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❆✉t❤❡♥t✐❝❛t✐♦♥

❙❤❛♥❦❛r ❆♣r✐❧ ✶✶✱ ✷✵✶✼

❖✉t❧✐♥❡

♦✈❡r✈✐❡✇

❖✈❡r✈✐❡✇ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s✐❝s ❆✉t❤❡♥t✐❝❛t✐♥❣ ❤✉♠❛♥s ❙t♦r✐♥❣ ♣❛ss✇♦r❞s ❛t s❡r✈❡rs ❙❝❛❧✐♥❣ t♦ ♠❛♥② ✉s❡rs ❛♥❞ ❞♦♠❛✐♥s ❑❉❈✿ ❑❡② ❉✐str✐❜✉t✐♦♥ ❈❡♥t❡r ❈❆✿ ❈❡rt✐✜❝❛t✐♦♥ ❆✉t❤♦r✐t②

❆✉t❤❡♥t✐❝❛t✐♦♥ ❢r❛♠❡✇♦r❦

♦✈❡r✈✐❡✇

Pr✐♥❝✐♣❛❧s ❛r❡ ❛♣♣❧✐❝❛t✐♦♥ ❝❧✐❡♥ts ❛♥❞ s❡r✈❡rs ✐♥t❡r❛❝t✐♥❣ ♦✈❡r ❚❈P✴❯❉P ✐♥ ❛♥ ✐♥s❡❝✉r❡ ♥❡t✇♦r❦ ✭❡❣✱ ■♥t❡r♥❡t✮ Pr✐♥❝✐♣❛❧s ❡st❛❜❧✐s❤ s❡ss✐♦♥s✱ ❡①❝❤❛♥❣❡ ❞❛t❛✱ ❝❧♦s❡ s❡ss✐♦♥s ❆tt❛❝❦s ◆❡t✇♦r❦ ❛tt❛❝❦s✿ ❧✐st❡♥✱ ✐♥t❡r❝❡♣t ♠s❣s✱ r❡s❡♥❞ ♠♦❞✐❢❡❞ ♠s❣s ❊♥❞♣♦✐♥t✿ ♠❛❧✐❝✐♦✉s✴❝♦♠♣r♦♠✐s❡❞ ✉s❡r ❆✉t❤❡♥t✐❝❛t✐♦♥ ❣♦❛❧s✿ ❊♥s✉r❡ t❤❛t s❡ss✐♦♥ ♣❡❡rs ❛r❡ ✇❤♦ t❤❡② s❛② t❤❡② ❛r❡ ❊st❛❜❧✐s❤ s❡ss✐♦♥ ❦❡②✭s✮ ❢♦r ❞❛t❛ ❝♦♥✜❞❡♥t✐❛❧✐t②✴✐♥t❡❣r✐t② ❜❡tt❡r t♦ ✉s❡ t❡♠♣♦r❛r② ❦❡②s t❤❛♥ ❧♦♥❣✲t❡r♠ ❦❡②s

❚❈P✲❜❛s❡❞ s❡ss✐♦♥✿ ✇✐t❤♦✉t ❛✉t❤❡♥t✐❝❛t✐♦♥

♦✈❡r✈✐❡✇ recv( ) send(data) data recv( )

client server tcp tcp

data send(data)

• pen

connect(x2) <ip addr, port #> close( ) close( ) accept( ) accepting

• pen

closing closed connecting

• pen

closing closed

• pen to x1

tcp closing [x1, x2, ..., [data] ] x1 B [x2, x1 ..., [data] ] [x1, x2, ACK, ...] x2 [x2, x1, SYNACK, ...] [x1, x2, SYN, ...] A

❚❈P✲❜❛s❡❞ s❡ss✐♦♥✿ ❛tt❛❝❦s

♦✈❡r✈✐❡✇

endpoint attacks network attacks

data recv( ) send(data)

client server tcp tcp endpoint attacks

data recv( ) send(data) <ip addr, port #> close( ) close( )

• pen

accepting

• pen

closing closed connecting

• pen

closing closed connect(x2) accept( )

• pen to x1

[x2, x1, SYNACK, ...] x1 B [x1, x2, SYN, ...] [x1, x2, ACK, ...] [x1, x2, ..., [data] ] x2 tcp closing [x2, x1 ..., [data] ] A

❚❈P✲❜❛s❡❞ s❡ss✐♦♥✿ ✇✐t❤ ❛✉t❤❡♥t✐❝❛t✐♦♥

♦✈❡r✈✐❡✇

client server tcp tcp

• pen

connect(x2) accept( )

• pen to x1

<ip addr, port #> close( ) close( ) closed accepting

• pen

closing closed connecting

• pen

closing x2 x1 B A tcp closing tcp conn establishment tcp data exchange authenticated secure data exchange

❉✐❝t✐♦♥❛r② ✭❛❦❛ ♣❛ss✇♦r❞✲❣✉❡ss✐♥❣✮ ❛tt❛❝❦s

♦✈❡r✈✐❡✇

❲❡❛❦ s❡❝r❡t ✭❛❦❛ ❧♦✇✲q✉❛❧✐t② s❡❝r❡t✮ ❝♦♠❡s ❢r♦♠ ❛ s♣❛❝❡ s♠❛❧❧ ❡♥♦✉❣❤ ❢♦r ❛ ❜r✉t❡✲❢♦r❝❡ s❡❛r❝❤ ❡❣✿ ♣❛ss✇♦r❞s✱ ❛♥❞ ❦❡②s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡♠ ❙tr♦♥❣ s❡❝r❡t ✭❛❦❛ ❤✐❣❤✲q✉❛❧✐t② s❡❝r❡t✮✿ ♥♦t ✇❡❛❦ ❡❣✿ ❦❡② ✇✐t❤ ✶✷✽ r❛♥❞♦♠ ❜✐ts ❉✐❝t✐♦♥❛r② ❛tt❛❝❦s ✭❛❦❛ ♣❛ss✇♦r❞✲❣✉❡ss✐♥❣ ❛tt❛❝❦s✮

• ✐✈❡♥ ❝✐♣❤❡rt❡①t ❢r♦♠ str✉❝t✉r❡❞ ♣❧❛✐♥t❡①t ❛♥❞ ✇❡❛❦ ❦❡②✱

❞❡❝r②♣t ✇✐t❤ ❡✈❡r② ♣♦ss✐❜❧❡ ❦❡② ✉♥t✐❧ str✉❝t✉r❡ ❛♣♣❡❛rs ❖♥❧✐♥❡ ❛tt❛❝❦✿ ✐♥t❡r❛❝t ✇✐t❤ ❛✉t❤❡♥t✐❝❛t♦r ❛t ❡✈❡r② ❣✉❡ss ❉❡❢❡♥s❡✿ ❧✐♠✐t ♥✉♠❜❡r✴❢r❡q✉❡♥❝② ♦❢ ❛tt❡♠♣ts ❖✤✐♥❡ ❛tt❛❝❦✿ ✐♥t❡r❛❝t ✇✐t❤ ❛✉t❤❡♥t✐❝❛t♦r ❥✉st ♦♥❝❡ ❉❡❢❡♥s❡✿ ❞♦♥✬t ❡①♣♦s❡ r❡❧❡✈❛♥t ❝✐♣❤❡rt❡①t

❈♦♥✈❡♥t✐♦♥s✿ ❈r②♣t♦

♦✈❡r✈✐❡✇

❙②♠♠❡tr✐❝ ❝r②♣t♦ E(key, msg)✿ ❡♥❝r②♣t msg ✇✐t❤ key

/ / ✐♥❝❧✉❞❡s ❛♥② ■❱

D(key, ctx)✿ ❞❡❝r②♣t ctx ✇✐t❤ key

/ / ✐♥❝❧✉❞❡s ❛♥② ■❱

❍❛s❤ H(msg)✿ ❤❛s❤ ♦❢ msg

/ / ❡❣✱ ❙❍❆✲✶

H(key, msg)✿ ❦❡②❡❞✲❤❛s❤

/ / ❡❣✱ ❍▼❆❈✲❙❍❆✲✶

❆s②♠♠❡tr✐❝ ❝r②♣t♦

/ / ♣✉❜❧✐❝✲❦❡② ♣❛✐r [sk, pk]

EP(pk, msg)✿ ❡♥❝r②♣t msg ✭✇✐t❤ ♣✉❜❧✐❝ ❦❡②✮ DP(sk, msg)✿ ❞❡❝r②♣t msg ✭✇✐t❤ s❡❝r❡t ❦❡②✮ Sgn(sk, msg)✿ s✐❣♥❛t✉r❡ ♦❢ msg ✭✉s✐♥❣ s❡❝r❡t ❦❡②✮ Vfy(pk, msg, s)✿ ✈❡r✐❢② s✐❣♥❛t✉r❡ s ♦❢ msg ✭✇✐t❤ ♣✉❜❧✐❝ ❦❡②✮

❈♦♥✈❡♥t✐♦♥s✿ ◆♦♥❝❡s

♦✈❡r✈✐❡✇

◆♦♥❝❡✿ ♥❡✇ ✈❛❧✉❡

/ / ♥❡✇ ❂ ♥❡✈❡r ❜❡❢♦r❡ s❡❡♥

❈❛♥ ❜❡ ♣r❡❞✐❝t❛❜❧❡ ♦r r❛♥❞♦♠ Pr❡❞✐❝t❛❜❧❡✿ ❣✐✈❡♥ ♦♥❡ ✈❛❧✉❡✱ ❛tt❛❝❦❡r ❝❛♥ ❣✉❡ss t❤❡ ♥❡①t ♦♥❡ ❘❛♥❞♦♠✿ ♥♦t ♣r❡❞✐❝t❛❜❧❡ /

/ ♣❤②s✐❝❛❧ r❛♥❞♦♠♥❡ss✱ ❝r②♣t♦ ♦✉t♣✉t

❖✉t❧✐♥❡

❛✉t❤ ❜❛s✐❝

❖✈❡r✈✐❡✇ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s✐❝s ❆✉t❤❡♥t✐❝❛t✐♥❣ ❤✉♠❛♥s ❙t♦r✐♥❣ ♣❛ss✇♦r❞s ❛t s❡r✈❡rs ❙❝❛❧✐♥❣ t♦ ♠❛♥② ✉s❡rs ❛♥❞ ❞♦♠❛✐♥s ❑❉❈✿ ❑❡② ❉✐str✐❜✉t✐♦♥ ❈❡♥t❡r ❈❆✿ ❈❡rt✐✜❝❛t✐♦♥ ❆✉t❤♦r✐t②

A, B s❤❛r❡ s②♠♠❡tr✐❝ ❦❡② k

❛✉t❤ ❜❛s✐❝

❝❧✐❡♥t A ✭❦❡② k ❢♦r s❡r✈❡r B✮ s❡r✈❡r B ✭❤❛s ❦❡② k ❢♦r ✉s❡r A✮ s❡♥❞ [A, B, ❝♦♥♥] r❝✈ [A, B, ❝♦♥♥] cB ← r❛♥❞♦♠ /

/ s❡r✈❡r ❝❤❛❧❧❡♥❣❡

s❡♥❞ [B, A, cB] r❝✈ [B, A, cB] cA ← r❛♥❞♦♠ /

/ ❝❧✐❡♥t ❝❤❛❧❧❡♥❣❡

rB ← E(k, cB) /

/ ❝❧✐❡♥t r❡s♣♦♥s❡

s❡♥❞ [cA, rB] r❝✈ [cA, rB] ✐❢ ✭rB = E(k, cB)✮ ❋❆■▲ rA ← E(k, cA) /

/ s❡r✈❡r r❡s♣♦♥s❡

s❡♥❞ [rA] s❡ss✐♦♥ ❦❡② ← Func(cA, cB, k) r❝✈ [rA] ✐❢ ✭rA = E(k, cA)✮ ❋❆■▲ s❡ss✐♦♥ ❦❡② ← Func(cA, cB, k)

A, B s❤❛r❡ s②♠♠❡tr✐❝ ❦❡② k ✭❝♦♥t✮

❛✉t❤ ❜❛s✐❝

▼❛♥② ✈❛r✐❛t✐♦♥s ♦❢ ❝❤❛❧❧❡♥❣❡✴r❡s♣♦♥s❡ ♦♣❡♥ ❝❤❛❧❧❡♥❣❡✱ ❡♥❝r②♣t❡❞ r❡s♣♦♥s❡

/ / cA → E(k, cA)

❡♥❝r②♣t❡❞ ❝❤❛❧❧❡♥❣❡ ❛♥❞ r❡s♣♦♥s❡

/ / E(k, cA) → E(k, cA + ✶)

❖✤✐♥❡ ❞✐❝t✐♦♥❛r② ❛tt❛❝❦ ✐❢ k ✐s ✇❡❛❦ ❛♥❞ ❛tt❛❝❦❡r ❝❛♥ ❡❛✈❡s❞r♦♣✱ ♦r ❛tt❛❝❦❡r ❝❛♥ ❛tt❛❝❤ t♦ B✬s ♥❡t ❛❞❞r❡ss ■❢ ❝❧✐❡♥t ✐ss✉❡s ❝❤❛❧❧❡♥❣❡ ✜rst ❛♥❞ k ✐s ✇❡❛❦✱ ❝❛♥ ❞♦ ♦✤✐♥❡ ❞✐❝t✐♦♥❛r② ❛tt❛❝❦ ✇✐t❤♦✉t ❛tt❛❝❦✐♥❣ ♥❡t✇♦r❦ ❛tt❛❝❦❡r s❡♥❞s ❝❤❛❧❧❡♥❣❡✱ ❣❡ts r❡s♣♦♥s❡

A, B ❤❛✈❡ ❡❛❝❤ ♦t❤❡r✬s ♣✉❜❧✐❝ ❦❡②

❛✉t❤ ❜❛s✐❝

❝❧✐❡♥t A ✭❤❛s [skA, pkA]✱ pkB✮ s❡r✈❡r B ✭❤❛s [skB, pkB]✱ pkA✮ cA ← r❛♥❞♦♠

/ / ❝❤❛❧❧❡♥❣❡

s❡♥❞ [A, B, ❝♦♥♥, EP(pkB, cA)] r❝✈ [A, B, ❝♦♥♥, yA] cA ← DP(skB, yA) cB ← r❛♥❞♦♠

/ / ❝❤❛❧❧❡♥❣❡

s❡♥❞ [B,A, EP(pkA, [cB, cA])] /

/r❡s♣

r❝✈ [B, A, yB] [cB, rA] ← DP(skA, yB) ✐❢ ✭rA = cA✮ ❋❆■▲ s❡♥❞ [EP(pkB, cB)]

/ / r❡s♣♦♥s❡

s❡ss✐♦♥ ❦❡② ← Func(cA, cB) r❝✈ [yB] rB ← DP(skB, yB) ✐❢ ✭rB = cB✮ ❋❆■▲ s❡ss✐♦♥ ❦❡② ← Func(cA, cB) ❙❛❢❡ ❢r♦♠ ❞✐❝t✐♦♥❛r② ❛tt❛❝❦

/ / ❛s②♠♠❡tr✐❝ ❦❡②s ❛❧✇❛②s str♦♥❣

A ❤❛s pkB❀ A, B s❤❛r❡ s②♠♠ ❦❡② k

❛✉t❤ ❜❛s✐❝

❝❧✐❡♥t A ✭❤❛s k✱ pkB✮ s❡r✈❡r B ✭❤❛s [skB, pkB]✱ k✮ cA ← r❛♥❞♦♠

/ / ❝❤❛❧❧❡♥❣❡

s❡♥❞ [A, B, ❝♦♥♥, EP(pkB, cA)] r❝✈ [A, B, ❝♦♥♥, yA] rA ← DP(skB, yA)

/ / r❡s♣♦♥s❡

cB ← r❛♥❞♦♠

/ / ❝❤❛❧❧❡♥❣❡

s❡♥❞ [B,A, cB, rA)]

/ / ♣❧❛✐♥t❡①t

r❝✈ [B, A, cB, rA] ✐❢ ✭rA = cA✮ ❋❆■▲ rB ← E(k, cB)

/ / r❡s♣♦♥s❡

s❡♥❞ [EP(pkB, rB)] s❡ss✐♦♥ ❦❡② ← Func(cA, cB, k) r❝✈ [yB] rB ← DP(skB, yB) ✐❢ ✭D(k, rB) = cB✮ ❋❆■▲ s❡ss✐♦♥ ❦❡② ← Func(cA, cB, k) ❲❛r♥✐♥❣✿ t❤❡ ❛❜♦✈❡ s❡ss✐♦♥ ❦❡② ✐s ✇❡❛❦ ✐❢ k ✐s ✇❡❛❦

/ / ❲❤②❄

❇❡tt❡r t♦ ✉s❡ ❉❍ t♦ ❣❡t ❛ str♦♥❣ s❡ss✐♦♥ ❦❡②

❆✉t❤❡♥t✐❝❛t❡❞ ❉✐✣❡✲❍❡❧♠❛♥

❛✉t❤ ❜❛s✐❝

❆✉t❤❡♥t✐❝❛t❡❞ ❉❍✿ ✐♥❝♦r♣♦r❛t❡ ❛ ♣r❡✲s❤❛r❡❞ ❦❡② ✐♥t♦ ❉❍ ■❢ A ❛♥❞ B s❤❛r❡ ❛ s②♠♠❡tr✐❝ ❦❡② k✱ ❤❡r❡ ❛r❡ t✇♦ ✇❛②s ✶✳❊♥❝r②♣t ❉❍ ♣✉❜❧✐❝ ❦❡②s ✇✐t❤ k A s❡♥❞s E(k, g SA ♠♦❞✲p) B s❡♥❞s E(k, g SB ♠♦❞✲p) s❤❛r❡❞ ❦❡②✿ g SA·SB ♠♦❞✲p ✷✳❉♦ ✉s✉❛❧ ❉❍✱ t❤❡♥ ❡①❝❤❛♥❣❡ ❦❡②❡❞✲❤❛s❤❡s ♦❢ ❉❍ ❦❡②✳ ❙❡❝✉r❡ ❛❣❛✐♥st ❞✐❝t✐♦♥❛r② ❛tt❛❝❦ ❡✈❡♥ ✐❢ k ✐s ✇❡❛❦✦ ■❢ A ❛♥❞ B ❤❛✈❡ ❡❛❝❤ ♦t❤❡r✬s ♣✉❜❧✐❝ ❦❡②✱ ❤❡r❡ ❛r❡ t✇♦ ✇❛②s ✶✳❊♥❝r②♣t ❉❍ q✉❛♥t✐t✐❡s ✇✐t❤ r❡❝❡✐✈❡r✬s ♣✉❜❧✐❝ ❦❡② ✷✳❙✐❣♥ ❉❍ q✉❛♥t✐t✐❡s ✇✐t❤ s❡♥❞❡r✬s ♣r✐✈❛t❡ ❦❡②

❙❡ss✐♦♥ ❑❡②s

❛✉t❤ ❜❛s✐❝

❙❤♦✉❧❞ ❞✐✛❡r ❢r♦♠ ❧♦♥❣✲t❡r♠ ❦❡② ✉s❡❞ ❢♦r ❛✉t❤❡♥t✐❝❛t✐♦♥ t♦ ❛✈♦✐❞ ❧♦♥❣✲t❡r♠ ❦❡② ✏✇❡❛r✐♥❣ ♦✉t✑ ✭♦✤✐♥❡ ❝r②♣t♦ ❛tt❛❝❦✮ ❙❤♦✉❧❞ ❜❡ ❢♦r❣♦tt❡♥ ❛❢t❡r s❡ss✐♦♥ ❡♥❞s ❙❤♦✉❧❞ ❜❡ ✉♥✐q✉❡ ❢♦r ❡❛❝❤ s❡ss✐♦♥ ✐❢ ❝♦♠♣r♦♠✐s❡❞✱ ♦♥❧② ❛✛❡❝ts ❞❛t❛ s❡♥t ✐♥ t❤❛t s❡ss✐♦♥ ❝❛♥ ❣✐✈❡ t♦ ✉♥tr✉st❡❞ s♦❢t✇❛r❡

/ / ❞❡❧❡❣❛t✐♦♥

❉❡❧❡❣❛t✐♦♥ A, B s❤❛r❡ ❦❡② k A ✇❛♥ts C t♦ ❛❝❝❡ss B ♦♥ A✬s ❜❡❤❛❧❢ ❚✇♦ s♦❧✉t✐♦♥s t♦ ❞❡❧❡❣❛t✐♦♥ ✶✳A ❣✐✈❡s C t❤❡ s❤❛r❡❞ ❦❡② k

/ / t❡rr✐❜❧❡✦

✷✳A ❣✐✈❡s C ❛ t✐❝❦❡t✿ E(k, [allowed ops, expiry time, ...])

❖✉t❧✐♥❡

❛✉t❤ ❤✉♠❛♥s

❖✈❡r✈✐❡✇ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s✐❝s ❆✉t❤❡♥t✐❝❛t✐♥❣ ❤✉♠❛♥s ❙t♦r✐♥❣ ♣❛ss✇♦r❞s ❛t s❡r✈❡rs ❙❝❛❧✐♥❣ t♦ ♠❛♥② ✉s❡rs ❛♥❞ ❞♦♠❛✐♥s ❑❉❈✿ ❑❡② ❉✐str✐❜✉t✐♦♥ ❈❡♥t❡r ❈❆✿ ❈❡rt✐✜❝❛t✐♦♥ ❆✉t❤♦r✐t②

■❞❡♥t✐❢②✐♥❣ ❛ ❤✉♠❛♥

❛✉t❤ ❤✉♠❛♥s

❲❤❛t ✇❡ ❦♥♦✇ ♣❛ss✇♦r❞✱ ❞❛t❡✲♦❢✲❜✐rt❤✱ ❛❞❞r❡ss✱ ❡t❝ ❈♦♥s✿ ❡①♣♦s❡❞ ✇❤❡♥ ✉s❡❞ ❲❤❛t ♣❤②s✐❝❛❧ ♦❜❥❡❝t ✇❡ ❤♦❧❞ ❜❛❞❣❡s✱ ❦❡②s✱ s♠❛rt ❝❛r❞ ✭✇✐t❤ str♦♥❣ ❝r②♣t♦✮ ❈♦♥s✿ ♦❜❥❡❝t ♠✉st ❜❡ ❞✐✣❝✉❧t t♦ ❢♦r❣❡✱ t❛♠♣❡r✱ r❡✈❡rs❡ ❡♥❣✐♥❡❡r ❲❤❛t ♣❤②s✐❝❛❧ ♣r♦♣❡rt② ✇❡ ❤❛✈❡ ✭❜✐♦♠❡tr✐❝s✮ ✜♥❣❡r♣r✐♥t✱ ❢❛❝❡✱ ✐r✐s ❈♦♥s✿ ♥♦t ❤❛r❞ t♦ ❢♦r❣❡ ❖t❤❡rs✿ ✇❤❡r❡ ✇❡ ❛r❡✱ ❤♦✇ ✇❡ r❡❛❝t✒ ✇❤❡r❡ ✇❡ tr❛✈❡❧✱ ❡t❝ ❈♦♥s✿ ❡❛s② t♦ ❢♦r❣❡ ❚②♣✐❝❛❧❧② ✉s❡ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♠❡t❤♦❞s ❡❣✿ ♣❛ss✇♦r❞✱ ❜r♦✇s❡r ✜♥❣❡r♣r✐♥t✱ ❧♦❝❛t✐♦♥✱ ✳✳✳

P❛ss✇♦r❞s

❛✉t❤ ❤✉♠❛♥s

❙❡tt✐♥❣ ❛ ♣❛ss✇♦r❞ A ❝❤♦♦s❡s ❛ ♣❛ss✇♦r❞ t❤❛t ✐s ❤❛r❞ t♦ ❣✉❡ss

/ / ❤♦✇ ❤❛r❞❄

A s❤❛r❡s ✐t s❡❝✉r❡❧② ✇✐t❤ B✱ ✇❤✐❝❤ st♦r❡s ✐t ▲♦❣❣✐♥❣ ✐♥ A ♣r♦✈✐❞❡s B t❤❡ ♣❛ss✇♦r❞❀ B ❝❤❡❝❦s ✐t A ✐s ❛✉t❤❡♥t✐❝❛t❡❞ ✐✛ ♠❛t❝❤ ■❢ ♥♦ ♠❛t❝❤✿ B ♠❛② ❞❡❧❛② ♥❡①t ❧♦❣✐♥ ❛tt❡♠♣t t♦ A ❘❡❝♦✈❡r✐♥❣ ❛ ❢♦r❣♦tt❡♥ ♣❛ss✇♦r❞ ❋❛❧❧✐♥❣ ❜❛❝❦ t♦ s♦♠❡ ♦t❤❡r ❢♦r♠ ❛✉t❤❡♥t✐❝❛t✐♦♥ ♣r❡✲s♣❡❝✐✜❡❞ ❡♠❛✐❧ ♦r ♣❤♦♥❡ ✈✐s✐t ♦✣❝❡ ✇✐t❤ ♣❤②s✐❝❛❧ ✐❞

❲❤❛t ❞♦ str♦♥❣ ♣❛ss✇♦r❞s ❣❡t ✉s

❛✉t❤ ❤✉♠❛♥s

❙tr♦♥❣ ♣❛ss✇♦r❞ ❍❛r❞ t♦ ❣✉❡ss❀ ✐♥❝❧✉❞❡s s②♠❜♦❧s✱ ♠✐①❡❞ ❝❛s❡✱ ❡t❝ ❉✐❝t✐♦♥❛r② ❛tt❛❝❦ ❞♦❛❜❧❡✱ ❜✉t ♠♦r❡ ✇♦r❦ t❤❛♥ ✇❡❛❦ ♣✇❞ ❖♥❧✐♥❡ ❞✐❝t✐♦♥❛r② ❛tt❛❝❦ ❉❡❢❡♥s❡✿ ❧✐♠✐t ♦♥ ♥✉♠❜❡r ♦❢ ✇r♦♥❣ ❧♦❣✐♥s ❚❛r❣t❡❞ ✈✐❝t✐♠✿ str♦♥❣ ♣✇❞ ❞♦❡s♥✬t ❤❡❧♣ ❆♥② ✈✐❝t✐♠ ✭st♦♣ ❛t ✜rst s✉❝❝❡ss✮✿ str♦♥❣ ♣✇❞ ❤❡❧♣s ❖✤✐♥❡ ❞✐❝t✐♦♥❛r② ❛tt❛❝❦ ❚❛r❣❡t❡❞ ✈✐❝t✐♠✿ str♦♥❣ ♣✇❞ ❞♦❡s♥✬t ❤❡❧♣ ✭✉♥❧❡ss ✈❡r② str♦♥❣✮ ❆♥② ✈✐❝t✐♠✿ str♦♥❣ ♣✇❞ ❤❡❧♣s ✭✐❢ ♠❛♥② ♦t❤❡rs ❤❛✈❡ ✇❡❛❦ ♣✇❞s✮

❖✉t❧✐♥❡

♣✇❞s❅sr✈r

❖✈❡r✈✐❡✇ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s✐❝s ❆✉t❤❡♥t✐❝❛t✐♥❣ ❤✉♠❛♥s ❙t♦r✐♥❣ ♣❛ss✇♦r❞s ❛t s❡r✈❡rs ❙❝❛❧✐♥❣ t♦ ♠❛♥② ✉s❡rs ❛♥❞ ❞♦♠❛✐♥s ❑❉❈✿ ❑❡② ❉✐str✐❜✉t✐♦♥ ❈❡♥t❡r ❈❆✿ ❈❡rt✐✜❝❛t✐♦♥ ❆✉t❤♦r✐t②

❙t♦r✐♥❣ ♣❛ss✇♦r❞s ❛t s❡r✈❡r

♣✇❞s❅sr✈r

❆ss✉♠❡ ❛♥ ❛tt❛❝❦❡r t❤❛t ❤❛s ❛❝❝❡ss t♦ s❡r✈❡r ✜❧❡s②st❡♠ ❆tt❡♠♣t ✶✿ st♦r❡ [usr, pwd] ♣❛✐rs ✐♥ ♣❧❛✐♥t❡①t ✜❧❡✿ ✇♦rt❤❧❡ss ❆tt❡♠♣t ✷✿ st♦r❡ [usr, pwd] ♣❛✐rs ✐♥ ❡♥❝r②♣t❡❞ ✜❧❡ ✇♦rt❤❧❡ss ✐❢ ❡♥❝r②♣t✐♦♥ ❦❡② ✐s ❛❧s♦ ✐♥ ✜❧❡s②st❡♠ ❆tt❡♠♣t ✸✿ st♦r❡ ❤❛s❤❡s ♦❢ ♣❛ss✇♦r❞s st♦r❡ [usr, h] ♣❛✐rs ✐♥ ♣❧❛✐♥t❡①t ✜❧❡✱ ✇❤❡r❡ h = H(pwd) ✇❤❡♥ A ❧♦❣s ✐♥ ✇✐t❤ pwd✱ ❝❤❡❝❦ ✐❢ H(pwd) = h

• ♦♦❞✿ pwd ✐s ♥❡✈❡r ✐♥ ✜❧❡s②st❡♠✱ ♦♥❧② ❜r✐❡✢② ✐♥ ♠❡♠♦r②

❇❛❞✿ ✈✉❧♥❡r❛❜❧❡ t♦ ❞✐❝t✐♦♥❛r② ❛tt❛❝❦ ❛tt❛❝❦❡r ♣r❡❝♦♠♣✉t❡s {H(pi)} ❢♦r ❝❛♥❞✐❞❛t❡ ♣✇❞s p✶, p✷, ... ❝❤❡❝❦s ❡❛❝❤ H(pi) ❛❣❛✐♥st t❤❡ h✬s ♦❢ ❛❧❧ ✉s❡rs

❙t♦r✐♥❣ ♣❛ss✇♦r❞s ❛t s❡r✈❡r ✭❝♦♥t✮

♣✇❞s❅sr✈r

❆tt❡♠♣t ✹✿ st♦r❡ ❤❛s❤❡s ♦❢ s❛❧t❡❞ ♣❛ss✇♦r❞s s❛❧t ✐s ❛ r❛♥❞♦♠ ♥♦♥❝❡✱ ❞✐✛❡r❡♥t ❢♦r ❡❛❝❤ ✉s❡r st♦r❡ [usr, salt, h] tr✐♣❧❡s✱ ✇❤❡r❡ h = H(saltpwd) ✇❤❡♥ A ❧♦❣s ✐♥ ✇✐t❤ pwd✱ ❝❤❡❝❦ ✐❢ H(saltpwd) = h ❉✐❝t✐♦♥❛r② ❛tt❛❝❦ st✐❧❧ ❞♦❛❜❧❡ ❜✉t ♠♦r❡ ✇♦r❦ ❝❛♥❞✐❞❛t❡ ❤❛s❤❡s {H(pi)} ❝❛♥♥♦t ❜❡ ♣r❡❝♦♠♣✉t❡❞ ❡❛❝❤ ❝❛♥❞✐❞❛t❡ ❤❛s❤ ❛♣♣❧✐❡s ♦♥❧② t♦ ♦♥❡ ✉s❡r ❆tt❡♠♣t ✺✿ st♦r❡ k✲❢♦❧❞ ❤❛s❤❡s ♦❢ s❛❧t❡❞ ♣❛ss✇♦r❞s st♦r❡ [usr, salt, h] tr✐♣❧❡s✱ ✇❤❡r❡ h = Hk(saltpwd) Hk(x) = H(H(· · · H(x) · · · )) k t✐♠❡s

/ / s❧♦✇ ❤❛s❤

❉✐❝t✐♦♥❛r② ❛tt❛❝❦ st✐❧❧ ❞♦❛❜❧❡ ❜✉t ✇♦r❦ ✐♥❝r❡❛s❡s k t✐♠❡s

❖✉t❧✐♥❡

s❝❛❧✐♥❣

❖✈❡r✈✐❡✇ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s✐❝s ❆✉t❤❡♥t✐❝❛t✐♥❣ ❤✉♠❛♥s ❙t♦r✐♥❣ ♣❛ss✇♦r❞s ❛t s❡r✈❡rs ❙❝❛❧✐♥❣ t♦ ♠❛♥② ✉s❡rs ❛♥❞ ❞♦♠❛✐♥s ❑❉❈✿ ❑❡② ❉✐str✐❜✉t✐♦♥ ❈❡♥t❡r ❈❆✿ ❈❡rt✐✜❝❛t✐♦♥ ❆✉t❤♦r✐t②

❙❝❛❧✐♥❣ t♦ N ✉s❡rs

s❝❛❧✐♥❣

◆❛✐✈❡ ❛♣♣r♦❛❝❤ ✉s✐♥❣ s②♠♠❡tr✐❝ ❦❡②s ❊✈❡r② ♣r✐♥❝✐♣❛❧ s❤❛r❡s ❛ ❦❡② ✇✐t❤ ❡✈❡r② ♦t❤❡r ♣r✐♥❝✐♣❛❧ ◆♦t s❝❛❧❛❜❧❡ N✷ st♦r❛❣❡ ❛t ❡❛❝❤ ♣r✐♥❝✐♣❛❧ N ❝♦st ❢♦r ❛❞❞✐♥❣✴r❡♠♦✈✐♥❣ ♣r✐♥❝✐♣❛❧ ◆❛✐✈❡ ❛♣♣r♦❛❝❤ ✉s✐♥❣ ❛s②♠♠❡tr✐❝ ❦❡②s ❤❛s s✐♠✐❧❛r ♣r♦❜❧❡♠s ❙②♠♠❡tr✐❝✲❦❡② s♦❧✉t✐♦♥✿ ❦❡② ❞✐str✐❜✉t✐♦♥ ❝❡♥t❡r ✭❑❉❈✮ ❆s②♠♠❡tr✐❝✲❦❡② s♦❧✉t✐♦♥✿ ❝❡rt✐✜❝❛t✐♦♥ ❛✉t❤♦r✐t② ✭❈❆✮ ❇r✐♥❣s ✉♣ ♥❡✇ ❛tt❛❝❦s ✐♥✈♦❧✈✐♥❣ ♥♦✲❧♦♥❣❡r✲✈❛❧✐❞ ♠❛st❡r ❦❡②s ❛ ❚❖❈❚❖❯ ✈✉❧♥❡r❛❜✐❧✐t② ❉♦♠❛✐♥✿ s❡t ♦❢ ♣r✐♥❝✐♣❛❧s ❝♦✈❡r❡❞ ❜② ♦♥❡ ❑❉❈ ♦r ❈❆

❖✉t❧✐♥❡

❦❞❝ s❝❛❧✐♥❣

❖✈❡r✈✐❡✇ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s✐❝s ❆✉t❤❡♥t✐❝❛t✐♥❣ ❤✉♠❛♥s ❙t♦r✐♥❣ ♣❛ss✇♦r❞s ❛t s❡r✈❡rs ❙❝❛❧✐♥❣ t♦ ♠❛♥② ✉s❡rs ❛♥❞ ❞♦♠❛✐♥s ❑❉❈✿ ❑❡② ❉✐str✐❜✉t✐♦♥ ❈❡♥t❡r ❈❆✿ ❈❡rt✐✜❝❛t✐♦♥ ❆✉t❤♦r✐t②

❉♦♠❛✐♥ ✇✐t❤ ❛ ❑❉❈

❦❞❝ s❝❛❧✐♥❣

❑❉❈ ✐s ❛ s♣❡❝✐❛❧ ♣r✐♥❝✐♣❛❧ ✐♥ t❤❡ ❞♦♠❛✐♥ ✭❂ ♥❡t✇♦r❦ ✉s✉❛❧❧②✮ ❊✈❡r② ♦t❤❡r ♣r✐♥❝✐♣❛❧ z s❤❛r❡s ❛ ♠❛st❡r ❦❡②✱ s❛② kz✱ ✇✐t❤ ❑❉❈ A✲B s❡ss✐♦♥✿ A ❣❡ts ❬s❡ss✐♦♥ ❦❡②✱ t✐❝❦❡t ❢♦r B❪ ❢r♦♠ ❑❉❈ ❝❧✐❡♥t A ✭❤❛s kA✮ ❑❉❈ ✭❤❛s kA✱ kB✮ s❡r✈❡r B ✭❤❛s kB✮ s❡♥❞ [A, B] t♦ ❑❉❈ r❝✈ [A, B] S ← random /

/ s❡ss✐♦♥ ❦❡②

tA ← E(kA, [A, B, S]) tB ← E(kB, [A, B, S]) s❡♥❞ [tA, tB] t♦ A r❝✈ [tA, tB] ·, ·, S ← D(kA, tA) s❡♥❞ [A, B, tB] r❝✈ [A, B, tB] ·, ·, S ← D(kB, tB) ❆❜♦✈❡ ✐s ✐♥❝♦♠♣❧❡t❡✿ ❡❣✱ ✈✉❧♥❡r❛❜❧❡ t♦ r❡♣❧❛② ♦❢ S

❚r✉st ♠♦❞❡❧✱ ♣r♦s✱ ❝♦♥s

❦❞❝ s❝❛❧✐♥❣

❚r✉st t❤❡ ❑❉❈ t♦ ♥♦t ✐ss✉❡ ✇❡❛❦ ❦❡②s✱ r❡✉s❡ ❦❡②s✱ r❡❛❞ ♠s❣s✱ ✐♠♣❡rs♦♥❛t❡ ♦t❤❡rs✱ ❡t❝ ❣♦ ♦✤✐♥❡ ❆❞✈❛♥t❛❣❡s ♦❢ ❑❉❈ ❆❞❞✐♥❣ ♥❡✇ ♣r✐♥❝✐♣❛❧ D✿ ♦♥❡ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ D ❛♥❞ ❑❉❈ ❘❡✈♦❝❛t✐♦♥ ♦❢ ♣r✐♥❝✐♣❛❧ D✿ ❞❡❛❝t✐✈❛t❡ D✬s ♠❛st❡r ❦❡② ❛t ❑❉❈ ❉✐s❛❞✈❛♥t❛❣❡s ♦❢ ❑❉❈ ❑❉❈ ❝♦♠♣r♦♠✐s❡ ♠❛❦❡s t❤❡ ❡♥t✐r❡ ♥❡t✇♦r❦ ✈✉❧♥❡r❛❜❧❡✳ ❑❉❈ ❢❛✐❧✉r❡ ♠❡❛♥s ♥♦ ♥❡✇ s❡ss✐♦♥s ❝❛♥ ❜❡ st❛rt❡❞✳ ❑❉❈ ❝❛♥ ❜❡ ❛ ♣❡r❢♦r♠❛♥❝❡ ❜♦tt❧❡♥❡❝❦✳ ❘❡♣❧✐❝❛t✐♥❣ ❑❉❈ ✜①❡s t❤❡ ❧❛st t✇♦ ❞✐s❛❞✈❛♥t❛❣❡s✱ ❜✉t t❤❡♥ ♥❡❡❞ t♦ ♣r♦t❡❝t r❡♣❧✐❝❛s ❛♥❞ ❦❡❡♣ t❤❡♠ ✐♥ s②♥❝ ✐❢ ♠❛st❡r ❦❡② ❝❤❛♥❣❡s✱ ♥❡❡❞ t♦ ❤❛♥❞❧❡ t✐❝❦❡ts ✐ss✉❡❞ ✇✐t❤ ♦❧❞ ❦❡②

❈r♦ss✲❞♦♠❛✐♥ s❡ss✐♦♥

❦❞❝ s❝❛❧✐♥❣

A✬s ❑❉❈ ✐s X B✬s ❑❉❈ ✐s Y X, Y s❤❛r❡ ❦❡② kXY A✿ s❡♥❞ [A, B.Y ] t♦ X X✿ ❣❡♥❡r❛t❡ s❡ss✐♦♥ ❦❡② kAY

/ / ❢♦r A✕Y s❡ss✐♦♥

tXA ← E(kAX, [A, Y , kAY ])

/ / kAX✿ A✲X ❦❡②

tXY ← E(kXY , [A, Y , kAY ])

/ / kXY ✿ X✲Y ❦❡②

s❡♥❞ [tXA, tXY ] t♦ A A✿ ❡①tr❛❝t kAY ❢r♦♠ tXA❀ s❡♥❞ [A.X, B, tXY ] t♦ Y Y ✿ ❡①tr❛❝t kAY ❢r♦♠ tXY ❣❡♥❡r❛t❡ s❡ss✐♦♥ ❦❡② kAB

/ / ❢♦r A✲B s❡ss✐♦♥

tYA ← E(kAY , [A, Y , kAB]) tYB ← E(kBY , [A, Y , kAB])

/ / kBY ✿ B✲Y ❦❡②

s❡♥❞ [tYA, tYB] t♦ A A✿ ❡①tr❛❝t kAB ❢r♦♠ tYA❀ s❡♥❞ [A, B, tYB] t♦ Y B✿ ❡①tr❛❝t kAB ❢r♦♠ tYB

/ / A, B ♥♦✇ s❤❛r❡ kAB

❈r♦ss✲❞♦♠❛✐♥s s❡ss✐♦♥ A.X✶✲X✷✲· · · ✲B.XN

❦❞❝ s❝❛❧✐♥❣

A ❣❡ts ❬s❡ss✐♦♥ ❦❡② kA,X✷✱ t✐❝❦❡t tX✶,X✷❪ ❢r♦♠ X✶ A ❣❡ts ❬s❡ss✐♦♥ ❦❡② kA,X✸✱ t✐❝❦❡t tX✷,X✸❪ ❢r♦♠ X✷ · · · A ❣❡ts ❬s❡ss✐♦♥ ❦❡② kA,B✱ t✐❝❦❡t tXN,B❪ ❢r♦♠ XN A s❡♥❞s ❬t✐❝❦❡t tXN,B❪ t♦ B ❇❡tt❡r✿ A ♣❛ss❡s ❛❧♦♥❣ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❑❉❈s tr❛✈❡rs❡❞✱ s♦ t❤❛t B s❡❡s t❤❡ ❡♥t✐r❡ ❑❉❈✲❝❤❛✐♥ r❛t❤❡r t❤❛♥ ❥✉st XN

❑❡r❜❡r♦s✿ ❯♥✐① ❑❉❈ ✐♠♣❧❡♠❡♥❛t✐♦♥

❦❞❝ s❝❛❧✐♥❣

❑❡r❜❡r♦s ✶ ❑❡r❜❡r♦s ✷ ❈♦♠♠♦♥❧② ✉s❡❞ ✐♥ ❡♥t❡r♣r✐s❡✲❧❡✈❡❧ ♥❡t✇♦r❦s ❍❛♥❞❧❡s ❈❤❛♥❣✐♥❣ ♠❛st❡r ❦❡②s ❚✐❝❦❡ts✿ ❧♦♥❣✲❧✐✈❡❞✱ ♣♦st✲❞❛t❡❞✱ ❞❡❧❡❣❛t✐♦♥✱ ❡t❝ ❍❛♥❞❧❡s ✈❛r✐❡t② ♦❢ ❝r②♣t♦✱ ❤✇ ❛r❝❤✐t❡❝t✉r❡✱ ❡t❝ ❈♦♠♣❡♥s❛t❡s ❢♦r ✇❡❛❦ ❦❡②s ✭❤✉♠❛♥ ✉s❡rs✮ ❳✲s❡r✈❡rs ❈r♦ss✲❞♦♠❛✐♥s ❛✉t❤❡♥t✐❝❛t✐♦♥ ❧♦ts ♠♦r❡

❖✉t❧✐♥❡

❝❛ s❝❛❧✐♥❣

❖✈❡r✈✐❡✇ ❆✉t❤❡♥t✐❝❛t✐♦♥ ❜❛s✐❝s ❆✉t❤❡♥t✐❝❛t✐♥❣ ❤✉♠❛♥s ❙t♦r✐♥❣ ♣❛ss✇♦r❞s ❛t s❡r✈❡rs ❙❝❛❧✐♥❣ t♦ ♠❛♥② ✉s❡rs ❛♥❞ ❞♦♠❛✐♥s ❑❉❈✿ ❑❡② ❉✐str✐❜✉t✐♦♥ ❈❡♥t❡r ❈❆✿ ❈❡rt✐✜❝❛t✐♦♥ ❆✉t❤♦r✐t②

❉♦♠❛✐♥ ✇✐t❤ ❛ ❈❆ ✕ ✶

❝❛ s❝❛❧✐♥❣

❊✈❡r② ♣r✐♥❝✐♣❛❧ z ❤❛s ❛ ♣✉❜❧✐❝✲❦❡② ♣❛✐r [skz, pkz] ❡①❝❡♣t s♦♠❡ ❤✉♠❛♥ ♣r✐♥❝✐♣❛❧s ♠❛② ✉s❡ ♣❛ss✇♦r❞s ❈❆ ✐s ❛ s♣❡❝✐❛❧ ♣r✐♥❝✐♣❛❧✱ s❛② ✇✐t❤ ✐❞ X ❈❆ ✐s tr✉st❡❞ t♦ ❝r❡❛t❡ ❝♦rr❡❝t ❝❡rt✐✜❝❛t❡s ❈❆ ✐ss✉❡s ❛ ❝❡rt✐✜❝❛t❡ ❢♦r ❡✈❡r② z✿ [z, pkz, expiry time, · · · , sgn] sgn✿ ❈❆✬s s✐❣♥❛t✉r❡ ♦❢ t❤❡ ❝❡rt✐✜❝❛t❡

/ / ✉s✐♥❣ skX

❝❡rt✐✜❝❛t❡ ✐s t②♣✐❝❛❧❧② ❧♦♥❣✲❧✐✈❡❞

/ / ❡❣✱ ♠♦♥t❤s✱ ②❡❛rs

❈❆ ❝❛♥ r❡✈♦❦❡ z✬s ❝❡rt✐✜❝❛t❡ ❜❡❢♦r❡ ❡①♣✐r② ✐❢ ♥❡❡❞❡❞ ❡❣✿ skz ❤❛s ❜❡❝♦♠❡ ❡①♣♦s❡❞✱ z ❧❡❛✈❡s t❤❡ ❞♦♠❛✐♥✱ ❡t❝ ❊✈❡r② z ❤❛s ❈❆✬s ♣✉❜❧✐❝ ❦❡② s♦ z ❝❛♥ ✈❡r✐❢② ❝❡rt✐✜❝❛t❡s ❛♥❞ t❤❡✐r st❛t✉s ✭r❡✈♦❦❡❞ ♦r ♥♦t✮

❉♦♠❛✐♥ ✇✐t❤ ❛ ❈❆ ✕ ✷

❝❛ s❝❛❧✐♥❣

❚♦ ❛❝q✉✐r❡ y✬s ♣✉❜❧✐❝ ❦❡② ❣❡t y✬s ❝❡rt✐✜❝❛t❡ ❛♥❞ ❛♥❞ ✈❡r✐❢②

/ / ✉s✐♥❣ pkX

❣❡t ❝❡rt✐✜❝❛t❡✬s st❛t✉s ❛♥❞ ✈❡r✐❢② ❝❛♥ ❣❡t t❤❡s❡ ❢r♦♠ ❛♥②✇❤❡r❡

/ / ❡❣✱ y✱ ❛ s❡r✈❡r✱ ❈❆

❈❆ ♠❛❦❡s ❝❡rt✐✜❝❛t❡ st❛t✉s ✐♥❢♦ ❛✈❛✐❧❛❜❧❡ ✐♥ t✇♦ ✇❛②s P❡r✐♦❞✐❝❛❧❧② ✐ss✉❡s ❛ ❝❡rt✐✜❝❛t❡ r❡✈♦❝❛t✐♦♥ ❧✐st ✭❈❘▲✮ ❧✐st ♦❢ ❛❧❧ r❡✈♦❦❡❞ ✉♥❡①♣✐r❡❞ ❝❡rt✐✜❝❛t❡s✱ s✐❣♥❡❞ ❜② ❈❆ ✉♥❡①♣✐r❡❞ ❝❡rt✐✜❝❛t❡ ✈❛❧✐❞ ✐❢ ✐t✬s ♥♦t ✐♥ ❛ r❡❝❡♥t✲❡♥♦✉❣❤ ❈❘▲ ❖♥ ❞❡♠❛♥❞✿ ✐ss✉❡s ❛ ❝❡rt✐✜❝❛t❡✬s st❛t✉s ✭❈❙✮ ❖♥❧✐♥❡ ❈❡rt✐✜❝❛t❡ ❙t❛t✉s Pr♦t♦❝♦❧ ✭❖❈❙P✮ ❈❆ ✭♦r ✐ts ❛❣❡♥t✮ ♠✉st ❜❡ ♦♥❧✐♥❡ ❛♥❞ r❡s♣♦♥s✐✈❡

❈❡rt✐✜❝❛t❡s

❝❛ s❝❛❧✐♥❣

❈❡rt✐✜❝❛t❡ ❢♦r Z ✐ss✉❡❞ ❜② X s❡r✐❛❧ ♥✉♠❜❡r

/ / ❢♦r ❈❘▲

✐ss✉❡r✿ X✬s ♥❛♠❡✱ ❛❞❞r❡ss✱ ✳✳✳ s✉❜❥❡❝t✿ Z✬s ♥❛♠❡✱ ❛❞❞r❡ss✱ ✳✳✳ s✉❜❥❡❝t✬s ♣✉❜❧✐❝✲❦❡②✿ pkZ ❡①♣✐r② t✐♠❡

/ / ❧♦♥❣✲❧✐✈❡❞✿ ♠♦♥t❤✱ ②❡❛r✱ ✳✳✳

❝❡rt✐✜❝❛t❡✬s ❝❛♣❛❜✐❧✐t✐❡s

/ / ❡❣✱ ❝❛♥ Z ✐ss✉❡ ❝❡rt✐✜❝❛t❡s❄

✳✳✳ X✬s s✐❣♥❛t✉r❡ ♦♥ ❛❜♦✈❡

❈❘▲s ❛♥❞ ❈❙s

❝❛ s❝❛❧✐♥❣

❈❘▲ ✐ss✉❡❞ ❜② X ✐ss✉❡r✿ X✬s ♥❛♠❡✱ ❛❞❞r❡ss✱ ✳✳✳ ✐ss✉❡ t✐♠❡

/ / ❢r❡q✉❡♥t✿ ❤♦✉r❧②✱ ❞❛✐❧②✱ ✳✳✳

❧✐st ♦❢ s❡r✐❛❧ ♥✉♠❜❡rs ♦❢ ❛❧❧ r❡✈♦❦❡❞ ✉♥❡①♣✐r❡❞ ❝❡rt✐✜❝❛t❡s X✬s s✐❣♥❛t✉r❡ ♦♥ ❛❜♦✈❡ ❈❘▲ ✐s t②♣✐❝❛❧❧② ❤✉❣❡

/ / ❜✉r❞❡♥ ♦♥ ❝❧✐❡♥t

❈❡rt✐✜❝❛t❡ st❛t✉s ✭❈❙✮ ♦❢ ❛ ❝❡rt✐✜❝❛t❡ ✐ss✉❡❞ ❜② X s❡r✐❛❧ ♥✉♠❜❡r ♦❢ ❝❡rt✐✜❝❛t❡ ✐ss✉❡r✿ X✬s ♥❛♠❡✱ ❛❞❞r❡ss✱ ✳✳✳ ✐ss✉❡ t✐♠❡

/ / s❤♦✉❧❞ ❜❡ r❡❝❡♥t

st❛t✉s✿ st✐❧❧ ✈❛❧✐❞ ♦r ♥♦ ❧♦♥❣❡r ✈❛❧✐❞

/ / ❛s ♦❢ ✐ss✉❡ t✐♠❡

❖❈❙P t❛❦❡s t✐♠❡

/ / ❜✉r❞❡♥ ♦♥ ❝❧✐❡♥t

❖❈❙P st❛♣❧✐♥❣✿ s❡r✈❡r ♣r♦✈✐❞❡s ❈❙ ✭❛♥❞ ❝❡rt✐✜❝❛t❡✮ t♦ ❝❧✐❡♥t

❖❜t❛✐♥✐♥❣ Z✬s ♣✉❜❧✐❝✲❦❡②

❝❛ s❝❛❧✐♥❣

❉♦ st❡♣ ✶ ❛♥❞ ❡✐t❤❡r st❡♣ ✷ ♦r st❡♣ ✸ ✶✳ ❖❜t❛✐♥ ❛ ❝❡rt✐✜❝❛t❡ ❢♦r Z ✐ss✉❡❞ ❜② X✳ ❈❤❡❝❦ t❤❛t t❤❡ ❝❡rt✐✜❝❛t❡ ❤❛s ♥♦t ❡①♣✐r❡❞✳ ❱❡r✐❢② t❤❡ ❝❡rt✐✜❝❛t❡✬s s✐❣♥❛t✉r❡✳

/ / ✉s✐♥❣ pkX

✷✳ ❖❜t❛✐♥ ❛ r❡❝❡♥t✲❡♥♦✉❣❤ ❈❘▲ ✐ss✉❡❞ ❜② X t❤❛t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ t❤❡ ❝❡rt✐✜❝❛t❡✬s s❡r✐❛❧ ♥✉♠❜❡r✳ ❱❡r✐❢② t❤❡ ❈❘▲✬s s✐❣♥❛t✉r❡✳

/ / ✉s✐♥❣ pkX

♦r ✸✳ ❖❜t❛✐♥ ❛ r❡❝❡♥t✲❡♥♦✉❣❤ ❈❙ ✭❝❡rt✐✜❝❛t❡ st❛t✉s✮ ✐ss✉❡❞ ❜② X t❤❛t ✐♥❞✐❝❛t❡s t❤❡ ❝❡rt✐✜❝❛t❡ ✐s st✐❧❧ ✈❛❧✐❞ ❱❡r✐❢② t❤❡ ❈❙✬s s✐❣♥❛t✉r❡✳

/ / ✉s✐♥❣ pkX

❍❛♥❞❧✐♥❣ ✉s❡rs ✇✐t❤♦✉t ♣✉❜❧✐❝ ❦❡②s

❝❛ s❝❛❧✐♥❣

❈♦♥s✐❞❡r ❝❧✐❡♥t A ❛♥❞ s❡r✈❡r B✱ ✇❤❡r❡ B ❤❛s ♣✉❜❧✐❝ ❦❡② A ❞♦❡s ♥♦t ❤❛✈❡ ❛ ♣✉❜❧✐❝ ❦❡② A s❤❛r❡s pwd ✇✐t❤ B A✕B s❡ss✐♦♥ ❡st❛❜❧✐s❤♠❡♥t A ♦❜t❛✐♥s B✬s ♣✉❜❧✐❝ ❦❡②

/ / ✉s✐♥❣ st❛♥❞❛r❞ ♣r♦❝❡❞✉r❡

A s❡♥❞s EP(pkB, pwd) t♦ B

❚r✉st ♠♦❞❡❧✱ ♣r♦s✱ ❝♦♥s

❝❛ s❝❛❧✐♥❣

❚r✉st t❤❡ ❈❆ t♦ ❝♦rr❡❝t❧② ✈❡t ♣r✐♥❝✐♣❛❧s ❜❡ ♦♥❧✐♥❡ t♦ ❤❛♥❞❧❡ ❖❈❙P r❡q✉❡sts ❈❆ ✐s t❤❡ tr✉st r♦♦t

/ / ✐ts ♣✉❜❧✐❝ ❦❡② ✐s ♥♦t ✈❡r✐✜❡❞

❆❞✈❛♥t❛❣❡s ♦❢ ❈❆ ✭✈s ❑❉❈✮ ❈❆ ❝❛♥ ❜❡ ♦✤✐♥❡

/ / ✐❢ s❡♣❛r❛t❡ ❖❈❙P s❡r✈❡r

❈❆ ❞♦❡s ♥♦t ♣❛rt✐❝✐♣❛t❡ ✐♥ A✕B s❡ss✐♦♥ ❈❆ ❝❛♥♥♦t ❞❡❝r②♣t A✕B s❡ss✐♦♥ ✭❜✉t ✐t ❝❛♥ ✐♠♣❡rs♦♥❛t❡ ❛ ♣r✐♥❝✐♣❛❧ ✈✐❛ ❢❛❧s❡ ❝❡rt✐✜❝❛t❡✮ ❈❆ ❢❛✐❧✉r❡ ❞♦❡s ♥♦t st♦♣ ♥❡✇ s❡ss✐♦♥s ✉♥t✐❧ ❝❡rts ❡①♣✐r❡ ❉✐s❛❞✈❛♥t❛❣❡s ❚✐♠❡❧② r❡✈♦❝❛t✐♦♥ ✐s ❡①♣❡♥s✐✈❡

/ / s❧♦♣♣✐❧② ❞♦♥❡ ✐♥ ■♥t❡r♥❡t

❈❡rt✐✜❝❛t❡ ❝❤❛✐♥s✿ ❝r♦ss✐♥❣ ❞♦♠❛✐♥s

❝❛ s❝❛❧✐♥❣

❍♦✇ ❞♦❡s A ✈❡r✐❢② B✬s ♣✉❜❧✐❝ ❦❡② ✐❢ A ❤❛s ❛ ❝❡rt✐✜❝❛t❡ ✐ss✉❡❞ ❜② ❈❆ X

/ / certXA

B ❤❛s ❛ ❝❡rt✐✜❝❛t❡ ✐ss✉❡❞ ❜② ❈❆ Y

/ / certYB

❙♦❧✉t✐♦♥✿ X ✐ss✉❡s ❛ ❝❡rt✐✜❝❛t❡ ❢♦r Y

/ / certXY

A ✈❡r✐✜❡s pkY ✉s✐♥❣ certXY ❛♥❞ csXY

/ / csXY ✿ r❡✈♦❝❛t✐♦♥ ✐♥❢♦

A ✈❡r✐✜❡s pkB ✉s✐♥❣ pkY ✱ certYB✱ csYB [certXY , csXY ], [certYB, csYB] ✐s ❛ ❝❡rt✐✜❝❛t❡ ❝❤❛✐♥ ❈❡rt✐✜❝❛t❡ ❝❤❛✐♥✿ [cert✶, cs✶], [cert✷, cs✷], · · · , [certn, csn] [certj, csj] ✈❡r✐✜❡s ♣✉❜❧✐❝✲❦❡② ♦❢ certj+✶✬s ✐ss✉❡r cert✶✬s ✐ss✉❡r ✐s t❤❡ ❛♥❝❤♦r ♦❢ t❤❡ ❝❤❛✐♥ certn✬s s✉❜❥❡❝t ✐s t❤❡ t❛r❣❡t ♦❢ t❤❡ ❝❤❛✐♥ A ❝❛♥ ✉s❡ t❤❡ ❝❤❛✐♥ ✐❢ t❤❡ ❛♥❝❤♦r ✐s ❛ tr✉st r♦♦t ♦❢ A

P❑■✿ P✉❜❧✐❝✲❑❡② ■♥❢r❛str✉❝t✉r❡

❝❛ s❝❛❧✐♥❣

P❑■ ✐s ❤✐❡r❛r❝❤✐❝❛❧ ❚♦♣✲❧❡✈❡❧ ❈❆s ❱❡r✐s✐❣♥✱ ❈♦♠♦❞♦✱ ❚❤❛✇t❡✱ ❡t❝ ❚❤❡✐r ♣✉❜❧✐❝ ❦❡②s ❛r❡ ♣r❡✲❝♦♥✜❣✉r❡❞ ✐♥ ❖❙✴❜r♦✇s❡rs✴✳✳✳ ▼✐❞✲❧❡✈❡❧ ❈❆s ❍❛✈❡ ❝❡rt✐✜❝❛t❡s ❢r♦♠ t♦♣✲❧❡✈❡❧✴♠✐❞✲❧❡✈❡❧ ❈❆s ■ss✉❡ ❝❡rt✐✜❝❛t❡s ❘❡♣✉t❛❜❧❡ ❛♥❞ ♥♦t

/ / ❝❡rt✐✜❝❛t❡s ❢♦r ✩✶✵

▲♦✇✲❧❡✈❡❧ ❈❆s

/ / ✐♥❞✐✈✐❞✉❛❧s ❛♥❞ s♠❛❧❧ ♦r❣❛♥✐③❛t✐♦♥s

▼❛② ♥♦t ❤❛✈❡ ❝❡rt✐✜❝❛t❡s ✐ss✉❡❞ ❜② ♦t❤❡rs ▼❛② ✐ss✉❡ ❝❡rt✐✜❝❛t❡s ❢♦r ✐♥t❡r♥❛❧ ✉s❡✱ ❛❝❝❡♣t❡❞ ♦♥ ❢❛✐t❤

❲❡❜ ♦❢ ❚r✉st

❝❛ s❝❛❧✐♥❣

◆♦♥✲❤✐❡r❛r❝❤✐❝❛❧ P❑■ ♣✐♦♥❡❡r❡❞ ❜② P●P ❆♥②♦♥❡ ❝❛♥ ✐ss✉❡ ❝❡rt✐✜❝❛t❡s ❢♦r ♣❡♦♣❧❡ t❤❡② ❦♥♦✇ ❉✐r❡❝t❡❞ ❣r❛♣❤ ♦❢ ❝❡rt✐✜❝❛t❡ ❝❤❛✐♥s ❝❛♥ ❤❛✈❡ ❝②❝❧❡s ❍♦✇ t♦ ❞❡❝✐❞❡ ✇❤❡t❤❡r t♦ tr✉st ❛ ❝❡rt✐✜❝❛t❡ ❝❤❛✐♥❄ ❛♥❝❤♦r ❛♥❞ ✐♥t❡r♠❡❞✐❛t❡s ❧❡♥❣t❤ ♦❢ ❝❤❛✐♥

/ / s❤♦rt❡r ✐s ❜❡tt❡r

❤♦✇ ♠❛♥② ♦t❤❡r ❝❤❛✐♥s ❡♥❞ ❛t t❤❡ s❛♠❡ t❛r❣❡t ✳✳✳ ❍♦✇ t♦ ❞❡❝✐❞❡ ✇❤❡t❤❡r t♦ ✐ss✉❡ ❛ ❝❡rt✐✜❝❛t❡ ❢♦r s♦♠❡♦♥❡❄ r❡♣✉t❛t✐♦♥✱ ❛♣♣❡❛r❛♥❝❡✱ ✳✳✳ ❄❄❄