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✹✶✹✲❙✶✼ ❈r②♣t♦ ❙❤❛♥❦❛r ❆♣r✐❧ ✶✽✱ ✷✵✶✼

❖✉t❧✐♥❡ ♦✈❡r✈✐❡✇ ❖✈❡r✈✐❡✇ ❙②♠♠❡tr✐❝ ❈r②♣t♦ ❇❧♦❝❦ ❈✐♣❤❡r ❊♥❝r②♣t✐♦♥ ▼♦❞❡s ❢♦r ❱❛r✐❛❜❧❡✲s✐③❡ ▼❡ss❛❣❡s ▼❡ss❛❣❡ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❈♦❞❡s ✭▼❆❈s✮ ▼❆❈ ❛♥❞ ❈♦♥✜❞❡♥t✐❛❧✐t② ❆s②♠♠❡tr✐❝ ❈r②♣t♦ ✭❛❦❛ P✉❜❧✐❝✲❑❡② ❈r②♣t♦✮ ■♥tr♦❞✉❝t✐♦♥ ❆ ▲✐tt❧❡ ❇✐t ♦❢ ◆✉♠❜❡r ❚❤❡♦r② ❘❙❆ ❉✐✣❡✲❍❡❧♠❛♥

❖✈❡r✈✐❡✇ ♦✈❡r✈✐❡✇ ❈r②♣t♦ ✐s ❡✈❡r②✇❤❡r❡ ❝♦♠♠✉♥✐❝❛t✐♦♥s✿ ❤tt♣s✱ ■Ps❡❝✱ ✽✵✷✳✶✶✱ ❲P❆✷✱ ✳✳✳ ✜❧❡s ♦♥ ❞✐s❦✿ ❇✐t❧♦❝❦❡r✱ ❋✐❧❡❱❛✉❧t✱ ✳✳✳ ✉s❡r ❛✉t❤❡♥t✐❝❛t✐♦♥✿ ❑❡r❜❡r♦s✱ ✳✳✳ . . . ❈r②♣t♦ ❡♥❛❜❧❡s s❡❝✉r❡ ❞❛t❛ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❛♥❞ st♦r❛❣❡ ❈♦♥✜❞❡♥t✐❛❧✐t②✿ ♦♥❧② t❤❡ ✐♥t❡♥❞❡❞ r❡❝❡✐✈❡r ❝❛♥ r❡❛❞ t❤❡ ❞❛t❛ ■♥t❡❣r✐t②✿ t❤❡ ✐♥t❡♥❞❡❞ r❡❝❡✐✈❡r ❞❡t❡❝ts ❛♥② ❝❤❛♥❣❡s t♦ t❤❡ ❞❛t❛ ❆✉t❤❡♥t✐❝❛t✐♦♥✿ ❞❛t❛ r❡❝❡✐✈❡❞ ✇❛s s❡♥t ❜② t❤❡ s♣❡❝✐✜❡❞ s❡♥❞❡r ◆♦♥✲r❡♣✉❞✐❛t✐♦♥✿ t❤✐r❞ ♣❛rt② ❝❛♥ ✈❡r✐❢② t❤❛t t❤❡ ❞❛t❛ ✇❛s s❡♥t ❜② t❤❡ s♣❡❝✐✜❡❞ s❡♥❞❡r

❙②♠♠❡tr✐❝ ❛♥❞ ❆s②♠♠❡tr✐❝ ❈r②♣t♦ ♦✈❡r✈✐❡✇ ❑❡② ❣❡♥❡r❛t✐♦♥✿ ❣❡♥❡r❛t❡ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ❦❡②s ❊♥❝r②♣t✐♦♥ E ✿ ♣❧❛✐♥t❡①t ✰ ❡♥❝r②♣t✐♦♥ ❦❡② − → ❝✐♣❤❡rt❡①t ❉❡❝r②♣t✐♦♥ D ✿ ♣❧❛✐♥t❡①t ← − ❝✐♣❤❡rt❡①t ✰ ❞❡❝r②♣t✐♦♥ ❦❡② ❙②♠♠❡tr✐❝ ❝r②♣t♦ ❡♥❝r②♣t✐♦♥ ❦❡② = ❞❡❝r②♣t✐♦♥ ❦❡② ❡❣✱ ❆❊❙✱ ▼❉✺✱ ❙❍❆✲✶✱ ❙❍❆✲✷✺✻✱ ✳✳✳ ❢❛st ❆s②♠♠❡tr✐❝ ✭❛❦❛ ♣✉❜❧✐❝✲❦❡②✮ ❝r②♣t♦ ❡♥❝r②♣t✐♦♥ ❦❡② � = ❞❡❝r②♣t✐♦♥ ❦❡② ❡❣✱ ❘❙❆✱ ❉❍✱ ❉❙❙✱ ✳✳✳ ✈❡r② s❧♦✇

❉❡s✐r❡❞ ♣r♦♣❡rt✐❡s ♦❢ ❝r②♣t♦ ❢✉♥❝t✐♦♥s ♦✈❡r✈✐❡✇ ❈♦rr❡❝t♥❡ss ❋♦r ❛♥② ❡♥❝r②♣t✐♦♥ ❦❡② key E ❛♥❞ ✐ts ❞❡❝r②♣t✐♦♥ ❦❡② key D ✿ ✐❢ E ( key E , ptxt ) r❡t✉r♥s ctxt t❤❡♥ D ( key D , ctxt ) r❡t✉r♥s ptxt ❙❡❝✉r✐t②✿ ❆ss✉♠✐♥❣ ❦❡②s ❛r❡ ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② r❛♥❞♦♠❧② ●✐✈❡♥ ❝②♣❤❡rt❡①t✱ ❤❛r❞ t♦ ❣❡t ♣❧❛✐♥t❡①t✳ ●✐✈❡♥ ♣❧❛✐♥t❡①t ❛♥❞ ❝✐♣❤❡rt❡①t✱ ❤❛r❞ t♦ ❣❡t ❦❡②✳ ❍❛r❞✿ r❡q✉✐r❡s ❜r✉t❡✲❢♦r❝❡ s❡❛r❝❤ ♦❢ ❦❡②✲s♣❛❝❡ ✭❡❣✱ ✷ ✶✷✽ ❦❡②s✮ ❆tt❛❝❦❡r ♠♦❞❡❧s ✭❢r♦♠ ✇❡❛❦❡st t♦ str♦♥❣❡st✮ ❈✐♣❤❡rt❡①t✲♦♥❧② ❛tt❛❝❦ ❑♥♦✇♥ ♣❧❛✐♥t❡①t ❛tt❛❝❦✿ ♦♥❡ ♠❛t❝❤✐♥❣ ♣❛✐r ❈❤♦s❡♥ ♣❧❛✐♥t❡①t ❛tt❛❝❦✿ ❡♥❝r②♣t✐♦♥ ♦r❛❝❧❡ ❈❤♦s❡♥ ❝✐♣❤❡rt❡①t ❛tt❛❝❦✿ ❡♥❝r②♣t✐♦♥ ♦r❛❝❧❡ ✰ ❞❡❝r②♣t✐♦♥ ♦r❛❝❧❡

❆❝❤✐❡✈✐♥❣ s❡❝✉r❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♦✈❡r✈✐❡✇ A ❛♥❞ B s❡♣❛r❛t❡❞ ❜② ✐♥s❡❝✉r❡ ❝❤❛♥♥❡❧✱ s❤❛r❡ s❡❝r❡t ❦❡② k ✳ ❈♦♥✜❞❡♥t✐❛❧✐t②✿ A s❡♥❞s E ( k , plaintext ) B r❡❝❡✐✈❡s ❛♥❞ ❞♦❡s D ( k , ciphertext ) ■♥t❡❣r✐t②✿ mac ✿ E ( k , hash ( plaintext )) A s❡♥❞s [ plaintext , mac ] B r❡❝❡✐✈❡s ❛♥❞ ✈❡r✐✜❡s mac ❆✉t❤❡♥t✐❝❛t✐♦♥✿ A s❡♥❞s ❛ r❛♥❞♦♠ r A t♦ B ✱ ❛♥❞ ❡①♣❡❝ts E ( k , r A ) ❜❛❝❦ B s❡♥❞s ❛ r❛♥❞♦♠ r B t♦ A ✱ ❛♥❞ ❡①♣❡❝ts E ( k , r B ) ❜❛❝❦

❖✉t❧✐♥❡ s②♠♠ ❖✈❡r✈✐❡✇ ❙②♠♠❡tr✐❝ ❈r②♣t♦ ❇❧♦❝❦ ❈✐♣❤❡r ❊♥❝r②♣t✐♦♥ ▼♦❞❡s ❢♦r ❱❛r✐❛❜❧❡✲s✐③❡ ▼❡ss❛❣❡s ▼❡ss❛❣❡ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❈♦❞❡s ✭▼❆❈s✮ ▼❆❈ ❛♥❞ ❈♦♥✜❞❡♥t✐❛❧✐t② ❆s②♠♠❡tr✐❝ ❈r②♣t♦ ✭❛❦❛ P✉❜❧✐❝✲❑❡② ❈r②♣t♦✮ ■♥tr♦❞✉❝t✐♦♥ ❆ ▲✐tt❧❡ ❇✐t ♦❢ ◆✉♠❜❡r ❚❤❡♦r② ❘❙❆ ❉✐✣❡✲❍❡❧♠❛♥

❖✉t❧✐♥❡ ❜❧♦❝❦ ❝✐♣❤❡r s②♠♠ ❖✈❡r✈✐❡✇ ❙②♠♠❡tr✐❝ ❈r②♣t♦ ❇❧♦❝❦ ❈✐♣❤❡r ❊♥❝r②♣t✐♦♥ ▼♦❞❡s ❢♦r ❱❛r✐❛❜❧❡✲s✐③❡ ▼❡ss❛❣❡s ▼❡ss❛❣❡ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❈♦❞❡s ✭▼❆❈s✮ ▼❆❈ ❛♥❞ ❈♦♥✜❞❡♥t✐❛❧✐t② ❆s②♠♠❡tr✐❝ ❈r②♣t♦ ✭❛❦❛ P✉❜❧✐❝✲❑❡② ❈r②♣t♦✮ ■♥tr♦❞✉❝t✐♦♥ ❆ ▲✐tt❧❡ ❇✐t ♦❢ ◆✉♠❜❡r ❚❤❡♦r② ❘❙❆ ❉✐✣❡✲❍❡❧♠❛♥

■❞❡❛❧ ❜❧♦❝❦ ❝✐♣❤❡r ❜❧♦❝❦ ❝✐♣❤❡r s②♠♠ ❋✐①❡❞✲s✐③❡ ♠❡ss❛❣❡s ♦❢ d ❜✐ts ✭❡❣✱ ✻✹✱ ✶✷✽✮ ❋✐①❡❞✲s✐③❡ ❦❡②s ♦❢ k ❜✐ts ✭❡❣✱ ✶✷✽✱ ✷✺✻✮ ❛♥② r❛♥❞♦♠ k ❜✐ts ✐s ❛ ✈❛❧✐❞ ❦❡② ❊♥❝r②♣t✐♦♥ E : d ✲❜✐t ♠s❣ ✰ k ✲❜✐t ❦❡② − → d ✲❜✐t ♦✉t♣✉t ❚♦ ❞❡❝r②♣t✱ E ♠✉st ❜❡ ✶✲✶ ♠❛♣♣✐♥❣ ♦❢ ♠s❣s t♦ ♦✉t♣✉ts ❚♦ ❜❡ s❡❝✉r❡✱ E ♠✉st ❜❡ ✏r❛♥❞♦♠✑ E ( key , msg ) ❣✐✈❡s ♥♦ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t key ♦r msg ▼s❣s ❛♥❞ ❦❡②s t❤❛t ❞✐✛❡r ✭❡✈❡♥ ✐❢ ♦♥❧② s❧✐❣❤t❧②✮ ♠❛♣ t♦ ♦✉t♣✉ts t❤❛t ❞✐✛❡r r❛♥❞♦♠❧② ❑❡② s✐③❡ k ❧❛r❣❡ ❡♥♦✉❣❤ s♦ t❤❛t s❡❛r❝❤✐♥❣ ✷ k ✐s ✐♥❢❡❛s✐❜❧❡ ❈❧❡❛r❧②✱ E ❝❛♥♥♦t ❜❡ ❛ ✏s✐♠♣❧❡✑ ❢✉♥❝t✐♦♥✱ ❡❣✱ msg ⊕ key

■♠♣❧❡♠❡♥t❛t✐♦♥✿ d ✲❜✐t ❞❛t❛✱ k ✲❜✐t ❦❡② ❜❧♦❝❦ ❝✐♣❤❡r s②♠♠ ◆❛✐✈❡ ❛♣♣r♦❛❝❤ / ✷ d × d ❜✐ts ❚❛❜❧❡ ♦❢ ❛ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ♦❢ d ✲❜✐t str✐♥❣s / E ( i ) ✐s i t❤ r♦✇ ♦❢ t❛❜❧❡ ❙❡❝✉r❡ ❜✉t ✐♠♣r❛❝t✐❝❛❧ / t❛❜❧❡ ✐ts❡❧❢ ✐s t❤❡ ❦❡②✦ / Pr❛❝t✐❝❛❧ ❛♣♣r♦❛❝❤✿ ❧♦❝❛❧✐③❡❞ s❝r❛♠❜❧✐♥❣ ❛♥❞ ❣❧♦❜❛❧ ♣❡r♠✉t❛t✐♦♥s ●❡♥❡r❛t❡ n ✏r♦✉♥❞ ❦❡②s✑ ❢r♦♠ t❤❡ ❦❡② / n s♠❛❧❧✱ ❡❣✱ ✶✵ / ❘❡♣❡❛t n t✐♠❡s / ✷ p ✐s ♠❛♥❛❣❡❛❜❧❡ P❛rt✐t✐♦♥ d ✲❜✐t str✐♥❣ ✐♥t♦ p ✲❜✐t ❝❤✉♥❦s / ❙❝r❛♠❜❧❡ ❡❛❝❤ p ✲❜✐t ❝❤✉♥❦ ✉s✐♥❣ ✷ p × p t❛❜❧❡s / t❛❜❧❡✬s ♣❡r♠✉t❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ r♦✉♥❞✲ n ❦❡② / P❡r♠✉t❡ t❤❡ r❡s✉❧t✐♥❣ d ✲❜✐t str✐♥❣ ❉❡❝r②♣t✐♦♥ ✐s s✐♠✐❧❛r / ♦❢t❡♥ r❡✉s❡ s❛♠❡ ❤❛r❞✇❛r❡ /

❊①❛♠ ❉❊❙✿ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞ ❜❧♦❝❦ ❝✐♣❤❡r s②♠♠ ❖❧❞ st❛♥❞❛r❞ ♥♦ ❧♦♥❣❡r ✉s❡❞✿ ✺✻✲❜✐t ❦❡②s✱ ✻✹✲❜✐t t❡①t