UC/Garbled Searchable Symmetric Encryption Kaoru Kurosawa Ibaraki University, Japan

UC/Garbled Searchable Symmetric Encryption

Kaoru Kurosawa Ibaraki University, Japan

I will talk about

(1) UC-Secure Searchable Symmetric Encryption　　　 A preliminary version = FC 2012　　　 Final version = ePrint 2015/251

(2) Garbled Searchable Symmetric Encryption　　　 FC 2014

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Curtmola, Garay, Kamara and Ostrovsky (2006)

• defined privacy of SSE schemes • as follows.

3

In the store phase,

E(D1), , E(D⋯ N), E(Index)

the server learns |D1|, …, |DN| and |{keywords}|

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In the search phase,

This means that the server knows the corresponding indexes {3, 6, 10}

E(keyword)

C(keyword)=( E(D3), E(D6), E(D10) )

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We call

these information• |D1|, …, |DN| and |{keywords}|• corresponding indexes {3, 6, 10}

The minimum leakage

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The Privacy definition

• requires that the server should not be able to learn any more information

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In the Real Game

D = {D1, …, DN}W={set of keywords}Index

Distinguisher

E(D1), , E(D⋯ N) E{ Index }

Challenger

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In the Simulation Game

D = {D1, …, DN}W={set of keywords}Index

Distinguisher

Somehow returns E(D1), , E(D⋯ N) E{ Index }

ChallengerSimulator

the minimum leakage|D1|, …, |DN| and |{keywords}|

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In the search phase of the real game

keyword

Distinguisher

E(keyword)

Challenger

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In the simulation game,

keyword

Distinguisher

Somehow returns E(keyword)

ChallengerSimulator

the minimum leakage {3, 6, 10}

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Def. of Curtmola et al.

• Privacy is satisfied if• there exists a simulator such that

the real game ≈ the simulation game

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We now define

• reliability and strong reliability • UC security • Prove a weak equivalence　 (1) UC-secure → 　 privacy + reliability　 (2) privacy + strong reliability → UC-secure• Show an efficient UC-secure SSE scheme

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We now define

• reliability and strong reliability • UC security • Prove a weak equivalence　 (1) UC-secure → 　 privacy + reliability　 (2) privacy + strong reliability → UC-secure• Finally an efficient UC-secure SSE scheme

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A malicious server• tries to forge some files, delete some files,• or replace E(D3) with E(D100).

Client Server

E(keyword)

E(D3), E(D6), E(D10)E(D100)

Malicious

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Consider an adversary (A1,A2) s.t.

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A1 A2Client

A1 gives the inputs to the client

A2 runs the protocolwith the client

Adversary

server

If A2 is honest,

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A1 A2Client

keyword w E(w)

D(w) = {files which contain w}[C(w), Tag]

Reliability is satisfied if

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A1 A2Client

keyword w E(w)

D(w)’≠ D(w)with negligible probabilityfor any (A1,A2)

Strong reliability is satisfied if

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A1 A2Client

keyword w E(w)

[C(w)’, Tag’]≠ [C(w), Tag] acceptswith negligible probabilityfor any (A1,A2)

We then define

• Reliability, strong reliability• UC security • Prove a weak equivalence　 (1) UC-secure → 　 privacy + reliability　 (2) privacy + strong reliability → UC-secure• Finally an efficient UC-secure SSE scheme

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In the ideal world,

dummyClient

Ideal Functionality

FSSE

Environment

Z

D={D1, …, DN} W={set of keywords}Index

D={D1, …, DN} W={set of keywords}Index 21

FSSE sends the minimum leakage

dummyClient

Ideal Functionality

FSSE

Environment

Z


UC adversary

S

|D1|, …, |DN||{keywords}|

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In the search phase

dummyClient

Ideal Functionality

FSSE

Environment

Z

keyword

keyword

UC adversary

S

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FSSE sends the minimum leakage

dummyClient

Ideal Functionality

FSSE

Environment

Z

keyword

keyword

UC adversary

S

{3,6,10}

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S returns

dummyClient

Ideal Functionality

FSSE

Environment

Z

keyword

keyword

UC adversary

S

{3,6,10} Accept or Reject

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If S returns Reject,then FSSE sends Reject

dummyClient

Ideal Functionality

FSSE

Environment

Z

keyword

keyword

UC adversary

S

{3,6,10} Reject

Reject

Reject

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If S returns Accept,FSSE sends D(w)={D3,D6,D10}

dummyClient

Ideal Functionality

FSSE

Environment

Z

keyword

keyword

UC adversary

S

{3,6,10} Accept

D(w)={D3,D6,D10}

D(w)={D3,D6,D10}

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Also S and Z can interact freely

dummyClient

Ideal Functionality

FSSE

Environment

ZUC adversary

S

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This is an ideal world

Because(Correctness.) The dummy client outputs reject or D(w) correctly (Security.) The UC adversary S learns only the minimum leakage.

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Client Server

Environment

Z

Z gives the inputs to the client

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In the real world

the client and the server runthe real protocol

A can corrupt the server andcommunicate with Z freely

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Client Server

Environment

ZAdversary

A

corrupt

We say that

• An SSE scheme is UC-secure if for any adversary A, there exists a UC-adversary S such that 　 Pr[Z 1 in the real]⇒ ≈ Pr[Z 1 in the ideal]⇒

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We define

• reliability (unforgeability) strong reliability (strong unforgeability) UC security • Prove a weak equivalence　 (1) UC-secure → 　 privacy + reliability　 (2) privacy + strong reliability → UC-secure• Finally an efficient UC-secure SSE scheme

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Suppose that

• There exists an SSE scheme• which is UC-secure

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In the real world,

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Client Server

Environment

ZAdversary

A

Consider A who relays everything to Z

E(keyword)

E(keyword)

E(keyword)

keyword

The real world = the real game of privacy

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Client Server

distinguisher

ZAdversary

A

challenger

E(keyword)

E(keyword)

E(keyword)

keyword

In the ideal world,

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dummyclient

FSSE

Environment

ZUC adversary

S

There exists S which simulates Afrom the minimum leakage

Minimum leakage

keyword

E(keyword)

The ideal world = the ideal game of privacy

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dummyclient

FSSE

distinguisher

ZUC adversary

S

Minimum leakage

challenger

simulatorE(keyword)

keyword

keyword

Therefore

• if the SSE scheme is UC secure,• then privacy is satisfied.

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Nextfor a reliability adversary (A1,A2),

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A1 A2Client

Adversary

Consider (Z,A) s.t.

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Client Server

Z=A1

Adversary

A=A2

In the corresponding ideal world,

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dummyClient

FSSE

ZUC Adversary

S

The dummy client never outputs D(w)’ ≠ D(w)from the definition of FSSE

wD(w) or reject

D(w) or reject

Hence

• In the real world,• the client outputs D(w)’ ≠ D(w)• with negligible probability.• Therefore• Reliability is satisfied

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We define

• reliability (unforgeability) strong reliability (strong unforgeability) UC security • Prove a weak equivalence　 (1) UC-secure → 　 privacy + reliability　 (2) privacy + strong reliability → UC-secure• Finally an efficient UC-secure SSE scheme

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Suppose that

• There exists an SSE scheme• Which satisfies privacy and strong reliability

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Game 0 = Real world

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Client Server

ZAdversary

A

keyword wD(w) orreject

E(w)

C(w), Tag

In Game 1,

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Client Server

ZAdversary

A

w

E(w)

[C(w)’, Tag’]≠[C(w), Tag]

If A instructs the server to return an invalid message

E(w)

Game 1

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Client Server

ZAdversary

A

wreject

E(w)

reject

Then the server returns reject to the client,And the client sends reject to Z

[C(w)’, Tag’]≠[C(w), Tag]

E(w)

Game 1

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Client Server

ZAdversary

A

wD(w)

E(w)

accept

[C(w), Tag]

Otherwise the server returns accept to the clientand the client outputs D(w) = {files which contain the keyword w}

• Game 1 and Game 0 are indistinguishable• Because • the SSE scheme satisfies strong reliability.

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Client ２

Z A

server

Client １acceptor reject

D(w) or reject

E(w)

In Game 2,

w51

• From a view point of Z,• Game 2 and Game 1 are the same

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Client ２

serverZ A

Simulatorof privacy

Client １

Minimum leakage

acceptreject

In Game 3,

E(w)

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Client ２

serverZ A

Simulatorof privacy

Client １

Minimum leakage

acceptreject

distinguisher

challenger

Game 3 = simulation game of privacy

E(w)keyword

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Client ２

serverZ A

Client １

acceptreject

distinguisher

challenger

Game 2 = real game of privacy

E(w)

keyword

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Therefore

• Game 3 and Game 2 are indistinguishable• Because • the SSE scheme satisfies privacy

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Client ２

serverZ A

simulatorS0

Client １

Minimum leakage

acceptreject

UC adversary S

FSSE

Finally Game 3 = the ideal world

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Namely

• Game 0 = the real world• Game 3 = the ideal world• and Z cannot distinguish them• Therefore the SSE scheme is UC-secure.

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We define

• reliability (unforgeability) strong reliability (strong unforgeability) UC security • Prove a weak equivalence　 (1) UC-secure → 　 privacy + reliability　 (2) privacy + strong reliability → UC-secure• show an efficient UC-secure SSE scheme

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Consider this example

D1 D2 D3 D4 D5Austin 1 0 1 0 1Boston 0 1 0 1 0

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The client computes

E(D1) E(D2) E(D3) E(D4) E(D5)PRP(Austin) ( 1 0 1 0 1)PRP(Boston) ( 0 1 0 1 0)

where PRP means pseudorandom permutation

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and adds


+PRF(Austin)+PRF(Boston)

where PRF means pseudorandom function.

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The client stores this table



　　　　　　　　＋

TagA=MAC( PRP(Austin), E(D1), E(D3), E(D5) ) TagB=MAC(PRP(Boston), E(D2), E(D4))

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For a keyword Austin, the client sends

E(Austin)

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The server decrypts (10101)



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And returns

E(D1), E(D3), E(D5 ), TagA

E(Austin)={PRP(Austin), PRF(Austin)}

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The client accepts if

E(D1), E(D3), E(D5 ),

TagA=MAC(PRP(Austin), E(D1), E(D3), E(D5 ))

PRP(Austin) and PRF(Austin)

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Theorem

• The above SSE scheme satisfies privacy and strong reliability if E is CPA-secure

Corollary• The above SSE scheme is UC-secure

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So far,

• single keyword search SSE schemes.

Next• multiple keyword search SSE schemes.

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Wang et al. (2008)

• Showed a multiple keyword SSE scheme• for AND search.

At CRYPTO 2013,

• Cash, Jarecki, Jutla, Krawczyk, Rosu, and Steiner showed an SSE scheme

• which can support any search formula f (in the random oracle model).

• The comm. overhead is sublinear in N,• where N=the number of files.

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However,

• the search formula f is revealed to the server and• the search phase requires 2 rounds.

Search phase

Search formula

Cash et al. 2 rounds revealed

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In their scheme,

If 「 Japan AND Crypto 」 is searched,the following information is leaked to the server

the search formula ＝ AND the search result of Japan or that of Crypto and some more information （ see Sec.5.3 of their paper ）

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Kurosawa (FC 2014)

• even the search formula f is kept secret.• the search phase requires only 1 round.

Search phase

Search formula

Cash et al.

2 rounds revealed

Proposed 1 round secret74

In my scheme

only the following information is leaked (other than the minimum leakage)• The topological circuit f- • (π(j1), …, π(jc)),

　 where π is a random permutation and {wj1, …, wjc} are the queried keywords

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XOR

AND

1

OR

4

2

3

If this the search formula f,

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1

4

2

3

This is the topological circuit f-

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On the other hand,

• The communication overhead is O(N)• While it is sublinear in N in Cash et al’s scheme• where N=the number of files.

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The proposed SSE scheme

• is based on Yao’s garbled circuit.

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A garbled circuit of f

• is an encoding garble(f) such that• one can compute f(X) • from garble(f) and label(X) without learning anything on f and X.

garble(f)label(X) f(X)

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x1= 0

x2= 1

Consider f(x1,x2)= (x1 and x2)

x1 x2 x30 0 00 1 01 0 11 1 1

x3= 0

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garble(f) is an encoded truth tableby random strings

x1 x2 x3

A0 B0 H(A0,B0)+ 0

A0 B1 H(A0,B1)+ 0

A1 B0 H(A1,B0)+ 0

A1 B1 H(A1,B1)+ 1

A0

B1

x3= 0

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label(X) is these random strings

x1 x2 x3

A0 B0 H(A0,B0)+ 0

A0 B1 H(A0,B1)+ 0

A1 B0 H(A1,B0)+ 0

A1 B1 H(A1,B1)+ 1

A0

B1

x3= 0

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In this example,x3=0 is obtained by computing H(A0,B1)

x1 x2 x3

A0 B0 H(A0,B0)+ 0

A0 B1 H(A0,B1)+ 0

A1 B0 H(A1,B0)+ 0

A1 B1 H(A1,B1)+ 1

A0

B1

x3= 0

label(X)garble(f)

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High level overview of the proposed scheme

w1 w2 w3

D1 1 1 1D2 1 0 0

keywords

files

Consider this example.

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Let

w1 w2 w3

D1 (1 1 1)=X1

D2 (1 0 0)=X2

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The client computes

w1 w2 w3

D1 label(X1)D2 label(X2)

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The client also computes

PRP(w1) PRP(w2) PRP(w3)E(D1) label(X1)E(D2) label(X2)

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and sends

PRP(w1) PRP(w2) PRP(w3)E(D1) label(X1)E(D2) label(X2)

Server89


• Suppose that the client wants to search on f(w1,w2,w3)=w1 w⋀ 2 w⋀ 3

• He computes the garbled circuits of f: Γ1 for D1 and

Γ2 for D2.

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PRP(w1), …, PRP(w3) Γ1

Γ2

The client sends

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Γ2

The server has this tablePRP(w1) PRP(w2) PRP(w3)

E(D1) label(X1)E(D2) label(X2)

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Γ2

The server computes f(X1) fromPRP(w1) PRP(w2) PRP(w3)


label(X1) Γ1 f(X1)=1

garbled circuit93


Γ2

Similarly she computes f(X2)PRP(w1) PRP(w2) PRP(w3)


Γ2 label(X2) f(X2)=0

garbled circuit94

The server returns E(D1)

If f(X1)=1 and f(X2)=0,

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However, if

• label(X) is reused, then some information on (f, X) is leaked.

garble(f)label(X) f(X)

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We use counter as an additional input to H

x1 x2 x3

A0 B0 H(counter, A0,B0)+ 0




A0

B1

x3= 0

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Formally

Bellare et al. (2012)defined Kurosawa（ 2014）

extended them togarbling schemes extended garbling

schemesInput-circuit privacy label reusable privacy

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Label reusable privacy

• Even if label(X) is reused for multiple garbled circuits Γ1, Γ2, …. ,

• no information on X and (f1,f2, … )

are leaked, where Γi is a garbled circuit of fi

Theorem 1

• Our construction satisfies label reusable privacy in the random oracle model

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Theorem 2

If the underlying extended garbling scheme satisfies label reusable privacy

only the following information is leaked (other than the minimum leakage)

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• The topological circuit f- • (π(j1), …, π(jc)),

　 where π is a random permutation and {wj1, …, wjc} are the queried keywords

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Communication overheadof the proposed scheme

• Let m = # of files c = # of search keywords s = # of gates of f• In the search phase, the com. overhead is |counter|+(c+4m(s-1))×128+4m bits

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If # of search keywords is 2

• The communication overhead is |counter|+256+ 4× （ # of files ） bits

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Computer simulation

• We used a computer such as follows. 2.4GHz CPU and 32G byte RAM OS = CentOS 6.5 C++ and NTL library

• The total # of keywords is 20.

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The running time of the clientin the search phase

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The running time of the serverin the search phase

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Summary

(1) UC-Secure Searchable Symmetric Encryption　　　 A preliminary version = FC 2012　　　 Final version = ePrint 2015/251

(2) Garbled Searchable Symmetric Encryption　　　 FC 2014

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Open problem (1)

• Construct a multiple keyword SSE scheme such that

• The communication overhead is sublinear in N• And the leakage is as small as possible• In the standard model

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Open problem (2)

• In all the known single keyword SSE schemes, E(keyword) is deterministic

• Hence if the client sends E(keyword) twice,• This search pattern is leaked.• So • construct a UC-secure scheme such that • Even the search pattern is kept secret

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Open problem (3)

• Prove the tight equivalence between • UC security and some stand alone security

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Thank you !

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Documents

UC/Garbled Searchable Symmetric Encryption Kaoru Kurosawa Ibaraki University, Japan