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❳❖❘ ♦❢ P❘Ps ✐♥ ❛ ◗✉❛♥t✉♠ ❲♦r❧❞ ❇❛rt ▼❡♥♥✐♥❦✱ ❆❧❛♥ ❙③❡♣✐❡♥✐❡❝ ❘❛❞❜♦✉❞ ❯♥✐✈❡rs✐t② ✭❚❤❡ ◆❡t❤❡r❧❛♥❞s✮✱ ❑❯ ▲❡✉✈❡♥ ✭❇❡❧❣✐✉♠✮ P◗❈r②♣t♦ ✷✵✶✼ ❏✉♥❡ ✷✻✱ ✷✵✶✼ ✶ ✴ ✶✼

▲✉❜②✲❘❛❝❦♦✛ ✴ ❋❡✐st❡❧ ◆♦✇ ■♥tr♦❞✉❝t✐♦♥ P❘P P❘❋ ✷ ✴ ✶✼

◆♦✇ ■♥tr♦❞✉❝t✐♦♥ ▲✉❜②✲❘❛❝❦♦✛ ✴ ❋❡✐st❡❧ P❘P P❘❋ ✷ ✴ ✶✼

■♥tr♦❞✉❝t✐♦♥ ▲✉❜②✲❘❛❝❦♦✛ ✴ ❋❡✐st❡❧ P❘P P❘❋ ◆♦✇ ✷ ✴ ✶✼

❙❡❝✉r✐t② ❜♦✉♥❞✿ ❈❚❘ ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s✿ ✐s ❛ s❡❝✉r❡ P❘P ✭t②♣✐❝❛❧❧② ✮ ◆✉♠❜❡r ♦❢ ❡♥❝r②♣t❡❞ ❜❧♦❝❦s ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · E k E k E k m 1 m 2 m ℓ c 1 c 2 c ℓ ✸ ✴ ✶✼

❈❚❘ ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s✿ ✐s ❛ s❡❝✉r❡ P❘P ✭t②♣✐❝❛❧❧② ✮ ◆✉♠❜❡r ♦❢ ❡♥❝r②♣t❡❞ ❜❧♦❝❦s ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · E k E k E k m 1 m 2 m ℓ c 1 c 2 c ℓ • ❙❡❝✉r✐t② ❜♦✉♥❞✿ � q � Adv cpa CTR [ E ] ( q, t ) ≤ Adv prp / 2 n E ( q, t ) + 2 ✸ ✴ ✶✼

❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · E k E k E k m 1 m 2 m ℓ c 1 c 2 c ℓ • ❙❡❝✉r✐t② ❜♦✉♥❞✿ � q � Adv cpa CTR [ E ] ( q, t ) ≤ Adv prp / 2 n E ( q, t ) + 2 • ❈❚❘ [ E ] ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s✿ • E k ✐s ❛ s❡❝✉r❡ P❘P ✭t②♣✐❝❛❧❧② t ≪ 2 κ ✮ • ◆✉♠❜❡r ♦❢ ❡♥❝r②♣t❡❞ ❜❧♦❝❦s q ≪ 2 n/ 2 ✸ ✴ ✶✼

❙❡❝✉r✐t② ❜♦✉♥❞✿ ❈❚❘ ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s ✐s ❛ s❡❝✉r❡ P❘❋ ❇✐rt❤❞❛② ❜♦✉♥❞ s❡❝✉r✐t② ❧♦ss ❞✐s❛♣♣❡❛r❡❞ ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ ❋✉♥❝t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · F k F k F k m 1 m 2 m ℓ c 1 c 2 c ℓ ✹ ✴ ✶✼

❈❚❘ ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s ✐s ❛ s❡❝✉r❡ P❘❋ ❇✐rt❤❞❛② ❜♦✉♥❞ s❡❝✉r✐t② ❧♦ss ❞✐s❛♣♣❡❛r❡❞ ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ ❋✉♥❝t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · F k F k F k m 1 m 2 m ℓ c 1 c 2 c ℓ • ❙❡❝✉r✐t② ❜♦✉♥❞✿ Adv cpa CTR [ F ] ( q ) ≤ Adv prf F ( q ) ✹ ✴ ✶✼

❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ ❋✉♥❝t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · F k F k F k m 1 m 2 m ℓ c 1 c 2 c ℓ • ❙❡❝✉r✐t② ❜♦✉♥❞✿ Adv cpa CTR [ F ] ( q ) ≤ Adv prf F ( q ) • ❈❚❘ [ F ] ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s F k ✐s ❛ s❡❝✉r❡ P❘❋ • ❇✐rt❤❞❛② ❜♦✉♥❞ s❡❝✉r✐t② ❧♦ss ❞✐s❛♣♣❡❛r❡❞ ✹ ✴ ✶✼

s❡❝✉r✐t② ❬❇■✾✾✱▲✉❝✵✵✱P❛t✵✽❪ ❇♦✉♥❞ ♣r❡s❡r✈❡❞ ❢♦r ❬❈▲P✶✹✱▼P✶✺❪ ❳❖❘ ♦❢ P❘Ps x E k 1 E k 2 XoP ( k , x ) ✺ ✴ ✶✼

❇♦✉♥❞ ♣r❡s❡r✈❡❞ ❢♦r ❬❈▲P✶✹✱▼P✶✺❪ ❳❖❘ ♦❢ P❘Ps x E k 1 E k 2 XoP ( k , x ) • min { 2 κ , 2 n } s❡❝✉r✐t② ❬❇■✾✾✱▲✉❝✵✵✱P❛t✵✽❪ ✺ ✴ ✶✼

❳❖❘ ♦❢ P❘Ps x E k 1 E k 2 E k r − 1 E k r · · · XoP r ( k , x ) • min { 2 κ , 2 n } s❡❝✉r✐t② ❬❇■✾✾✱▲✉❝✵✵✱P❛t✵✽❪ • ❇♦✉♥❞ ♣r❡s❡r✈❡❞ ❢♦r r ≥ 3 ❬❈▲P✶✹✱▼P✶✺❪ Adv prf XoP ( q, t ) ≤ r · Adv prp E ( q, t ) + q/ 2 n ✺ ✴ ✶✼

s❡❝✉r✐t② ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ ❳♦P 0 � n +1 1 � n +1 0 � n +2 1 � n +2 0 � n + ℓ 1 � n + ℓ E k E k · · · · · · E k E k E k E k m 1 m 2 m ℓ c 2 c ℓ c 1 • ❙❡❝✉r✐t② ❜♦✉♥❞✿ Adv cpa CTR [ XoP ] ( q, t ) ≤ Adv prf XoP ( q, t ) ✻ ✴ ✶✼

s❡❝✉r✐t② ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ ❳♦P 0 � n +1 1 � n +1 0 � n +2 1 � n +2 0 � n + ℓ 1 � n + ℓ E k E k · · · · · · E k E k E k E k m 1 m 2 m ℓ c 2 c ℓ c 1 • ❙❡❝✉r✐t② ❜♦✉♥❞✿ Adv cpa CTR [ XoP ] ( q, t ) ≤ Adv prf XoP ( q, t ) ≤ Adv prp E (2 q, t ) + q/ 2 n ✻ ✴ ✶✼

❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ ❳♦P 0 � n +1 1 � n +1 0 � n +2 1 � n +2 0 � n + ℓ 1 � n + ℓ E k E k · · · · · · E k E k E k E k m 1 m 2 m ℓ c 2 c ℓ c 1 • ❙❡❝✉r✐t② ❜♦✉♥❞✿ Adv cpa CTR [ XoP ] ( q, t ) ≤ Adv prf XoP ( q, t ) ≤ Adv prp E (2 q, t ) + q/ 2 n • min { 2 κ , 2 n } s❡❝✉r✐t② ✻ ✴ ✶✼

❙✐♠♦♥✴❙❤♦r P♦❧②✲t✐♠❡ ♣❡r✐♦❞ ✜♥❞✐♥❣ ❯s❡❞ t♦ ❛tt❛❝❦ ❊✈❡♥✲▼❛♥s♦✉r✱ ❈❇❈✲▼❆❈✱ ✳ ✳ ✳ ◗✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ❦❡②❡❞ ♣r✐♠✐t✐✈❡ ●r♦✈❡r ✏❍❛❧✈❡s t❤❡ ❦❡② s✐③❡✑ ◆♦ q✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ♥❡❡❞❡❞ ❚❤✐s ✇♦r❦✿ ♥♦ q✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ◗✉❛♥t✉♠ ❙❡❝✉r✐t② ❆♥❛❧②s✐s❄ ✼ ✴ ✶✼

●r♦✈❡r ✏❍❛❧✈❡s t❤❡ ❦❡② s✐③❡✑ ◆♦ q✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ♥❡❡❞❡❞ ❚❤✐s ✇♦r❦✿ ♥♦ q✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ◗✉❛♥t✉♠ ❙❡❝✉r✐t② ❆♥❛❧②s✐s❄ ❙✐♠♦♥✴❙❤♦r • P♦❧②✲t✐♠❡ ♣❡r✐♦❞ ✜♥❞✐♥❣ • ❯s❡❞ t♦ ❛tt❛❝❦ ❊✈❡♥✲▼❛♥s♦✉r✱ ❈❇❈✲▼❆❈✱ ✳ ✳ ✳ • ◗✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ❦❡②❡❞ ♣r✐♠✐t✐✈❡ ✼ ✴ ✶✼

❚❤✐s ✇♦r❦✿ ♥♦ q✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ◗✉❛♥t✉♠ ❙❡❝✉r✐t② ❆♥❛❧②s✐s❄ ❙✐♠♦♥✴❙❤♦r • P♦❧②✲t✐♠❡ ♣❡r✐♦❞ ✜♥❞✐♥❣ • ❯s❡❞ t♦ ❛tt❛❝❦ ❊✈❡♥✲▼❛♥s♦✉r✱ ❈❇❈✲▼❆❈✱ ✳ ✳ ✳ • ◗✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ❦❡②❡❞ ♣r✐♠✐t✐✈❡ ●r♦✈❡r • ✏❍❛❧✈❡s t❤❡ ❦❡② s✐③❡✑ • ◆♦ q✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ♥❡❡❞❡❞ ✼ ✴ ✶✼

◗✉❛♥t✉♠ ❙❡❝✉r✐t② ❆♥❛❧②s✐s❄ ❙✐♠♦♥✴❙❤♦r • P♦❧②✲t✐♠❡ ♣❡r✐♦❞ ✜♥❞✐♥❣ • ❯s❡❞ t♦ ❛tt❛❝❦ ❊✈❡♥✲▼❛♥s♦✉r✱ ❈❇❈✲▼❆❈✱ ✳ ✳ ✳ • ◗✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ ❦❡②❡❞ ♣r✐♠✐t✐✈❡ ●r♦✈❡r • ✏❍❛❧✈❡s t❤❡ ❦❡② s✐③❡✑ • ◆♦ q✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ♥❡❡❞❡❞ ❚❤✐s ✇♦r❦✿ ♥♦ q✉❛♥t✉♠ ✐♥t❡r❛❝t✐♦♥ ✼ ✴ ✶✼

◗✉❛♥t✉♠ ❙❡❝✉r✐t② ❆♥❛❧②s✐s ♦❢ ❳♦P ❆♣♣❧✐❝❛t✐♦♥ ♦❢ s✉❜s✐st❡♥❝❡✿ s❡❝✉r✐t② ❑❡② ❘❡❝♦✈❡r② ❆tt❛❝❦ ♦♥ ❳♦P ❆tt❛❝❦ ✐♥ ❝♦♠♣❧❡①✐t② ✭✐♠♣r♦✈❡s ♦✈❡r ●r♦✈❡r✮ ❘❡❧✐❡s ♦♥ ❝❧❛✇✲✜♥❞✐♥❣ ❛❧❣♦r✐t❤♠ ❖✉r ❈♦♥tr✐❜✉t✐♦♥ ❈❧❛ss✐❝❛❧ ❱❡rs✉s ◗✉❛♥t✉♠ Pr♦♦❢s • ❋♦r♠❛❧✐③❛t✐♦♥ ♦❢ t②♣❡s ♦❢ ❞✐st✐♥❣✉✐s❤❡rs • ❊①♣♦s✐t✐♦♥ ♦❢ ❤♦✇ ❝❧❛ss✐❝❛❧ ♣r♦♦❢s s✉❜s✐st q✉❛♥t✉♠❧② • ❆♣♣❧✐❝❛❜❧❡ t♦ ♠②r✐❛❞ ❝r②♣t♦❣r❛♣❤✐❝ s❝❤❡♠❡s ✽ ✴ ✶✼