Formal Veri cation of Cryptographic Protocol Tutorial on BAN Logic

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❋♦r♠❛❧ ❱❡r✐✜❝❛t✐♦♥ ♦❢ ❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧ ✫

❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝

❆✈✐♥❛♥t❛ ❚❛r✐❣❛♥

❈❡♥t❡r ❢♦r ■♥❢♦r♠❛t✐♦♥ ❙❡❝✉r✐t② ❘❡s❡❛r❝❤

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❙❡❝✉r✐t② ✫ ❙❛❢❡t②

❙❡❝✉r✐t② ✐s ❢✉♥ ✐❞❡❛ ♦❢ ✿ ❛tt❛❝❦✐♥❣ ❛♥❞ ❞❡❢❡♥s✐♥❣ ♠❛♥❛❣✐♥❣ ❤✉♠❛♥✬s ♣r♦❜❧❡♠ ✭✉s❡r ✲ ❛tt❛❝❦❡r ✲ ❞❡s✐❣♥❡r✮ ❙❛❢❡t② ✐s ❛❧s♦ ❢✉♥ ✐❞❡❛ ♦❢ ✿

♣r❡✈❡♥t✐♥❣ ❢❛✐❧✉r❡s r❡❛❝t✐♥❣ ✉♣♦♥ ❢❛✐❧✉r❡s

♠❛♥❛❣✐♥❣ ❤✉♠❛♥ ❛♥❞ ❡♥✈✐r♦♥♠❡♥t❛❧

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❈r②♣t♦❣r❛♣❤② ✭r❡✈✐❡✇✮

❆❧❣♦r✐t❤♠ t♦ ♣r♦t❡❝t s❡❝r❡❝② ♦❢ ❞❛t❛ ❆❧s♦ ✉s❡❞ t♦ ❣❛✐♥ ✿

❛✉t❤❡♥t✐❝❛t✐♦♥ ✐♥t❡❣r✐t② ♥♦♥ r❡♣✉❞✐❛t✐♦♥

■♥❝❧✉❞❡s ✿ ❛❧❣♦r✐t❤♠ ❛♥❞ ❦❡②✭s✮ ■♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ r❡q✉✐r❡s ❛ ♣r♦t♦❝♦❧ ✭❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧✮

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❙②♠♠❡tr✐❝ ❈r②♣t✳

A

_7→

B

✿ ④

M

⑥

_K ❛❜

Pr✐♥❝✐♣❛❧ A s❡♥❞s B ♠❡ss❛❣❡ M

❡♥❝r②♣t❡❞ ✇✐t❤ s❤❛r❡❞✲❦❡②K_❛❜

❑❡② ✐s s❤❛r❡❞ ❜❡t✇❡❡♥ ✷ ♣r✐♥❝✐♣❛❧s ◆❡❡❞s ◆✷ _{❦❡②s ❢♦r ◆ ♣r✐♥❝✐♣❛❧s}

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❆s②♠♠❡tr✐❝✴P✉❜❧✐❝ ❑❡② ❈r②♣t✳

A

_7→

B

✿ ④

M

⑥

_K ❜

Pr✐♥❝✐♣❛❧ A s❡♥❞s B ♠❡ss❛❣❡ M

❡♥❝r②♣t❡❞ ✇✐t❤ B✬s ♣✉❜❧✐❝✲❦❡② K_❜

❖♥❧② ✇✐t❤ ♣r✐✈❛t❡✲❦❡② K−✶

❜ ✱B ❝❛♥ ❞❡❝r②♣t M

Pr✐♥❝✐♣❛❧ ❤❛s ✐ts ♦✇♥K ✇❤✐❝❤ ✐s ♣✉❜❧✐s❤❡❞ ❛♥❞ K−✶ ✇❤✐❝❤

♠✉st ❜❡ ❦❡❡♣❡❞ s❡❝r❡t

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❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❈r②♣t♦❣r❛♣❤② ❆❧❣♦r✐t❤♠ ❆❝❤✐❡✈✐♥❣ s❡❝✉r✐t② ♣r♦♣❡rt✐❡s ✭❛✉t❤❡♥t✐❝❛t✐♦♥✱ s❡❝r❡❝②✱ ❡t❝✳✮

❊①❛♠♣❧❡ ✿

◆❡❡❞❤❛♠✲❙❝❤r♦❡❞❡r ✭❛✉t❤❡♥t✐❝❛t✐♦♥✮ ❑❡r❜❡r♦s ✭❛✉t❤❡♥t✐❝❛t✐♦♥✮

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❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧

▼♦r❡ ❡①❛♠♣❧❡ ✿ ❑❡r❜❡r♦s Pr♦t♦❝♦❧ ▼✶ A _7→S ✿ ❆_,❇

▼✷ S _7→A ✿ _{❚_s_,▲_,❇_,❑_❛❜_,_{❚_s_,▲_,❑_❛❜_,❆_}_❑ ❜s}❑❛s ▼✸ A _7→B ✿ _{❚_s_,▲_,❑_❛❜_,❆_}_❑

❜s✱{❆,❚❛}❑❛❜ ▼✹ B_7→A ✿ _{❚_❛₊✶_}_❑

❛❜

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❈r②♣t♦❣r❛♣❤✐❝ Pr♦t♦❝♦❧

Pr♦❜❧❡♠ ✿

❲r♦♥❣ ❞❡s✐❣♥ ❝♦✉❧❞ ❧❡❛❞ t♦ ✢❛✇ ◆❡❡❞❤❛♠✲❙❝❤r♦❡❞❡r Pr♦t♦❝♦❧

❙❙▲✈✶✳✵

❲r♦♥❣ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❝♦✉❧❞ ❧❡❛❞ t♦ ✈✉❧♥❡r❛❜✐❧✐t② P❛❞❞✐♥❣ ♣r♦❜❧❡♠ ✐♥ ❙❙▲✱ ❙❙❍✱ ❛♥❞ ❲❚▲❙

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❋♦r♠❛❧ ▼❡t❤♦❞ ❢♦r ❈r②♣t✳ Pr♦t♦❝♦❧ ■■

❚❤❡r❡ ❛r❡ t✇♦ ❞❡✈❡❧♦♣♠❡♥t ❛♣♣r♦❛❝❤ ✿

❊①t❡♥t✐♦♥ ❢r♦♠ ♠❡t❤♦❞ ✉s❡❞ ✐♥ ❝♦♠♠✉♥✐❝❛t✐♦♥ ◆❡✇❧② ❞❡✈❡❧♦♣❡❞ ♠❡t❤♦❞

❋♦✉r ❝❧❛ss✐✜❝❛t✐♦♥s ✿

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●❡♥❡r❛❧ P✉r♣♦s❡ ❚♦♦❧s

❚r❡❛t❡❞ ❛s ♦r❞✐♥❛r② ❝♦♠♠✉♥✐❝❛t✐♦♥ ♣r♦t♦❝♦❧

❆❞✈❡rs❛r② ✐s ❡①♣❧✐❝✐t✱ ❝❛♣❛❜❧❡ ✐♥ r❡❛❞✱ ✐♥t❡r❝❡♣t✱ ❛♥❞ ♠♦❞✐❢② ♠❡ss❛❣❡s

▼❡t❤♦❞ ✿ ❋❙▼✱ ❈❙P✱ ❋❉❘✱ P❡tr✐ ◆❡ts ❊①❛♠♣❧❡ ✿ ▲♦t♦s✱ ■♥❛ ❏♦✱ ▼✉r♣❤②

A INTRUDER B

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❯s✐♥❣ ❊①♣❡rt ❙②st❡♠

■♥✈❡st✐❣❛t❡ ❡✈❡r② ♣♦ss✐❜❧❡ s❝❡♥❛r✐♦ ♦❢ ❆tt❛❝❦ ✲ ❋❧❛✇ ✲ ❉❡❢❡♥❝❡

◆❡❡❞s t♦ ❞❡✜♥❡ ✐♥s❡❝✉r❡ st❛t❡s ❛♥❞ s❡❛r❝❤ ♣❛t❤s t♦ t❤❡♠ ▼♦r❡ s✉❝❝❡ss❢✉❧ t❤❛♥ ●❡♥❡r❛❧ P✉r♣♦s❡ ❚♦♦❧s

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❆❧❣❡❜r❛✐❝ ❆♣♣r♦❛❝❤

❈❛♣❛❜✐❧t✐❡s ✐♥ ♠♦❞❡❧✐♥❣ ❦♥♦✇❧❡❞❣❡ ✇❤✐❝❤ r❡♣r❡s❡♥ts

❝♦♠♣♦♥❡♥t ✐♥ ❝r②♣t♦❣r❛♣❤✐❝ ♦♣❡r❛t✐♦♥ ✭◆♦♥❝❡✱ ❑❡②✭s✮✱ ❛♥❞ ♦❧❞ ♠❡ss❛❣❡s✮

❊①❛♠♣❧❡ ✿

❉♦❧❡✈ ✲ ❨❛♦ ✭t❡r♠ r❡✲✇r✐t✐♥❣ s②st❡♠s✮

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▲♦❣✐❝ ❇❛s❡❞

❖♥❡ s❡❡s ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦t♦❝♦❧ ❛s ❞✐str✐❜✉t❡❞ ❛❧❣♦r✐t❤♠ ❉❡✈❡❧♦♣ ❧♦❣✐❝s ❢r♦♠ ♠♦❞❛❧ ❧♦❣✐❝

❚❤❡r❡ ❛r❡ ✐♥❢❡r❡♥❝❡ r✉❧❡s

●♦❛❧ ✐s t♦ ❞❡r✐✈❡❞ st❛t❡♠❡♥ts ✇❤✐❝❤ r❡♣r❡s❡♥ts ❝♦rr❡❝t ❝♦♥❞✐t✐♦♥

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❇♦♦❦s✱ P❛♣❡rs✱ ❛♥❞ ▲✐♥❦s

❙❝❤♥❡✐❞❡r ❡t✳ ❛❧✳ ✏▼♦❞❡❧❧✐♥❣ ❛♥❞ ❆♥❛❧②s✐s ♦❢ ❙❡❝✉r✐t② Pr♦t♦❝♦❧s✑

❘♦ss ❆♥❞❡rs♦♥✱ ✏❙❡❝✉r✐t② ❊♥❣✐♥❡❡r✐♥❣✑ ❉♦♥❛❧❞ ▼❛❝❦❡♥③✐❡✱ ✏▼❡❝❤❛♥✐③✐♥❣ Pr♦♦❢✑ ▼❛rt✐♥ ❆❜❛❞✐✬s ♣❛♣❡rs ❛t

❤tt♣✿✴✴✇✇✇✳❝s❡✳✉❝s❝✳❡❞✉✴⑦❛❜❛❞✐✴❛❧❧♣❛♣❡rs✳❤t♠❧★❥s❞s P❛♣❡rs ❛t

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❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ■■■

P♦st✉❧❛t❡s

▼❡ss❛❣❡ ▼❡❛♥✐♥❣ ✿

P|≡◗←→❑ P, P⊳{❳}_❑

P |≡◗|∼❳ ✭✶✮

P|≡7→❑ ◗, P⊳{❳}_❑−✶

P |≡◗|∼❳ ✭✷✮

P |≡◗⇋❨ P, P⊳h❳i_❨

P |≡◗|∼❳ ✭✸✮

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❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ❱

▼♦r❡ ❞❡❝♦♠♣♦s✐t✐♦♥s

P⊳(❳,❨)

P⊳❳ ✭✶✵✮

P⊳h❳i_❨

P⊳❳ ✭✶✶✮

P|≡◗←→❑ P, P⊳{❳}_❑

P⊳❳ ✭✶✷✮

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❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ❱■

P|≡7→❑ ◗, P⊳{❳}_❑−✶

P⊳❳ ✭✶✹✮

◆♦♥❝❡ ✐♥✢✉❡♥❝❡

P |≡♯(❨)

P|≡♯(❨,❳) ✭✶✺✮

❙❤❛r❡❞✲❑❡② ❈♦♠♠✉t❛t✐✈❡

P |≡❘←→❑ ❘′

P |≡❘′_←→❑ _❘ ✭✶✻✮

P |≡◗|≡❘←→❑ ❘′

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❚✉t♦r✐❛❧ ♦♥ ❇❆◆ ▲♦❣✐❝ ❱■■

❙❤❛r❡❞✲❙❡❝r❡t ❈♦♠♠✉t❛t✐✈❡

P|≡❘⇋❳ ❘′

P|≡❘′⇋❳ ❘

✭✶✽✮

P |≡◗|≡❘⇋❳ ❘′

P |≡◗|≡❘′⇋❳ ❘

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❙t❛❣❡s

❙t❛❣❡s ✐♥ ✉s✐♥❣ ❇❆◆ ♣r♦♦❢ ✿

✶ ❚r❛♥s❢♦r♠ ♠❡ss❛❣❡ ✐♥t♦ ✐❞❡❛❧✐③❡❞ ❧♦❣✐❝❛❧ ❢♦r♠✉❧❛

❙❦✐♣ t❤❡ ♠❡ss❛❣❡ ♣❛rts t❤❛t ❞♦ ♥♦t ❝♦♥tr✐❜✉t❡ t♦ t❤❡ r❡❝❡✐✈❡r✬s ❜❡❧✐❡❢s

✷ ❙t❛t❡ ❛ss✉♠♣t✐♦♥s ❛❜♦✉t ♦r✐❣✐♥❛❧ ♠❡ss❛❣❡

✸ ▼❛❦❡ ❛♥♥♦t❛t❡❞ ✐❞❡❛❧✐③❡❞ ♣r♦t♦❝♦❧s ❢♦r ❡❛❝❤ ♣r♦t♦❝♦❧

st❛t❡♠❡♥t ✇✐t❤ ❛ss❡rt✐♦♥s

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■❞❡❛❧✐③❛t✐♦♥

❚♦ ❢♦r♠❛❧✐③❡ ❛♥❞ r❡♠♦✈❡ ❛♠❜✐❣✉✐t② ✐♥ ♣r♦t♦❝♦❧ ❜✐t str✐♥❣ ❙❦✐♣ t❤❡ ♠❡ss❛❣❡ ♣❛rts t❤❛t ❞♦ ♥♦t ❝♦♥tr✐❜✉t❡ t♦ t❤❡ r❡❝❡✐✈❡r✬s ❜❡❧✐❡❢s

❊①❛♠♣❧❡ ✿

❆→❇:{❆,◆_❛,❑_❛❜}_❑_❛❜

t♦

❇⊳{◆_❛,❆←→❑❛❜ ❇}_❑

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Pr♦✈✐♥❣ s✐♠♣❧❡ ♣r♦t♦❝♦❧ ■■

❲❡ ❤❛✈❡ ❛ss✉♠♣t✐♦♥s ✿

❆|≡❆←→❑❛s ❙ ❇|≡❇←→❑❜s ❙ ❙ |≡❇←→❑❜s ❙ ❙|≡❆←→❑❛❜ ❇ ❇|≡❙ Z⇒❆←→❑ ❇ ❆|≡❙ Z⇒❇|∼❳

❆|≡♯(◆_❛) ❆|≡♯(◆_❝)

❙ |≡❆←→❑❛s ❙ ❆|≡❙ Z⇒❆←→❑ ❇

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▲✐♠✐t❛t✐♦♥s

❇❆◆ ❞♦❡s ♥♦t ❝❛t❝❤ ❛❧❧ ♣r♦t♦❝♦❧ ✢❛✇s ❋❛❧s❡✲♣♦s✐t✐✈❡s ❝❛♥ r❡s✉❧t

❆ ♣r✐♥❝✐♣❛❧✬s ❜❡❧✐❡❢s ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❛t ❧❛t❡r st❛❣❡s ♦❢ t❤❡ ♣r♦t♦❝♦❧

◆♦ ❞✐✈✐s✐♦♥ ♦❢ t✐♠❡ ✐♥ ♣r♦t♦❝♦❧ r✉♥

Pr♦✈✐❞❡s ❛ ♣r♦♦❢ ♦❢ tr✉st ♦♥ ♣❛rt ♦❢ ♣r✐♥❝✐♣❧❡s✱ ❜✉t ♥♦t ❛ ♣r♦♦❢ ♦❢ s❡❝✉r✐t②

(32)

❚❤❡ ❊♥❞

❚❤❛♥❦ ❨♦✉

❱✐❡❧❡♥ ❉❛♥❦

❚❛❝❦ så ▼②❝❦❡t

▼❛t✉r ◆✉✇✉♥