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❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s ❈■▲✿ ❆ Pr♦♦❢ ❙②st❡♠ ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ●✐❧❧❡s ❇❛rt❤❡ 3 ✱ ▼❛r✐♦♥ ❉❛✉❜✐❣♥❛r❞ 2 ✱ ❇r✉❝❡ ❑❛♣r♦♥ 1 ❛♥❞ ❨❛ss✐♥❡ ▲❛❦❤♥❡❝❤ 2 1 ❯♥✐✈❡rs✐t② ♦❢ ❱✐❝t♦r✐❛ 2 ❱❊❘■▼❆●✱ ❯♥✐✈❡rs✐té ❞❡ ●r❡♥♦❜❧❡✱ ❈◆❘❙ 3 ■▼❉❊❆✱ ▼❛❞r✐❞ 19 th ❏✉♥❡✱ ✷✵✵✾ ❚❤✐s ✇♦r❦ ✐s ♣❛rt✐❛❧❧② s✉♣♣♦rt❡❞ ❜② t❤❡ ❆◆❘ ♣r♦❥❡❝t ❙❈❆▲P ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s ❆✐♠ Pr♦✈❛❜❧❡ s❡❝✉r✐t② ♣r♦✈✐❞❡s ❣✉❛r❛♥t❡❡s✱ ❜✉t✳✳✳ Pr♦❜❧❡♠s ✿ ♥♦✇❛❞❛②s✱ ♦♥❡ s❝❤❡♠❡ ❂ ♦♥❡ ♣r♦♦❢✱ ♣r♦♦❢s ❛r❡ ✐♥tr✐❝❛t❡✱ ❛♥❞ t❤❡r❡❢♦r❡ s♦♠❡✇❤❛t ✉♥r❡❧✐❛❜❧❡✳✳✳ ❖✉r ❧♦♥❣✲t❡r♠ ❣♦❛❧ ✐s t♦ ♣r♦✈❡ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s s❡❝✉r❡ ❜② ❡♥❛❜❧✐♥❣ ❈♦♠♣✉t❡r✲❆✐❞❡❞ ❈r②♣t♦❣r❛♣❤✐❝ Pr♦♦❢s ❛t t❤❡ ❧❡✈❡❧ ♦❢ ❛❜str❛❝t ❝♦♥str✉❝t✐♦♥s ❛♥❞ t❤❡✐r ✐♠♣❧❡♠❡♥t❛t✐♦♥s✳ ❊①✐st✐♥❣ ❛♣♣r♦❛❝❤❡s✿ ❣❛♠❡✲❜❛s❡❞ t❡❝❤♥✐q✉❡s✱ ❍♦❛r❡ ❧♦❣✐❝s✱ ❛♣♣❧✐❡❞ ♣✐✲❝❛❧❝✉❧✉s✳✳✳ ▼♦st s❡❝✉r✐t② ❝r✐t❡r✐❛ r❡❧② ♦♥ t❤❡ ❝♦♥❝❡♣t ♦❢ ✐♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②✳ ❍❡♥❝❡ ♦✉r ❝✉rr❡♥t s✉❜❣♦❛❧✿ ❞❡s✐❣♥✐♥❣ ❛ ✈❡rs❛t✐❧❡ s②st❡♠ ♦❢ ✐♥❢❡r❡♥❝❡ r✉❧❡s t♦ ♣r♦✈❡ ✐♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②✳ ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❢ ❉✐str✐❜✉t✐♦♥s ❆❞✈❛♥t❛❣❡ ♦❢ ❛♥ ❛❞✈❡rs❛r② ✐♥ ❞✐st✐♥❣✉✐s❤✐♥❣ D 0 ❛♥❞ D 1 ❆❞✈ ( η, A ) = | Pr[ A O i ( D 1 ) = 1] − Pr[ A O i ( D 0 ) = 1] | ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❢ t✇♦ ❞✐str✐❜✉t✐♦♥s D 0 ❛♥❞ D 1 ❛r❡ ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ✐✛ sup A ( ❆❞✈ ( η, A )) ✐s ❛ ♥❡❣❧✐❣✐❜❧❡ ❢✉♥❝t✐♦♥ ✐♥ η ✳ ❚❤✐s ✐s ❞❡♥♦t❡❞ D 0 ∼ D 1 ✳ ❞❡❢✳ ✿ f ( η ) ♥❡❣❧✐❣✐❜❧❡ ✐✛ ∀ k ≥ 0 , η k × f ( η ) η : ∞ − → 0 ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s ❖✉t❧✐♥❡ ✶ ❖✉r ❋r❛♠❡✇♦r❦✿ ❈♦♠♣✉t❛t✐♦♥❛❧ ❋r❛♠❡s ✷ ❈■▲✿ ❚❤❡ ■♥❢❡r❡♥❝❡ ❙②st❡♠ ✸ ❘❡❛s♦♥✐♥❣ ❲✐t❤ ❖r❛❝❧❡s ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s ❖✉t❧✐♥❡ ✶ ❖✉r ❋r❛♠❡✇♦r❦✿ ❈♦♠♣✉t❛t✐♦♥❛❧ ❋r❛♠❡s ✷ ❈■▲✿ ❚❤❡ ■♥❢❡r❡♥❝❡ ❙②st❡♠ ✸ ❘❡❛s♦♥✐♥❣ ❲✐t❤ ❖r❛❝❧❡s ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s ❲❤❛t ❞❡✜♥❡s t❤❡ ❞✐str✐❜✉t✐♦♥s ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥❄ ❚❤❡ ■◆❉✲❈P❆ ❣❛♠❡ ❢♦r ❛ s❝❤❡♠❡ ( K , E , D ) r ✶ ❦❡②s ❛r❡ ❞r❛✇♥ ✿ ( pk, sk ) ← K ( η ) ✱ ✷ A 1 ( pk ) ❝❤♦s❡s ❛ ♣❛✐r ♦❢ ♠❡ss❛❣❡s ❛♥❞ ♦✉t♣✉ts t❤❡♠ ♣❧✉s s♦♠❡ st❛t❡ ✐♥❢♦r♠❛t✐♦♥✿ ( s, m 0 , m 1 ) ✱ ✸ b ✐s ❝❤♦s❡♥ ❛t r❛♥❞♦♠✱ m b ✐s ❝✐♣❤❡r❡❞✿ y = E ( m b ) ✱ ✹ A 2 ( s, pk, m 0 , m 1 , y ) ❞❡❝✐❞❡s ✇❤✐❝❤ ♠❡ss❛❣❡ ✇❛s ❡♥❝r②♣t❡❞✳ ❢r❡s❤ r❛♥❞♦♠ ✈❛❧✉❡s ❛r❡ ❞r❛✇♥ ✕ ✭❦❡② ♣❛✐rs✮ ❛❞✈❡rs❛r② ❝❛❧❧s ❛r❡ ♠❛❞❡ ✕ ✭ A 1 ✮ A 2 ❣❡ts ❛s ❛♥ ✐♥♣✉t ❛ t✉♣❧❡ ( s, m 0 , m 1 , y ) ✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♣r❡✈✐♦✉s ❝♦♠♣✉t❛t✐♦♥s A 1 ❛♥❞ A 2 ❝❛♥ q✉❡r② ♦r❛❝❧❡s✳ ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s ■♥tr♦❞✉❝t✐♦♥ ❚♦ ❈♦♠♣✉t❛t✐♦♥❛❧ ❋r❛♠❡ ❙②♥t❛① ❆ ❢r❛♠❡ ✐s ❛ ❞✐str✐❜✉t✐♦♥ ❞❡♥♦t❡❞ s = ν� x.ν� a. ( u 1 , . . . , u m ) | I 1 / O 1 , . . . , I n / O n ✇❤❡r❡✿ r r ✶ � x st❛♥❞s ❢♦r x 1 ← U 1 , . . . , x k ← U k ✳ ❚❤♦s❡ r❡♣r❡s❡♥t ❢r❡s❤ ❞r❛✇✐♥❣s ✐♥ ✐♥❞❡♣❡♥❞❡♥t ❞✐str✐❜✉t✐♦♥s✳ r ← A � ✷ � O a r❡♣r❡s❡♥ts ❛ ❧✐st ♦❢ ❛❞✈❡rs❛r② ❝❛❧❧s a i i ( in i ) ✳ ❚❤❡ ✐♥♣✉ts ✭ in i ✬s✮ ❝❛♥ ❞❡♣❡♥❞ ♦♥ x k ✬s ❛♥❞ ♣r❡❝❡❞✐♥❣ a j ✬s✳ ✸ ( u 1 , . . . , u m ) ❛r❡ ❡①♣r❡ss✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥ � x ❛♥❞ � a ✳ ✹ I j ✐s t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♦r❛❝❧❡ O j ✳ ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s ■♥t❡r❛❝t✐♦♥ ✇✐t❤ ❛♥ ❛❞✈❡rs❛r② ❋r❛♠❡ s ❋r❡s❤ ❞r❛✇s ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♦r❛❝❧❡ O 1 ❆❞✈❡rs❛r② ❝❛❧❧s ν� x.ν� y. ( u 1 , . . . , u m ) | I 1 / O 1 , . . . , I n / O n q✉❡r✐❡s ✐♥♣✉t ♦✉t♣✉t ( T O 1 , . . . , T O n ) ❆❞✈❡rs❛r② A ♦✉t♣✉t R ❡①✳ ✿ ❡✈❡♥t ❘❂✶✳ ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s ❋♦r♠❛❧ ❈♦♠♣✉t❛t✐♦♥❛❧ ❋r❛♠❡ ❙❡♠❛♥t✐❝s ▲❡t s = ν� x.ν� a. ( u 1 , . . . , u m ) | I 1 / O 1 , . . . , I n / O n ❜❡ ❛ ❢r❛♠❡✱ ❛♥❞ � A = ( A 1 , . . . , A p , A ) ✳ ❚❤❡② ❞❡✜♥❡ � A|| s ✱ t❤❡ r❡s✉❧t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦♥ a, � u, R, � ( � x,� I, � T O ) ✱ ❛s ❢♦❧❧♦✇s✿ ✶ ❋♦r ❡❛❝❤ i ✱ ❛ ✈❛❧✉❡ ˆ x i ✐s ❞r❛✇♥ ✐♥ U i ❛♥❞ ❛ss✐❣♥❡❞ t♦ x i ✳ ← A � r j ( in j ) ♠❡❛♥s ˆ ✷ ❋♦r ❡❛❝❤ j ✱ a j O in j ✐s ❝♦♠♣✉t❡❞✱ A j ❣❡ts ✐t ❛s ❛♥ ✐♥♣✉t✱ ♣♦ss✐❜❧② ❝❛❧❧s ♦r❛❝❧❡s � O ✭✐♠♣❧❡♠❡♥t❡❞ ❢♦❧❧♦✇✐♥❣ � I ✮✱ ❛♥❞ ♦✉t♣✉ts ❛♥ ❛♥s✇❡r ˆ a j ✱ ✇❤✐❝❤ ✐s ❛ss✐❣♥❡❞ t♦ a j ✳ ✸ ❈❛❧❧s t♦ ♦r❛❝❧❡s✿ ❛♥②t✐♠❡ ❛♥ ❛❞✈❡rs❛r② q✉❡r✐❡s O k ( bs ) ✱ ✐t ˆ ❣❡ts I k ( bs ) ❛♥❞ ❛s ❛ s✐❞❡ ❡✛❡❝t✱ T O := T O :: [( k, bs, I k ( bs ))] ✳ ✹ ❱❛❧✉❡s ( ˆ u 1 , . . . , ˆ u m ) ❛r❡ ❝♦♠♣✉t❡❞ ❢♦r ❡①♣r❡ss✐♦♥s ( u 1 , . . . , u m ) ✱ ❛♥❞ ❣✐✈❡♥ ❛s ❛♥ ✐♥♣✉t t♦ A ✳ ✺ ❆❢t❡r s♦♠❡ ♣♦❧②t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥❝❧✉❞✐♥❣ ♣♦ss✐❜❧❡ ♦r❛❝❧❡ ❝❛❧❧s✱ A ♦✉t♣✉ts ❛ ❜✐tstr✐♥❣ ˆ R ❛ss✐❣♥❡❞ t♦ ✈❛r✐❛❜❧❡ R ✱ ❛♥❞ � ˆ ˆ T O ✱ ✇❤❡r❡ T O k ✐s t❤❡ ❧✐st ♦❢ ❛❧❧ q✉❡r✐❡s t♦ O k ✳ ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②