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A Framework for Achieving KDM-CCA Secure Public-Key Encryption Fuyuki Kitagawa (Tokyo Institute of Technology) Keisuke Tanaka (Tokyo Institute of Technology)

A Framework for Achieving KDM-CCA Secure Public-Key … · Previous works on KDM-CCA F-KDM-CCA PKE F-KDM-CPA PKE 1. [CCS09] +NIZK NY paradigm 2. [Hof13] Circ-CCA PKE More efficient

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AFrameworkforAchievingKDM-CCASecurePublic-KeyEncryption

FuyukiKitagawa (TokyoInstituteofTechnology)KeisukeTanaka(TokyoInstituteofTechnology)

SecuritynotionsforPKE

• Ithasbeenconsidered“IND-CCAsecurity =standard”

u takesactiveadversariesintoconsiderationu impliesnon-malleability

SecuritynotionsforPKE

• Ithasbeenconsidered“IND-CCAsecurity =standard”

à Onetypicalexampleisencryptingsecretkeys

u takesactiveadversariesintoconsiderationu impliesnon-malleability

• INDsecurityfallsshortifanadversarycanobtainsideinformationofsecretstates

e.g.)Anonymouscredential,hard-diskencryption,FHE…

SecuritynotionsforPKE

• Ithasbeenconsidered“IND-CCAsecurity =standard”

à Onetypicalexampleisencryptingsecretkeys

Keydependentmessage(KDM)security[BRS02]

u takesactiveadversariesintoconsiderationu impliesnon-malleability

• INDsecurityfallsshortifanadversarycanobtainsideinformationofsecretstates

e.g.)Anonymouscredential,hard-diskencryption,FHE…

KDM securityF-KDMgame(1userver.)

KDMquery:functionfinF

encryptionoff(sk) or0|f(sk)|

accordingtob

F-KDM-CPAsecure⇒

Challenger AdversaryA

b←{0,1}

Guess b

A cannot guess b correctlywith prob. greater than 1/2

KDM security

• Theadversarycanalsomakeadecryptionquery⇒ F-KDM-CCAsecure

F-KDMgame(1userver.)

KDMquery:functionfinF

encryptionoff(sk) or0|f(sk)|

accordingtob

F-KDM-CPAsecure⇒

Challenger AdversaryA

b←{0,1}

Guess b

A cannot guess b correctlywith prob. greater than 1/2

Ourfocus

PreviousworksonKDM-CCA

F-KDM-CCAPKE

F-KDM-CPAPKE

1.[CCS09]+NIZK

NYparadigm

F:anyfunctionclass

Concreteinstantiation:Aff-KDM-CCAPKEfromDDHonpairing

PreviousworksonKDM-CCA

F-KDM-CCAPKE

F-KDM-CPAPKE

1.[CCS09]+NIZK

NYparadigm

2.[Hof13]Circ-CCAPKE

Moreefficientthan[CCS09]DCR

+DDHonpairing

F:anyfunctionclass

Concreteinstantiation:Aff-KDM-CCAPKEfromDDHonpairing

PreviousworksonKDM-CCA

F-KDM-CCAPKE

F-KDM-CPAPKE

1.[CCS09]+NIZK

NYparadigm

2.[Hof13]Circ-CCAPKE

Moreefficientthan[CCS09]DCR

+DDHonpairing

3.[HLL16](basedon[LLJ15])

Aff-KDM-CCAPKEDDH+DCR

onaspecificgroup poly-KDM-CCAPKE

w/opairingandefficient

F:anyfunctionclass

Concreteinstantiation:Aff-KDM-CCAPKEfromDDHonpairing

Openproblem

ALLexistingKDM-CCAsecurePKErelyon

NIZKor

pairingor

multipleassumptions(DDH+DCR)

ConstructionbasedonasingleassumptionusingneitherNIZKnorpairing??

???

Especially,allschemesw/oNIZKareproposedunderDDH+DCRonaspecificgroupà Removingoneofthemseemstobedifficult

KDM-CCAforallfunctions[App11]showed

All-KDM-CCAPKE

P-KDM-CCAPKE

P-KDM:KDMsecurityw.r.t.projectionfunctions

Only[CCS09]’sschemeiscompatiblewiththistransformationà NeedNIZKorpairing

All-KDM-CCAsecurePKEusingneitherNIZKnorpairing??

+Garbledcircuit

???

Thiswork

Aff-KDM-CCAPKE

Projectivehashfunctions

1.AframeworkachievingKDM-CCAsecurityin1usersetting

IND-CCAPKE

Homomorphic

+

Thiswork

Aff-KDM-CCAPKE

ProjectivehashfunctionsDDH

1.AframeworkachievingKDM-CCAsecurityin1usersetting

IND-CCAPKE

Homomorphic

+DCR

QR

Thiswork

Aff-KDM-CCAPKE

ProjectivehashfunctionsDDH

All-KDM-CCAPKE

[App14]

1.AframeworkachievingKDM-CCAsecurityin1usersetting

IND-CCAPKE

Homomorphic

+DCR

QR

Thiswork

Aff-KDM-CCAPKE

ProjectivehashfunctionsDDH

All-KDM-CCAPKE

[App14]

1.AframeworkachievingKDM-CCAsecurityin1usersetting

2.KDM-CCAsecurityinmultiusersettingofconcreteinstantiations

IND-CCAPKE

Homomorphic

+DCR

QR

Reducedgoal

KDM-CPA/IND-CCAPKE

KDM-CCAPKE

IND-CCAPKE

+

Reducedgoal

Weessentiallyshow

Reducedgoal

KDM-CPA/IND-CCAPKE

KDM-CCAPKE

IND-CCAPKE

+

Reducedgoal

MucheasiertoconstructthanKDM-CCAPKEJ

Weessentiallyshow

Projectivehashfunctions

Homomorphic

Reducedgoal

KDM-CPA/IND-CCAPKE

KDM-CCAPKE

IND-CCAPKE

+

Reducedgoal

MucheasiertoconstructthanKDM-CCAPKEJ

Weessentiallyshow

Triplemodeproof[MTY11]

E(pk, f(sk))<latexit 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Standardmode

Hidemode

Itisdifficulttodirectlyproveitbasedonthesecrecyofsecret-key…L

E(pk, 0)<latexit 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sha1_base64="tpc8j8DK99Yu74CQLOsr8QeyiMs=">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</latexit>

• FrameworkforprovingKDM-CPA

Triplemodeproof[MTY11]

E(pk, f(sk))<latexit 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Sim(pk, f)<latexit 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Standardmode

Fakemode

Hidemode

Itisdifficulttodirectlyproveitbasedonthesecrecyofsecret-key…L

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• FrameworkforprovingKDM-CPA

Triplemodeproof[MTY11]

E(pk, f(sk))<latexit 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Usingsecrecyofencryptionrandomness(1)

Standardmode

Fakemode

Hidemode

Itisdifficulttodirectlyproveitbasedonthesecrecyofsecret-key…L

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• FrameworkforprovingKDM-CPA

Triplemodeproof[MTY11]

E(pk, f(sk))<latexit 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Usingsecrecyofencryptionrandomness

Usingsecrecyofsecret-key

(1)

(2)

Standardmode

Fakemode

Hidemode

Itisdifficulttodirectlyproveitbasedonthesecrecyofsecret-key…L

E(pk, 0)<latexit 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CompleteentireproofJ

• FrameworkforprovingKDM-CPA

Triplemodeproof[MTY11]

E(pk, f(sk))<latexit 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Sim(pk, f)<latexit 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Usingsecrecyofencryptionrandomness

Usingsecrecyofsecret-key

(1)

(2)

Standardmode

Fakemode

Hidemode

Itisdifficulttodirectlyproveitbasedonthesecrecyofsecret-key…L

ReductiondoesnotneeditJE(pk, 0)

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CompleteentireproofJ

• FrameworkforprovingKDM-CPA

ExtensiontoCCAsetting

Usingsecrecyofencryptionrandomness

Usingsecrecyofsecret-key

à Reductionknowssecret-keyà ThisstepgoesthroughwhenprovingKDM-CCAJ

àReductiondoesnotknowsecret-keyà ThisstepfailswhenprovingKDM-CCAL

E(pk, f(sk))<latexit 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E(pk, 0)<latexit 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(1)

(2)

Standardmode

Hidemode

Fakemode

ExtensiontoCCAsetting

Usingsecrecyofencryptionrandomness

Usingsecrecyofsecret-key

à Reductionknowssecret-keyà ThisstepgoesthroughwhenprovingKDM-CCAJ

àReductiondoesnotknowsecret-keyà ThisstepfailswhenprovingKDM-CCAL

E(pk, f(sk))<latexit 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E(pk, 0)<latexit 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(1)

(2)

Standardmode

Hidemode

Fakemode

Weneednewtechnique

Firsttry

Innerscheme:KDM-CPA

Outerscheme:IND-CCA

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(1)

(2)

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Ecca(pkcca,E(pk, 0))<latexit 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sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit>

Ecca(pkcca, Sim(pk, f))<latexit 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sha1_base64="MfIHB0Xg9VAs+SW1GWDxtKCxBIE=">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</latexit><latexit 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Standardmode

Hidemode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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UsingIND-CCAofouterscheme

Shownusingtriplemodeproof

Firsttry

Innerscheme:KDM-CPA

Outerscheme:IND-CCA

⇡ <latexit 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(1)

(2)

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Ecca(pkcca,E(pk, 0))<latexit 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sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit>

Ecca(pkcca, Sim(pk, f))<latexit 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sha1_base64="MfIHB0Xg9VAs+SW1GWDxtKCxBIE=">AAAPLniclZdPb9s2FMDVdn86b/XS7bgLscBAMjiG5DoNeigQrBhQYBjWYU3boQoCSqZszZQoiHQCQdDX2XkfZNcF2GHYdR9jJEXKpEwbi4siAt9P7z0+vj9UVOCUMt//6979Bx98+NHHDz8ZfPrZo+HnB4+/eEPJuozRRUwwKd9FkCKc5uiCpQyjd0WJYBZh9DZavRDyt9eopCnJX7OqQJcZXORpksaQ8aWrg/Mwg2xJk/q7qzqOYdMc6YVipVbGeuXnNDPFzRgkx8dXB4f+xJc/sP0QqIdDT/1eXT1+dBvOSbzOUM5iDCl9H0wLdln/CkuWxhg1g3BNUQHjFVygWu6wASO+NAcJKfn/nAG5anEwo7TKIk5K//oyseiUscUSlYRyoyXK0U1Msgzm82/CBGYpruYogWvM6pAm6pErGYEfc1yBNAFsiYAIfusUaRf03kBKASMgQoDCnAKKyjQxTb9/HVzW4kWUx81gNBi1WridNE/F4XBHxUrIFoQtwXMg/pwBGkOM5vwUFpShwlcMNZjA3wFF+6BAQXgfNNWa9qp60lF7sZnCvt0HnYrIDEZ3+PFAYsJ1gAzG/GTv+C5/+yUqEc+wDFEA86pVAyqyBjhdocmdveEaZbpaJzsCL3745XtwFFcwP+Fb5fkCTyqEMbk5iTBPkONBKHkk360F19RxVq2aOhj74l9jEzhdLJmJ+ZNTJ1hAHldsk1M3SknSUxnsINUWOnAcOKB2fxtGUq5tbINyM86NuNjpNiy24kIDBysPwMLGQQ+Zw3K1KGFlU/7k7LQHuqA+I/fsAqd98maZMtSjekiJ5uYpbEeYlDBfmEp4aHecwzYqouU4Cl6zJUbMOn4HFpXkxs5PHjCH8RhmCNvZOXly5jYdi2lnuSjgLWxRImRV0I4tW5wqoZ2514Pb2tiVfT1YVZIzATNCaQ9/OuPszOF1m9TA4fc2S1dVhNebA50IpcFTRxbljNnkM3Gis5kr4UgFu8Pi836sotbXCK97xp+1x+9MFAMMnL1EHpat73SHZQyvUT5HZS+hpi72OiVmHqv0dB1psS4LbFl353yR5iu72Ha15D7pdFBkUh8MNKlHpRo4vBV0I0fenWKS81Ck/MKBgBxOQnUo210dmr2vNRrKdqMlbe9REq5ar4uG02hNa7RRtO5omZxa0GaqkojpogVy0qh1NU+0SI8XJW1zXgtVW9c+iB7TOSEbjpK03UyLVG9TMpGdWtJlauc9rDbO8y6t1vUM0LJuJih519Y1sOnzepe6zrt9doWvCFWxWq4L2LRgyjdVYRJmkDfXBNtLwwmjD5qMHT5zPihqc7HQkHHVsBjLnNlKLUqkuQ3JxNc+qaruHNJVriPXjqQucmpCKWlb51qoql5bN+0aFv+PV/rG1JnVNyhDbmevcTExGCtCm+lhEKYfXVNo7npT5iHT3y2OS+rd7rn8SzDof/dtP7yZTgLesn7yD89n6pvwofeV97V35AXemXfuvfReeRde7P3m/eH96d0Ofx/eDv8e/tOi9++pd770rN/w3/8ASQpJnQ==</latexit><latexit 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Standardmode

Hidemode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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UsingIND-CCAofouterschemeReductioncanuse

skofinnerschemedecryptionoracleforouterschemeà SimulatedecryptionoracleJ

Shownusingtriplemodeproof

Firsttry

Innerscheme:KDM-CPA

Outerscheme:IND-CCA

⇡ <latexit 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(1)

(2)

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Ecca(pkcca,E(pk, 0))<latexit 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sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit>

Ecca(pkcca, Sim(pk, f))<latexit 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sha1_base64="MfIHB0Xg9VAs+SW1GWDxtKCxBIE=">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</latexit><latexit 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Standardmode

Hidemode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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UsingIND-CCAofouterschemeReductioncanuse

skofinnerschemedecryptionoracleforouterschemeà SimulatedecryptionoracleJ

But,thisideahasaproblem…

Shownusingtriplemodeproof

Howtomaintainskcca??

Innerscheme:KDM-CPA

Outerscheme:IND-CCA

⇡ <latexit 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(1)

(2)

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Ecca(pkcca,E(pk, 0))<latexit 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sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">AAAPLHiclZfditw2FICd9C+dNpNNe9kb0WVgt8wO9mQ2Sy5aQkMhUEpTyCYp8bLIHo3HHdkylmYXY/w2ve6D9LahUEpv+xyVZMkjeTRDZ0JYo/P5nKOj8yNHBU4p8/2/7tx97/0PPvzo3seDTz69P3xw9PCzV5SsyxhdxgST8k0EKcJpji5ZyjB6U5QIZhFGr6PVMyF/fYNKmpL8JasKdJXBJE8XaQwZX7o++ibMIFvSRf3ddR3HsGlO9EKxUivjDjGFzRj4p6fXR8f+xJc/sP0QqIdjT/1eXD+8/2c4J/E6QzmLMaT0bTAt2FX9CyxZGmPUDMI1RQWMVzBBtdxfA0Z8aQ4WpOT/cwbkqsXBjNIqizgp/evLxKJTxpIlKgnlRkuUo9uYZBnM51+FC5iluJqjBVxjVod0oR65khH4MccVSBeALREQoW+dIu2C3htIKWAERAhQmFNAUZkuTNNvXwZXtXgR5XEzGA1GrRZuJ81TcTTcUbESsoSwJfgaiD8XgMYQozk/hYQyVPiKoQYT+DugaB8UKAjvg6Za015VjzpqLzZT2Lf7oHMRmcHogB8PJCZcB8hgzE/2wHf5289RiXiGZYgCmFetGlCRNcDpCk0O9oZrlOlqnewIPPvh5+/BSVzB/IxvlecLPKsQxuT2LMI8QU4HoeSRfLcWXFPHWbVq6mDsi3+NTeA0WTIT8yfnTrCAPK7YJqdulJJFT2Wwg1Rb6MBx4IDa/W0YSbm2sQ3KzTg34mKn27DYigsNHKw8AAsbBz1kDstVUsLKpvzJxXkPdEF9Ru7ZBU775O0yZahH9ZASzc1T2I4wKWGemEp4aHecwzYqouU4Cl6zJUbMOn4HFpXk1s5PHjCH8RhmCNvZOXl04TYdi1lnuSjgLSwpEbIqaMeWLU6V0M7c68FtbezKvh6sKsmZgBmhtIc/nnF25vC6TWrg8HubpasqwuvNgU6E0uCxI4tyxmzyiTjR2cyVcKSC3WHxeT9WUetrhDc940/a43cmigEGzl4iD8vWd77DMoY3KJ+jspdQUxd7kxIzj1V6uo60WJcFtqy7c75I85VdbLtacp90OigyqQ8GmtSjUg0c3gq6kSPvTjHJeShSfuFAQA4noTqU7a4Ozd7XGg1lu9GStvcoCVet10XDabSmNdooWne0TE4taDNVScR00QI5adS6midapMeLkrY5r4WqrWsfRI/pnJANR0nabqZFqrcpmchOLekytfMeVhvneZdW63oGaFk3E5S8a+sa2PR5vUtd590+u8JXhKpYLdcFbFow5ZuqMAkzyJtrgu2l4YTRB03GDp85HxS1uVhoyLhqWIxlzmylFiXS3IZk4mufVFV3Dukq15FrR1IXOTWhlLStcy1UVa+tm3YNi//HK31j6szqG5Qht7PXuJgYjBWhzfQwCNOPrik0h96Uecj0d4vjknrYPZd/CQb9777th1fTScBb1k/+8dOZ+ia8533hfemdeIF34T31nnsvvEsv9n71fvf+8N4Nfxu+G/49/KdF795R73zuWb/hv/8Bq6hIbw==</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">AAAPLHiclZfditw2FICd9C+dNpNNe9kb0WVgt8wO9mQ2Sy5aQkMhUEpTyCYp8bLIHo3HHdkylmYXY/w2ve6D9LahUEpv+xyVZMkjeTRDZ0JYo/P5nKOj8yNHBU4p8/2/7tx97/0PPvzo3seDTz69P3xw9PCzV5SsyxhdxgST8k0EKcJpji5ZyjB6U5QIZhFGr6PVMyF/fYNKmpL8JasKdJXBJE8XaQwZX7o++ibMIFvSRf3ddR3HsGlO9EKxUivjDjGFzRj4p6fXR8f+xJc/sP0QqIdjT/1eXD+8/2c4J/E6QzmLMaT0bTAt2FX9CyxZGmPUDMI1RQWMVzBBtdxfA0Z8aQ4WpOT/cwbkqsXBjNIqizgp/evLxKJTxpIlKgnlRkuUo9uYZBnM51+FC5iluJqjBVxjVod0oR65khH4MccVSBeALREQoW+dIu2C3htIKWAERAhQmFNAUZkuTNNvXwZXtXgR5XEzGA1GrRZuJ81TcTTcUbESsoSwJfgaiD8XgMYQozk/hYQyVPiKoQYT+DugaB8UKAjvg6Za015VjzpqLzZT2Lf7oHMRmcHogB8PJCZcB8hgzE/2wHf5289RiXiGZYgCmFetGlCRNcDpCk0O9oZrlOlqnewIPPvh5+/BSVzB/IxvlecLPKsQxuT2LMI8QU4HoeSRfLcWXFPHWbVq6mDsi3+NTeA0WTIT8yfnTrCAPK7YJqdulJJFT2Wwg1Rb6MBx4IDa/W0YSbm2sQ3KzTg34mKn27DYigsNHKw8AAsbBz1kDstVUsLKpvzJxXkPdEF9Ru7ZBU775O0yZahH9ZASzc1T2I4wKWGemEp4aHecwzYqouU4Cl6zJUbMOn4HFpXk1s5PHjCH8RhmCNvZOXl04TYdi1lnuSjgLSwpEbIqaMeWLU6V0M7c68FtbezKvh6sKsmZgBmhtIc/nnF25vC6TWrg8HubpasqwuvNgU6E0uCxI4tyxmzyiTjR2cyVcKSC3WHxeT9WUetrhDc940/a43cmigEGzl4iD8vWd77DMoY3KJ+jspdQUxd7kxIzj1V6uo60WJcFtqy7c75I85VdbLtacp90OigyqQ8GmtSjUg0c3gq6kSPvTjHJeShSfuFAQA4noTqU7a4Ozd7XGg1lu9GStvcoCVet10XDabSmNdooWne0TE4taDNVScR00QI5adS6midapMeLkrY5r4WqrWsfRI/pnJANR0nabqZFqrcpmchOLekytfMeVhvneZdW63oGaFk3E5S8a+sa2PR5vUtd590+u8JXhKpYLdcFbFow5ZuqMAkzyJtrgu2l4YTRB03GDp85HxS1uVhoyLhqWIxlzmylFiXS3IZk4mufVFV3Dukq15FrR1IXOTWhlLStcy1UVa+tm3YNi//HK31j6szqG5Qht7PXuJgYjBWhzfQwCNOPrik0h96Uecj0d4vjknrYPZd/CQb9777th1fTScBb1k/+8dOZ+ia8533hfemdeIF34T31nnsvvEsv9n71fvf+8N4Nfxu+G/49/KdF795R73zuWb/hv/8Bq6hIbw==</latexit>

Ecca(pkcca, Sim(pk, f))<latexit 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sha1_base64="MfIHB0Xg9VAs+SW1GWDxtKCxBIE=">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</latexit><latexit 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Standardmode

Hidemode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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Shownusingtriplemodeproof

Howtomaintainskcca??

Innerscheme:KDM-CPA

Outerscheme:IND-CCA

⇡ <latexit 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(1)

(2)

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Ecca(pkcca,E(pk, 0))<latexit 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sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit>

Ecca(pkcca, Sim(pk, f))<latexit 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sha1_base64="MfIHB0Xg9VAs+SW1GWDxtKCxBIE=">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</latexit><latexit 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Standardmode

Hidemode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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Ifwemaintainskcca asapartofsecret-key

f(skkskcca)<latexit 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isencrypted

Shownusingtriplemodeproof

Howtomaintainskcca??

Innerscheme:KDM-CPA

Outerscheme:IND-CCA

⇡ <latexit 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(1)

(2)

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Ecca(pkcca,E(pk, 0))<latexit 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sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">AAAPLHiclZfditw2FICd9C+dNpNNe9kb0WVgt8wO9mQ2Sy5aQkMhUEpTyCYp8bLIHo3HHdkylmYXY/w2ve6D9LahUEpv+xyVZMkjeTRDZ0JYo/P5nKOj8yNHBU4p8/2/7tx97/0PPvzo3seDTz69P3xw9PCzV5SsyxhdxgST8k0EKcJpji5ZyjB6U5QIZhFGr6PVMyF/fYNKmpL8JasKdJXBJE8XaQwZX7o++ibMIFvSRf3ddR3HsGlO9EKxUivjDjGFzRj4p6fXR8f+xJc/sP0QqIdjT/1eXD+8/2c4J/E6QzmLMaT0bTAt2FX9CyxZGmPUDMI1RQWMVzBBtdxfA0Z8aQ4WpOT/cwbkqsXBjNIqizgp/evLxKJTxpIlKgnlRkuUo9uYZBnM51+FC5iluJqjBVxjVod0oR65khH4MccVSBeALREQoW+dIu2C3htIKWAERAhQmFNAUZkuTNNvXwZXtXgR5XEzGA1GrRZuJ81TcTTcUbESsoSwJfgaiD8XgMYQozk/hYQyVPiKoQYT+DugaB8UKAjvg6Za015VjzpqLzZT2Lf7oHMRmcHogB8PJCZcB8hgzE/2wHf5289RiXiGZYgCmFetGlCRNcDpCk0O9oZrlOlqnewIPPvh5+/BSVzB/IxvlecLPKsQxuT2LMI8QU4HoeSRfLcWXFPHWbVq6mDsi3+NTeA0WTIT8yfnTrCAPK7YJqdulJJFT2Wwg1Rb6MBx4IDa/W0YSbm2sQ3KzTg34mKn27DYigsNHKw8AAsbBz1kDstVUsLKpvzJxXkPdEF9Ru7ZBU775O0yZahH9ZASzc1T2I4wKWGemEp4aHecwzYqouU4Cl6zJUbMOn4HFpXk1s5PHjCH8RhmCNvZOXl04TYdi1lnuSjgLSwpEbIqaMeWLU6V0M7c68FtbezKvh6sKsmZgBmhtIc/nnF25vC6TWrg8HubpasqwuvNgU6E0uCxI4tyxmzyiTjR2cyVcKSC3WHxeT9WUetrhDc940/a43cmigEGzl4iD8vWd77DMoY3KJ+jspdQUxd7kxIzj1V6uo60WJcFtqy7c75I85VdbLtacp90OigyqQ8GmtSjUg0c3gq6kSPvTjHJeShSfuFAQA4noTqU7a4Ozd7XGg1lu9GStvcoCVet10XDabSmNdooWne0TE4taDNVScR00QI5adS6midapMeLkrY5r4WqrWsfRI/pnJANR0nabqZFqrcpmchOLekytfMeVhvneZdW63oGaFk3E5S8a+sa2PR5vUtd590+u8JXhKpYLdcFbFow5ZuqMAkzyJtrgu2l4YTRB03GDp85HxS1uVhoyLhqWIxlzmylFiXS3IZk4mufVFV3Dukq15FrR1IXOTWhlLStcy1UVa+tm3YNi//HK31j6szqG5Qht7PXuJgYjBWhzfQwCNOPrik0h96Uecj0d4vjknrYPZd/CQb9777th1fTScBb1k/+8dOZ+ia8533hfemdeIF34T31nnsvvEsv9n71fvf+8N4Nfxu+G/49/KdF795R73zuWb/hv/8Bq6hIbw==</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit>

Ecca(pkcca, Sim(pk, f))<latexit 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sha1_base64="MfIHB0Xg9VAs+SW1GWDxtKCxBIE=">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</latexit><latexit 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Standardmode

Hidemode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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Ifwemaintainskcca asapartofsecret-key

f(skkskcca)<latexit 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isencrypted

à Thereisacircularityinvolvingouterscheme

Shownusingtriplemodeproof

NotKDMsecure

Howtomaintainskcca??

Innerscheme:KDM-CPA

Outerscheme:IND-CCA

⇡ <latexit 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(1)

(2)

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Ecca(pkcca,E(pk, 0))<latexit 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sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit>

Ecca(pkcca, Sim(pk, f))<latexit 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sha1_base64="MfIHB0Xg9VAs+SW1GWDxtKCxBIE=">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</latexit><latexit 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Standardmode

Hidemode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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Ifwemaintainskcca asapartofsecret-key

f(skkskcca)<latexit 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isencrypted

à Thereisacircularityinvolvingouterscheme

Shownusingtriplemodeproof

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Thecircularityremainsaftercompletingstep(1)à Weneedtoremoveit

touse IND-CCAofouterscheme

NotKDMsecure

Howtomaintainskcca??

Innerscheme:KDM-CPA

Outerscheme:IND-CCA

⇡ <latexit 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(1)

(2)

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Ecca(pkcca,E(pk, 0))<latexit 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sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">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</latexit><latexit sha1_base64="mEvhQwCNx4bBlmms7GFOYYac6Xk=">AAAPLHiclZfditw2FICd9C+dNpNNe9kb0WVgt8wO9mQ2Sy5aQkMhUEpTyCYp8bLIHo3HHdkylmYXY/w2ve6D9LahUEpv+xyVZMkjeTRDZ0JYo/P5nKOj8yNHBU4p8/2/7tx97/0PPvzo3seDTz69P3xw9PCzV5SsyxhdxgST8k0EKcJpji5ZyjB6U5QIZhFGr6PVMyF/fYNKmpL8JasKdJXBJE8XaQwZX7o++ibMIFvSRf3ddR3HsGlO9EKxUivjDjGFzRj4p6fXR8f+xJc/sP0QqIdjT/1eXD+8/2c4J/E6QzmLMaT0bTAt2FX9CyxZGmPUDMI1RQWMVzBBtdxfA0Z8aQ4WpOT/cwbkqsXBjNIqizgp/evLxKJTxpIlKgnlRkuUo9uYZBnM51+FC5iluJqjBVxjVod0oR65khH4MccVSBeALREQoW+dIu2C3htIKWAERAhQmFNAUZkuTNNvXwZXtXgR5XEzGA1GrRZuJ81TcTTcUbESsoSwJfgaiD8XgMYQozk/hYQyVPiKoQYT+DugaB8UKAjvg6Za015VjzpqLzZT2Lf7oHMRmcHogB8PJCZcB8hgzE/2wHf5289RiXiGZYgCmFetGlCRNcDpCk0O9oZrlOlqnewIPPvh5+/BSVzB/IxvlecLPKsQxuT2LMI8QU4HoeSRfLcWXFPHWbVq6mDsi3+NTeA0WTIT8yfnTrCAPK7YJqdulJJFT2Wwg1Rb6MBx4IDa/W0YSbm2sQ3KzTg34mKn27DYigsNHKw8AAsbBz1kDstVUsLKpvzJxXkPdEF9Ru7ZBU775O0yZahH9ZASzc1T2I4wKWGemEp4aHecwzYqouU4Cl6zJUbMOn4HFpXk1s5PHjCH8RhmCNvZOXl04TYdi1lnuSjgLSwpEbIqaMeWLU6V0M7c68FtbezKvh6sKsmZgBmhtIc/nnF25vC6TWrg8HubpasqwuvNgU6E0uCxI4tyxmzyiTjR2cyVcKSC3WHxeT9WUetrhDc940/a43cmigEGzl4iD8vWd77DMoY3KJ+jspdQUxd7kxIzj1V6uo60WJcFtqy7c75I85VdbLtacp90OigyqQ8GmtSjUg0c3gq6kSPvTjHJeShSfuFAQA4noTqU7a4Ozd7XGg1lu9GStvcoCVet10XDabSmNdooWne0TE4taDNVScR00QI5adS6midapMeLkrY5r4WqrWsfRI/pnJANR0nabqZFqrcpmchOLekytfMeVhvneZdW63oGaFk3E5S8a+sa2PR5vUtd590+u8JXhKpYLdcFbFow5ZuqMAkzyJtrgu2l4YTRB03GDp85HxS1uVhoyLhqWIxlzmylFiXS3IZk4mufVFV3Dukq15FrR1IXOTWhlLStcy1UVa+tm3YNi//HK31j6szqG5Qht7PXuJgYjBWhzfQwCNOPrik0h96Uecj0d4vjknrYPZd/CQb9777th1fTScBb1k/+8dOZ+ia8533hfemdeIF34T31nnsvvEsv9n71fvf+8N4Nfxu+G/49/KdF795R73zuWb/hv/8Bq6hIbw==</latexit>

Ecca(pkcca, Sim(pk, f))<latexit 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sha1_base64="MfIHB0Xg9VAs+SW1GWDxtKCxBIE=">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</latexit><latexit 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Standardmode

Hidemode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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Ifwemaintainskcca asapartofsecret-key

f(skkskcca)<latexit 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isencrypted

à Thereisacircularityinvolvingouterscheme

WecandoitifinnerschemeisalsoIND-CCA

Shownusingtriplemodeproof

f(·kskcca)<latexit 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Thecircularityremainsaftercompletingstep(1)à Weneedtoremoveit

touse IND-CCAofouterscheme

NotKDMsecure

Removecircularity

Ecca(pkcca,E(pk,m))<latexit 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sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">AAAPLHiclZfditw2FICd9C+dNpNNe9kb0WVgt8wO9mQ2Sy5aQkMhUEpTyCYp8bLIHo3HHdkylmYXY/w2ve6D9LahUEpv+xyVZMkjeTRDZ0JYo/P5nKOj8yNHBU4p8/2/7tx97/0PPvzo3seDTz69P3xw9PCzV5SsyxhdxgST8k0EKcJpji5ZyjB6U5QIZhFGr6PVMyF/fYNKmpL8JasKdJXBJE8XaQwZX7o++ibMIFvSRf3ddR3HsGlO9EKxUivjDjGFzRhkp6fXR8f+xJc/sP0QqIdjT/1eXD+8/2c4J/E6QzmLMaT0bTAt2FX9CyxZGmPUDMI1RQWMVzBBtdxfA0Z8aQ4WpOT/cwbkqsXBjNIqizgp/evLxKJTxpIlKgnlRkuUo9uYZBnM51+FC5iluJqjBVxjVod0oR65khH4MccVSBeALREQoW+dIu2C3htIKWAERAhQmFNAUZkuTNNvXwZXtXgR5XEzGA1GrRZuJ81TcTTcUbESsoSwJfgaiD8XgMYQozk/hYQyVPiKoQYT+DugaB8UKAjvg6Za015VjzpqLzZT2Lf7oHMRmcHogB8PJCZcB8hgzE/2wHf5289RiXiGZYgCmFetGlCRNcDpCk0O9oZrlOlqnewIPPvh5+/BSVzB/IxvlecLPKsQxuT2LMI8QU4HoeSRfLcWXFPHWbVq6mDsi3+NTeA0WTIT8yfnTrCAPK7YJqdulJJFT2Wwg1Rb6MBx4IDa/W0YSbm2sQ3KzTg34mKn27DYigsNHKw8AAsbBz1kDstVUsLKpvzJxXkPdEF9Ru7ZBU775O0yZahH9ZASzc1T2I4wKWGemEp4aHecwzYqouU4Cl6zJUbMOn4HFpXk1s5PHjCH8RhmCNvZOXl04TYdi1lnuSjgLSwpEbIqaMeWLU6V0M7c68FtbezKvh6sKsmZgBmhtIc/nnF25vC6TWrg8HubpasqwuvNgU6E0uCxI4tyxmzyiTjR2cyVcKSC3WHxeT9WUetrhDc940/a43cmigEGzl4iD8vWd77DMoY3KJ+jspdQUxd7kxIzj1V6uo60WJcFtqy7c75I85VdbLtacp90OigyqQ8GmtSjUg0c3gq6kSPvTjHJeShSfuFAQA4noTqU7a4Ozd7XGg1lu9GStvcoCVet10XDabSmNdooWne0TE4taDNVScR00QI5adS6midapMeLkrY5r4WqrWsfRI/pnJANR0nabqZFqrcpmchOLekytfMeVhvneZdW63oGaFk3E5S8a+sa2PR5vUtd590+u8JXhKpYLdcFbFow5ZuqMAkzyJtrgu2l4YTRB03GDp85HxS1uVhoyLhqWIxlzmylFiXS3IZk4mufVFV3Dukq15FrR1IXOTWhlLStcy1UVa+tm3YNi//HK31j6szqG5Qht7PXuJgYjBWhzfQwCNOPrik0h96Uecj0d4vjknrYPZd/CQb9777th1fTScBb1k/+8dOZ+ia8533hfemdeIF34T31nnsvvEsv9n71fvf+8N4Nfxu+G/49/KdF795R73zuWb/hv/8BIwNIrA==</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">AAAPLHiclZfditw2FICd9C+dNpNNe9kb0WVgt8wO9mQ2Sy5aQkMhUEpTyCYp8bLIHo3HHdkylmYXY/w2ve6D9LahUEpv+xyVZMkjeTRDZ0JYo/P5nKOj8yNHBU4p8/2/7tx97/0PPvzo3seDTz69P3xw9PCzV5SsyxhdxgST8k0EKcJpji5ZyjB6U5QIZhFGr6PVMyF/fYNKmpL8JasKdJXBJE8XaQwZX7o++ibMIFvSRf3ddR3HsGlO9EKxUivjDjGFzRhkp6fXR8f+xJc/sP0QqIdjT/1eXD+8/2c4J/E6QzmLMaT0bTAt2FX9CyxZGmPUDMI1RQWMVzBBtdxfA0Z8aQ4WpOT/cwbkqsXBjNIqizgp/evLxKJTxpIlKgnlRkuUo9uYZBnM51+FC5iluJqjBVxjVod0oR65khH4MccVSBeALREQoW+dIu2C3htIKWAERAhQmFNAUZkuTNNvXwZXtXgR5XEzGA1GrRZuJ81TcTTcUbESsoSwJfgaiD8XgMYQozk/hYQyVPiKoQYT+DugaB8UKAjvg6Za015VjzpqLzZT2Lf7oHMRmcHogB8PJCZcB8hgzE/2wHf5289RiXiGZYgCmFetGlCRNcDpCk0O9oZrlOlqnewIPPvh5+/BSVzB/IxvlecLPKsQxuT2LMI8QU4HoeSRfLcWXFPHWbVq6mDsi3+NTeA0WTIT8yfnTrCAPK7YJqdulJJFT2Wwg1Rb6MBx4IDa/W0YSbm2sQ3KzTg34mKn27DYigsNHKw8AAsbBz1kDstVUsLKpvzJxXkPdEF9Ru7ZBU775O0yZahH9ZASzc1T2I4wKWGemEp4aHecwzYqouU4Cl6zJUbMOn4HFpXk1s5PHjCH8RhmCNvZOXl04TYdi1lnuSjgLSwpEbIqaMeWLU6V0M7c68FtbezKvh6sKsmZgBmhtIc/nnF25vC6TWrg8HubpasqwuvNgU6E0uCxI4tyxmzyiTjR2cyVcKSC3WHxeT9WUetrhDc940/a43cmigEGzl4iD8vWd77DMoY3KJ+jspdQUxd7kxIzj1V6uo60WJcFtqy7c75I85VdbLtacp90OigyqQ8GmtSjUg0c3gq6kSPvTjHJeShSfuFAQA4noTqU7a4Ozd7XGg1lu9GStvcoCVet10XDabSmNdooWne0TE4taDNVScR00QI5adS6midapMeLkrY5r4WqrWsfRI/pnJANR0nabqZFqrcpmchOLekytfMeVhvneZdW63oGaFk3E5S8a+sa2PR5vUtd590+u8JXhKpYLdcFbFow5ZuqMAkzyJtrgu2l4YTRB03GDp85HxS1uVhoyLhqWIxlzmylFiXS3IZk4mufVFV3Dukq15FrR1IXOTWhlLStcy1UVa+tm3YNi//HK31j6szqG5Qht7PXuJgYjBWhzfQwCNOPrik0h96Uecj0d4vjknrYPZd/CQb9777th1fTScBb1k/+8dOZ+ia8533hfemdeIF34T31nnsvvEsv9n71fvf+8N4Nfxu+G/49/KdF795R73zuWb/hv/8BIwNIrA==</latexit>

(pk, pkcca,E(pk, skcca))<latexit 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sk<latexit 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Public-key:

Secret-key:

Encryption:

• Wemaintainskcca asapartofpublic-keyafterencryptedbyinnerscheme

Decryption:Reject E(pk, skcca)<latexit 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sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">AAAPBXiclZdPi9w2FMCd9F86baab9tBDL6LLwKbsDvZkNksOhdBQCJTSFLJJSrwsskYz445sGUuzizE+94P02txKr/0WhX6bSrLkkTyaoTMhrNH7+b2np/dHTgqSMh6G/965+977H3z40b2PB598en/42dGDz18xui4RvkSU0PJNAhkmaY4vecoJflOUGGYJwa+T1TMpf32DS5bS/CWvCnyVwUWezlMEuVi6PvoyziBfsnn9/UmxOmWr6xoh2Dxsro+Ow3GofmD7IdIPx4H+vbh+cP+feEbROsM5RwQy9jaaFPyq/hWWPEUEN4N4zXAB0QoucK0cb8BILM3AnJbif86BWnU4mDFWZYkglZt9mVz0yvhiiUvKhNES5/gW0SyD+eybeA6zlFQzPIdrwuuYzfWjUDICP+WkAukc8CUGMqatU7RdMHsDKQOcggQDBnMGGC7TuW367cvoqpYv4hw1g9Fg1GoRdtI8lTEXjsqVmC8oX4JvgfxzARiCBM/EYSwYx0WoGWYxUbgDSvZBkYbIPmhiNO1V9aij9mJTjX23DzqXkRmMDviJQBIqdIAMInGyB74r3n6OSywyLMMMwLxq1YCKrgFJV3h8sDdCo0pX52RH4NmPv/wATlAF8zOxVZEv8KzChNDbs4SIBHk4iBWP1bu15JoaZdWqqaPTUP5rXIKkiyW3sXB87gULKOJKXHLiRxmd91RGO0i9hQ48jTxQu78NoyjfNrZBtRnvRnzsZBuWW/GhkYdVB+Bgp1EPmcFytShh5VLh+OK8B/qgPqP27AMnffJ2mXLco3pIiWf2KWxHmJYwX9hKRGh3nMM2KqPlOQpRsyXB3Dl+D5aU9NbNTxEwj3EEM0zc7Bw/uvCbRnKIOS5KeAtblBg7FbRjyw6nS2hn7vXgtjZ2ZV8P1pXkTcCMMtbDH08FO/V43SY18Pi9zbJVlZD15kDHUmn02JNFOecu+USe6HTqSzhawe6wxLw/1VHra4Q3PeNP2uP3JooFRt5eog7L1Xe+wzKBNzif4bKXUBMfe5NSO491evqOtFiXBXGs+3O+SPOVW2y7WnKf9DooM6kPRoY0o1IPHNEKupGj7k6I5iIUqbhwYKCGk1Qdq3ZXx3bva43Gqt0YSdt7tESoNuuy4TRG0xpvFK07WiWnEbSZqiVyuhiBmjR6Xc8TIzLjRUvbnDdC3daND7LHdE6ohqMlbTczIt3btExmp5F0mdp5D6uN86JL63UzA4ysmwla3rV1A2z6vNmlqfNun13ha0JXrJGbArYt2PJNVdiEHeTNNcH10nLC6oM244bPng+a2lwsDGRdNRzGMWe3UoeSae5CKvGNT7qqO4dMlZvItSOpi5yeUFra1rkR6qo31m27lsX/45W5MXVmzQ3KkrvZa11MLMaJ0GZ6WITtR9cUmkNvyiJk5rvFc0k97J4rvgSj/nff9sOryTgSzz+Hx0+n+pvwXvBV8HVwEkTBRfA0eB68CC4DFDTB78Efwbvhb8N3wz+Hf7Xo3Tv6nS8C5zf8+z9PfTfX</latexit>

Removecircularity

Standardmode

Ecca(pkcca,E(pk, f(sk)))<latexit 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sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit>

Ecca(pkcca,E(pk,m))<latexit 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sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit>

(pk, pkcca,E(pk, skcca))<latexit 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sk<latexit 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sha1_base64="FSdmGBQzq2NBh47JzuOu4aDN3Vk=">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</latexit><latexit 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Public-key:

Secret-key:

Encryption:

• Wemaintainskcca asapartofpublic-keyafterencryptedbyinnerscheme

Ecca(pkcca,E(pk, f(sk)))<latexit 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sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit><latexit sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit><latexit sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit><latexit sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit>

Decryption:Reject E(pk, skcca)<latexit 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sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit>

Removecircularity

Standardmode

Ecca(pkcca,E(pk, f(sk)))<latexit 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sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit>

Ecca(pkcca,E(pk,m))<latexit 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sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit>

(pk, pkcca,E(pk, skcca))<latexit 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Public-key:

Secret-key:

Encryption:

• Wemaintainskcca asapartofpublic-keyafterencryptedbyinnerscheme

⇡ <latexit 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(1)

Ecca(pkcca, Sim(pk, f))<latexit 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Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit><latexit sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit><latexit sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit><latexit sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit>

Usingthepropertyofinnerscheme

Decryption:Reject E(pk, skcca)<latexit 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sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit>

Removecircularity

Standardmode

Ecca(pkcca,E(pk, f(sk)))<latexit 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sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit>

Ecca(pkcca,E(pk,m))<latexit 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sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit>

(pk, pkcca,E(pk, skcca))<latexit 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Public-key:

Secret-key:

Encryption:

• Wemaintainskcca asapartofpublic-keyafterencryptedbyinnerscheme

ThereisnocircularityJ

⇡ <latexit 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(1)

Ecca(pkcca, Sim(pk, f))<latexit 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Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit><latexit sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit><latexit sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit><latexit sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit>

Usingthepropertyofinnerscheme

Decryption:Reject E(pk, skcca)<latexit 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sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit><latexit sha1_base64="0YnKQ21Ek3tLSkIwhOLJ8f/Cr/E=">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</latexit>

Remainingproofstrategy

⇡ <latexit 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(1)

Ecca(pkcca, Sim(pk, f))<latexit 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Standardmode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit>

Ecca(pkcca,E(pk,m))<latexit 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sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit>

(pk, pkcca,E(pk, skcca))<latexit 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Public-key:

Secret-key:

Encryption:

Usingthepropertyofinnerscheme

Eliminateitbeforestep(2)usingIND-CCA ofinnerscheme

ThereisnocircularityJ

• Wemaintainskcca asapartofpublic-keyafterencryptedbyinnerscheme

Reductionneedstosimulatedecryptionoracle

Remainingproofstrategy

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(1)

(2)

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Ecca(pkcca, Sim(pk, f))<latexit 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sha1_base64="MfIHB0Xg9VAs+SW1GWDxtKCxBIE=">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</latexit><latexit 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Standardmode

Hidemode

Fakemode

Ecca(pkcca,E(pk, f(sk)))<latexit 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sha1_base64="aE0133W3sNYhVKxYnlkJ9rxdsw8=">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</latexit>

Ecca(pkcca,E(pk,m))<latexit 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sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit><latexit sha1_base64="e9BxvZ9LIHo6n+3tfqMZ1LmCzls=">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</latexit>

(pk, pkcca,E(pk, skcca))<latexit 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Public-key:

Secret-key:

Encryption:

Usingthepropertyofinnerscheme

UsingIND-CCAofouterscheme

CompleteentireproofJ

Eliminateitbeforestep(2)usingIND-CCA ofinnerscheme

ThereisnocircularityJ

• Wemaintainskcca asapartofpublic-keyafterencryptedbyinnerscheme

Reductionneedstosimulatedecryptionoracle

Reducedgoal

KDM-CPA/IND-CCAPKE

KDM-CCAPKE

IND-CCAPKE

+

Outerscheme

Innerscheme

KDM-CPAisprovedviatriplemodeproof

Doublelayeredconstruction

Reducedgoal

Projectivehashfunctions

Homomorphic

KDM-CPA/IND-CCAPKE

KDM-CCAPKE

IND-CCAPKE

+

Outerscheme

Innerscheme

KDM-CPAisprovedviatriplemodeproof

Doublelayeredconstruction

InnerschemefromPHF• Extend[Wee16]thatisageneralizationof[BHHO08,BG10]

HomomorphicandSmoothPHF

KDM-CPAPKE

Usingthesecrecyofencryptionrandomness

InnerschemefromPHF

HomomorphismprojectivepropertySubsetmembershipproblem

E(pk, f(sk))<latexit 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Sim(pk, f)<latexit 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• Extend[Wee16]thatisageneralizationof[BHHO08,BG10]

• KDM-CPAofWee’s schemecanbeshownusingtriplemodeproof

HomomorphicandSmoothPHF

KDM-CPAPKE

Usingthesecrecyofencryptionrandomness

InnerschemefromPHF

HomomorphismprojectivepropertySubsetmembershipproblem

SmoothnessSubsetmembershipproblem

E(pk, f(sk))<latexit 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E(pk, 0)<latexit 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• Extend[Wee16]thatisageneralizationof[BHHO08,BG10]

• KDM-CPAofWee’s schemecanbeshownusingtriplemodeproof

HomomorphicandSmoothPHF

KDM-CPAPKE

• ItisalsoIND-CPA

Usingthesecrecyofencryptionrandomness

InnerschemefromPHF

HomomorphismprojectivepropertySubsetmembershipproblem

ByusinguniversalPHF,wecanupgradeitintoIND-CCAJSmoothness

Subsetmembershipproblem

E(pk, f(sk))<latexit 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E(pk, 0)<latexit 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• Extend[Wee16]thatisageneralizationof[BHHO08,BG10]

• KDM-CPAofWee’s schemecanbeshownusingtriplemodeproof

HomomorphicandSmoothPHF

UniversalPHF+

[CS02]KDM-CPA

PKE

AlsoIND-CCA

• ItisalsoIND-CPA

Ourgenericconstruction

Projectivehashfunctions

Homomorphic

KDM-CPA/IND-CCAPKE

KDM-CCAPKE

IND-CCAPKE

+

Outerscheme

Innerscheme

KDM-CPAisprovedviatriplemodeproof

Doublelayeredconstruction

Instantiationsandmulti-usersecurity

Wecaninstantiateinnerschemebasedoninstantiationsof[Wee16]DDH[BHHO08]DCR/QR[BG10]

Instantiations

Instantiationsandmulti-usersecurity

Wecaninstantiateinnerschemebasedoninstantiationsof[Wee16]DDH[BHHO08]DCR/QR[BG10]

WeobtainKDM-CCAPKEbasedonDDHDCRQR

Instantiations

Instantiationsandmulti-usersecurity

Wecaninstantiateinnerschemebasedoninstantiationsof[Wee16]DDH[BHHO08]DCR/QR[BG10]

WeobtainKDM-CCAPKEbasedonDDHDCRQR

OurgenericconstructionachievesonlyKDM-CCAin1-usersettingHowever,wecanproveKDM-CCAinmulti-usersettingofinstantiations

1-user/multi-user

Instantiations

Instantiationsandmulti-usersecurity

Wecaninstantiateinnerschemebasedoninstantiationsof[Wee16]DDH[BHHO08]DCR/QR[BG10]

WeobtainKDM-CCAPKEbasedonDDHDCRQR

OurgenericconstructionachievesonlyKDM-CCAin1-usersettingHowever,wecanproveKDM-CCAinmulti-usersettingofinstantiations

1-user/multi-user

Instantiations

Usingsimilartechniqueasthem,wecanprovemulti-usersecurityJ

Ourinstantiationsareextensionsof[BHHO08,BG10]

Summary

Aff-KDM-CCAPKE

ProjectivehashfunctionsDDH

All-KDM-CCAPKE

[App14]

1.AframeworkachievingKDM-CCAsecurityin1usersetting

2.KDM-CCAsecurityinmultiusersettingofconcreteinstantiations

IND-CCAPKE

Homomorphic

+DCR

QR