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❚❲❊❆❑❊❨ ❜❧❡♥❞✐♥❣ ❬❏◆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 ✶✵ ✴ ✹✽

■♥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❡ ❛❞✈❛♥❝❡❞ ✶✵ ✴ ✹✽

❚✇♦✲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❦ ✶✶ ✴ ✹✽

❚✇♦✲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❦ • � E k ( t, m ) = � E k ( t ⊕ C, m ⊕ C ) ✶✶ ✴ ✹✽

❚✇♦✲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❦ • � E k ( t, m ) = � E k ( t ⊕ C, m ⊕ C ) • ❙♦♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ h ✶✶ ✴ ✹✽

❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r② ■♥t✉✐t✐♦♥✿ ❉❡s✐❣♥ 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 ■♥t✉✐t✐♦♥✿ ❉❡s✐❣♥ 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 • � E − 1 k ( t, c ) ⊕ � E − 1 k ( t ⊕ C, c ) = h ⊗ C ✶✶ ✴ ✹✽

■♥t✉✐t✐♦♥✿ ❉❡s✐❣♥ 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 • � E − 1 k ( t, c ) ⊕ � 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♦❜❧❡♠ ■♥t✉✐t✐♦♥✿ ❉❡s✐❣♥ 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♦❜❧❡♠ ■♥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 ✶✷ ✴ ✹✽

▼❛❥♦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 ✶✷ ✴ ✹✽

▼❛❥♦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♦❜❧❡♠ ✶✷ ✴ ✹✽

■♥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♦❜❧❡♠ ✶✷ ✴ ✹✽

❙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 • � E k s❤♦✉❧❞ ✏❧♦♦❦ ❧✐❦❡✑ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t • ❈♦♥s✐❞❡r ❛❞✈❡rs❛r② D t❤❛t ♠❛❦❡s q ❡✈❛❧✉❛t✐♦♥s ♦❢ � E k ✶✸ ✴ ✹✽

❙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 • � E k s❤♦✉❧❞ ✏❧♦♦❦ ❧✐❦❡✑ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t • ❈♦♥s✐❞❡r ❛❞✈❡rs❛r② D t❤❛t ♠❛❦❡s q ❡✈❛❧✉❛t✐♦♥s ♦❢ � E k • ❙t❡♣ ✶✿ • ❍♦✇ ♠❛♥② ❡✈❛❧✉❛t✐♦♥s ❞♦❡s D ♥❡❡❞ ❛t ♠♦st❄ ❙t❡♣ ✶✿ • ❇♦✐❧s ❞♦✇♥ t♦ ✜♥❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❦s ✶✸ ✴ ✹✽

■♥t✉✐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 D t❤❛t ♠❛❦❡s q ❡✈❛❧✉❛t✐♦♥s ♦❢ � 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 ✶✸ ✴ ✹✽

❋♦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 ✶✹ ✴ ✹✽

❯♥❧✐❦❡❧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 ′ ) ✶✹ ✴ ✹✽

■♠♣❧✐❝❛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 ✶✹ ✴ ✹✽

❙❝❤❡♠❡ ❝❛♥ ❜❡ ❜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 ′ ✶✹ ✴ ✹✽

■♥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 ✶✹ ✴ ✹✽

❚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❡ ✐♥✈♦❧✈❡❞ ✶✺ ✴ ✹✽

❆❧❧ ❝♦♥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 ✶✺ ✴ ✹✽

■♥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 ✶✺ ✴ ✹✽

❖✉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✐♦♥ ✶✻ ✴ ✹✽

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❦✐♥❣ ✐♥ ❖❈❇✷✲▲✐❦❡ ❈♦♥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 ✷✵ ✴ ✹✽

❯♣❞❛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 ✷✵ ✴ ✹✽

❯♣❞❛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 ✷✵ ✴ ✹✽

❯♣❞❛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 ✷✵ ✴ ✹✽

❯♣❞❛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 ✷✵ ✴ ✹✽

❯♣❞❛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 ✷✵ ✴ ✹✽

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 ✷✵ ✴ ✹✽

❙✐♥❣❧❡ ❳❖❘ ▲♦❣❛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❧✐♥❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡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✐♦♥ ✷✷ ✴ ✹✽

❈♦♠❜✐♥❡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 ϕ γ 2 ◦ ϕ β 0 � = 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 ϕ γ 2 ◦ ϕ β 0 � = 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 ϕ γ 2 ◦ ϕ β 0 � = 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 ϕ γ 2 ◦ ϕ β 0 � = 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 ϕ γ 2 ◦ ϕ β 0 � = 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 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 � 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♦✉♥❞ ❇▲❆❑❊✷❜ ✷✻ ✴ ✹✽

❆♣♣❧✐❝❛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 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 � 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♦✉♥❞ ❇▲❆❑❊✷❜ ✷✼ ✴ ✹✽

❖✉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✐♦♥ ✷✽ ✴ ✹✽

■❢ ✐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 ❜♦✉♥❞ ✷✾ ✴ ✹✽

❇❡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❝❛❞✐♥❣ ✭♥♦✇✮ ✷✾ ✴ ✹✽

✿ 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] ✸✵ ✴ ✹✽

❈❛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 ✸✵ ✴ ✹✽

✿ 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 ✸✶ ✴ ✹✽

❈❛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 ❬❇❑▲✰✶✷❪ ✸✶ ✴ ✹✽

■♠♣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] ✸✷ ✴ ✹✽

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] ✸✷ ✴ ✹✽

❖✉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✐♦♥ ✸✸ ✴ ✹✽

✐♠♣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 ❣❛♣ ✸✹ ✴ ✹✽

✐♠♣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✐♦♥✮ ✸✹ ✴ ✹✽

❝❛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✐♦♥✮ ✸✹ ✴ ✹✽

❚✐❣❤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] ✸✹ ✴ ✹✽

❖✉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✐♦♥ ✸✺ ✴ ✹✽

●❡♥❡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 ✸✻ ✴ ✹✽

■♠♣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 ✸✻ ✴ ✹✽

❙✉♣♣♦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 ✸✼ ✴ ✹✽

◆❡❝❡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 ✸✼ ✴ ✹✽

❙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 ✸✼ ✴ ✹✽

■♠♣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 ′ ) ✸✼ ✴ ✹✽

❇✉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 ✸✽ ✴ ✹✽

❈❤♦♦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 ✸✽ ✴ ✹✽

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 ✸✽ ✴ ✹✽

❊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 ✸✽ ✴ ✹✽

■♠♣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 ✸✽ ✴ ✹✽

❊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 � � � � π = 1 D � E k = 1 D � • ❆♥❛❧②s✐s ❝♦♥s✐sts ♦❢ ♣r♦♣❡r❧② ❜♦✉♥❞✐♥❣ Pr ❛♥❞ Pr ✸✾ ✴ ✹✽

■♠♣r♦✈❡❞ ❆tt❛❝❦✿ ❱❡r✐✜❝❛t✐♦♥ ❚❤❡♦r❡t✐❝❛❧ ❱❡r✐✜❝❛t✐♦♥ • ❆ss✉♠✐♥❣ n ≥ 27 ✱ t❤❡ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ♦❢ D ✐s ❛t ❧❡❛st 1 / 2 � � � � π = 1 D � E k = 1 D � • ❆♥❛❧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 ✸✾ ✴ ✹✽

❖✉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✐♦♥ ✹✵ ✴ ✹✽

❈❛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 ✹✶ ✴ ✹✽