Key Policy Attribute-based Proxy Re-encryptionwith Matrix Access Structure

Keying Li

Department of MathematicsXidian University

Xi’an 710071, P.R.ChinaEmail: likeying818@163.com

Yinghui Zhang

State Key Laboratory of Integrated Service Networks (ISN)Xidian University

Xi’an 710071, P.R.ChinaEmail: prrd2007@163.com

Hua Ma

Department of MathematicsXidian University

Xi’an 710071, P.R.ChinaEmail: ma hua@126.com

Abstract—Cloud computing has achieved rapid development.The cloud server even provides unlimited storage and powerfulcomputing capability as services. A lot of attribute-basedschemes have been constructed for cloud computing to comeinto practical applications. To our knowledge, there seems noflexible key policy attribute-based proxy re-encryption (KP-AB-PRE) scheme in the literature, which is a promisingcryptographic primitive. In this paper, we propose a KP-AB-PRE scheme, in which the cloud server can function as theproxy. In the proposed scheme, matrix access structure is usedfor the key policy. Furthermore, our construction enjoys thedesirable properties of unidirectionality, non-interactivity, andmulti-Use, and the secret key security is guaranteed.

Keywords-key policy; attribute-based encryption; proxy re-encryption; matrix access structure;

I. INTRODUCTION

Cloud computing is a promising computing paradigm

which recently has drawn extensive attention from both

academia and industry [1], [2]. Especially, cloud computing

is used to provide data storage for internet businesses. It

has become a great solution for providing a flexible, on-

demand, and dynamically scalable computing infrastructure

for many applications. The businesses could utilize these

characteristics to increase revenue [3]. As compared to

building their own infrastructures, users are able to save their

investments significantly by migrating businesses into the

cloud. With the increasing development of cloud computing

technologies, in the near future more and more businesses

will be moved into the cloud [4].

As promising as it is, this paradigm also has many

new challenges for data security and access control when

users outsource sensitive data for sharing on cloud servers.

These challenges come from the fact that cloud servers are

generally operated by commercial providers which are very

likely to be outside of the trusted domain. Data confidential

against cloud servers is hence frequently desired when users

outsource data for storage and computing in the cloud

servers.

Now we describe an application scenario. The data owner

encrypts the message M with the attributes set S1 using

KP-ABE algorithm [11] that the key policy is constructed

by LSSS access structure (M1, ρ1). Then he sends the

ciphertext to the cloud server. U11 can decrypt the ciphertext

if the attributes associated with the ciphertext satisfy his

key’s access structure. But U1 is on a business trip, he has

no time to deal with the ciphertext. Then he empower his

secretary U2 to deal with the ciphertext. In this scenario:

a) U1 don’t want to let anyone know his private key.

b) the cloud proxy updates the ciphertext.

c) U2 can get the message.

Our Contribution. We present a key policy attribute-based

proxy re-encryption (KP-AB-PRE) scheme. The key policy

realized in our scheme is matrix access structure, and the

proxy can convert the KP-ABE ciphertext under pki into

ciphertext under pkj with the help of transform key. Our

scheme inherits the following properties of PRE mentioned

in [5], [6]:

− Unidirectionality. The converting can only from User1

to User2.

− Non-interactivity. The private key generator (PKG) can

compute the transform key without the participation of User1

or User2.

− Multi-Use. The cloud proxy can re-encrypt a ciphertext

multiple times, e.g. re-encrypt from User1 to User2, and then

re-encrypt the result from User2 to User3. In this process, the

computation would increase, but not exponent increasing.

Our scheme has the other properties:

Secret Key Security. The cloud proxy cannot obtain

User1’s secret key even collude with User2.

Re-encryption Control. The encryptor can decide

whether the ciphertext can be re-encrypted [12].

Related Work. Attribute-based encryption was first

proposed by Sahai and Waters [7]. Attribute based

encryption is classified as Ciphertext Policy-Attribute Based

Encryption (CP-ABE) [8] and Key Policy-Attribute Based

Encryption(KP-ABE) [9]. The access structure including

AND and OR gates, tree access structure, and LSSS

1U1 could also be a user group that every user has the common attributes

2013 5th International Conference on Intelligent Networking and Collaborative Systems

978-0-7695-4988-0/13 $26.00 © 2013 IEEE

DOI 10.1109/INCoS.2013.17

46

access structure. The notion of PRE was first introduced

by Mambo and Okamoto [10]. Green, Hohenberger, and

Waters [11] first presented outsourcing the decryption of

abe ciphertexts in the cloud environments. Luo, Hu, and

Chen [12] presented a novel ciphertext policy attribute-

based proxy re-encryption (CP-AB-PRE) scheme. For the

purpose of data confidentiality and fine-grained access

control in cloud computing environments, Yu et al. [4]

put forward a system model using Key Policy-Attribute

Based Encryption (KP-ABE) and Proxy Re-Encryption

(PRE). Do, Song, and Park [13] propose system model that

store and divide data file into header, body. In addition,

their scheme selectively delegate decryption right using

Type-based Proxy re-encryption. Zhao, Feng, et al. [17]

raised Attribute-Based Conditional Proxy Re-Encryption

with Chosen-Ciphertext Security. Mizuno and Doi [18]

come up with Hybrid Proxy Re-encryption Scheme for

Attribute-Based Encryption.

Organization. The paper is organized as follows. We give

necessary background knowledge and assumptions in Sec-

tion 2. We present our scheme and security model, then

construct and give security analysis in Section 3. We discuss

the scheme and the follow-up work in Section 4. In Section

5, we give the conclusions of our work.

II. PRELIMINARIES

A. Bilinear Maps

Let G and GT be two multiplicative cyclic groups of

prime order p. Let g be a generator of G and e : G×G→ GT

be a bilinear map with the properties:

1. Bilinearity: for all u, v ∈ G and a, b ∈ Zp, we have

e(ua, vb) = e(u, v)ab.

2. Non-degeneracy: e(g, g) �= 1. We say that G is a

bilinear group if the group operation in G and the bilinear

map e: G×G→ GT are both efficiently computable.

B. Access Structure

Definition 1 (Access Structure [14]) Let {P1, P2, · · · ,Pn} be a set of parties. A collection A ⊆ 2{P1,P2,··· ,Pn} is

monotone if ∀B,C : if B ∈ A and B ⊆ C then C ∈ A. An

access structure (respectively, monotone access structure) is

a collection (resp., monotone collection) A of non-empty

subsets of {P1, P2, · · · , Pn}, i.e., A ⊆ 2{P1,P2,··· ,Pn}\{∅}The sets in A are called the authorized sets, and the sets not

in A are called the unauthorized sets.

C. LSSS and Monotone Span Programs [9]

LSSS has a close relation with a linear algebraic model

of computation called monotone span programs (MSP) [15].

It has been shown that the existence of an efficient LSSS

for some access structure is equivalent to the existence of a

small monotone span program for the characteristic function

of that access structure [14], [15].

Using Access Trees. Some prior ABE works (e.g. [9])

described access formulas in binary trees. Using standard

techniques [14] one can convert any monotonic boolean

formula into an LSSS representation. An access tree of lnodes can be converted into an LSSS matrix of l rows.

D. The Bilinear Diffie-Hellman (BDH) ProblemDefinition 2 DBDH AssumptionThe decisional BDH assumption [7], [16] is that no

probabilistic polynomial-time algorithm B can distinguish

the tuple (A = ga;B = gb;C = gc; e(g; g)abc) from the

tuple (A = ga;B = gb;C = gc; e(g; g)z) with more than a

negligible advantage.

III. PROXY RE-ENCRYPT KP-ABE CIPHERTEXT

A. Algorithms of KP-AB-PREIn our scheme, from the system level, there are six

algorithms as follows:

Setup(λ,U). This algorithm takes the security parameter

κ and a universe description of attributes as input and then

generates a public key PK, a master secret key MSK.Encrypt(PK;M;S1). This algorithm takes as input a

message M , a set of attributes S1, and PK. It output the

ciphertext CT.KeyGen(MSK;(M1; ρ1). This algorithm takes as input an

access structure A1, the master key MSK and the public

parameters. It outputs a decryption key SK1.TransformKey(MSK,S2). It firstly call the KeyGen al-

gorithm, output the transform key. Then call the Encryptalgorithm to encrypt gαd under the attributes set S2.

ReEnc(TK,CT). This algorithm takes as input the TK,

CT that is associated with S1. At final, it output the updated

ciphertext-CT ′.Decryption(SK1, SK2;CT,CT ′). This algorithm takes

as input secret key and the ciphertext. Output the message

M.

Fig.1 KP-AB-PRE System

B. Security Model for Our SchemeThrough the analysis of the above algorithms, we need to

construct two security models [12]. One is for the system

level and the other is for the private key.

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1) Selective-Policy Model for KP-AB-PRE:

Init: The adversary declares the set of attributes S∗, that

he wishes to be challenged upon. Then he commits to the

challenge key policy A∗1.

Setup: The challenger runs the Setup algorithm and gives

PK to A.

Phase 1: A makes the queries as follows.

− Extract(S∗1 ): A submits an attribute list S∗

1 for a

KeyGen query where S∗1 � A

∗1, the challenger gives the

adversary the secret key SKS∗1

.

− TKExtract(SKS∗1,A∗

1): A submits SKS∗1

and access

structure A∗1 for a TK query, the challenger gives the

adversary the transform key TKS∗1

.

Challenge: A submits two equal-length messages M0,

M1 to the challenger. The challenger flips a random coin

b, then Encrypt(PK,Mb, S∗1 ) and compute ReEnc(TK,CT),

gives the 1st level ciphertext to the adversary.

Phase 2: Phase 1 is repeated.

Guess: A outputs a guess b′ of b.

The advantage of A in this game is defined as

AdvA = |Pr[b′ = b]− 1/2|.

2) Selective Secret Key Security Model:

Init: The adversary declares the set of attributes S∗ that

he wishes to be challenged upon. Then he commits to the

challenge ciphertext policy A∗1.

Setup: The challenger runs the Setup algorithm and gives

PK to A.

Phase 1: A makes the queries as follows.

− Extract(S∗1 ): A submits an attribute list S∗

1 for a

KeyGen query where S∗1 �= S∗, the challenger computes

the secret key SKS∗1

.

− TKExtract(S∗1 ,A

∗1): A submits S∗

1 and access structure

A∗1 for a TK query, the challenger gives the adversary the

transform key TKS∗1

.

Output: A outputs SKS∗ for the attribute list S∗, then

A succeeds.

The advantage of A in this game is defined as AdvA =Pr[Asucceeds].

C. The Proposed Scheme

The first three algorithms is the same as in [11]:

Setup(λ,U). The setup algorithm takes as input a universe

description U and the security parameter. Let U = {0, 1}∗.

It then chooses a group G of prime order p, a generator gand a hash function F that maps {0, 1}∗ to G. Furthermore,

it randomly chooses values α ∈ Zp and g1, h ∈ G. The

authority sets α as the master secret key. MSK=α. The public

key is published as

PK = g; g1; gα;h;F

Encrypt(PK;M;S1). The encryption algorithm takes as

input the public parameters PK, a message M , and a set of

attributes S1. It chooses a random s1 ∈ Zp. The 2nd-level2

ciphertext is published as CT = (S1;C0) where

CT0 = M · e(g, h)αs1 ;C1 = gs1 ;C ′1 = gs11 ; {Cx =

F (x)s1}x∈S1

KeyGen(MSK;(M1;ρ1)). The KeyGen algorithm takes as

input MSK and security parameter. Furthermore, it takes as

input an LSSS access structure (M1; ρ1). The function ρ1associates rows of M1 to attributes. Let M1 be an l1×n1 ma-

trix. It first chooses a random vector −→v1 = (α, y2, · · · , yn) ∈Znp , which are used to share the encryption exponent α. For

i = 1 to l, it calculates λ1,i =−→v1 ·M1,i, where M1,i is the

vector corresponding to the ith row of M . In addition, the

algorithm chooses random r11, · · · , r1l ∈ Zp. The SK1 is

published as :

(D11 = hλ11 · F (ρ1(1))r11 , R11 = gr11), · · · , (D1l =

hλ1l · F (ρ1(l))r1l , R1l = gr1l)

along with a description of (M1; ρ1).

TransformKey(SK1,(M1, ρ1)). The Transform key algo-

rithm calls the KeyGen algorithm, then chooses random d ∈Zp, and compute g

λ1,id1 , gαd. Then encrypt gαd with the pub-

lic key (attributes) of User2 using the Encrypt(PK;gαd;S2)algorithm. It output CT1 = EnS2

(gαd) and the TK as:

(D′11 = hλ11 · F (ρ1(1))

r′11 · gλ11d1 , R11 = gr

′11), · · · ,

(D′1l = hλ1l · F (ρ1(l))

r′1l · gλ1ld1 , R1l = gr

′1l)

ReEnc(TK,CT). The ReEnc algorithm takes as input

the CT, public parameters PK. The Re-encryption algorithm

then takes as input S1. Suppose that S1 satisfies the access

structure (M1; ρ1) and let I1 ⊂ {1, 2, · · · , l1} be defined as

I1 = {i : ρ1(i) ∈ S1}. Then, let {ωi ∈ Zp}i∈I1 be a set of

constants such that if λ1i are valid shares of any secret αaccording to M1, then

∑i∈I1

ωiλ1i = α. It calculate CT2

as follow:

CT2 =e(C1,

∏i∈I1

D′ωi1i )

∏i∈I1

e(R1i, Cωi

ρ1(i))

=e(gs1 ,

∏i∈I1

(hλ1iωi · F (ρ1(i))r′1iωi · gλ1idωi

1 ))

(∏

i∈I1e(gr

′1i , F (ρ1(i))

s1ωi))

= e(g, h)s1αe(g, g1)

s1αd

The 1st-level CT ′:CT0 = M · e(g, h)αs1 ; CT1 = EnS2

(gαd); C1 = gs1 ;

C ′1 = gs11 ;CT2

Decryption(SK1, SK2;CT,CT’). U1 can decrypt the ci-

phertext if the attributes associated with the ciphertext satisfy

2In our proxy re-encrypt KP-ABE CT scheme, a 2nd-level ciphertextis an original ABE ciphertext and a 1st-level ciphertext is a transformedciphertext.

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his key’s access structure, and if the attributes satisfy U2’

access structure he could get the message.

Dec2(SK1, CT ). The decryption algorithm takes as input

a private key SK1 and CT. Suppose that S1 satisfies the ac-

cess structure (M1; ρ1). The decryption algorithm computes

ct2 =e(C1,

∏i∈I1

Dωi1i )∏

i∈I1e(R1i, C

ωi

ρ1(i))= e(g, h)

s1α

then get M = CT0/ct2Dec1(SK2, CT ′). User2 decrypts CT1 using his secret

key skj to get gαd. Next, calculate CT3 = e(gαd, gs11 ) =

e(g, g1)s1αd. Finally, it calculate CT0 · CT3/CT2 = M .

D. Security Analysis

Theory 1. If there is an adversary who breaks our scheme

in selective the transform key security model to get User1’s

SKS1, then he can solve discrete logarithm problem.

Proof. In the security model, the simulator B runs A.

The adversary A commits to a challenge attribute list S′.To provide a public key PK to A, B generate PK =g; gα

′;F1, · · · , FS′ ;h. A makes queries.

− Extract(S∗1 ): A submits an attribute list S∗

1 for a

KeyGen query where S∗1 �= S

′, B randomly choose

λ′11, · · · , λ′

1l,r11, · · · , r1l, computes the secret key SKS∗1

:

(D11 = hλ′11 ·F (ρ1(1))

r11 , R1 = gr11), · · · , (D1l = hλ′1l ·

F (ρ1(l))r1l , R1l = gr1l)

− TKExtract(S∗1 ,A∗

1): A submits an attribute list S∗1 for

a transform key query, B runs the TransformKey algorithm.

TK : (D′11 = hλ′

11 ·F (ρ1(1))r′11 · gλ′

11d∗

1 , R′1 = gr

′11),· · · ,

(D′1l = hλ′

1l · F (ρ1(l))r′1l · gλ′

1ld∗

1 , R′1 = gr

′1l)

If the proxy collude with the User2, he can get gα′d∗

from

User2. If he want to get SKS∗1

, he must firstly compute

gλ′11d

∗

1 , · · · , gλ′1ld

∗

1 , it equal to solve discrete logarithm

problem, and he knows nothing about the random parameter

r11, · · · , r1l.Theory 2. Our KP-ABPRE scheme is a selectively CPA-

secure construction, as the GPSW KP-ABE scheme [7] is

selectively CPA-secure.

Proof.3 If there exists a polynomial-time adversary A,

that can break our scheme in the Selective-Policy model

with advantage ε, it can can play the Decisional BDH game

with advantage ε/2.

Init: Given a DBDH tuple [g, ga, gb, gc, Z]. The simulator

B runs A. A gives the key policy A∗1 to B.

Setup: B randomly choose α′ r←− Zp, g, h ∈ G set A =gα

′. Then it gives the PK to A.

Phase 1: A makes the following queries.

− Extract(S∗1 ): A submits an attribute list S∗

1 for a

KeyGen query where S∗1 � A

∗1. B choose λ′

11, · · · , λ′1l

3The security of the 2nd level ciphertext and CT1 have been provedsecurity [11]. We only need to prove the security of 1st level ciphertext.

satisfy that ∃ ω′i, i ∈ I1,

∑i∈I1

λ′1iω

′i = α′. The secret

key SK∗1 is:

(D11 = hλ′11 ·F (ρ1(1))

r′11 , R11 = gr′11),· · · ,(D1l = hλ′

1l ·F (ρ1(l))

r′1l , R1l = gr′1l)

− TKExtract(SKS∗1,A∗

1): A submits SKS∗1

and access

structure A∗1 for a TK query. It randomly choose d′ ∈ ZP .

Finally B gives the adversary C = gd′

and the transform

key TK’:

(D′11 = hλ′

11 · F (ρ1(1))r′11 · gλ′

11d′

1 , R11 = gr′11), · · · ,

(D′1l = hλ′

1l · F (ρ1(l))r′1l · gλ′

1ld′

1 , R1l = gr′1l)

Challenge: A submits two challenge messages M0 and

M1. Then B flips a random coin b ∈ {0, 1} and returns Athe ciphertext as CT0 = Mb · e(g;h)α

′s1 ; B = C ′1 = gs11 ;

CT2 = Z · e(g;h)α′s1

Phase 2: Phase 1 is repeated.

Guess: A outputs a guess b′ of b. B outputs 1 if and only

if b′ = b. The advantage of breaking DBDH assumption is

AdvA = |Pr[b′ = b]− 1/2| = 12ε

IV. DISCUSSIONS

A. Multi-Use

To realize the Multi-Use property, the form of the CT1 is

CT1 = gαd1 · e(g;h)αs2 ; C ′2 = gs2 ; C ′

x = F (x)s2

x∈S2. The

User2’s secret key is :(D21 = hλ21 · F (ρ2(1))r21 , R21 =

gr21), · · · , (D2l = hλ2l · F (ρ2(l))r2l , R2l = gr2l) along

with a description of (M2; ρ2).

B. Re-encryption Control

Note that if the encryptor does not provide gs11 in cipher-

text, the original decryption is not affected but the decryption

of re-encrypted ciphertext cannot go on. That’s because gs11is only used in decrypting re-encrypted step, so he can

control whether the ciphertext can be re-encrypted.

C. Construction of CCA-Secure KP-AB-PRE

Peikert and Waters [20] first put forward Lossy trapdoor

functions (LTFs), in particular as a means to construct

chosen-ciphertext (CCA) secure public-key encryption (P-

KE) schemes. After that, it drawn extensive attention by a

lot of cryptography scholars. We may construct the CCA-

Secure KP-AB-PRE scheme with the help of it. We can also

reference the work of Zhao, Feng [17]. But it is not an easy

work, we need more time to research.

V. CONCLUSIONS

We present a key policy attribute-based proxy re-

encryption(KP-AB-PRE)scheme, in which the proxy can

be the cloud server. In our scheme, we use matrix access

structure to realize the key policy. The secret key size,

encryption, and decryption time scales linearly with the

complexity of the access formula. Our work result can also

inherit some properties of PRE. In addition, our scheme

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has secret key security property, the cloud proxy cannot

obtain the secret key information even collude with the

User. Cloud computing is a promising computing paradigm

which has drawn extensive attention from both academia

and commerce. A lot of security work need to do.

ACKNOWLEDGEMENTS

We would like to thank Xiaofeng Chen for the sug-

gestions to improve this paper. Also, we are grateful to

the anonymous referees for their invaluable suggestion-

s. This work is supported by the Fundamental Research

Funds for the Central Universities (K50511010001 and

JY10000901034), the National Natural Science Foundation

of China (No.61070249) and the Graduate Student Innova-

tion Fund of Xidian University (No.K50513100015).

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Keying Li, master of Faculty of science, Xidian University,

Xi’an, China. His research interests cover the attributes

based encryption, cloud computing, lossy trapdoor function,

lossy encryption, e-cash payment.

Yinghui Zhang, received his B.S. (2007) and M.S.

(2010) from Nanchang Hangkong University and Xidian

University, both in Mathematics. Currently, He is working

toward the Ph.D. degree in Cryptography, Xidian University.

His research interests are in the areas of cloud computing

security and cryptography.

Hua Ma, professor of Faculty of science, Xidian University,

Xi’an, China. Her research directions including The Theory

and technology in e-commerce security, Design and analysis

of fast public key cryptography, Theory and technology of

the network security.

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