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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✐❞

19th ❏✉♥❡✱ ✷✵✵✾

❚❤✐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❤✐♥❣ D0 ❛♥❞ D1 ❆❞✈(η, A) = | Pr[AOi(D1) = 1] − Pr[AOi(D0) = 1]| ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❢ t✇♦ ❞✐str✐❜✉t✐♦♥s D0 ❛♥❞ D1 ❛r❡ ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ✐✛ supA(❆❞✈(η, A)) ✐s ❛ ♥❡❣❧✐❣✐❜❧❡ ❢✉♥❝t✐♦♥ ✐♥ η✳ ❚❤✐s ✐s ❞❡♥♦t❡❞ D0 ∼ D1✳ ❞❡❢✳ ✿ f(η) ♥❡❣❧✐❣✐❜❧❡ ✐✛ ∀k ≥ 0, ηk × 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)

✶ ❦❡②s ❛r❡ ❞r❛✇♥ ✿ (pk, sk)

r

← K(η)✱

✷ A1(pk) ❝❤♦s❡s ❛ ♣❛✐r ♦❢ ♠❡ss❛❣❡s ❛♥❞ ♦✉t♣✉ts t❤❡♠ ♣❧✉s

s♦♠❡ st❛t❡ ✐♥❢♦r♠❛t✐♦♥✿ (s, m0, m1)✱

✸ b ✐s ❝❤♦s❡♥ ❛t r❛♥❞♦♠✱ mb ✐s ❝✐♣❤❡r❡❞✿ y = E(mb)✱ ✹ A2(s, pk, m0, m1, y) ❞❡❝✐❞❡s ✇❤✐❝❤ ♠❡ss❛❣❡ ✇❛s ❡♥❝r②♣t❡❞✳

❢r❡s❤ r❛♥❞♦♠ ✈❛❧✉❡s ❛r❡ ❞r❛✇♥ ✕ ✭❦❡② ♣❛✐rs✮ ❛❞✈❡rs❛r② ❝❛❧❧s ❛r❡ ♠❛❞❡ ✕ ✭A1✮ A2 ❣❡ts ❛s ❛♥ ✐♥♣✉t ❛ t✉♣❧❡ (s, m0, m1, y)✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♣r❡✈✐♦✉s ❝♦♠♣✉t❛t✐♦♥s A1 ❛♥❞ A2 ❝❛♥ 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.(u1, . . . , um)|I1/O1, . . . , In/On ✇❤❡r❡✿

✶

x st❛♥❞s ❢♦r x1

r

← U1, . . . , xk

r

← Uk✳ ❚❤♦s❡ r❡♣r❡s❡♥t ❢r❡s❤ ❞r❛✇✐♥❣s ✐♥ ✐♥❞❡♣❡♥❞❡♥t ❞✐str✐❜✉t✐♦♥s✳

✷

a r❡♣r❡s❡♥ts ❛ ❧✐st ♦❢ ❛❞✈❡rs❛r② ❝❛❧❧s ai

r

← A

O i (ini)✳ ❚❤❡

✐♥♣✉ts ✭ini✬s✮ ❝❛♥ ❞❡♣❡♥❞ ♦♥ xk✬s ❛♥❞ ♣r❡❝❡❞✐♥❣ aj✬s✳

✸ (u1, . . . , um) ❛r❡ ❡①♣r❡ss✐♦♥s ❞❡♣❡♥❞✐♥❣ ♦♥

x ❛♥❞ a✳

✹ Ij ✐s t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♦r❛❝❧❡ Oj✳ ❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

■♥t❡r❛❝t✐♦♥ ✇✐t❤ ❛♥ ❛❞✈❡rs❛r②

✐♥♣✉t q✉❡r✐❡s ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♦r❛❝❧❡ O1 (u1, . . . , um)| I1/O1, . . . , In/On

❆❞✈❡rs❛r② A

❋r❛♠❡ s ν x.ν y. ❋r❡s❤ ❞r❛✇s ❆❞✈❡rs❛r② ❝❛❧❧s ♦✉t♣✉t R ♦✉t♣✉t (TO1, . . . , TOn)

❡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.(u1, . . . , um)|I1/O1, . . . , In/On ❜❡ ❛ ❢r❛♠❡✱ ❛♥❞

• A = (A1, . . . , Ap, A)✳

❚❤❡② ❞❡✜♥❡ A||s✱ t❤❡ r❡s✉❧t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦♥ ( x, a, I, u, R, TO)✱ ❛s ❢♦❧❧♦✇s✿

✶ ❋♦r ❡❛❝❤ i✱ ❛ ✈❛❧✉❡ ˆ

xi ✐s ❞r❛✇♥ ✐♥ Ui ❛♥❞ ❛ss✐❣♥❡❞ t♦ xi✳

✷ ❋♦r ❡❛❝❤ j✱ aj

r

← A

O j (inj) ♠❡❛♥s ˆ

inj ✐s ❝♦♠♣✉t❡❞✱ Aj ❣❡ts ✐t ❛s ❛♥ ✐♥♣✉t✱ ♣♦ss✐❜❧② ❝❛❧❧s ♦r❛❝❧❡s O ✭✐♠♣❧❡♠❡♥t❡❞ ❢♦❧❧♦✇✐♥❣

• I✮✱ ❛♥❞ ♦✉t♣✉ts ❛♥ ❛♥s✇❡r ˆ

aj✱ ✇❤✐❝❤ ✐s ❛ss✐❣♥❡❞ t♦ aj✳

✸ ❈❛❧❧s t♦ ♦r❛❝❧❡s✿ ❛♥②t✐♠❡ ❛♥ ❛❞✈❡rs❛r② q✉❡r✐❡s Ok(bs)✱ ✐t

❣❡ts Ik(bs) ❛♥❞ ❛s ❛ s✐❞❡ ❡✛❡❝t✱ TO := ˆ TO :: [(k, bs, Ik(bs))]✳

✹ ❱❛❧✉❡s ( ˆ

u1, . . . , ˆ um) ❛r❡ ❝♦♠♣✉t❡❞ ❢♦r ❡①♣r❡ss✐♦♥s (u1, . . . , um)✱ ❛♥❞ ❣✐✈❡♥ ❛s ❛♥ ✐♥♣✉t t♦ A✳

✺ ❆❢t❡r s♦♠❡ ♣♦❧②t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ✐♥❝❧✉❞✐♥❣ ♣♦ss✐❜❧❡ ♦r❛❝❧❡

❝❛❧❧s✱ A ♦✉t♣✉ts ❛ ❜✐tstr✐♥❣ ˆ R ❛ss✐❣♥❡❞ t♦ ✈❛r✐❛❜❧❡ R✱ ❛♥❞

• ˆ

TO✱ ✇❤❡r❡ ˆ TOk ✐s t❤❡ ❧✐st ♦❢ ❛❧❧ q✉❡r✐❡s t♦ Ok✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

❈♦♥❞✐t✐♦♥❛❧ ■♥❞✐st✐❣✉✐s❤❛❜✐❧✐t② ♦❢ ❋r❛♠❡s E → s ∼ t

❈♦♥❞✐t✐♦♥❛❧ ❆❞✈❛♥t❛❣❡ ▲❡t A ❜❡ ❛ ❧✐st ♦❢ ❛❞✈❡rs❛r✐❡s✳ ■ts ❛❞✈❛♥t❛❣❡ ✐♥ ❞✐st✐♥❣✉✐s❤✐♥❣ s ❛♥❞ t ❣✐✈❡♥ E ✐s✿ ❆❞✈( A, s, t, η) = | Pr[( ˆ x, ˆ a, ˆ u, ˆ I, ˆ R, ˆ TO)

r

← ( A s) : ˆ R = 1|E]− Pr[( ˆ x, ˆ a, ˆ u, ˆ I, ˆ R, ˆ TO)

r

← ( A t) : ˆ R = 1|E]| E → s ∼ t s ✐s ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ t ❣✐✈❡♥ E ✐✛ ∀ A✱ ❆❞✈( A, s, t, η) ✐s ♥❡❣❧✐❣✐❜❧❡ ✐♥ η✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

❈♦♥❞✐t✐♦♥❛❧ ◆❡❣❧✐❣✐❜✐❧✐t② ♦❢ ❊✈❡♥ts E2 → s : E1

❲❡ ❝❛♥ ❞❡✜♥❡ ♥❡❣❧✐❣✐❜✐❧✐t② ❢♦r ❛♥② ❡✈❡♥t E1 ❞❡♣❡♥❞✐♥❣ ♦♥ ✈❛r✐❛❜❧❡s ✐♥ x, a, u, R, TO, I ♦r O✳ E2 → s : E1 ▲❡t A ❜❡ ❛ ❧✐st ♦❢ ❛❞✈❡rs❛r✐❡s✳ ❊✈❡♥t A ✐s ♥❡❣❧✐❣✐❜❧❡ ✐♥ s ✐✛ | Pr[α

r

← ( A s) : E1(α)|E2(α)] ✐s ♥❡❣❧✐❣✐❜❧❡ ✐♥ η✳

❇❛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

❚✇♦ ✇❛②s ♦❢ r❡❛❞✐♥❣ ❛ r✉❧❡

❢r♦♠ t♦♣ t♦ ❜♦tt♦♠✿ v[s/y] ∼ v[t/y] s ∼ t t❤❡ ♦t❤❡r ✇❛② r♦✉♥❞ ✭❛s ❢♦r r❡❞✉❝t✐♦♥✐st ♣r♦♦❢s✮✿

v[s/y]∼v[t/y] s∼t

s✉♣♣♦s❡ t❤❡r❡ ✐s ❛♥ ❛❞✈❡rs❛r② t❤❛t ❜r❡❛❦s t❤❡ ❝♦♥❝❧✉s✐♦♥✱ t❤❡♥ t❤❡r❡ ✐s ❛ ✇❛② t♦ ♠♦❞✐❢② ✐t t♦ ❜r❡❛❦ t❤❡ ♣r❡♠✐s❡✦

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

❚❤❡ ❙✉❜st✐t✉t✐♦♥ ❘✉❧❡s

▲❡t s ❜❡ ❛ ❢r❛♠❡✱ ❛♥❞ ❧❡t v ❜❡ ❛ ♣♦❧②✲t✐♠❡ t❡r♠ ✇✐t❤ ❛ ❢r❡❡ ✈❛r✐❛❜❧❡ y✳ ❙✉❜st✐t✉t✐♦♥ ♦❢ s t♦ y ✐s ♣❡r❢♦r♠❡❞ ❛✈♦✐❞✐♥❣ ♥❛♠❡ ❝❛♣t✉r❡ ✭❡✈❡♥ ❢♦r ♦r❛❝❧❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s✮✳ A → s ∼ t ❙✉❜ A → v[s/y] ∼ v[t/y] A → s : E1 ◆❡❣❙✉❜ A → v(s) : E1 ■❞❡❛ ♦❢ t❤❡ r❡❞✉❝t✐♦♥✿ t❤❡ ❝♦♥t❡①t ✐s ♣♦❧②t✐♠❡ s✐♠✉❧❛t❛❜❧❡✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ❜② ❈❛s❡ ❙t✉❞②

❚❤❡ ✉s❡ ♦❢ t❤✐s r✉❧❡ ♠♦t✐✈❛t❡s t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❝♦♥❞✐t✐♦♥❛❧ r❡❛s♦♥✐♥❣✳

(captured by an indistinguishability statement) the scheme is secure

• ne specific query was made to an oracle

... was not made to the oracle

E → s ∼ t s : ¬E t : ¬E ❈❙ s ∼ t ■♥t✉✐t✐✈❡❧②✱ ❡✐t❤❡r E ❤♦❧❞s ❛♥❞ s ∼ t✱ ♦r ¬E ❤♦❧❞s✱ ❜✉t t❤✐s ❤❛♣♣❡♥s ✇✐t❤ ♥❡❣❧✐❣✐❜❧❡ ♣r♦❜❛❜✐❧✐t②✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

■♠♣♦rt✐♥❣ ❊①t❡r♥❛❧ ❘❡❛s♦♥✐♥❣ ✿ ✶✳❊q✉❛❧✐t②

❉❡❢✳✿ s =X t ✐✛ [α

r

← s : ΠX(α)], = [α

r

← t : ΠX(α)] ✇❤❡r❡ ΠX ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♥ X✳ A → s =R t ❯◆■❱ A → s ∼ t ✳✳✳❜❡❝❛✉s❡ t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❛♥② ❛❞✈❡rs❛r② ✐s ♥✉❧❧✳ A → s : E(X) A → s =X t ◆❡❣❯◆■❱ A → t : E(X) ✳✳✳❜❡❝❛✉s❡ E ❞❡♣❡♥❞s ❡①❝❧✉s✐✈❡❧② ♦♥ ✈❛r✐❛❜❧❡s ✐♥ X✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

■♠♣♦rt✐♥❣ ❊①t❡r♥❛❧ ❘❡❛s♦♥✐♥❣ ✿ ✷✳▲♦❣✐❝❛❧ ❉✐s❥✉♥❝t✐♦♥

p

i=1(Ai[

I/ O] → Bi[ I/ O]) ⇒ (A[ I/ O] → B[ I/ O]) A1 → s : B1 ✳ ✳ ✳ Ap → s : Bp ❯❈❘ A → s : B ■❢ ∀i, Pr[Bi|Ai] ✐s ♥❡❣❧✐❣✐❜❧❡✱ t❤❡♥ Pr[B|A] ✐s ♥❡❣❧✐❣✐❜❧❡ ✭✇✐t❤ ✉♥✐✈❡rs❛❧ q✉❛♥t✐✜❝❛t✐♦♥ ♦❢ ❢r❡❡ ✈❛r✐❛❜❧❡s✱ ❡①❝❡♣t ❢♦r ♦r❛❝❧❡ ♥❛♠❡s t❤❛t ✇❡ r❡♣❧❛❝❡ ❜② ✐♠♣❧❡♠❡♥t❛t✐♦♥s✮✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

■♠♣♦rt✐♥❣ ❊①t❡r♥❛❧ ❘❡❛s♦♥✐♥❣ ✿ ✷✳▲♦❣✐❝❛❧ ❉✐s❥✉♥❝t✐♦♥

p

i=1(Ai[

I/ O] → Bi[ I/ O]) ⇒ (A[ I/ O] → B[ I/ O]) A1 → s : B1 ✳ ✳ ✳ Ap → s : Bp ❯❈❘ A → s : B ■❢ ∀i, Pr[Bi|Ai] ✐s ♥❡❣❧✐❣✐❜❧❡✱ t❤❡♥ Pr[B|A] ✐s ♥❡❣❧✐❣✐❜❧❡ ∀i, Pr[Bi|Ai] ❛❝t✉❛❧❧② ✐s ♥❡❣❧✐❣✐❜❧❡

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

■♠♣♦rt✐♥❣ ❊①t❡r♥❛❧ ❘❡❛s♦♥✐♥❣ ✿ ✷✳▲♦❣✐❝❛❧ ❉✐s❥✉♥❝t✐♦♥

p

i=1(Ai[

I/ O] → Bi[ I/ O]) ⇒ (A[ I/ O] → B[ I/ O]) A1 → s : B1 ✳ ✳ ✳ Ap → s : Bp ❯❈❘ A → s : B

❯s❡❢✉❧ r✉❧❡s t❤❛t ✇❡ ❝❛♥ ❞❡r✐✈❡ ❢r♦♠ ❯❈❘✿ A → s : B ❯❈❘ s : A ∧ B (A → B) ⇒ (A ∧ B) A → s : B1 ✳ ✳ ✳ A → s : Bp ❯❈❘ A → s : p

i=1(Bi)

(p

i=1(A → Bi)) ⇒ (A → p i=1(Bi))

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

❆ ❘❡❞✉❝t✐♦♥ ❘✉❧❡ ✭❞❡❞✉❝✐❜❧❡ ❢r♦♠ ♦t❤❡rs✮

v ✐s ❛ ♣r♦❜❛❜✐❧✐st✐❝ ♣♦❧②✲t✐♠❡ t❡r♠ t♦ ❜❡ ❡①❤✐❜✐t❡❞ ✇❤❡♥ ❛♣♣❧②✐♥❣ t❤❡ r✉❧❡✳ A → s : E1 A(v(α)) ⇒ A(α) A → s : E2 ∧ ¬E1 ◦ v ◆❡❣❘❊❉ A → s : E2 ■❞❡❛✿ s✐♠♣❧❡ r❡❞✉❝t✐♦♥ ❜② ❡♠❜❡❞❞✐♥❣ ❛♥ ❛❞✈❡rs❛r② ❛❣❛✐♥st t❤❡ ❝♦♥❝❧✉s✐♦♥ ❛♥❞ ❛♣♣❧②✐♥❣ v t♦ ✐ts ♦✉t♣✉t✿

❖✉t♣✉t α s❛t✐s❢②✐♥❣ E2|A ❖✉t♣✉t v(α) s❛t✐s❢②✐♥❣ E1|A E2(α) ⇒ E1(v(α)) A(α) ⇒ A(v(α))

❇❛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

❈❤❛♥❣✐♥❣ ❆♥s✇❡rs t♦ ◗✉❡r✐❡s

❖♥❡ ♦❢t❡♥ ♥❡❡❞s t♦ r❡❞✉❝❡ ❜r❡❛❦✐♥❣ ❛ s❝❤❡♠❡ t♦ s♦❧✈✐♥❣ ❛ ❤❛r❞ ♣r♦❜❧❡♠ ✭❡✳❣✳✱ ✐♥✈❡rt✐♥❣ ❛ ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥✮✳ ❚♦ tr✐❝❦ t❤❡ ❛❞✈❡rs❛r② ❛❣❛✐♥st t❤❡ s❝❤❡♠❡ ✐♥t♦ s♦❧✈✐♥❣ t❤❡ ❤❛r❞ ♣r♦❜❧❡♠✱ r❡♣❧❛❝❡ t❤❡ ❛♥s✇❡r t♦ s♦♠❡ q✉❡r② ❜② s♦♠❡ ❜✐tstr✐♥❣ r❡❧❛t❡❞ t♦ t❤❡ ❝❤❛❧❧❡♥❣❡ t♦ t❤❡ ❤❛r❞ ♣r♦❜❧❡♠✳ ■♥ ♦✉r ❢r❛♠❡✇♦r❦✱ ✐t tr❛♥s❧❛t❡s ✐♥ ❝❤❛♥❣✐♥❣ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛♥ ♦r❛❝❧❡ ✭s❛②✱ O1✮ ♦♥ ♦♥❡ ❡①♣r❡ss✐♦♥ e✳ ❋♦r ♥❡❣❧✐❣✐❜✐❧✐t②✱ t✇♦ ❝❛s❡s✿ t❤❡ ❡✈❡♥t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ❤❛s t❤❡ ❢♦r♠ e / ∈ TO1 ∧ ... ♦r t❤❡ ❢♦r♠ e ∈ TO1 ∧ ...✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

■❢ t❤❡ ❡✈❡♥t ❝♦♥t❛✐♥s ✬e ✐s ♥♦t q✉❡r✐❡❞✬✳✳✳

e ✐s ❛♥ ❡①♣r❡ss✐♦♥ ♣♦ss✐❜❧② ❞❡♣❡♥❞✐♥❣ ♦♥ x, a ♦r ui✬s ♦r ✐s R✳ A → s|I1/O1 : e / ∈ TO1 ∧ E( x, y, R, TO) q = e ⇒ I1(q) = I′

1(q) ✭✯✮

◆❡❣❖❘∀ A → s|I′

1/O1 : e /

∈ TO1 ∧ E( x, y, R, TO) ✇✐t❤✿ ✭✯✮ q = e ⇒ I1(q) = I′

1(q) ♠❡❛♥✐♥❣ t❤❛t t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

②✐❡❧❞ t❤❡ s❛♠❡ r❡s✉❧t ♦♥ ❛♥② q✉❡r② ❜✉t e✳ ❆❧❧ ✈❛r✐❛❜❧❡s ♦❝❝✉r✐♥❣ ✐♥ t❤❡ st❛t❡♠❡♥t ❛r❡ q✉❛♥t✐✜❡❞ ✉♥✐✈❡rs❛❧❧②✳ ■♥t✉✐t✐✈❡❧②✱ r✉♥♥✐♥❣ ❛♥ ❛❞✈❡rs❛r② ✐♥ t❤❡ ✜rst ♦r s❡❝♦♥❞ ❝♦♥t❡①t ❧❡❛❞s t♦ t❤❡ s❛♠❡ ❡①❡❝✉t✐♦♥✳✳✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

■❢ t❤❡ ❡✈❡♥t ❝♦♥t❛✐♥s ✬e ✐s q✉❡r✐❡❞✬✳✳✳

s|I1/O1 : e ∈ TO1 ∧ E( x, y, TO) E ✐s TO✲♣r❡✜① ❝❧♦s❡❞ q = e ⇒ I1(q) = I′

1(q)

◆❡❣❖❘∃ s|I′

1/O1 : e ∈ TO1 ∧ E(

x, y, TO) q = e❄ ◆❖ I1/O1 O1(q) ❨❊❙ Ok(q) . . . A B ❙❚❖P✱ E ❤♦❧❞s ✭♣r❡✜①✲❝❧♦s❡❞✮

e ∈ TO1 ✐s ❛s❝❡rt❛✐♥❛❜❧❡ ✭✐✳❡✳ t❤❡ ❛❞✈❡rs❛r② ❝❛♥ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t ❤♦❧❞s✮✱ ✐❢ ♥♦t✱ ❞r❛✇ ✇❤❡♥ t♦ st♦♣ ❛t r❛♥❞♦♠✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

❚✇♦ ▼♦r❡ ❘✉❧❡s

A → s|I1/O1 : e ∈ TO1 q = e ⇒ I1(q) = I′

1(q)

❖❘ A → s|I1/O1 ∼ s|I′

1/O1

■❞❡❛✿ q✉❡r②✐♥❣ e ❤❛s s❛♠❡ ♣r♦❜❛❜✐❧✐t② ✐♥ ❜♦t❤ ❝♦♥t❡①ts✳ s : E′ E(TO) ⇒ ∃TO′ TO · E′(TO′) ❚❊▼P s : E ✇❤❡r❡ TO′ TO ❞❡♥♦t❡s TO′ ♣r❡✜① ❢r♦♠ TO✳ ■❞❡❛✿ st♦♣ t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ ❛❞✈❡rs❛r✐❡s ❛❣❛✐♥st s : E ♦♥❝❡ E′ ✐s ❢✉❧✜❧❧❡❞✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

❋r❛♠❡✇♦r❦ ❈■▲ ❖r❛❝❧❡s

❈♦♥❝❧✉s✐♦♥

❈■▲ ✐s ❛ s②st❡♠ t♦ ♣r♦✈❡ ✐♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ ❢r❛♠❡s✱ ✐♥ ✇❤✐❝❤ ✇❡ ❝❛♥ ♣r♦✈❡✿ ❛♥② ❛s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✇❡ ❝♦✉❧❞ ♣r♦✈❡ ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s ❢♦r♠❛❧✐s♠ ✭❍♦❛r❡ ❧♦❣✐❝ ❬❈❈❙✵✽❪✮✱ ❊❧●❛♠❛❧✱ ❍❛s❤❡❞ ❊❧✲●❛♠❛❧ ✐♥ t❤❡ ❘❖▼ ♦r st❛♥❞❛r❞ ♠♦❞❡❧✱ ❖❆❊P✱ s✐❣♥❛t✉r❡ s❝❤❡♠❡s ✭❋❉❍ ✐s ✜♥✐s❤❡❞ ❛♥❞ P❙❙ ✐s ♥❡❛r❧② ❝♦♥❝❧✉❞❡❞✮ ❋♦r♠❛❧✐③❛t✐♦♥ ♦❢ ❈■▲ ✐♥ ❈♦q ✐s ♣r♦❣r❡ss✐♥❣ ✭❙❈❆▲P Pr♦❥❡❝t✮✳

❇❛rt❤❡✱ ❉❛✉❜✐❣♥❛r❞✱ ❑❛♣r♦♥✱ ▲❛❦❤♥❡❝❤ Pr♦✈✐♥❣ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②