rt ttt rt - PowerPoint PPT Presentation

●❡♥❡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 ❊✫▼ ▼t❊ ❊t▼ 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 ✐♥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 ❊✫▼ ▼t❊ ❊t▼ 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 � t ✶✶ ✴ ✺✼

▼♦st s❡❝✉r❡ ✈❛r✐❛♥t ❈✐♣❤❡rt❡①t ✐♥t❡❣r✐t② ●❡♥❡r✐❝ ❈♦♠♣♦s✐t✐♦♥ • ●❡♥❡r✐❝ ❝♦♥str✉❝t✐♦♥s ❢♦r ❆❊✿ • ❊♥❝ ✰ ▼❆❈ ❂ ❆❊ • ❇❡❧❧❛r❡ ❛♥❞ ◆❛♠♣r❡♠♣r❡ ✭✷✵✵✵✮✿ ✸ ❜❛s✐❝ ❛♣♣r♦❛❝❤❡s ❊✫▼ ▼t❊ ❊t▼ 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 � t • P❛❞❞✐♥❣ ♦r❛❝❧❡ ❛tt❛❝❦ ✶✶ ✴ ✺✼

●❡♥❡r✐❝ ❈♦♠♣♦s✐t✐♦♥ • ●❡♥❡r✐❝ ❝♦♥str✉❝t✐♦♥s ❢♦r ❆❊✿ • ❊♥❝ ✰ ▼❆❈ ❂ ❆❊ • ❇❡❧❧❛r❡ ❛♥❞ ◆❛♠♣r❡♠♣r❡ ✭✷✵✵✵✮✿ ✸ ❜❛s✐❝ ❛♣♣r♦❛❝❤❡s ❊✫▼ ▼t❊ ❊t▼ 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 � t • P❛❞❞✐♥❣ ♦r❛❝❧❡ • ❈✐♣❤❡rt❡①t ✐♥t❡❣r✐t② ❛tt❛❝❦ ✶✶ ✴ ✺✼

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 � 1 N � 2 N � 3 N � ( m + 1) • ❊t▼ ❞❡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 ✶✷ ✴ ✺✼

❲❤❛t ❤❛♣♣❡♥s ✐❢ ♥♦♥❝❡ ✐s r❡✲✉s❡❞❄ ●❈▼ ❢♦r 96 ✲❜✐t ♥♦♥❝❡ N • ▼❝●r❡✇ ❛♥❞ ❱✐❡❣❛ ✭✷✵✵✹✮ N � 1 N � 2 N � 3 N � ( m + 1) • ❊t▼ ❞❡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 ✐♥ ❙❲ ✶✷ ✴ ✺✼

●❈▼ ❢♦r 96 ✲❜✐t ♥♦♥❝❡ N • ▼❝●r❡✇ ❛♥❞ ❱✐❡❣❛ ✭✷✵✵✹✮ N � 1 N � 2 N � 3 N � ( m + 1) • ❊t▼ ❞❡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❡❞❄ ✶✷ ✴ ✺✼

■♥❤❡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 • ▼t❊ ❞❡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 ✶✸ ✴ ✺✼

●❈▼✲❙■❱ T +0 T +1 T +( m − 1) N • ●✉❡r♦♥ ❛♥❞ ▲✐♥❞❡❧❧ ✭✷✵✶✺✮ KeyGen E k E K E K E K • ▼t❊ ❞❡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 ✶✸ ✴ ✺✼

❖✉t❧✐♥❡ ●❡♥❡r✐❝ ❈♦♠♣♦s✐t✐♦♥ ▲✐♥❦ ❲✐t❤ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ ◆♦♥❝❡✲❘❡✉s❡ ❈♦♥❝❧✉s✐♦♥ ✶✹ ✴ ✺✼

❚✇❡❛❦✿ ✢❡①✐❜✐❧✐t② t♦ t❤❡ ❝✐♣❤❡r ❊❛❝❤ t✇❡❛❦ ❣✐✈❡s ❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♦♥ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs k m c E ✶✺ ✴ ✺✼

❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs k � m c E t • ❚✇❡❛❦✿ ✢❡①✐❜✐❧✐t② t♦ t❤❡ ❝✐♣❤❡r • ❊❛❝❤ t✇❡❛❦ ❣✐✈❡s ❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♦♥ ✶✺ ✴ ✺✼

tr✐❡s t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦r❛❝❧❡ ✐t ❝♦♠♠✉♥✐❝❛t❡s ✇✐t❤ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡r ❙❡❝✉r✐t② � E k � π IC tweakable blockcipher random tweakable permutation distinguisher D • � E k s❤♦✉❧❞ ❧♦♦❦ ❧✐❦❡ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t • ❉✐✛❡r❡♥t t✇❡❛❦s − → ♣s❡✉❞♦✲✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ✶✻ ✴ ✺✼

❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡r ❙❡❝✉r✐t② � E k π � IC tweakable blockcipher random tweakable permutation distinguisher D • � 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❤ � �� � � � π − 1 = 1 � � E k , � � E − 1 Adv stprp D � π, � ( D ) = � Pr D = 1 − Pr � k � E ✶✻ ✴ ✺✼

✐♥ ❈❆❊❙❆❘ ❑■❆❙❯✱ ❈❇❆✱ ❈❖❇❘❆✱ ✐❋❡❡❞✱ 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❡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 ❛❧✳ ❬❘❇❇❑✵✶✱❘♦❣✵✹✱❑❘✶✶❪ ✶✽ ✴ ✺✼

❚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 � E • ❚✇❡❛❦ ( N, tweak ) ✐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 ❛❧✳ ❬❘❇❇❑✵✶✱❘♦❣✵✹✱❑❘✶✶❪ • ■♥t❡r♥❛❧❧② ❜❛s❡❞ ♦♥ t✇❡❛❦❛❜❧❡ ❜❧♦❝❦❝✐♣❤❡r � 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 [ � ✶✽ ✴ ✺✼

❊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 � 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 [ � AE [ � ✶✽ ✴ ✺✼

❊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 � 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 [ � AE [ � � E ✶✽ ✴ ✺✼

❚❛❣ ❢♦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 ✶✾ ✴ ✺✼

❚❛❣ ❢♦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 ✶✾ ✴ ✺✼

❚❛❣ ❢♦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 [ � π ] = $ ✶✾ ✴ ✺✼

❞❡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 [ � π ] = $ • ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❡❤❛✈❡s ❧✐❦❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ ✶✾ ✴ ✺✼

❞❡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 [ � π ] = $ • ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❡❤❛✈❡s ❧✐❦❡ r❛♥❞♦♠ ❢✉♥❝t✐♦♥ • ❚❛❣ ❢♦r❣❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❛t ♠♦st 1 / (2 n − 1) ✶✾ ✴ ✺✼

❞❡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 [ � 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 [ � ✶✾ ✴ ✺✼

❊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 [ � 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 [ � ✶✾ ✴ ✺✼

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

❜❧❡♥❞ ✐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✐♦♥❄ ✷✸ ✴ ✺✼

■♥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❡ ✷✸ ✴ ✺✼

❋♦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✦ ✷✹ ✴ ✺✼

❚❲❊❆❑❊❨ ❜❧❡♥❞✐♥❣ ❬❏◆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 • � k ( t, c ) ⊕ � E − 1 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 • � 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♦❜❧❡♠ ■♥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 ♣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❡ ✐♥✈♦❧✈❡❞ ✷✾ ✴ ✺✼

❆❧❧ ❝♦♥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 ✷✾ ✴ ✺✼

■♥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 ✷✾ ✴ ✺✼

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

■♥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 ✸✺ ✴ ✺✼

❉✐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 ✸✻ ✴ ✺✼

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 ✸✻ ✴ ✺✼

❉✐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 ) ✸✻ ✴ ✺✼

■♥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 ) ✸✻ ✴ ✺✼

❙✐♥❣❧❡ ❳❖❘ ▲♦❣❛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✐♥❣✲✉♣ ❬❑❘✶✶❪ ✸✼ ✴ ✺✼