ( t, m ) = � E k ( t ⊕ C, m ⊕ C ) • ❙♦♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ h • ❙t✐❧❧ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ❛❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ � E − 1 k ✷✶ ✴ ✺✸

❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r② ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ k h ⊗ t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♦❢ t✇❡❛❦ ❛♥❞ st❛t❡ ❞♦❡s ♥♦t ✇♦r❦ • � E k ( t, m ) = � E k ( t ⊕ C, m ⊕ C ) • ❙♦♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ h • ❙t✐❧❧ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ❛❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ � E − 1 k • � k ( t, c ) ⊕ � E − 1 E − 1 k ( t ⊕ C, c ) = h ⊗ C ✷✶ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ k h ⊗ t h ⊗ t m c E • ❙✐♠♣❧❡ ❜❧❡♥❞✐♥❣ ♦❢ t✇❡❛❦ ❛♥❞ st❛t❡ ❞♦❡s ♥♦t ✇♦r❦ • � E k ( t, m ) = � E k ( t ⊕ C, m ⊕ C ) • ❙♦♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ h • ❙t✐❧❧ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ❛❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦ � E − 1 k • � k ( t, c ) ⊕ � E − 1 E − 1 k ( t ⊕ C, c ) = h ⊗ C • ❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r② ✷✶ ✴ ✺✸

▼❛❥♦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♦❜❧❡♠ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ k h ⊗ t h ⊗ t m c E • ❚✇♦✲s✐❞❡❞ s❡❝r❡t ♠❛s❦✐♥❣ s❡❡♠s t♦ ✇♦r❦ • ❈❛♥ ✇❡ ❣❡♥❡r❛❧✐③❡❄ ✷✷ ✴ ✺✸

▼❛❥♦r✐t② ♦❢ t✇❡❛❦❛❜❧❡ ❜❧♦❝❦❝✐♣❤❡rs ❢♦❧❧♦✇ ♠❛s❦✲ ✲♠❛s❦ ♣r✐♥❝✐♣❧❡ ❱❛r✐❛t✐♦♥ ✐♥ ♠❛s❦✐♥❣❄ ❉❡♣❡♥❞s ♦♥ ❢✉♥❝t✐♦♥s ❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ ❄ ❯s✉❛❧❧② ♥♦ ♣r♦❜❧❡♠ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ 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 ✷✷ ✴ ✺✸

▼❛❥♦r✐t② ♦❢ t✇❡❛❦❛❜❧❡ ❜❧♦❝❦❝✐♣❤❡rs ❢♦❧❧♦✇ ♠❛s❦✲ ✲♠❛s❦ ♣r✐♥❝✐♣❧❡ ❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ ❄ ❯s✉❛❧❧② ♥♦ ♣r♦❜❧❡♠ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ 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 ✷✷ ✴ ✺✸

▼❛❥♦r✐t② ♦❢ t✇❡❛❦❛❜❧❡ ❜❧♦❝❦❝✐♣❤❡rs ❢♦❧❧♦✇ ♠❛s❦✲ ✲♠❛s❦ ♣r✐♥❝✐♣❧❡ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ 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♦❜❧❡♠ ✷✷ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞ 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♦❜❧❡♠ ✷✷ ✴ ✺✸

❙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 ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • � E k s❤♦✉❧❞ ✏❧♦♦❦ ❧✐❦❡✑ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t • ❈♦♥s✐❞❡r ❛❞✈❡rs❛r② A t❤❛t ♠❛❦❡s q ❡✈❛❧✉❛t✐♦♥s ♦❢ � E k ✷✸ ✴ ✺✸

❙t❡♣ ✸❜✿ ❍♦✇ ♠❛♥② ❡✈❛❧✉❛t✐♦♥s ❞♦❡s ♥❡❡❞ ❛t ❧❡❛st❄ ❙t❡♣ ✸❜✿ ❇♦✐❧s ❞♦✇♥ t♦ ♣r♦✈❛❜❧❡ s❡❝✉r✐t② ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • � E k s❤♦✉❧❞ ✏❧♦♦❦ ❧✐❦❡✑ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t • ❈♦♥s✐❞❡r ❛❞✈❡rs❛r② A t❤❛t ♠❛❦❡s q ❡✈❛❧✉❛t✐♦♥s ♦❢ � E k • ❙t❡♣ ✸❛✿ • ❍♦✇ ♠❛♥② ❡✈❛❧✉❛t✐♦♥s ❞♦❡s A ♥❡❡❞ ❛t ♠♦st❄ ❙t❡♣ ✸❛✿ • ❇♦✐❧s ❞♦✇♥ t♦ ✜♥❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❦s ✷✸ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸✿ ❆♥❛❧②s✐s f 1 ( t ) f 2 ( t ) E k /P m c • � E k s❤♦✉❧❞ ✏❧♦♦❦ ❧✐❦❡✑ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t • ❈♦♥s✐❞❡r ❛❞✈❡rs❛r② A t❤❛t ♠❛❦❡s q ❡✈❛❧✉❛t✐♦♥s ♦❢ � E k • ❙t❡♣ ✸❛✿ • ❍♦✇ ♠❛♥② ❡✈❛❧✉❛t✐♦♥s ❞♦❡s A ♥❡❡❞ ❛t ♠♦st❄ ❙t❡♣ ✸❛✿ • ❇♦✐❧s ❞♦✇♥ t♦ ✜♥❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❦s • ❙t❡♣ ✸❜✿ • ❍♦✇ ♠❛♥② ❡✈❛❧✉❛t✐♦♥s ❞♦❡s A ♥❡❡❞ ❛t ❧❡❛st❄ ❙t❡♣ ✸❜✿ • ❇♦✐❧s ❞♦✇♥ t♦ ♣r♦✈❛❜❧❡ s❡❝✉r✐t② ✷✸ ✴ ✺✸

❋♦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 ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦ f 1 ( t ) f 2 ( t ) E k /P m c ✷✹ ✴ ✺✸

❯♥❧✐❦❡❧② t♦ ❤❛♣♣❡♥ ❢♦r r❛♥❞♦♠ ❢❛♠✐❧② ♦❢ ♣❡r♠✉t❛t✐♦♥s ■♠♣❧✐❝❛t✐♦♥ st✐❧❧ ❤♦❧❞s ✇✐t❤ ❞✐✛❡r❡♥❝❡ ①♦r❡❞ t♦ ❙❝❤❡♠❡ ❝❛♥ ❜❡ ❜r♦❦❡♥ ✐♥ ❡✈❛❧✉❛t✐♦♥s ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦ 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 ′ ) ✷✹ ✴ ✺✸

■♠♣❧✐❝❛t✐♦♥ st✐❧❧ ❤♦❧❞s ✇✐t❤ ❞✐✛❡r❡♥❝❡ ①♦r❡❞ t♦ ❙❝❤❡♠❡ ❝❛♥ ❜❡ ❜r♦❦❡♥ ✐♥ ❡✈❛❧✉❛t✐♦♥s ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦ 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 ✷✹ ✴ ✺✸

❙❝❤❡♠❡ ❝❛♥ ❜❡ ❜r♦❦❡♥ ✐♥ ❡✈❛❧✉❛t✐♦♥s ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦ 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 ′ ✷✹ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦ 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 ✷✹ ✴ ✺✸

❚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 ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❜✿ ❙❡❝✉r✐t② Pr♦♦❢ f 1 ( t ) f 2 ( t ) E k /P m c • ❚❤❡ ❢✉♥ st❛rts ❤❡r❡✦ • ▼♦r❡ t❡❝❤♥✐❝❛❧ ❛♥❞ ♦❢t❡♥ ♠♦r❡ ✐♥✈♦❧✈❡❞ ✷✺ ✴ ✺✸

❆❧❧ ❝♦♥str✉❝t✐♦♥s ✐♥ t❤✐s ♣r❡s❡♥t❛t✐♦♥✿ s❡❝✉r❡ ✉♣ t♦ ❡✈❛❧✉❛t✐♦♥s ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❜✿ ❙❡❝✉r✐t② Pr♦♦❢ 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 ✷✺ ✴ ✺✸

❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❜✿ ❙❡❝✉r✐t② Pr♦♦❢ 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 ✷✺ ✴ ✺✸

❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ • ❙t❛t❡ ♦❢ t❤❡ ❆rt • ■♠♣r♦✈❡❞ ❊✣❝✐❡♥❝② • ■♠♣r♦✈❡❞ ❙❡❝✉r✐t② ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥ ✷✻ ✴ ✺✸

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 ✷✼ ✴ ✺✸

❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡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 ✷✼ ✴ ✺✸

❖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 ✷✽ ✴ ✺✸

❯s❡❞ ✐♥ ❖❈❇✷✱ ✶✹ ❈❆❊❙❆❘ ❝❛♥❞✐❞❛t❡s✱ ❛♥❞ ❳❚❙ P❡r♠✉t❛t✐♦♥✲❜❛s❡❞ ✈❛r✐❛♥ts ✐♥ ▼✐♥❛❧♣❤❡r ❛♥❞ Prøst ✭❣❡♥❡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✐♠♣❧✐✜❡❞✮ ✷✾ ✴ ✺✸

P❡r♠✉t❛t✐♦♥✲❜❛s❡❞ ✈❛r✐❛♥ts ✐♥ ▼✐♥❛❧♣❤❡r ❛♥❞ Prøst ✭❣❡♥❡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✱ ❛♥❞ ❳❚❙ ✷✾ ✴ ✺✸

P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✭❳❊❳✮ • XEX ❜② ❘♦❣❛✇❛② ❬❘♦❣✵✹❪ ✿ 2 α 3 β 7 γ · ( k � N ⊕ P ( k � 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 ❛♥❞ Prøst ✭❣❡♥❡r❛❧✐③❡❞ ❜② ❈♦❣❧✐❛t✐ ❡t ❛❧✳ ❬❈▲❙✶✺❪ ✮ ✷✾ ✴ ✺✸

❯♣❞❛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 ) ✸✵ ✴ ✺✸

❯♣❞❛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 ) ✸✵ ✴ ✺✸

❯♣❞❛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 ) ✸✵ ✴ ✺✸

❯♣❞❛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 ) ✸✵ ✴ ✺✸

❯♣❞❛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 ) ✸✵ ✴ ✺✸

❯♣❞❛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 ) ✸✵ ✴ ✺✸

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 ✸✵ ✴ ✺✸

❙✐♥❣❧❡ ❳❖❘ ▲♦❣❛r✐t❤♠✐❝ ❛♠♦✉♥t ♦❢ ✜❡❧❞ ❞♦✉❜❧✐♥❣s ✭♣r❡❝♦♠♣✉t❡❞✮ ▼♦r❡ ❡✣❝✐❡♥t t❤❛♥ ♣♦✇❡r✐♥❣✲✉♣ ❬❑❘✶✶❪ ●r❛② ❈♦❞❡ ▼❛s❦✐♥❣ • ❖❈❇✶ ❛♥❞ ❖❈❇✸ ✉s❡ ●r❛② ❈♦❞❡s✿ � � α ⊕ ( α ≫ 1) · E k ( N ) E k m c • ( α, N ) ✐s t✇❡❛❦ • ❯♣❞❛t✐♥❣✿ G ( α ) = G ( α − 1) ⊕ 2 ntz ( α ) ✸✶ ✴ ✺✸

●r❛② ❈♦❞❡ ▼❛s❦✐♥❣ • ❖❈❇✶ ❛♥❞ ❖❈❇✸ ✉s❡ ●r❛② ❈♦❞❡s✿ � � 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✐♥❣✲✉♣ ❬❑❘✶✶❪ ✸✶ ✴ ✺✸

❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ • ❙t❛t❡ ♦❢ t❤❡ ❆rt • ■♠♣r♦✈❡❞ ❊✣❝✐❡♥❝② • ■♠♣r♦✈❡❞ ❙❡❝✉r✐t② ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥ ✸✷ ✴ ✺✸

❈♦♠❜✐♥❡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 � k ) P m c • ϕ i ❛r❡ ✜①❡❞ ▲❋❙❘s✱ ( α, β, γ, N ) ✐s t✇❡❛❦ ✭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 � 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 ✸✸ ✴ ✺✸

▼❛s❦❡❞ ❊✈❡♥✲▼❛♥s♦✉r ✭ MEM ✮ • 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 ❛❞✈❛♥t❛❣❡s ♦❢✿ • P♦✇❡r✐♥❣✲✉♣ ♠❛s❦✐♥❣ • ❲♦r❞✲❜❛s❡❞ ▲❋❙❘s • ❙✐♠♣❧❡r✱ ❝♦♥st❛♥t✲t✐♠❡ ✭❜② ❞❡❢❛✉❧t✮✱ ♠♦r❡ ❡✣❝✐❡♥t ✸✸ ✴ ✺✸

❙❛♠♣❧❡ ▲❋❙❘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 ❝❤♦✐❝❡ ♦❢ ▲❋❙❘ ✸✹ ✴ ✺✸

❲♦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)) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✴ ✺✸

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 ✸✹ ✴ ✺✸

✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹ s♦❧✈❡❞ ❜② r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❘♦❣❛✇❛② ❬❘♦❣✵✹❪ ❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮ s♦❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▼◆✶✻❪ MEM ✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❦✐♥❣ ❣♦❡s ✇❡❧❧ ❛s ❧♦♥❣ ❛s 0 � = ϕ γ ′ 2 ◦ ϕ β ′ ϕ γ 2 ◦ ϕ β 1 ◦ ϕ α ′ 1 ◦ ϕ α 0 ❢♦r ❛♥② ( α, β, γ ) � = ( α ′ , β ′ , γ ′ ) • ❈❤❛❧❧❡♥❣❡✿ s❡t ♣r♦♣❡r ❞♦♠❛✐♥ ❢♦r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✸✺ ✴ ✺✸

s♦❧✈❡❞ ❜② r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❘♦❣❛✇❛② ❬❘♦❣✵✹❪ ❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮ s♦❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▼◆✶✻❪ MEM ✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❦✐♥❣ ❣♦❡s ✇❡❧❧ ❛s ❧♦♥❣ ❛s 0 � = ϕ γ ′ 2 ◦ ϕ β ′ ϕ γ 2 ◦ ϕ β 1 ◦ ϕ α ′ 1 ◦ ϕ α 0 ❢♦r ❛♥② ( α, β, γ ) � = ( α ′ , β ′ , γ ′ ) • ❈❤❛❧❧❡♥❣❡✿ s❡t ♣r♦♣❡r ❞♦♠❛✐♥ ❢♦r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹ ✸✺ ✴ ✺✸

r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮ s♦❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▼◆✶✻❪ MEM ✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❦✐♥❣ ❣♦❡s ✇❡❧❧ ❛s ❧♦♥❣ ❛s 0 � = ϕ γ ′ 2 ◦ ϕ β ′ ϕ γ 2 ◦ ϕ β 1 ◦ ϕ α ′ 1 ◦ ϕ α 0 ❢♦r ❛♥② ( α, β, γ ) � = ( α ′ , β ′ , γ ′ ) • ❈❤❛❧❧❡♥❣❡✿ s❡t ♣r♦♣❡r ❞♦♠❛✐♥ ❢♦r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹ � �� � s♦❧✈❡❞ ❜② ❘♦❣❛✇❛② ❬❘♦❣✵✹❪ ✸✺ ✴ ✺✸

s♦❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▼◆✶✻❪ MEM ✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❦✐♥❣ ❣♦❡s ✇❡❧❧ ❛s ❧♦♥❣ ❛s 0 � = ϕ γ ′ 2 ◦ ϕ β ′ ϕ γ 2 ◦ ϕ β 1 ◦ ϕ α ′ 1 ◦ ϕ α 0 ❢♦r ❛♥② ( α, β, γ ) � = ( α ′ , β ′ , γ ′ ) • ❈❤❛❧❧❡♥❣❡✿ s❡t ♣r♦♣❡r ❞♦♠❛✐♥ ❢♦r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹ � �� � � �� � s♦❧✈❡❞ ❜② r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❘♦❣❛✇❛② ❬❘♦❣✵✹❪ ❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮ ✸✺ ✴ ✺✸

MEM ✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣ • ■♥t✉✐t✐✈❡❧②✱ ♠❛s❦✐♥❣ ❣♦❡s ✇❡❧❧ ❛s ❧♦♥❣ ❛s 0 � = ϕ γ ′ 2 ◦ ϕ β ′ ϕ γ 2 ◦ ϕ β 1 ◦ ϕ α ′ 1 ◦ ϕ α 0 ❢♦r ❛♥② ( α, β, γ ) � = ( α ′ , β ′ , γ ′ ) • ❈❤❛❧❧❡♥❣❡✿ s❡t ♣r♦♣❡r ❞♦♠❛✐♥ ❢♦r ( α, β, γ ) • ❘❡q✉✐r❡s ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠s ✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹ � �� � � �� � s♦❧✈❡❞ ❜② r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❘♦❣❛✇❛② ❬❘♦❣✵✹❪ ❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮ � �� � s♦❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▼◆✶✻❪ ✸✺ ✴ ✺✸

❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❖PP A 0 A 1 A a –1 ⊕ M i M 0 M 1 M d –1 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 ) ϕ 0 ( L ) ϕ 1 ( L ) P P P P P P P 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 ) ϕ 0 ( L ) ϕ 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♦✉♥❞ ❇▲❆❑❊✷❜ ✸✻ ✴ ✺✸

❆♣♣❧✐❝❛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♦✉♥❞ ❇▲❆❑❊✷❜ ✸✼ ✴ ✺✸

❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ • ❙t❛t❡ ♦❢ t❤❡ ❆rt • ■♠♣r♦✈❡❞ ❊✣❝✐❡♥❝② • ■♠♣r♦✈❡❞ ❙❡❝✉r✐t② ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥ ✸✽ ✴ ✺✸

✶ ✏❲❡❛❦✑ ✐♥s❡❝✉r❡ ✷ ✏◆♦r♠❛❧✑ s✐♥❣❧❡✲❦❡② s❡❝✉r❡ ✸ ✏❙tr♦♥❣✑ r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ❙❡❝✉r✐t② ♦❢ str♦♥❣❧② ❞❡♣❡♥❞s ♦♥ ❝❤♦✐❝❡ ♦❢ ❳P❳ • XPX ❜② ▼❡♥♥✐♥❦ ❬▼❡♥✶✻❪ ✿ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c • ( t 11 , t 12 , t 21 , t 22 ) ❢r♦♠ s♦♠❡ t✇❡❛❦ s❡t T ⊆ ( { 0 , 1 } n ) 4 • T ❝❛♥ ✭st✐❧❧✮ ❜❡ ❛♥② s❡t ✸✾ ✴ ✺✸

✶ ✏❲❡❛❦✑ ✐♥s❡❝✉r❡ ✷ ✏◆♦r♠❛❧✑ s✐♥❣❧❡✲❦❡② s❡❝✉r❡ ✸ ✏❙tr♦♥❣✑ r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ❳P❳ • XPX ❜② ▼❡♥♥✐♥❦ ❬▼❡♥✶✻❪ ✿ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c • ( 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 ✸✾ ✴ ✺✸

✷ ✏◆♦r♠❛❧✑ s✐♥❣❧❡✲❦❡② s❡❝✉r❡ ✸ ✏❙tr♦♥❣✑ r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ❳P❳ • XPX ❜② ▼❡♥♥✐♥❦ ❬▼❡♥✶✻❪ ✿ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c • ( 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 − → ✐♥s❡❝✉r❡ ✸✾ ✴ ✺✸

✸ ✏❙tr♦♥❣✑ r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ❳P❳ • XPX ❜② ▼❡♥♥✐♥❦ ❬▼❡♥✶✻❪ ✿ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c • ( 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 − → ✐♥s❡❝✉r❡ ✷ ✏◆♦r♠❛❧✑ T − → s✐♥❣❧❡✲❦❡② s❡❝✉r❡ ✸✾ ✴ ✺✸

❳P❳ • XPX ❜② ▼❡♥♥✐♥❦ ❬▼❡♥✶✻❪ ✿ t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) P m c • ( 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 − → ✐♥s❡❝✉r❡ ✷ ✏◆♦r♠❛❧✑ T − → s✐♥❣❧❡✲❦❡② s❡❝✉r❡ ✸ ✏❙tr♦♥❣✑ T − → r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ✸✾ ✴ ✺✸

✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s t 11 k ⊕ t 12 P ( k ) t 21 k ⊕ t 22 P ( k ) m P c ✹✵ ✴ ✺✸

✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s 0 k ⊕ 0 P ( k ) 0 k ⊕ 0 P ( k ) m P (0 , 0 , 0 , 0) ∈ T ✹✵ ✴ ✺✸

✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s 0 k ⊕ 0 P ( k ) 0 k ⊕ 0 P ( k ) m P P ( m ) (0 , 0 , 0 , 0) ∈ T = ⇒ XPX k ((0 , 0 , 0 , 0) , m ) = P ( m ) ✹✵ ✴ ✺✸

✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ❳P❳✿ ❲❡❛❦ ❚✇❡❛❦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 ✹✵ ✴ ✺✸

✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ❳P❳✿ ❲❡❛❦ ❚✇❡❛❦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 ) ✹✵ ✴ ✺✸

✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡ ✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ❳P❳✿ ❲❡❛❦ ❚✇❡❛❦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 ) · · · · · · · · · ✹✵ ✴ ✺✸