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r s t t r rrr Prt r rs rst
Pr❛t②❛② ▼✉❦❤❡r❥❡❡
❆❛r❤✉s ❯♥✐✈❡rs✐t②
❆s✐❛❝r②♣t ✷✵✶✸
❏♦✐♥t ✇♦r❦ ✇✐t❤
■✈❛♥ ❉❛♠❣år❞✱ ❙❡❜❛st✐❛♥ ❋❛✉st ✫ ❉❛♥✐❡❧❡ ❱❡♥t✉r✐
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❙t❛♥❞❛r❞ ❆♥❛❧②s✐s✿ ❇❧❛❝❦❜♦①
❛❞✈❡rs❛r② ✐♥♣✉t ♦✉t♣✉t
✐♠♣❧❡♠❡♥t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s
❛❞✈❡rs❛r② ❧❡❛❦❛❣❡ ✐♥♣✉t ♦✉t♣✉t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s
❛❞✈❡rs❛r② t❛♠♣❡r❡❞ ✐♥♣✉t t❛♠♣❡r❡❞ ♦✉t♣✉t ❧❡❛❦❛❣❡ t❛♠♣❡r✐♥❣ ❚❤✐s t❛❧❦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✷ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❙t❛♥❞❛r❞ ❆♥❛❧②s✐s✿ ❇❧❛❝❦❜♦① FK(·)
❛❞✈❡rs❛r② ✐♥♣✉t ♦✉t♣✉t
✐♠♣❧❡♠❡♥t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s
❛❞✈❡rs❛r② ❧❡❛❦❛❣❡ ✐♥♣✉t ♦✉t♣✉t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s
❛❞✈❡rs❛r② t❛♠♣❡r❡❞ ✐♥♣✉t t❛♠♣❡r❡❞ ♦✉t♣✉t ❧❡❛❦❛❣❡ t❛♠♣❡r✐♥❣ ❚❤✐s t❛❧❦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✷ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❙t❛♥❞❛r❞ ❆♥❛❧②s✐s✿ ❇❧❛❝❦❜♦① FK(·)
❛❞✈❡rs❛r② ✐♥♣✉t ♦✉t♣✉t
✐♠♣❧❡♠❡♥t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s FK(·)
❛❞✈❡rs❛r② ❧❡❛❦❛❣❡ ✐♥♣✉t ♦✉t♣✉t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s
❛❞✈❡rs❛r② t❛♠♣❡r❡❞ ✐♥♣✉t t❛♠♣❡r❡❞ ♦✉t♣✉t ❧❡❛❦❛❣❡ t❛♠♣❡r✐♥❣ ❚❤✐s t❛❧❦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✷ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❙t❛♥❞❛r❞ ❆♥❛❧②s✐s✿ ❇❧❛❝❦❜♦① FK(·)
❛❞✈❡rs❛r② ✐♥♣✉t ♦✉t♣✉t
✐♠♣❧❡♠❡♥t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s FK(·)
❛❞✈❡rs❛r② ❧❡❛❦❛❣❡ ✐♥♣✉t ♦✉t♣✉t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s
K(·)
❛❞✈❡rs❛r② t❛♠♣❡r❡❞ ✐♥♣✉t t❛♠♣❡r❡❞ ♦✉t♣✉t ❧❡❛❦❛❣❡ t❛♠♣❡r✐♥❣ ❚❤✐s t❛❧❦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✷ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❙t❛♥❞❛r❞ ❆♥❛❧②s✐s✿ ❇❧❛❝❦❜♦① FK(·)
❛❞✈❡rs❛r② ✐♥♣✉t ♦✉t♣✉t
✐♠♣❧❡♠❡♥t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s FK(·)
❛❞✈❡rs❛r② ❧❡❛❦❛❣❡ ✐♥♣✉t ♦✉t♣✉t
❘❡❛❧✐t②✿ P❤②s✐❝❛❧ ❆tt❛❝❦s
K(·)
❛❞✈❡rs❛r② t❛♠♣❡r❡❞ ✐♥♣✉t t❛♠♣❡r❡❞ ♦✉t♣✉t ❧❡❛❦❛❣❡ t❛♠♣❡r✐♥❣ ❚❤✐s t❛❧❦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✷ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❈❛♣t✉r❡s Pr❛❝t✐❝❛❧ ❆tt❛❝❦s
❇♦♥❡❤ ❡t ❛❧✳ ❬❥♦❝✬✵✶❪✿ ■♥❥❡❝t s✐♥❣❧❡ ✭r❛♥❞♦♠✮ ❢❛✉❧t t♦ t❤❡ s✐❣♥✐♥❣✲❦❡② ♦❢ s♦♠❡ t②♣❡ ♦❢ ❘❙❆✲s✐❣ ❢❛❝t♦r ❘❙❆✲♠♦❞✉❧✉s ✦ ❊①❛♠♣❧❡
❆♥❞❡rs♦♥ ✫ ❑✉❤♥ ❬✉s❡♥✐①✬✾✻❪ ❙❦♦r♦❜♦❣❛t♦✈ ❡t ❛❧✳ ❬❝❤❡s✬✵✷❪ ❈♦r♦♥ ❡t ❛❧✳ ❬❝❤❡s✬✵✾❪ ▼♦r❡ ✳✳✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✸ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❈❛♣t✉r❡s Pr❛❝t✐❝❛❧ ❆tt❛❝❦s
❇♦♥❡❤ ❡t ❛❧✳ ❬❥♦❝✬✵✶❪✿ ■♥❥❡❝t s✐♥❣❧❡ ✭r❛♥❞♦♠✮ ❢❛✉❧t t♦ t❤❡ s✐❣♥✐♥❣✲❦❡② ♦❢ s♦♠❡ t②♣❡ ♦❢ ❘❙❆✲s✐❣ ❢❛❝t♦r ❘❙❆✲♠♦❞✉❧✉s ✦ ❊①❛♠♣❧❡
❆♥❞❡rs♦♥ ✫ ❑✉❤♥ ❬✉s❡♥✐①✬✾✻❪ ❙❦♦r♦❜♦❣❛t♦✈ ❡t ❛❧✳ ❬❝❤❡s✬✵✷❪ ❈♦r♦♥ ❡t ❛❧✳ ❬❝❤❡s✬✵✾❪ ▼♦r❡ ✳✳✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✸ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❈❛♣t✉r❡s Pr❛❝t✐❝❛❧ ❆tt❛❝❦s
❇♦♥❡❤ ❡t ❛❧✳ ❬❥♦❝✬✵✶❪✿ ■♥❥❡❝t s✐♥❣❧❡ ✭r❛♥❞♦♠✮ ❢❛✉❧t t♦ t❤❡ s✐❣♥✐♥❣✲❦❡② ♦❢ s♦♠❡ t②♣❡ ♦❢ ❘❙❆✲s✐❣ = ⇒ ❢❛❝t♦r ❘❙❆✲♠♦❞✉❧✉s ✦ ❊①❛♠♣❧❡
❆♥❞❡rs♦♥ ✫ ❑✉❤♥ ❬✉s❡♥✐①✬✾✻❪ ❙❦♦r♦❜♦❣❛t♦✈ ❡t ❛❧✳ ❬❝❤❡s✬✵✷❪ ❈♦r♦♥ ❡t ❛❧✳ ❬❝❤❡s✬✵✾❪ ▼♦r❡ ✳✳✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✸ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❈❛♣t✉r❡s Pr❛❝t✐❝❛❧ ❆tt❛❝❦s
❇♦♥❡❤ ❡t ❛❧✳ ❬❥♦❝✬✵✶❪✿ ■♥❥❡❝t s✐♥❣❧❡ ✭r❛♥❞♦♠✮ ❢❛✉❧t t♦ t❤❡ s✐❣♥✐♥❣✲❦❡② ♦❢ s♦♠❡ t②♣❡ ♦❢ ❘❙❆✲s✐❣ = ⇒ ❢❛❝t♦r ❘❙❆✲♠♦❞✉❧✉s ✦ ❊①❛♠♣❧❡
❆♥❞❡rs♦♥ ✫ ❑✉❤♥ ❬✉s❡♥✐①✬✾✻❪ ❙❦♦r♦❜♦❣❛t♦✈ ❡t ❛❧✳ ❬❝❤❡s✬✵✷❪ ❈♦r♦♥ ❡t ❛❧✳ ❬❝❤❡s✬✵✾❪ ▼♦r❡ ✳✳✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✸ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❈❛♣t✉r❡s Pr❛❝t✐❝❛❧ ❆tt❛❝❦s
❇♦♥❡❤ ❡t ❛❧✳ ❬❥♦❝✬✵✶❪✿ ■♥❥❡❝t s✐♥❣❧❡ ✭r❛♥❞♦♠✮ ❢❛✉❧t t♦ t❤❡ s✐❣♥✐♥❣✲❦❡② ♦❢ s♦♠❡ t②♣❡ ♦❢ ❘❙❆✲s✐❣ = ⇒ ❢❛❝t♦r ❘❙❆✲♠♦❞✉❧✉s ✦ ❊①❛♠♣❧❡
❆♥❞❡rs♦♥ ✫ ❑✉❤♥ ❬✉s❡♥✐①✬✾✻❪ ❙❦♦r♦❜♦❣❛t♦✈ ❡t ❛❧✳ ❬❝❤❡s✬✵✷❪ ❈♦r♦♥ ❡t ❛❧✳ ❬❝❤❡s✬✵✾❪ · · · ▼♦r❡ ✳✳✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✸ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
▼❡♠♦r② ✫ ❈♦♠♣✉t❛t✐♦♥
✭■s❤❛✐ ❡t ❛❧✳ ❬❡✉r♦❝r②♣t✬✵✻❪✮
▼♦st ❣❡♥❡r❛❧ ♠♦❞❡❧ ❜✉t s❡✈❡r❛❧ ❧✐♠✐t❛t✐♦♥s✦
❖♥❧② ▼❡♠♦r②
✭●❡♥♥❛r♦ ❡t ❛❧✳ ❬t❝❝✬✵✹❪✮
❖✉r ❋♦❝✉s
◆❛t✉r❛❧ ❋✐rst ❙t❡♣✳ ▼✐❣❤t ❜❡ r❡❛s♦♥❛❜❧❡ ✐♥ ♣r❛❝t✐❝❡✿ ✕ ❖♥❧② ✐♠♣❧❡♠❡♥t t❛♠♣❡r✲♣r♦♦❢ ❝✐r❝✉✐ts ❜✉t ♥♦t ♠❡♠♦r②✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✹ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
F K ▼❡♠♦r② ✫ ❈♦♠♣✉t❛t✐♦♥
✭■s❤❛✐ ❡t ❛❧✳ ❬❡✉r♦❝r②♣t✬✵✻❪✮ A
▼♦st ❣❡♥❡r❛❧ ♠♦❞❡❧ ❜✉t s❡✈❡r❛❧ ❧✐♠✐t❛t✐♦♥s✦
❖♥❧② ▼❡♠♦r②
✭●❡♥♥❛r♦ ❡t ❛❧✳ ❬t❝❝✬✵✹❪✮
❖✉r ❋♦❝✉s
◆❛t✉r❛❧ ❋✐rst ❙t❡♣✳ ▼✐❣❤t ❜❡ r❡❛s♦♥❛❜❧❡ ✐♥ ♣r❛❝t✐❝❡✿ ✕ ❖♥❧② ✐♠♣❧❡♠❡♥t t❛♠♣❡r✲♣r♦♦❢ ❝✐r❝✉✐ts ❜✉t ♥♦t ♠❡♠♦r②✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✹ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
F K ▼❡♠♦r② ✫ ❈♦♠♣✉t❛t✐♦♥
✭■s❤❛✐ ❡t ❛❧✳ ❬❡✉r♦❝r②♣t✬✵✻❪✮ A
▼♦st ❣❡♥❡r❛❧ ♠♦❞❡❧ ❜✉t s❡✈❡r❛❧ ❧✐♠✐t❛t✐♦♥s✦
❖♥❧② ▼❡♠♦r②
✭●❡♥♥❛r♦ ❡t ❛❧✳ ❬t❝❝✬✵✹❪✮
❖✉r ❋♦❝✉s
◆❛t✉r❛❧ ❋✐rst ❙t❡♣✳ ▼✐❣❤t ❜❡ r❡❛s♦♥❛❜❧❡ ✐♥ ♣r❛❝t✐❝❡✿ ✕ ❖♥❧② ✐♠♣❧❡♠❡♥t t❛♠♣❡r✲♣r♦♦❢ ❝✐r❝✉✐ts ❜✉t ♥♦t ♠❡♠♦r②✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✹ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
F K ▼❡♠♦r② ✫ ❈♦♠♣✉t❛t✐♦♥
✭■s❤❛✐ ❡t ❛❧✳ ❬❡✉r♦❝r②♣t✬✵✻❪✮ A
▼♦st ❣❡♥❡r❛❧ ♠♦❞❡❧ ❜✉t s❡✈❡r❛❧ ❧✐♠✐t❛t✐♦♥s✦
F K ❖♥❧② ▼❡♠♦r②
✭●❡♥♥❛r♦ ❡t ❛❧✳ ❬t❝❝✬✵✹❪✮ A
❖✉r ❋♦❝✉s
◆❛t✉r❛❧ ❋✐rst ❙t❡♣✳ ▼✐❣❤t ❜❡ r❡❛s♦♥❛❜❧❡ ✐♥ ♣r❛❝t✐❝❡✿ ✕ ❖♥❧② ✐♠♣❧❡♠❡♥t t❛♠♣❡r✲♣r♦♦❢ ❝✐r❝✉✐ts ❜✉t ♥♦t ♠❡♠♦r②✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✹ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
F K ▼❡♠♦r② ✫ ❈♦♠♣✉t❛t✐♦♥
✭■s❤❛✐ ❡t ❛❧✳ ❬❡✉r♦❝r②♣t✬✵✻❪✮ A
▼♦st ❣❡♥❡r❛❧ ♠♦❞❡❧ ❜✉t s❡✈❡r❛❧ ❧✐♠✐t❛t✐♦♥s✦
F K ❖♥❧② ▼❡♠♦r②
✭●❡♥♥❛r♦ ❡t ❛❧✳ ❬t❝❝✬✵✹❪✮ A
❖✉r ❋♦❝✉s
◆❛t✉r❛❧ ❋✐rst ❙t❡♣✳ ▼✐❣❤t ❜❡ r❡❛s♦♥❛❜❧❡ ✐♥ ♣r❛❝t✐❝❡✿ ✕ ❖♥❧② ✐♠♣❧❡♠❡♥t t❛♠♣❡r✲♣r♦♦❢ ❝✐r❝✉✐ts ❜✉t ♥♦t ♠❡♠♦r②✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✹ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
s❡❝r❡t sk
Sign
❛❞✈❡rs❛r②
M σ = Signsk(M) (pk, σ)
❋✐① ❢❛♠✐❧② s❡❝r❡t ❛❞✈❡rs❛r②
❝❤♦♦s❡ ❛❞✈❡rs❛r② s❡❝r❡t
✭ ✮
❋♦r❣❡
❘❡q✉✐r❡♠❡♥t
❛✳❦✳❛✳ ❘❑❆✲♠♦❞❡❧ ◆❛t✉r❛❧ ●♦❛❧✿ ❇r♦❛❞❡♥
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✺ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
s❡❝r❡t ❛❞✈❡rs❛r②
(pk, σ, σ1, . . . , σi)
❋✐① ❢❛♠✐❧② Γ s❡❝r❡t sk
Sign
❛❞✈❡rs❛r②
Ti(·)
❝❤♦♦s❡ Ti(·) ← Γ
Sign
❛❞✈❡rs❛r②
M
s❡❝r❡t
ski
ski(M)
1 ≤ i ≤ t ✭t = poly(k)✮
❋♦r❣❡
❘❡q✉✐r❡♠❡♥t
❛✳❦✳❛✳ ❘❑❆✲♠♦❞❡❧ ◆❛t✉r❛❧ ●♦❛❧✿ ❇r♦❛❞❡♥
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✺ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
s❡❝r❡t ❛❞✈❡rs❛r②
(pk, σ, σ1, . . . , σi, . . . , σt)
❋✐① ❢❛♠✐❧② Γ s❡❝r❡t ❛❞✈❡rs❛r②
❝❤♦♦s❡
Sign
❛❞✈❡rs❛r② s❡❝r❡t
1 ≤ i ≤ t ✭t = poly(k)✮
❋♦r❣❡
M⋆, σ⋆
Pr[Vrfypk(M ⋆, σ⋆) = 1] ≤ negl
❘❡q✉✐r❡♠❡♥t
❛✳❦✳❛✳ ❘❑❆✲♠♦❞❡❧ ◆❛t✉r❛❧ ●♦❛❧✿ ❇r♦❛❞❡♥
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✺ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
s❡❝r❡t ❛❞✈❡rs❛r②
(pk, σ, σ1, . . . , σi, . . . , σt)
❋✐① ❢❛♠✐❧② Γ s❡❝r❡t ❛❞✈❡rs❛r②
❝❤♦♦s❡
Sign
❛❞✈❡rs❛r② s❡❝r❡t
1 ≤ i ≤ t ✭t = poly(k)✮
❋♦r❣❡
M⋆, σ⋆
Pr[Vrfypk(M ⋆, σ⋆) = 1] ≤ negl
❘❡q✉✐r❡♠❡♥t
❛✳❦✳❛✳ ❘❑❆✲♠♦❞❡❧ ◆❛t✉r❛❧ ●♦❛❧✿ ❇r♦❛❞❡♥
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✺ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
s❡❝r❡t ❛❞✈❡rs❛r②
(pk, σ, σ1, . . . , σi, . . . , σt)
❋✐① ❢❛♠✐❧② Γ s❡❝r❡t ❛❞✈❡rs❛r②
❝❤♦♦s❡
Sign
❛❞✈❡rs❛r② s❡❝r❡t
1 ≤ i ≤ t ✭t = poly(k)✮
❋♦r❣❡
M⋆, σ⋆
Pr[Vrfypk(M ⋆, σ⋆) = 1] ≤ negl
❘❡q✉✐r❡♠❡♥t
❛✳❦✳❛✳ ❘❑❆✲♠♦❞❡❧ ◆❛t✉r❛❧ ●♦❛❧✿ ❇r♦❛❞❡♥ Γ
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✺ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❋✐rst ♠♦❞❡❧✿ ❇❡❧❧❛r❡ ❛♥❞ ❑♦❤♥♦ ❬❡✉r♦❝r②♣t✬✵✸❪✳ ▼♦r❡
▲✉❝❦s❬❢s❡✬✵✹❪✱ ❇❈❬❝r②♣t♦✬✶✵❪✱ ❇❈▼❬❛s✐❛❝r②♣t✬✶✶❪✱ ❇P❚❬❛s✐❛❝r②♣t✬✶✷❪✱ ❆❍■❬✐❝s✬✶✶❪✱ ❇❘❬❢s❡✬✶✸❪✱ ●❖❘❬t❝❝✬✶✶❪✱ ❲❡❡❬♣❦❝✬✶✷❪✳✳✳
❘❑❆✭✐♥ s❤♦rt✮
✶ ❯♥r❡str✐❝t❡❞
✕ ❛♥② ♣♦❧②♥♦♠✐❛❧✳
✷ ❘❡str✐❝t❡❞
✳ ✕ ❈✉rr❡♥t✲st❛t❡✲♦❢✲❛rt ❝♦♥s✐❞❡rs ❛s ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s ❡✳❣✳ ❆✣♥❡ ❢✉♥❝t✐♦♥✴♣♦❧②♥♦♠✐❛❧ ♦✈❡r s♦♠❡ ✜❡❧❞✳ ❉r❛✇❜❛❝❦✦ ▼❛② ◆❖❚ ❜❡ r❡❛❧✐st✐❝ ✕ ❛r❜✐tr❛r② ❢❛✉❧t ♠❛② ♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✻ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❋✐rst ♠♦❞❡❧✿ ❇❡❧❧❛r❡ ❛♥❞ ❑♦❤♥♦ ❬❡✉r♦❝r②♣t✬✵✸❪✳ ▼♦r❡
▲✉❝❦s❬❢s❡✬✵✹❪✱ ❇❈❬❝r②♣t♦✬✶✵❪✱ ❇❈▼❬❛s✐❛❝r②♣t✬✶✶❪✱ ❇P❚❬❛s✐❛❝r②♣t✬✶✷❪✱ ❆❍■❬✐❝s✬✶✶❪✱ ❇❘❬❢s❡✬✶✸❪✱ ●❖❘❬t❝❝✬✶✶❪✱ ❲❡❡❬♣❦❝✬✶✷❪✳✳✳
❘❑❆✭✐♥ s❤♦rt✮
✶ ❯♥r❡str✐❝t❡❞
✕ ❛♥② ♣♦❧②♥♦♠✐❛❧✳
✷ ❘❡str✐❝t❡❞
✳ ✕ ❈✉rr❡♥t✲st❛t❡✲♦❢✲❛rt ❝♦♥s✐❞❡rs ❛s ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s ❡✳❣✳ ❆✣♥❡ ❢✉♥❝t✐♦♥✴♣♦❧②♥♦♠✐❛❧ ♦✈❡r s♦♠❡ ✜❡❧❞✳ ❉r❛✇❜❛❝❦✦ ▼❛② ◆❖❚ ❜❡ r❡❛❧✐st✐❝ ✕ ❛r❜✐tr❛r② ❢❛✉❧t ♠❛② ♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✻ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❋✐rst ♠♦❞❡❧✿ ❇❡❧❧❛r❡ ❛♥❞ ❑♦❤♥♦ ❬❡✉r♦❝r②♣t✬✵✸❪✳ ▼♦r❡
▲✉❝❦s❬❢s❡✬✵✹❪✱ ❇❈❬❝r②♣t♦✬✶✵❪✱ ❇❈▼❬❛s✐❛❝r②♣t✬✶✶❪✱ ❇P❚❬❛s✐❛❝r②♣t✬✶✷❪✱ ❆❍■❬✐❝s✬✶✶❪✱ ❇❘❬❢s❡✬✶✸❪✱ ●❖❘❬t❝❝✬✶✶❪✱ ❲❡❡❬♣❦❝✬✶✷❪✳✳✳
❘❑❆✭✐♥ s❤♦rt✮
✶ ❯♥r❡str✐❝t❡❞
✕ ❛♥② ♣♦❧②♥♦♠✐❛❧✳
✷ ❘❡str✐❝t❡❞
✳ ✕ ❈✉rr❡♥t✲st❛t❡✲♦❢✲❛rt ❝♦♥s✐❞❡rs ❛s ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s ❡✳❣✳ ❆✣♥❡ ❢✉♥❝t✐♦♥✴♣♦❧②♥♦♠✐❛❧ ♦✈❡r s♦♠❡ ✜❡❧❞✳ ❉r❛✇❜❛❝❦✦ ▼❛② ◆❖❚ ❜❡ r❡❛❧✐st✐❝ ✕ ❛r❜✐tr❛r② ❢❛✉❧t ♠❛② ♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✻ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❋✐rst ♠♦❞❡❧✿ ❇❡❧❧❛r❡ ❛♥❞ ❑♦❤♥♦ ❬❡✉r♦❝r②♣t✬✵✸❪✳ ▼♦r❡
▲✉❝❦s❬❢s❡✬✵✹❪✱ ❇❈❬❝r②♣t♦✬✶✵❪✱ ❇❈▼❬❛s✐❛❝r②♣t✬✶✶❪✱ ❇P❚❬❛s✐❛❝r②♣t✬✶✷❪✱ ❆❍■❬✐❝s✬✶✶❪✱ ❇❘❬❢s❡✬✶✸❪✱ ●❖❘❬t❝❝✬✶✶❪✱ ❲❡❡❬♣❦❝✬✶✷❪✳✳✳
❘❑❆✭✐♥ s❤♦rt✮
✶ ❯♥r❡str✐❝t❡❞ t ✕ ❛♥② ♣♦❧②♥♦♠✐❛❧✳ ✷ ❘❡str✐❝t❡❞
✳ ✕ ❈✉rr❡♥t✲st❛t❡✲♦❢✲❛rt ❝♦♥s✐❞❡rs ❛s ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s ❡✳❣✳ ❆✣♥❡ ❢✉♥❝t✐♦♥✴♣♦❧②♥♦♠✐❛❧ ♦✈❡r s♦♠❡ ✜❡❧❞✳ ❉r❛✇❜❛❝❦✦ ▼❛② ◆❖❚ ❜❡ r❡❛❧✐st✐❝ ✕ ❛r❜✐tr❛r② ❢❛✉❧t ♠❛② ♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✻ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❋✐rst ♠♦❞❡❧✿ ❇❡❧❧❛r❡ ❛♥❞ ❑♦❤♥♦ ❬❡✉r♦❝r②♣t✬✵✸❪✳ ▼♦r❡
▲✉❝❦s❬❢s❡✬✵✹❪✱ ❇❈❬❝r②♣t♦✬✶✵❪✱ ❇❈▼❬❛s✐❛❝r②♣t✬✶✶❪✱ ❇P❚❬❛s✐❛❝r②♣t✬✶✷❪✱ ❆❍■❬✐❝s✬✶✶❪✱ ❇❘❬❢s❡✬✶✸❪✱ ●❖❘❬t❝❝✬✶✶❪✱ ❲❡❡❬♣❦❝✬✶✷❪✳✳✳
❘❑❆✭✐♥ s❤♦rt✮
✶ ❯♥r❡str✐❝t❡❞ t ✕ ❛♥② ♣♦❧②♥♦♠✐❛❧✳ ✷ ❘❡str✐❝t❡❞ Γ✳ ✕ ❈✉rr❡♥t✲st❛t❡✲♦❢✲❛rt ❝♦♥s✐❞❡rs Γ ❛s ❛❧❣❡❜r❛✐❝
❢✉♥❝t✐♦♥s ❡✳❣✳ ❆✣♥❡ ❢✉♥❝t✐♦♥✴♣♦❧②♥♦♠✐❛❧ ♦✈❡r s♦♠❡ ✜❡❧❞✳ ❉r❛✇❜❛❝❦✦ ▼❛② ◆❖❚ ❜❡ r❡❛❧✐st✐❝ ✕ ❛r❜✐tr❛r② ❢❛✉❧t ♠❛② ♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✻ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❋✐rst ♠♦❞❡❧✿ ❇❡❧❧❛r❡ ❛♥❞ ❑♦❤♥♦ ❬❡✉r♦❝r②♣t✬✵✸❪✳ ▼♦r❡
▲✉❝❦s❬❢s❡✬✵✹❪✱ ❇❈❬❝r②♣t♦✬✶✵❪✱ ❇❈▼❬❛s✐❛❝r②♣t✬✶✶❪✱ ❇P❚❬❛s✐❛❝r②♣t✬✶✷❪✱ ❆❍■❬✐❝s✬✶✶❪✱ ❇❘❬❢s❡✬✶✸❪✱ ●❖❘❬t❝❝✬✶✶❪✱ ❲❡❡❬♣❦❝✬✶✷❪✳✳✳
❘❑❆✭✐♥ s❤♦rt✮
✶ ❯♥r❡str✐❝t❡❞ t ✕ ❛♥② ♣♦❧②♥♦♠✐❛❧✳ ✷ ❘❡str✐❝t❡❞ Γ✳ ✕ ❈✉rr❡♥t✲st❛t❡✲♦❢✲❛rt ❝♦♥s✐❞❡rs Γ ❛s ❛❧❣❡❜r❛✐❝
❢✉♥❝t✐♦♥s ❡✳❣✳ ❆✣♥❡ ❢✉♥❝t✐♦♥✴♣♦❧②♥♦♠✐❛❧ ♦✈❡r s♦♠❡ ✜❡❧❞✳ ❉r❛✇❜❛❝❦✦ ▼❛② ◆❖❚ ❜❡ r❡❛❧✐st✐❝ ✕ ❛r❜✐tr❛r② ❢❛✉❧t ♠❛② ♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✻ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❖✉r ●♦❛❧✿ ●♦✐♥❣ ❜❡②♦♥❞ ❛❧❣❡❜r❛✐❝ ❜❛rr✐❡r ❲❡ ✇❛♥t t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢♦r ✉♥r❡str✐❝t❡❞ ✳ ■♠♣♦ss✐❜✐❧✐t② ✕ ●❡♥♥❛r♦ ❡t ❛❧✳❬t❝❝✬✵✹❪ ❇♦t❤ ✉♥r❡str✐❝t❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ✐♠♣♦ss✐❜❧❡✦ ❖✉r ❙♦❧✉t✐♦♥✿ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣ ❯♥r❡str✐❝t❡❞ ✕ t❛♠♣❡r ✇✐t❤ ❛♥② ❡✣❝✐❡♥t ❢✉♥❝t✐♦♥ ✦ ❇♦✉♥❞❡❞ ✕ t❛♠♣❡r ♦♥❧② ❜♦✉♥❞❡❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✼ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❖✉r ●♦❛❧✿ ●♦✐♥❣ ❜❡②♦♥❞ ❛❧❣❡❜r❛✐❝ ❜❛rr✐❡r ❲❡ ✇❛♥t t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢♦r ✉♥r❡str✐❝t❡❞ Γ✳ ■♠♣♦ss✐❜✐❧✐t② ✕ ●❡♥♥❛r♦ ❡t ❛❧✳❬t❝❝✬✵✹❪ ❇♦t❤ ✉♥r❡str✐❝t❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ✐♠♣♦ss✐❜❧❡✦ ❖✉r ❙♦❧✉t✐♦♥✿ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣ ❯♥r❡str✐❝t❡❞ ✕ t❛♠♣❡r ✇✐t❤ ❛♥② ❡✣❝✐❡♥t ❢✉♥❝t✐♦♥ ✦ ❇♦✉♥❞❡❞ ✕ t❛♠♣❡r ♦♥❧② ❜♦✉♥❞❡❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✼ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❖✉r ●♦❛❧✿ ●♦✐♥❣ ❜❡②♦♥❞ ❛❧❣❡❜r❛✐❝ ❜❛rr✐❡r ❲❡ ✇❛♥t t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢♦r ✉♥r❡str✐❝t❡❞ Γ✳ ■♠♣♦ss✐❜✐❧✐t② ✕ ●❡♥♥❛r♦ ❡t ❛❧✳❬t❝❝✬✵✹❪ ❇♦t❤ ✉♥r❡str✐❝t❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ✐♠♣♦ss✐❜❧❡✦ ❖✉r ❙♦❧✉t✐♦♥✿ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣ ❯♥r❡str✐❝t❡❞ ✕ t❛♠♣❡r ✇✐t❤ ❛♥② ❡✣❝✐❡♥t ❢✉♥❝t✐♦♥ ✦ ❇♦✉♥❞❡❞ ✕ t❛♠♣❡r ♦♥❧② ❜♦✉♥❞❡❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✼ ✴ ✶✽
■♥tr♦❞✉❝t✐♦♥
❖✉r ●♦❛❧✿ ●♦✐♥❣ ❜❡②♦♥❞ ❛❧❣❡❜r❛✐❝ ❜❛rr✐❡r ❲❡ ✇❛♥t t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢♦r ✉♥r❡str✐❝t❡❞ Γ✳ ■♠♣♦ss✐❜✐❧✐t② ✕ ●❡♥♥❛r♦ ❡t ❛❧✳❬t❝❝✬✵✹❪ ❇♦t❤ ✉♥r❡str✐❝t❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ✐♠♣♦ss✐❜❧❡✦ ❖✉r ❙♦❧✉t✐♦♥✿ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣ ❯♥r❡str✐❝t❡❞ Γ ✕ t❛♠♣❡r ✇✐t❤ ❛♥② ❡✣❝✐❡♥t ❢✉♥❝t✐♦♥ ✦ ❇♦✉♥❞❡❞ t ✕ t❛♠♣❡r ♦♥❧② ❜♦✉♥❞❡❞ ♥✉♠❜❡r ♦❢ t✐♠❡s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✼ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ♥❡✇ ♠♦❞❡❧✿ ❇▲❚ ❇♦✉♥❞❡❞ ▲❡❛❦❛❣❡ ✫ ❚❛♠♣❡r✐♥❣ ✕ ❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣✳ ✕❆♥❛❧♦❣♦✉s t♦ ▲❡❛❦❛❣❡✲r❡s✐❧✐❡♥t ❈r②♣t♦❣r❛♣❤②
✲ ❍❖❚✦
❇▲❚✲s❡❝✉r❡ P✉❜❧✐❝✲❦❡② s❝❤❡♠❡s ❈♦♥❝r❡t❡ s❝❤❡♠❡s ❜❛s❡❞ ♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s ✭❉▲✱ ❋❛❝t♦r✐♥❣✮
✶ ■❉✲s❝❤❡♠❡s ❜❛s❡❞ ♦♥ s♦♠❡ ❝❧❛ss ♦❢
✲♣r♦t♦❝♦❧s ❛r❡ ❇▲❚ s❡❝✉r❡ ✕ ❡✳❣✳ ❖❦❛♠♦t♦✳
✷ ❇❍❍❖ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✐s ❇▲❚✲s❡❝✉r❡✳
▼♦r❡♦✈❡r✳✳✳
❇♦♦st t♦ ❈♦♥t✐♥✉♦✉s ❚❛♠♣❡r✐♥❣ ✉s✐♥❣ ✉♥t❛♠♣❡r❛❜❧❡ ❋❧♦♣♣② ◆❡✇ ❚❡❝❤♥✐q✉❡✿ ❘❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✽ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ♥❡✇ ♠♦❞❡❧✿ ❇▲❚ ❇♦✉♥❞❡❞ ▲❡❛❦❛❣❡ ✫ ❚❛♠♣❡r✐♥❣ ✕ ❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣✳ ✕❆♥❛❧♦❣♦✉s t♦ ▲❡❛❦❛❣❡✲r❡s✐❧✐❡♥t ❈r②♣t♦❣r❛♣❤②
✲ ❍❖❚✦
❇▲❚✲s❡❝✉r❡ P✉❜❧✐❝✲❦❡② s❝❤❡♠❡s ❈♦♥❝r❡t❡ s❝❤❡♠❡s ❜❛s❡❞ ♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s ✭❉▲✱ ❋❛❝t♦r✐♥❣✮
✶ ■❉✲s❝❤❡♠❡s ❜❛s❡❞ ♦♥ s♦♠❡ ❝❧❛ss ♦❢
✲♣r♦t♦❝♦❧s ❛r❡ ❇▲❚ s❡❝✉r❡ ✕ ❡✳❣✳ ❖❦❛♠♦t♦✳
✷ ❇❍❍❖ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✐s ❇▲❚✲s❡❝✉r❡✳
▼♦r❡♦✈❡r✳✳✳
❇♦♦st t♦ ❈♦♥t✐♥✉♦✉s ❚❛♠♣❡r✐♥❣ ✉s✐♥❣ ✉♥t❛♠♣❡r❛❜❧❡ ❋❧♦♣♣② ◆❡✇ ❚❡❝❤♥✐q✉❡✿ ❘❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✽ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ♥❡✇ ♠♦❞❡❧✿ ❇▲❚ ❇♦✉♥❞❡❞ ▲❡❛❦❛❣❡ ✫ ❚❛♠♣❡r✐♥❣ ✕ ❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣✳ ✕❆♥❛❧♦❣♦✉s t♦ ▲❡❛❦❛❣❡✲r❡s✐❧✐❡♥t ❈r②♣t♦❣r❛♣❤②
✲ ❍❖❚✦
❇▲❚✲s❡❝✉r❡ P✉❜❧✐❝✲❦❡② s❝❤❡♠❡s ❈♦♥❝r❡t❡ s❝❤❡♠❡s ❜❛s❡❞ ♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s ✭❉▲✱ ❋❛❝t♦r✐♥❣✮
✶ ■❉✲s❝❤❡♠❡s ❜❛s❡❞ ♦♥ s♦♠❡ ❝❧❛ss ♦❢
✲♣r♦t♦❝♦❧s ❛r❡ ❇▲❚ s❡❝✉r❡ ✕ ❡✳❣✳ ❖❦❛♠♦t♦✳
✷ ❇❍❍❖ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✐s ❇▲❚✲s❡❝✉r❡✳
▼♦r❡♦✈❡r✳✳✳
❇♦♦st t♦ ❈♦♥t✐♥✉♦✉s ❚❛♠♣❡r✐♥❣ ✉s✐♥❣ ✉♥t❛♠♣❡r❛❜❧❡ ❋❧♦♣♣② ◆❡✇ ❚❡❝❤♥✐q✉❡✿ ❘❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✽ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ♥❡✇ ♠♦❞❡❧✿ ❇▲❚ ❇♦✉♥❞❡❞ ▲❡❛❦❛❣❡ ✫ ❚❛♠♣❡r✐♥❣ ✕ ❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣✳ ✕❆♥❛❧♦❣♦✉s t♦ ▲❡❛❦❛❣❡✲r❡s✐❧✐❡♥t ❈r②♣t♦❣r❛♣❤②
✲ ❍❖❚✦
❇▲❚✲s❡❝✉r❡ P✉❜❧✐❝✲❦❡② s❝❤❡♠❡s ❈♦♥❝r❡t❡ s❝❤❡♠❡s ❜❛s❡❞ ♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s ✭❉▲✱ ❋❛❝t♦r✐♥❣✮
✶ ■❉✲s❝❤❡♠❡s ❜❛s❡❞ ♦♥ s♦♠❡ ❝❧❛ss ♦❢
✲♣r♦t♦❝♦❧s ❛r❡ ❇▲❚ s❡❝✉r❡ ✕ ❡✳❣✳ ❖❦❛♠♦t♦✳
✷ ❇❍❍❖ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✐s ❇▲❚✲s❡❝✉r❡✳
▼♦r❡♦✈❡r✳✳✳
❇♦♦st t♦ ❈♦♥t✐♥✉♦✉s ❚❛♠♣❡r✐♥❣ ✉s✐♥❣ ✉♥t❛♠♣❡r❛❜❧❡ ❋❧♦♣♣② ◆❡✇ ❚❡❝❤♥✐q✉❡✿ ❘❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✽ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ♥❡✇ ♠♦❞❡❧✿ ❇▲❚ ❇♦✉♥❞❡❞ ▲❡❛❦❛❣❡ ✫ ❚❛♠♣❡r✐♥❣ ✕ ❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣✳ ✕❆♥❛❧♦❣♦✉s t♦ ▲❡❛❦❛❣❡✲r❡s✐❧✐❡♥t ❈r②♣t♦❣r❛♣❤②
✲ ❍❖❚✦
❇▲❚✲s❡❝✉r❡ P✉❜❧✐❝✲❦❡② s❝❤❡♠❡s ❈♦♥❝r❡t❡ s❝❤❡♠❡s ❜❛s❡❞ ♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s ✭❉▲✱ ❋❛❝t♦r✐♥❣✮
✶ ■❉✲s❝❤❡♠❡s ❜❛s❡❞ ♦♥ s♦♠❡ ❝❧❛ss ♦❢
✲♣r♦t♦❝♦❧s ❛r❡ ❇▲❚ s❡❝✉r❡ ✕ ❡✳❣✳ ❖❦❛♠♦t♦✳
✷ ❇❍❍❖ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✐s ❇▲❚✲s❡❝✉r❡✳
▼♦r❡♦✈❡r✳✳✳
❇♦♦st t♦ ❈♦♥t✐♥✉♦✉s ❚❛♠♣❡r✐♥❣ ✉s✐♥❣ ✉♥t❛♠♣❡r❛❜❧❡ ❋❧♦♣♣② ◆❡✇ ❚❡❝❤♥✐q✉❡✿ ❘❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✽ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ♥❡✇ ♠♦❞❡❧✿ ❇▲❚ ❇♦✉♥❞❡❞ ▲❡❛❦❛❣❡ ✫ ❚❛♠♣❡r✐♥❣ ✕ ❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣✳ ✕❆♥❛❧♦❣♦✉s t♦ ▲❡❛❦❛❣❡✲r❡s✐❧✐❡♥t ❈r②♣t♦❣r❛♣❤②
✲ ❍❖❚✦
❇▲❚✲s❡❝✉r❡ P✉❜❧✐❝✲❦❡② s❝❤❡♠❡s ❈♦♥❝r❡t❡ s❝❤❡♠❡s ❜❛s❡❞ ♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s ✭❉▲✱ ❋❛❝t♦r✐♥❣✮
✶ ■❉✲s❝❤❡♠❡s ❜❛s❡❞ ♦♥ s♦♠❡ ❝❧❛ss ♦❢ Σ✲♣r♦t♦❝♦❧s ❛r❡ ❇▲❚
s❡❝✉r❡ ✕ ❡✳❣✳ ❖❦❛♠♦t♦✳
✷ ❇❍❍❖ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✐s ❇▲❚✲s❡❝✉r❡✳
▼♦r❡♦✈❡r✳✳✳
❇♦♦st t♦ ❈♦♥t✐♥✉♦✉s ❚❛♠♣❡r✐♥❣ ✉s✐♥❣ ✉♥t❛♠♣❡r❛❜❧❡ ❋❧♦♣♣② ◆❡✇ ❚❡❝❤♥✐q✉❡✿ ❘❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✽ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ♥❡✇ ♠♦❞❡❧✿ ❇▲❚ ❇♦✉♥❞❡❞ ▲❡❛❦❛❣❡ ✫ ❚❛♠♣❡r✐♥❣ ✕ ❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣✳ ✕❆♥❛❧♦❣♦✉s t♦ ▲❡❛❦❛❣❡✲r❡s✐❧✐❡♥t ❈r②♣t♦❣r❛♣❤②
✲ ❍❖❚✦
❇▲❚✲s❡❝✉r❡ P✉❜❧✐❝✲❦❡② s❝❤❡♠❡s ❈♦♥❝r❡t❡ s❝❤❡♠❡s ❜❛s❡❞ ♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s ✭❉▲✱ ❋❛❝t♦r✐♥❣✮
✶ ■❉✲s❝❤❡♠❡s ❜❛s❡❞ ♦♥ s♦♠❡ ❝❧❛ss ♦❢ Σ✲♣r♦t♦❝♦❧s ❛r❡ ❇▲❚
s❡❝✉r❡ ✕ ❡✳❣✳ ❖❦❛♠♦t♦✳
✷ ❇❍❍❖ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✐s ❇▲❚✲s❡❝✉r❡✳
▼♦r❡♦✈❡r✳✳✳
❇♦♦st t♦ ❈♦♥t✐♥✉♦✉s ❚❛♠♣❡r✐♥❣ ✉s✐♥❣ ✉♥t❛♠♣❡r❛❜❧❡ ❋❧♦♣♣② ◆❡✇ ❚❡❝❤♥✐q✉❡✿ ❘❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✽ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ♥❡✇ ♠♦❞❡❧✿ ❇▲❚ ❇♦✉♥❞❡❞ ▲❡❛❦❛❣❡ ✫ ❚❛♠♣❡r✐♥❣ ✕ ❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r✐♥❣✳ ✕❆♥❛❧♦❣♦✉s t♦ ▲❡❛❦❛❣❡✲r❡s✐❧✐❡♥t ❈r②♣t♦❣r❛♣❤②
✲ ❍❖❚✦
❇▲❚✲s❡❝✉r❡ P✉❜❧✐❝✲❦❡② s❝❤❡♠❡s ❈♦♥❝r❡t❡ s❝❤❡♠❡s ❜❛s❡❞ ♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s ✭❉▲✱ ❋❛❝t♦r✐♥❣✮
✶ ■❉✲s❝❤❡♠❡s ❜❛s❡❞ ♦♥ s♦♠❡ ❝❧❛ss ♦❢ Σ✲♣r♦t♦❝♦❧s ❛r❡ ❇▲❚
s❡❝✉r❡ ✕ ❡✳❣✳ ❖❦❛♠♦t♦✳
✷ ❇❍❍❖ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✐s ❇▲❚✲s❡❝✉r❡✳
▼♦r❡♦✈❡r✳✳✳
❇♦♦st t♦ ❈♦♥t✐♥✉♦✉s ❚❛♠♣❡r✐♥❣ ✉s✐♥❣ ✉♥t❛♠♣❡r❛❜❧❡ ❋❧♦♣♣② ◆❡✇ ❚❡❝❤♥✐q✉❡✿ ❘❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✽ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P ✇❛♥ts t♦ ❝♦♥✈✐♥❝❡ V t❤❛t P ❦♥♦✇s s❡❝r❡t sk ✇✳r✳t pk
♣r♦✈❡r✭P✮ ✈❡r✐❢✐❡r✭V✮
(sk, pk) (pk)
✐♥t❡r❛❝t
❉❡❝✐s✐♦♥
❛❝❝❡♣t✴r❡❥❡❝t
❛❝❝❡♣ts ✐❢ ❦♥♦✇s ❈♦rr❡❝t♥❡ss
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✾ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P ✇❛♥ts t♦ ❝♦♥✈✐♥❝❡ V t❤❛t P ❦♥♦✇s s❡❝r❡t sk ✇✳r✳t pk
♣r♦✈❡r✭P✮ ✈❡r✐❢✐❡r✭V✮
(sk, pk) (pk)
✐♥t❡r❛❝t
❉❡❝✐s✐♦♥
❛❝❝❡♣t✴r❡❥❡❝t
V ❛❝❝❡♣ts ✐❢ P ❦♥♦✇s sk ❈♦rr❡❝t♥❡ss
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✾ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
♣r♦✈❡r✭ ✮ ✈❡r✐❢✐❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
✐♥t❡r❛❝t
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡ ✐♥t❡r❛❝t
❚❛♠♣❡r t✐♠❡s✳✳✳
❈❤❛❧❧❡♥❣❡ P❤❛s❡
✐♥t❡r❛❝t
❉❡❝✐s✐♦♥
❛❝❝❡♣t✴r❡❥❡❝t
❛❝❝❡♣ts ❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✵ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
♣r♦✈❡r✭P✮ ✈❡r✐❢✐❡r✭ ✮ ❛❞✈❡rs❛r②✭A✮
(sk)
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
(pk)
✐♥t❡r❛❝t
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡ ✐♥t❡r❛❝t
❚❛♠♣❡r t✐♠❡s✳✳✳
❈❤❛❧❧❡♥❣❡ P❤❛s❡
✐♥t❡r❛❝t
❉❡❝✐s✐♦♥
❛❝❝❡♣t✴r❡❥❡❝t
❛❝❝❡♣ts ❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✵ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
♣r♦✈❡r✭P✮ ✈❡r✐❢✐❡r✭ ✮ ❛❞✈❡rs❛r②✭A✮
(sk)
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
(pk, Ψ)
✐♥t❡r❛❝t
Ψ(sk)
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡ ✐♥t❡r❛❝t
❚❛♠♣❡r t✐♠❡s✳✳✳
❈❤❛❧❧❡♥❣❡ P❤❛s❡
✐♥t❡r❛❝t
❉❡❝✐s✐♦♥
❛❝❝❡♣t✴r❡❥❡❝t
❛❝❝❡♣ts ❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✵ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
♣r♦✈❡r✭P✮ ✈❡r✐❢✐❡r✭ ✮ ❛❞✈❡rs❛r②✭A✮
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
(pk, Ψ, Ψ1, · · · , Ψi)
✐♥t❡r❛❝t
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡ Ti ✐♥t❡r❛❝t
Ti(sk)
ski) Ti(·) ( ski) ❚❛♠♣❡r i ≤ t t✐♠❡s✳✳✳
❈❤❛❧❧❡♥❣❡ P❤❛s❡
✐♥t❡r❛❝t
❉❡❝✐s✐♦♥
❛❝❝❡♣t✴r❡❥❡❝t
❛❝❝❡♣ts ❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✵ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
♣r♦✈❡r✭ ✮ ✈❡r✐❢✐❡r✭V✮
(pk)
❛❞✈❡rs❛r②✭A✮
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
(pk, Ψ, Ψ1, · · · , Ψi, · · · , Ψt)
✐♥t❡r❛❝t
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡ ✐♥t❡r❛❝t
❚❛♠♣❡r t✐♠❡s✳✳✳
❈❤❛❧❧❡♥❣❡ P❤❛s❡
✐♥t❡r❛❝t
❉❡❝✐s✐♦♥
❛❝❝❡♣t✴r❡❥❡❝t
Pr[V ❛❝❝❡♣ts] ≤ negl ❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✵ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❍❛r❞ ❘❡❧❛t✐♦♥ R R ✐s ❤❛r❞ ✐❢ ♥♦ PP❚ ❛❞✈❡rs❛r② ❝❛♥ ♦✉t♣✉t (y, x, x′) s✉❝❤ t❤❛t (y, x) ∈ R ∧ (y, x′) ∈ R✳
❍❛r❞ ❘❡❧❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭✐♥❢♦r♠❛❧✮
♣r♦✈❡r✭ ✮ ✈❡r✐❢✐❡r✭ ✮ ❛❝❝❡♣t✴r❡❥❡❝t
❉❡❝✐s✐♦♥
❚❤✐s ✐s ❇▲❚✲s❡❝✉r❡ ❢♦r ❚❤❡♦r❡♠✭✐♥❢♦r♠❛❧✮
Pr♦♣❡rt②✲✶✿ ❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss
❛❞✈❡rs❛r②✭ ✮ ✈❡r✐❢✐❡r✭ ✮
Pr♦♣❡rt②✲✷✿ ❍♦♥❡st ❱❡r✐✜❡r ❩❑
s✐♠✉❧❛t♦r✭
✮
✈❡r✐❢✐❡r✭ ✮
❈♦♠♣✉t❡
❉❡❝✐s✐♦♥
❛❝❝❡♣t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✶ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❍❛r❞ ❘❡❧❛t✐♦♥ ✐s ❤❛r❞ ✐❢ ♥♦ PP❚ ❛❞✈❡rs❛r② ❝❛♥ ♦✉t♣✉t s✉❝❤ t❤❛t ✳
❍❛r❞ ❘❡❧❛t✐♦♥ R ❉❡✜♥✐t✐♦♥ ✭✐♥❢♦r♠❛❧✮
♣r♦✈❡r✭P✮ ✈❡r✐❢✐❡r✭V✮
(y, x) (y) a c z V(y, (a, c, z)) ? = 1
❛❝❝❡♣t✴r❡❥❡❝t
❉❡❝✐s✐♦♥
❚❤✐s ✐s ❇▲❚✲s❡❝✉r❡ ❢♦r ❚❤❡♦r❡♠✭✐♥❢♦r♠❛❧✮
Pr♦♣❡rt②✲✶✿ ❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss
❛❞✈❡rs❛r②✭ ✮ ✈❡r✐❢✐❡r✭ ✮
Pr♦♣❡rt②✲✷✿ ❍♦♥❡st ❱❡r✐✜❡r ❩❑
s✐♠✉❧❛t♦r✭
✮
✈❡r✐❢✐❡r✭ ✮
❈♦♠♣✉t❡
❉❡❝✐s✐♦♥
❛❝❝❡♣t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✶ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❍❛r❞ ❘❡❧❛t✐♦♥ ✐s ❤❛r❞ ✐❢ ♥♦ PP❚ ❛❞✈❡rs❛r② ❝❛♥ ♦✉t♣✉t s✉❝❤ t❤❛t ✳
❍❛r❞ ❘❡❧❛t✐♦♥ R ❉❡✜♥✐t✐♦♥ ✭✐♥❢♦r♠❛❧✮
♣r♦✈❡r✭P✮ ✈❡r✐❢✐❡r✭V✮
(y, x) (y) a c z V(y, (a, c, z)) ? = 1
❛❝❝❡♣t✴r❡❥❡❝t
❉❡❝✐s✐♦♥
❚❤✐s ✐s ❇▲❚✲s❡❝✉r❡ ❢♦r t ≤ |x|
|y| − 1
❚❤❡♦r❡♠✭✐♥❢♦r♠❛❧✮
Pr♦♣❡rt②✲✶✿ ❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss
❛❞✈❡rs❛r②✭ ✮ ✈❡r✐❢✐❡r✭ ✮
Pr♦♣❡rt②✲✷✿ ❍♦♥❡st ❱❡r✐✜❡r ❩❑
s✐♠✉❧❛t♦r✭
✮
✈❡r✐❢✐❡r✭ ✮
❈♦♠♣✉t❡
❉❡❝✐s✐♦♥
❛❝❝❡♣t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✶ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❍❛r❞ ❘❡❧❛t✐♦♥ ✐s ❤❛r❞ ✐❢ ♥♦ PP❚ ❛❞✈❡rs❛r② ❝❛♥ ♦✉t♣✉t s✉❝❤ t❤❛t ✳
❍❛r❞ ❘❡❧❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭✐♥❢♦r♠❛❧✮
♣r♦✈❡r✭ ✮ ✈❡r✐❢✐❡r✭ ✮
(y) (y)
❛❝❝❡♣t✴r❡❥❡❝t
❉❡❝✐s✐♦♥
❚❤✐s ✐s ❇▲❚✲s❡❝✉r❡ ❢♦r ❚❤❡♦r❡♠✭✐♥❢♦r♠❛❧✮
Pr♦♣❡rt②✲✶✿ ❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss
❛❞✈❡rs❛r②✭A✮ ✈❡r✐❢✐❡r✭V✮
Extract V(y, (a, c, z)) = 1 ∧ V(y, (a, c′, z′)) = 1 a c, c′ z, z′ (y, x⋆) ∈ R : x⋆ ← Extract((a, c, z), (a, c′, z′))
Pr♦♣❡rt②✲✷✿ ❍♦♥❡st ❱❡r✐✜❡r ❩❑
s✐♠✉❧❛t♦r✭
✮
✈❡r✐❢✐❡r✭ ✮
❈♦♠♣✉t❡
❉❡❝✐s✐♦♥
❛❝❝❡♣t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✶ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❍❛r❞ ❘❡❧❛t✐♦♥ ✐s ❤❛r❞ ✐❢ ♥♦ PP❚ ❛❞✈❡rs❛r② ❝❛♥ ♦✉t♣✉t s✉❝❤ t❤❛t ✳
❍❛r❞ ❘❡❧❛t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭✐♥❢♦r♠❛❧✮
♣r♦✈❡r✭ ✮ ✈❡r✐❢✐❡r✭ ✮ ❛❝❝❡♣t✴r❡❥❡❝t
❉❡❝✐s✐♦♥
❚❤✐s ✐s ❇▲❚✲s❡❝✉r❡ ❢♦r ❚❤❡♦r❡♠✭✐♥❢♦r♠❛❧✮
Pr♦♣❡rt②✲✶✿ ❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss
❛❞✈❡rs❛r②✭ ✮ ✈❡r✐❢✐❡r✭ ✮
Pr♦♣❡rt②✲✷✿ ❍♦♥❡st ❱❡r✐✜❡r ❩❑
s✐♠✉❧❛t♦r✭Sim✮
(y)
✈❡r✐❢✐❡r✭V✮
(y) c ❈♦♠♣✉t❡ (a, z) ← Sim(c, y) a, c, z V(y, (a, c, z)) = 1
❉❡❝✐s✐♦♥
❛❝❝❡♣t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✶ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥ R
■❞❡❛
r❡❞✉❝t✐♦♥✭B✮ r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭C✮ ❛❞✈❡rs❛r②✭A✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t
✿ ❙❛♠♣❧❡ ✳
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛s② t♦ s✐♠✉❧❛t❡
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥
■❞❡❛
r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t ❘❡✇✐♥❞ t♦ ❡①tr❛❝t ✭❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss✮ ❘❡✇✐♥❞
r❡❞✉❝t✐♦♥✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛❝❤ t❛♠♣❡r✐♥❣ ❡①♣❡r✐♠❡♥t ♦✉t♣✉ts ♣♦❧②✲♠❛♥② tr❛♥s❝r✐♣ts✕❍❯●❊✦ ❚♦ Pr♦✈❡✿ ❚❤✐s ❞♦❡s ◆❖❚ ❧❡❛❦ ♠✉❝❤ ❛❜♦✉t ✳ ▼❛✐♥ ❈❤❛❧❧❡♥❣❡ ❋♦r ❡❛❝❤ t❛♠♣❡r✐♥❣✿ ▲❊❆❑ t❛♠♣❡r❡❞ ♣✉❜❧✐❝ st❛t❡♠❡♥t ❖♥❧② ❜✐ts ✕ ❙❍❖❘❚✦ ❚❡❝❤♥✐q✉❡
t❛♠♣❡r ✇✐t❤ ❧❡❛❦
❈❤♦♦s❡ r❛♥❞♦♠ ✭❍❱❩❑✮
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✷ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥ R
■❞❡❛
r❡❞✉❝t✐♦♥✭B✮ r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭C✮ ❛❞✈❡rs❛r②✭A✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t R
R ∧ (y, x⋆) ∈ R ∧ (x = x⋆) ❙❛♠♣❧❡ (y, x) ∈ R✳
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛s② t♦ s✐♠✉❧❛t❡
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥
■❞❡❛
r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t ❘❡✇✐♥❞ t♦ ❡①tr❛❝t ✭❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss✮ ❘❡✇✐♥❞
r❡❞✉❝t✐♦♥✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛❝❤ t❛♠♣❡r✐♥❣ ❡①♣❡r✐♠❡♥t ♦✉t♣✉ts ♣♦❧②✲♠❛♥② tr❛♥s❝r✐♣ts✕❍❯●❊✦ ❚♦ Pr♦✈❡✿ ❚❤✐s ❞♦❡s ◆❖❚ ❧❡❛❦ ♠✉❝❤ ❛❜♦✉t ✳ ▼❛✐♥ ❈❤❛❧❧❡♥❣❡ ❋♦r ❡❛❝❤ t❛♠♣❡r✐♥❣✿ ▲❊❆❑ t❛♠♣❡r❡❞ ♣✉❜❧✐❝ st❛t❡♠❡♥t ❖♥❧② ❜✐ts ✕ ❙❍❖❘❚✦ ❚❡❝❤♥✐q✉❡
t❛♠♣❡r ✇✐t❤ ❧❡❛❦
❈❤♦♦s❡ r❛♥❞♦♠ ✭❍❱❩❑✮
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✷ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥ R
■❞❡❛
r❡❞✉❝t✐♦♥✭ ✮ r❡❞✉❝t✐♦♥✭B✮ ❝❤❛❧❧❡♥❣❡r✭C✮ ❛❞✈❡rs❛r②✭A✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t
✿ ❙❛♠♣❧❡ ✳
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛s② t♦ s✐♠✉❧❛t❡ (y) (x, y) a c z
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥
■❞❡❛
r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t ❘❡✇✐♥❞ t♦ ❡①tr❛❝t ✭❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss✮ ❘❡✇✐♥❞
r❡❞✉❝t✐♦♥✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛❝❤ t❛♠♣❡r✐♥❣ ❡①♣❡r✐♠❡♥t ♦✉t♣✉ts ♣♦❧②✲♠❛♥② tr❛♥s❝r✐♣ts✕❍❯●❊✦ ❚♦ Pr♦✈❡✿ ❚❤✐s ❞♦❡s ◆❖❚ ❧❡❛❦ ♠✉❝❤ ❛❜♦✉t ✳ ▼❛✐♥ ❈❤❛❧❧❡♥❣❡ ❋♦r ❡❛❝❤ t❛♠♣❡r✐♥❣✿ ▲❊❆❑ t❛♠♣❡r❡❞ ♣✉❜❧✐❝ st❛t❡♠❡♥t ❖♥❧② ❜✐ts ✕ ❙❍❖❘❚✦ ❚❡❝❤♥✐q✉❡
t❛♠♣❡r ✇✐t❤ ❧❡❛❦
❈❤♦♦s❡ r❛♥❞♦♠ ✭❍❱❩❑✮
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✷ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥
■❞❡❛
r❡❞✉❝t✐♦♥✭ ✮ r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t
✿ ❙❛♠♣❧❡ ✳
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛s② t♦ s✐♠✉❧❛t❡
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥ R
■❞❡❛
r❡❞✉❝t✐♦♥✭B✮ ❝❤❛❧❧❡♥❣❡r✭C✮ ❛❞✈❡rs❛r②✭A✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t R ❘❡✇✐♥❞ A t♦ ❡①tr❛❝t x⋆ ✭❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss✮
y, x, x⋆
(y) (x, y) a c′ z′ ❘❡✇✐♥❞
r❡❞✉❝t✐♦♥✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛❝❤ t❛♠♣❡r✐♥❣ ❡①♣❡r✐♠❡♥t ♦✉t♣✉ts ♣♦❧②✲♠❛♥② tr❛♥s❝r✐♣ts✕❍❯●❊✦ ❚♦ Pr♦✈❡✿ ❚❤✐s ❞♦❡s ◆❖❚ ❧❡❛❦ ♠✉❝❤ ❛❜♦✉t ✳ ▼❛✐♥ ❈❤❛❧❧❡♥❣❡ ❋♦r ❡❛❝❤ t❛♠♣❡r✐♥❣✿ ▲❊❆❑ t❛♠♣❡r❡❞ ♣✉❜❧✐❝ st❛t❡♠❡♥t ❖♥❧② ❜✐ts ✕ ❙❍❖❘❚✦ ❚❡❝❤♥✐q✉❡
t❛♠♣❡r ✇✐t❤ ❧❡❛❦
❈❤♦♦s❡ r❛♥❞♦♠ ✭❍❱❩❑✮
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✷ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥
■❞❡❛
r❡❞✉❝t✐♦♥✭ ✮ r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t
✿ ❙❛♠♣❧❡ ✳
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛s② t♦ s✐♠✉❧❛t❡
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥
■❞❡❛
r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t ❘❡✇✐♥❞ t♦ ❡①tr❛❝t ✭❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss✮ ❘❡✇✐♥❞
r❡❞✉❝t✐♦♥✭B✮ ❛❞✈❡rs❛r②✭A✮
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ (x, y) (y) ❊❛❝❤ t❛♠♣❡r✐♥❣ ❡①♣❡r✐♠❡♥t ♦✉t♣✉ts ♣♦❧②✲♠❛♥② tr❛♥s❝r✐♣ts✕❍❯●❊✦ ❚♦ Pr♦✈❡✿ ❚❤✐s ❞♦❡s ◆❖❚ ❧❡❛❦ ♠✉❝❤ ❛❜♦✉t x✳ ▼❛✐♥ ❈❤❛❧❧❡♥❣❡ ❋♦r ❡❛❝❤ t❛♠♣❡r✐♥❣✿ ▲❊❆❑ t❛♠♣❡r❡❞ ♣✉❜❧✐❝ st❛t❡♠❡♥t ❖♥❧② ❜✐ts ✕ ❙❍❖❘❚✦ ❚❡❝❤♥✐q✉❡
t❛♠♣❡r ✇✐t❤ ❧❡❛❦
❈❤♦♦s❡ r❛♥❞♦♠ ✭❍❱❩❑✮
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✷ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥
■❞❡❛
r❡❞✉❝t✐♦♥✭ ✮ r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t
✿ ❙❛♠♣❧❡ ✳
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ ❊❛s② t♦ s✐♠✉❧❛t❡
❘❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❍❛r❞ ❘❡❧❛t✐♦♥
■❞❡❛
r❡❞✉❝t✐♦♥✭ ✮ ❝❤❛❧❧❡♥❣❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❍❛r❞♥❡ss ●❛♠❡ ✇✳r✳t ❘❡✇✐♥❞ t♦ ❡①tr❛❝t ✭❙♣❡❝✐❛❧ ❙♦✉♥❞♥❡ss✮ ❘❡✇✐♥❞
r❡❞✉❝t✐♦♥✭B✮ ❛❞✈❡rs❛r②✭A✮
❙✐♠✉❧❛t❡ ❇▲❚ ●❛♠❡ (x, y) (y) ❊❛❝❤ t❛♠♣❡r✐♥❣ ❡①♣❡r✐♠❡♥t ♦✉t♣✉ts ♣♦❧②✲♠❛♥② tr❛♥s❝r✐♣ts✕❍❯●❊✦ ❚♦ Pr♦✈❡✿ ❚❤✐s ❞♦❡s ◆❖❚ ❧❡❛❦ ♠✉❝❤ ❛❜♦✉t ✳ ▼❛✐♥ ❈❤❛❧❧❡♥❣❡ ❋♦r ❡❛❝❤ t❛♠♣❡r✐♥❣✿ ▲❊❆❑ t❛♠♣❡r❡❞ ♣✉❜❧✐❝ st❛t❡♠❡♥t ❖♥❧② |y| ❜✐ts ✕ ❙❍❖❘❚✦ ❚❡❝❤♥✐q✉❡
t❛♠♣❡r ✇✐t❤ T(·)
φ
← x ← T(x)
❧❡❛❦
y ❈❤♦♦s❡ r❛♥❞♦♠ c (a, c, z) ← Sim(c, y) ✭❍❱❩❑✮
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✷ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
✕ ❆❧✇❡♥ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✾❪✳ ❆❧s♦ ❇▲❚✲s❡❝✉r❡✕t❤✐s ✇♦r❦✳ ❆❞❞✐t✐♦♥❛❧❧② ❛❧❧♦✇s t❛♠♣❡r✐♥❣ ✇✐t❤ t❤❡ ♣✉❜❧✐❝ ♣❛r❛♠❡t❡rs ✭❡✳❣✳ ❝❤❛r❛❝t❡r✐st✐❝ ♣r✐♠❡ ✮ ✕ ❜✉t ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ ✦ ✕ ✐♠♣♦ss✐❜❧❡ ✇❤❡♥ t❛♠♣❡r✐♥❣ ❥♦✐♥t❧②✳ ❇▲❚✲s❡❝✉r❡ s✐❣♥❛t✉r❡s ✉s✐♥❣ ❋✐❛t✲❙❤❛♠✐r tr❛♥s❢♦r♠✳ ✕ r❡q✉✐r❡s r❛♥❞♦♠ ♦r❛❝❧❡s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✸ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
✕ ❆❧✇❡♥ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✾❪✳ ❆❧s♦ ❇▲❚✲s❡❝✉r❡✕t❤✐s ✇♦r❦✳ ❆❞❞✐t✐♦♥❛❧❧② ❛❧❧♦✇s t❛♠♣❡r✐♥❣ ✇✐t❤ t❤❡ ♣✉❜❧✐❝ ♣❛r❛♠❡t❡rs ✭❡✳❣✳ ❝❤❛r❛❝t❡r✐st✐❝ ♣r✐♠❡ ✮ ✕ ❜✉t ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ ✦ ✕ ✐♠♣♦ss✐❜❧❡ ✇❤❡♥ t❛♠♣❡r✐♥❣ ❥♦✐♥t❧②✳ ❇▲❚✲s❡❝✉r❡ s✐❣♥❛t✉r❡s ✉s✐♥❣ ❋✐❛t✲❙❤❛♠✐r tr❛♥s❢♦r♠✳ ✕ r❡q✉✐r❡s r❛♥❞♦♠ ♦r❛❝❧❡s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✸ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
✕ ❆❧✇❡♥ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✾❪✳ ❆❧s♦ ❇▲❚✲s❡❝✉r❡✕t❤✐s ✇♦r❦✳ ❆❞❞✐t✐♦♥❛❧❧② ❛❧❧♦✇s t❛♠♣❡r✐♥❣ ✇✐t❤ t❤❡ ♣✉❜❧✐❝ ♣❛r❛♠❡t❡rs ✭❡✳❣✳ ❝❤❛r❛❝t❡r✐st✐❝ ♣r✐♠❡ ✮ ✕ ❜✉t ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ ✦ ✕ ✐♠♣♦ss✐❜❧❡ ✇❤❡♥ t❛♠♣❡r✐♥❣ ❥♦✐♥t❧②✳ ❇▲❚✲s❡❝✉r❡ s✐❣♥❛t✉r❡s ✉s✐♥❣ ❋✐❛t✲❙❤❛♠✐r tr❛♥s❢♦r♠✳ ✕ r❡q✉✐r❡s r❛♥❞♦♠ ♦r❛❝❧❡s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✸ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
✕ ❆❧✇❡♥ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✾❪✳ ❆❧s♦ ❇▲❚✲s❡❝✉r❡✕t❤✐s ✇♦r❦✳ ❆❞❞✐t✐♦♥❛❧❧② ❛❧❧♦✇s t❛♠♣❡r✐♥❣ ✇✐t❤ t❤❡ ♣✉❜❧✐❝ ♣❛r❛♠❡t❡rs ✭❡✳❣✳ ❝❤❛r❛❝t❡r✐st✐❝ ♣r✐♠❡ p✮ ✕ ❜✉t ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ sk✦ ✕ ✐♠♣♦ss✐❜❧❡ ✇❤❡♥ t❛♠♣❡r✐♥❣ ❥♦✐♥t❧②✳ ❇▲❚✲s❡❝✉r❡ s✐❣♥❛t✉r❡s ✉s✐♥❣ ❋✐❛t✲❙❤❛♠✐r tr❛♥s❢♦r♠✳ ✕ r❡q✉✐r❡s r❛♥❞♦♠ ♦r❛❝❧❡s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✸ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
✕ ❆❧✇❡♥ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✾❪✳ ❆❧s♦ ❇▲❚✲s❡❝✉r❡✕t❤✐s ✇♦r❦✳ ❆❞❞✐t✐♦♥❛❧❧② ❛❧❧♦✇s t❛♠♣❡r✐♥❣ ✇✐t❤ t❤❡ ♣✉❜❧✐❝ ♣❛r❛♠❡t❡rs ✭❡✳❣✳ ❝❤❛r❛❝t❡r✐st✐❝ ♣r✐♠❡ p✮ ✕ ❜✉t ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ sk✦ ✕ ✐♠♣♦ss✐❜❧❡ ✇❤❡♥ t❛♠♣❡r✐♥❣ ❥♦✐♥t❧②✳ ❇▲❚✲s❡❝✉r❡ s✐❣♥❛t✉r❡s ✉s✐♥❣ ❋✐❛t✲❙❤❛♠✐r tr❛♥s❢♦r♠✳ ✕ r❡q✉✐r❡s r❛♥❞♦♠ ♦r❛❝❧❡s✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✸ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
❝❤❛❧❧❡♥❣❡r✭ ✮ ❛❞✈❡rs❛r②✭ ✮
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
❉❡❝
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡
❚❛♠♣❡r t✐♠❡s✳✳✳ ❉❡❝
❈❤❛❧❧❡♥❣❡ P❤❛s❡
❈❤♦♦s❡ ❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✹ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
❝❤❛❧❧❡♥❣❡r✭C✮ ❛❞✈❡rs❛r②✭A✮
(sk)
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
(pk) ❉❡❝
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡
❚❛♠♣❡r t✐♠❡s✳✳✳ ❉❡❝
❈❤❛❧❧❡♥❣❡ P❤❛s❡
❈❤♦♦s❡ ❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✹ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
❝❤❛❧❧❡♥❣❡r✭C✮ ❛❞✈❡rs❛r②✭A✮
(sk)
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
(pk, m) c m ← ❉❡❝(sk, c) m
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡
❚❛♠♣❡r t✐♠❡s✳✳✳ ❉❡❝
❈❤❛❧❧❡♥❣❡ P❤❛s❡
❈❤♦♦s❡ ❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✹ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
❝❤❛❧❧❡♥❣❡r✭C✮ ❛❞✈❡rs❛r②✭A✮
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
(pk, m, m1, · · · , mi) ❉❡❝
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡ Ti
❚❛♠♣❡r i ≤ t t✐♠❡s✳✳✳ Ti(sk) Ti(·) ( ski) ci
ski, ci)
❈❤❛❧❧❡♥❣❡ P❤❛s❡
❈❤♦♦s❡ ❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✹ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
P❛r❛♠❡t❡r
❝❤❛❧❧❡♥❣❡r✭C✮ ❛❞✈❡rs❛r②✭A✮
❯♥t❛♠♣❡r❡❞ ◗✉❡r②
(pk, m, m1, · · · , mi, · · · , mt) ❉❡❝
❚❛♠♣❡r✐♥❣
❝❤♦♦s❡
❚❛♠♣❡r t✐♠❡s✳✳✳ ❉❡❝
❈❤❛❧❧❡♥❣❡ P❤❛s❡
m0, mi ❈❤♦♦s❡ b $ ← {0, 1} c⋆ ← Enc(pk, mb) c⋆ b′
2
❘❡q✉✐r❡♠❡♥t
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✹ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❚✇♦ ❙t❡♣s✳
✶ ❆ ✇❡❛❦❡r ♠♦❞❡❧✿ ■◆❉✲❈P❆✲❇▲❚✳ ✷ ■◆❉✲❈P❆✲❇▲❚
■◆❉✲❈❈❆✲❇▲❚ ✕ ✉s✐♥❣ ❛ t❙❊✲◆■❩❑ ✭s✐♠✐❧❛r t♦ ❉♦❞✐s ❡t ❛❧✳❬❛s✐❛❝r②♣t✬✶✵❪✮ ❈♦♥❝r❡t❡ ■♥st❛♥t✐❛t✐♦♥ ❇❍❍❖ ✭❇♦♥❡❤ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✽❪✮ ✭❇♦✉♥❞❡❞✲❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥t✕ ◆❛♦r ✫ ❙❡❣❡✈ ❬❝r②♣t♦✬✵✾❪✮✳
■❞❡❛ ✭s✐♠✐❧❛r t♦ ■❉✮✿ s✐♠✉❧❛t❡ t❛♠♣❡r✐♥❣ q✉❡r✐❡s ❜② ❛❞❞✐t✐♦♥❛❧ ❧❡❛❦❛❣❡✳
▲✐♠✐t❛t✐♦♥✦
❈❛♥ ♥♦t t❛♠♣❡r✭♦r ❧❡❛❦✮ ❛❢t❡r ❝❤❛❧❧❡♥❣❡ ♣❤❛s❡✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✺ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❚✇♦ ❙t❡♣s✳
✶ ❆ ✇❡❛❦❡r ♠♦❞❡❧✿ ■◆❉✲❈P❆✲❇▲❚✳ ✷ ■◆❉✲❈P❆✲❇▲❚
■◆❉✲❈❈❆✲❇▲❚ ✕ ✉s✐♥❣ ❛ t❙❊✲◆■❩❑ ✭s✐♠✐❧❛r t♦ ❉♦❞✐s ❡t ❛❧✳❬❛s✐❛❝r②♣t✬✶✵❪✮ ❈♦♥❝r❡t❡ ■♥st❛♥t✐❛t✐♦♥ ❇❍❍❖ ✭❇♦♥❡❤ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✽❪✮ ✭❇♦✉♥❞❡❞✲❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥t✕ ◆❛♦r ✫ ❙❡❣❡✈ ❬❝r②♣t♦✬✵✾❪✮✳
■❞❡❛ ✭s✐♠✐❧❛r t♦ ■❉✮✿ s✐♠✉❧❛t❡ t❛♠♣❡r✐♥❣ q✉❡r✐❡s ❜② ❛❞❞✐t✐♦♥❛❧ ❧❡❛❦❛❣❡✳
▲✐♠✐t❛t✐♦♥✦
❈❛♥ ♥♦t t❛♠♣❡r✭♦r ❧❡❛❦✮ ❛❢t❡r ❝❤❛❧❧❡♥❣❡ ♣❤❛s❡✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✺ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❚✇♦ ❙t❡♣s✳
✶ ❆ ✇❡❛❦❡r ♠♦❞❡❧✿ ■◆❉✲❈P❆✲❇▲❚✳ ✷ ■◆❉✲❈P❆✲❇▲❚
■◆❉✲❈❈❆✲❇▲❚ ✕ ✉s✐♥❣ ❛ t❙❊✲◆■❩❑ ✭s✐♠✐❧❛r t♦ ❉♦❞✐s ❡t ❛❧✳❬❛s✐❛❝r②♣t✬✶✵❪✮ ❈♦♥❝r❡t❡ ■♥st❛♥t✐❛t✐♦♥ ❇❍❍❖ ✭❇♦♥❡❤ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✽❪✮ ✭❇♦✉♥❞❡❞✲❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥t✕ ◆❛♦r ✫ ❙❡❣❡✈ ❬❝r②♣t♦✬✵✾❪✮✳
■❞❡❛ ✭s✐♠✐❧❛r t♦ ■❉✮✿ s✐♠✉❧❛t❡ t❛♠♣❡r✐♥❣ q✉❡r✐❡s ❜② ❛❞❞✐t✐♦♥❛❧ ❧❡❛❦❛❣❡✳
▲✐♠✐t❛t✐♦♥✦
❈❛♥ ♥♦t t❛♠♣❡r✭♦r ❧❡❛❦✮ ❛❢t❡r ❝❤❛❧❧❡♥❣❡ ♣❤❛s❡✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✺ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❚✇♦ ❙t❡♣s✳
✶ ❆ ✇❡❛❦❡r ♠♦❞❡❧✿ ■◆❉✲❈P❆✲❇▲❚✳ ✷ ■◆❉✲❈P❆✲❇▲❚
■◆❉✲❈❈❆✲❇▲❚ ✕ ✉s✐♥❣ ❛ t❙❊✲◆■❩❑ ✭s✐♠✐❧❛r t♦ ❉♦❞✐s ❡t ❛❧✳❬❛s✐❛❝r②♣t✬✶✵❪✮ ❈♦♥❝r❡t❡ ■♥st❛♥t✐❛t✐♦♥ ❇❍❍❖ ✭❇♦♥❡❤ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✽❪✮ ✭❇♦✉♥❞❡❞✲❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥t✕ ◆❛♦r ✫ ❙❡❣❡✈ ❬❝r②♣t♦✬✵✾❪✮✳
■❞❡❛ ✭s✐♠✐❧❛r t♦ ■❉✮✿ s✐♠✉❧❛t❡ t❛♠♣❡r✐♥❣ q✉❡r✐❡s ❜② ❛❞❞✐t✐♦♥❛❧ ❧❡❛❦❛❣❡✳
▲✐♠✐t❛t✐♦♥✦
❈❛♥ ♥♦t t❛♠♣❡r✭♦r ❧❡❛❦✮ ❛❢t❡r ❝❤❛❧❧❡♥❣❡ ♣❤❛s❡✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✺ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❚✇♦ ❙t❡♣s✳
✶ ❆ ✇❡❛❦❡r ♠♦❞❡❧✿ ■◆❉✲❈P❆✲❇▲❚✳ ✷ ■◆❉✲❈P❆✲❇▲❚ =
⇒ ■◆❉✲❈❈❆✲❇▲❚ ✕ ✉s✐♥❣ ❛ t❙❊✲◆■❩❑ ✭s✐♠✐❧❛r t♦ ❉♦❞✐s ❡t ❛❧✳❬❛s✐❛❝r②♣t✬✶✵❪✮ ❈♦♥❝r❡t❡ ■♥st❛♥t✐❛t✐♦♥ ❇❍❍❖ ✭❇♦♥❡❤ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✽❪✮ ✭❇♦✉♥❞❡❞✲❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥t✕ ◆❛♦r ✫ ❙❡❣❡✈ ❬❝r②♣t♦✬✵✾❪✮✳
■❞❡❛ ✭s✐♠✐❧❛r t♦ ■❉✮✿ s✐♠✉❧❛t❡ t❛♠♣❡r✐♥❣ q✉❡r✐❡s ❜② ❛❞❞✐t✐♦♥❛❧ ❧❡❛❦❛❣❡✳
▲✐♠✐t❛t✐♦♥✦
❈❛♥ ♥♦t t❛♠♣❡r✭♦r ❧❡❛❦✮ ❛❢t❡r ❝❤❛❧❧❡♥❣❡ ♣❤❛s❡✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✺ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❚✇♦ ❙t❡♣s✳
✶ ❆ ✇❡❛❦❡r ♠♦❞❡❧✿ ■◆❉✲❈P❆✲❇▲❚✳ ✷ ■◆❉✲❈P❆✲❇▲❚ =
⇒ ■◆❉✲❈❈❆✲❇▲❚ ✕ ✉s✐♥❣ ❛ t❙❊✲◆■❩❑ ✭s✐♠✐❧❛r t♦ ❉♦❞✐s ❡t ❛❧✳❬❛s✐❛❝r②♣t✬✶✵❪✮ ❈♦♥❝r❡t❡ ■♥st❛♥t✐❛t✐♦♥ ❇❍❍❖ ✭❇♦♥❡❤ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✽❪✮ ✭❇♦✉♥❞❡❞✲❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥t✕ ◆❛♦r ✫ ❙❡❣❡✈ ❬❝r②♣t♦✬✵✾❪✮✳
■❞❡❛ ✭s✐♠✐❧❛r t♦ ■❉✮✿ s✐♠✉❧❛t❡ t❛♠♣❡r✐♥❣ q✉❡r✐❡s ❜② ❛❞❞✐t✐♦♥❛❧ ❧❡❛❦❛❣❡✳
▲✐♠✐t❛t✐♦♥✦
❈❛♥ ♥♦t t❛♠♣❡r✭♦r ❧❡❛❦✮ ❛❢t❡r ❝❤❛❧❧❡♥❣❡ ♣❤❛s❡✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✺ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❚✇♦ ❙t❡♣s✳
✶ ❆ ✇❡❛❦❡r ♠♦❞❡❧✿ ■◆❉✲❈P❆✲❇▲❚✳ ✷ ■◆❉✲❈P❆✲❇▲❚ =
⇒ ■◆❉✲❈❈❆✲❇▲❚ ✕ ✉s✐♥❣ ❛ t❙❊✲◆■❩❑ ✭s✐♠✐❧❛r t♦ ❉♦❞✐s ❡t ❛❧✳❬❛s✐❛❝r②♣t✬✶✵❪✮ ❈♦♥❝r❡t❡ ■♥st❛♥t✐❛t✐♦♥ ❇❍❍❖ ✭❇♦♥❡❤ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✽❪✮ ✭❇♦✉♥❞❡❞✲❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥t✕ ◆❛♦r ✫ ❙❡❣❡✈ ❬❝r②♣t♦✬✵✾❪✮✳
■❞❡❛ ✭s✐♠✐❧❛r t♦ ■❉✮✿ s✐♠✉❧❛t❡ t❛♠♣❡r✐♥❣ q✉❡r✐❡s ❜② ❛❞❞✐t✐♦♥❛❧ ❧❡❛❦❛❣❡✳
▲✐♠✐t❛t✐♦♥✦
❈❛♥ ♥♦t t❛♠♣❡r✭♦r ❧❡❛❦✮ ❛❢t❡r ❝❤❛❧❧❡♥❣❡ ♣❤❛s❡✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✺ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❚✇♦ ❙t❡♣s✳
✶ ❆ ✇❡❛❦❡r ♠♦❞❡❧✿ ■◆❉✲❈P❆✲❇▲❚✳ ✷ ■◆❉✲❈P❆✲❇▲❚ =
⇒ ■◆❉✲❈❈❆✲❇▲❚ ✕ ✉s✐♥❣ ❛ t❙❊✲◆■❩❑ ✭s✐♠✐❧❛r t♦ ❉♦❞✐s ❡t ❛❧✳❬❛s✐❛❝r②♣t✬✶✵❪✮ ❈♦♥❝r❡t❡ ■♥st❛♥t✐❛t✐♦♥ ❇❍❍❖ ✭❇♦♥❡❤ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✽❪✮ ✭❇♦✉♥❞❡❞✲❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥t✕ ◆❛♦r ✫ ❙❡❣❡✈ ❬❝r②♣t♦✬✵✾❪✮✳
■❞❡❛ ✭s✐♠✐❧❛r t♦ ■❉✮✿ s✐♠✉❧❛t❡ t❛♠♣❡r✐♥❣ q✉❡r✐❡s ❜② ❛❞❞✐t✐♦♥❛❧ ❧❡❛❦❛❣❡✳
▲✐♠✐t❛t✐♦♥✦
❈❛♥ ♥♦t t❛♠♣❡r✭♦r ❧❡❛❦✮ ❛❢t❡r ❝❤❛❧❧❡♥❣❡ ♣❤❛s❡✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✺ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ❚✇♦ ❙t❡♣s✳
✶ ❆ ✇❡❛❦❡r ♠♦❞❡❧✿ ■◆❉✲❈P❆✲❇▲❚✳ ✷ ■◆❉✲❈P❆✲❇▲❚ =
⇒ ■◆❉✲❈❈❆✲❇▲❚ ✕ ✉s✐♥❣ ❛ t❙❊✲◆■❩❑ ✭s✐♠✐❧❛r t♦ ❉♦❞✐s ❡t ❛❧✳❬❛s✐❛❝r②♣t✬✶✵❪✮ ❈♦♥❝r❡t❡ ■♥st❛♥t✐❛t✐♦♥ ❇❍❍❖ ✭❇♦♥❡❤ ❡t ❛❧✳ ❬❝r②♣t♦✬✵✽❪✮ ✭❇♦✉♥❞❡❞✲❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥t✕ ◆❛♦r ✫ ❙❡❣❡✈ ❬❝r②♣t♦✬✵✾❪✮✳
■❞❡❛ ✭s✐♠✐❧❛r t♦ ■❉✮✿ s✐♠✉❧❛t❡ t❛♠♣❡r✐♥❣ q✉❡r✐❡s ❜② ❛❞❞✐t✐♦♥❛❧ ❧❡❛❦❛❣❡✳
▲✐♠✐t❛t✐♦♥✦
❈❛♥ ♥♦t t❛♠♣❡r✭♦r ❧❡❛❦✮ ❛❢t❡r ❝❤❛❧❧❡♥❣❡ ♣❤❛s❡✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✺ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
❇▲❚✲s❡❝✉r❡
❊❛❝❤ ✉s❡r ❤❛s ❛
❇▲❚✲s❡❝✉r❡
❍♦❧❞s s❡❝r❡t ✉♣❞❛t❡ ❦❡② ❙❛♠♣❧❡ ❢r❡s❤ r❛♥❞♦♠♥❡ss ❯♣❞❛t❡
✶
■❉✿ ❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✿ ❛♥② ❇▲❚✲■❉ ❝♠❛✲❙✐❣ ✲■❉✳
✷
P❑❊✿ ❆ ❝♦♥❝r❡t❡ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ❇❍❍❖✳ ❘❡s✉❧ts ▼♦r❡ ♦♥ ✿
❜② ❆❣r❛✇❛❧ ❡t ❛❧✳ ❬❛s✐❛❝r②♣t ✷✵✶✸❪ ❚♦♠♦rr♦✇✳✳✳
❍❛♥❣ ❖◆ ✦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✻ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
CS(sk)
❇▲❚✲s❡❝✉r❡
❊❛❝❤ ✉s❡r ❤❛s ❛
❇▲❚✲s❡❝✉r❡
❍♦❧❞s s❡❝r❡t ✉♣❞❛t❡ ❦❡② ❙❛♠♣❧❡ ❢r❡s❤ r❛♥❞♦♠♥❡ss ❯♣❞❛t❡
A(pk)
✶
■❉✿ ❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✿ ❛♥② ❇▲❚✲■❉ ❝♠❛✲❙✐❣ ✲■❉✳
✷
P❑❊✿ ❆ ❝♦♥❝r❡t❡ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ❇❍❍❖✳ ❘❡s✉❧ts ▼♦r❡ ♦♥ ✿
❜② ❆❣r❛✇❛❧ ❡t ❛❧✳ ❬❛s✐❛❝r②♣t ✷✵✶✸❪ ❚♦♠♦rr♦✇✳✳✳
❍❛♥❣ ❖◆ ✦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✻ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
CS(sk)
❇▲❚✲s❡❝✉r❡
❊❛❝❤ ✉s❡r ❤❛s ❛
❇▲❚✲s❡❝✉r❡
❍♦❧❞s s❡❝r❡t ✉♣❞❛t❡ ❦❡② uk ❙❛♠♣❧❡ ❢r❡s❤ r❛♥❞♦♠♥❡ss r ❯♣❞❛t❡ (sk′) ← Refresh(uk, r)
Floppy
A(pk) A(pk)
✶
■❉✿ ❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✿ ❛♥② ❇▲❚✲■❉ ❝♠❛✲❙✐❣ ✲■❉✳
✷
P❑❊✿ ❆ ❝♦♥❝r❡t❡ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ❇❍❍❖✳ ❘❡s✉❧ts ▼♦r❡ ♦♥ ✿
❜② ❆❣r❛✇❛❧ ❡t ❛❧✳ ❬❛s✐❛❝r②♣t ✷✵✶✸❪ ❚♦♠♦rr♦✇✳✳✳
❍❛♥❣ ❖◆ ✦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✻ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
CS(sk)
❇▲❚✲s❡❝✉r❡
❊❛❝❤ ✉s❡r ❤❛s ❛
CS(sk ′)
❇▲❚✲s❡❝✉r❡
❍♦❧❞s s❡❝r❡t ✉♣❞❛t❡ ❦❡② uk ❙❛♠♣❧❡ ❢r❡s❤ r❛♥❞♦♠♥❡ss r ❯♣❞❛t❡ (sk′) ← Refresh(uk, r)
Floppy
A(pk) A(pk) A(pk)
✶
■❉✿ ❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✿ ❛♥② ❇▲❚✲■❉ ❝♠❛✲❙✐❣ ✲■❉✳
✷
P❑❊✿ ❆ ❝♦♥❝r❡t❡ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ❇❍❍❖✳ ❘❡s✉❧ts ▼♦r❡ ♦♥ ✿
❜② ❆❣r❛✇❛❧ ❡t ❛❧✳ ❬❛s✐❛❝r②♣t ✷✵✶✸❪ ❚♦♠♦rr♦✇✳✳✳
❍❛♥❣ ❖◆ ✦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✻ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
CS(sk)
❇▲❚✲s❡❝✉r❡
❊❛❝❤ ✉s❡r ❤❛s ❛
CS(sk ′)
❇▲❚✲s❡❝✉r❡
❍♦❧❞s s❡❝r❡t ✉♣❞❛t❡ ❦❡② uk ❙❛♠♣❧❡ ❢r❡s❤ r❛♥❞♦♠♥❡ss r ❯♣❞❛t❡ (sk′) ← Refresh(uk, r)
Floppy
A(pk) A(pk) A(pk)
✶
■❉✿ ❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✿ ❛♥② ❇▲❚✲■❉ + ❝♠❛✲❙✐❣ = ⇒ Floppy✲■❉✳
✷
P❑❊✿ ❆ ❝♦♥❝r❡t❡ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ❇❍❍❖✳ ❘❡s✉❧ts ▼♦r❡ ♦♥ ✿
❜② ❆❣r❛✇❛❧ ❡t ❛❧✳ ❬❛s✐❛❝r②♣t ✷✵✶✸❪ ❚♦♠♦rr♦✇✳✳✳
❍❛♥❣ ❖◆ ✦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✻ ✴ ✶✽
❖✉r ❈♦♥tr✐❜✉t✐♦♥
CS(sk)
❇▲❚✲s❡❝✉r❡
❊❛❝❤ ✉s❡r ❤❛s ❛
CS(sk ′)
❇▲❚✲s❡❝✉r❡
❍♦❧❞s s❡❝r❡t ✉♣❞❛t❡ ❦❡② uk ❙❛♠♣❧❡ ❢r❡s❤ r❛♥❞♦♠♥❡ss r ❯♣❞❛t❡ (sk′) ← Refresh(uk, r)
Floppy
A(pk) A(pk) A(pk)
✶
■❉✿ ❆ ❣❡♥❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✿ ❛♥② ❇▲❚✲■❉ + ❝♠❛✲❙✐❣ = ⇒ Floppy✲■❉✳
✷
P❑❊✿ ❆ ❝♦♥❝r❡t❡ ❝♦♥str✉❝t✐♦♥ ❢r♦♠ ❇❍❍❖✳ ❘❡s✉❧ts ▼♦r❡ ♦♥ Floppy✿
❜② ❆❣r❛✇❛❧ ❡t ❛❧✳ ❬❛s✐❛❝r②♣t ✷✵✶✸❪ ❚♦♠♦rr♦✇✳✳✳
❍❛♥❣ ❖◆ ✦
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✻ ✴ ✶✽
❈♦♥❝❧✉s✐♦♥
❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r ❘❡s✐❧✐❡♥❝❡✳ P✉❜❧✐❝✲❑❡② ❙❝❤❡♠❡s ❜❛s❡❞ ♦♥ ❙t❛♥❞❛r❞ ❆ss✉♠♣t✐♦♥s✳ ◆❡✇ t❡❝❤♥✐q✉❡s t♦ r❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡✳ ❖♣❡♥ ◗✉❡st✐♦♥s✳
❇♦♦st✐♥❣ t♦ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ♠♦❞❡❧ ✇✐t❤♦✉t ✳ P♦st✲❝❤❛❧❧❡♥❣❡ t❛♠♣❡r✐♥❣ ❢♦r ❇▲❚✲P❑❊✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✼ ✴ ✶✽
❈♦♥❝❧✉s✐♦♥
❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r ❘❡s✐❧✐❡♥❝❡✳ P✉❜❧✐❝✲❑❡② ❙❝❤❡♠❡s ❜❛s❡❞ ♦♥ ❙t❛♥❞❛r❞ ❆ss✉♠♣t✐♦♥s✳ ◆❡✇ t❡❝❤♥✐q✉❡s t♦ r❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡✳ ❖♣❡♥ ◗✉❡st✐♦♥s✳
❇♦♦st✐♥❣ t♦ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ♠♦❞❡❧ ✇✐t❤♦✉t ✳ P♦st✲❝❤❛❧❧❡♥❣❡ t❛♠♣❡r✐♥❣ ❢♦r ❇▲❚✲P❑❊✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✼ ✴ ✶✽
❈♦♥❝❧✉s✐♦♥
❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r ❘❡s✐❧✐❡♥❝❡✳ P✉❜❧✐❝✲❑❡② ❙❝❤❡♠❡s ❜❛s❡❞ ♦♥ ❙t❛♥❞❛r❞ ❆ss✉♠♣t✐♦♥s✳ ◆❡✇ t❡❝❤♥✐q✉❡s t♦ r❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡✳ ❖♣❡♥ ◗✉❡st✐♦♥s✳
❇♦♦st✐♥❣ t♦ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ♠♦❞❡❧ ✇✐t❤♦✉t ✳ P♦st✲❝❤❛❧❧❡♥❣❡ t❛♠♣❡r✐♥❣ ❢♦r ❇▲❚✲P❑❊✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✼ ✴ ✶✽
❈♦♥❝❧✉s✐♦♥
❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r ❘❡s✐❧✐❡♥❝❡✳ P✉❜❧✐❝✲❑❡② ❙❝❤❡♠❡s ❜❛s❡❞ ♦♥ ❙t❛♥❞❛r❞ ❆ss✉♠♣t✐♦♥s✳ ◆❡✇ t❡❝❤♥✐q✉❡s t♦ r❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡✳ ❖♣❡♥ ◗✉❡st✐♦♥s✳
❇♦♦st✐♥❣ t♦ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ♠♦❞❡❧ ✇✐t❤♦✉t ✳ P♦st✲❝❤❛❧❧❡♥❣❡ t❛♠♣❡r✐♥❣ ❢♦r ❇▲❚✲P❑❊✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✼ ✴ ✶✽
❈♦♥❝❧✉s✐♦♥
❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r ❘❡s✐❧✐❡♥❝❡✳ P✉❜❧✐❝✲❑❡② ❙❝❤❡♠❡s ❜❛s❡❞ ♦♥ ❙t❛♥❞❛r❞ ❆ss✉♠♣t✐♦♥s✳ ◆❡✇ t❡❝❤♥✐q✉❡s t♦ r❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡✳ ❖♣❡♥ ◗✉❡st✐♦♥s✳
❇♦♦st✐♥❣ t♦ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ♠♦❞❡❧ ✇✐t❤♦✉t ✳ P♦st✲❝❤❛❧❧❡♥❣❡ t❛♠♣❡r✐♥❣ ❢♦r ❇▲❚✲P❑❊✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✼ ✴ ✶✽
❈♦♥❝❧✉s✐♦♥
❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r ❘❡s✐❧✐❡♥❝❡✳ P✉❜❧✐❝✲❑❡② ❙❝❤❡♠❡s ❜❛s❡❞ ♦♥ ❙t❛♥❞❛r❞ ❆ss✉♠♣t✐♦♥s✳ ◆❡✇ t❡❝❤♥✐q✉❡s t♦ r❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡✳ ❖♣❡♥ ◗✉❡st✐♦♥s✳
❇♦♦st✐♥❣ t♦ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ♠♦❞❡❧ ✇✐t❤♦✉t Floppy✳ P♦st✲❝❤❛❧❧❡♥❣❡ t❛♠♣❡r✐♥❣ ❢♦r ❇▲❚✲P❑❊✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✼ ✴ ✶✽
❈♦♥❝❧✉s✐♦♥
❋✐rst ♠♦❞❡❧ ♦❢ ❇♦✉♥❞❡❞ ❚❛♠♣❡r ❘❡s✐❧✐❡♥❝❡✳ P✉❜❧✐❝✲❑❡② ❙❝❤❡♠❡s ❜❛s❡❞ ♦♥ ❙t❛♥❞❛r❞ ❆ss✉♠♣t✐♦♥s✳ ◆❡✇ t❡❝❤♥✐q✉❡s t♦ r❡❞✉❝❡ t❛♠♣❡r✲r❡s✐❧✐❡♥❝❡ ❢r♦♠ ❧❡❛❦❛❣❡✲r❡s✐❧✐❡♥❝❡✳ ❖♣❡♥ ◗✉❡st✐♦♥s✳
❇♦♦st✐♥❣ t♦ ❝♦♥t✐♥✉♦✉s t❛♠♣❡r✐♥❣ ♠♦❞❡❧ ✇✐t❤♦✉t Floppy✳ P♦st✲❝❤❛❧❧❡♥❣❡ t❛♠♣❡r✐♥❣ ❢♦r ❇▲❚✲P❑❊✳
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✼ ✴ ✶✽
❈♦♥❝❧✉s✐♦♥
◗✉❡st✐♦♥✭s✮❄
P✳ ▼✉❦❤❡r❥❡❡ ✭❆❛r❤✉s ❈r②♣t♦✮ ❇▲❚ ❆s✐❛❝r②♣t ✷✵✶✸ ✶✽ ✴ ✶✽