Fully Secure Multi-Authority Ciphertext-Policy Attribute-Based

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Fully Secure Multi-authority
Ciphertext-Policy Attribute-Based Encryption
without Random Oracles
Zhen Liu1,2
1 Shanghai
Jiao Tong University, Shanghai, China
2 City University of Hong Kong, Hong Kong SAR, China
Joint work with Zhenfu Cao, Qiong Huang, Duncan S. Wong,
and Tsz Hon Yuen
16th European Symposium on Research in Computer Security (ESORICS) 2011,
12-14 September 2011, Leuven, Belgium
Outline
Introduction
History
Motivation
Our Results
Background
Our scheme
2
Introduction: What is CP-ABE?
 CP-ABE is a tool for implementing fine-grained access
control over encrypted data, and is conceptually similar to
traditional access control methods such as Role-Based
Access Control.
 A user is described by a set of descriptive attributes, and a
corresponding private key is issued to the user by an
authority.
 During encryption, an encryptor associates an access policy
over attributes with the ciphertext.
 If and only if the attributes of a user satisfy the access policy
of the ciphertext, the user can decrypt the ciphertext .
3
Introduction: What is CP-ABE?
 ,  , …
ℎ ,  , …
 ,  , …
1980, 1981 , …
……
…..



Dept.: CS, EE, …
Type: PhD Stud., Alumni, …
Gender: Male, Female
Birth Year: 1980, 1981, …
……
……

 = (, , )
 satisfies 
 = {, ℎ}
M
Storage Server
(Untrusted)
AND
 = {, ℎ}
OR
CS
PDH
ALUMNI
 =   (ℎ  )
 does not satisfy 

4 
Introduction: What is CP-ABE?
-- Collusion-resistant
If none of the users can decrypt a ciphertext individually,
they still can’t even if they work together.

 = {, ℎ}
AND
OR
CS
PDH
ALUMNI
 =   (ℎ  )

 = {, }
5
Introduction: What is CP-ABE?
-- Definition
•
•
•
•
 ,  ⟶ , .
 , ,  ⟶ .  is implicitly included in .
(, , ) ⟶  .
 ,  ,  ⟶   ⊥. If and only if  satisfies , 
can be recovered.
6
Introduction: Why needs MA-CP-ABE?
 It might not be realistic to have one single authority to
manage all attributes. [SW05]
 E.g., an encryptor may want to share data with users who are
computer science alumni of University X and currently working as an
engineer for Company Y. i.e., the access policy is  =
.   .   . 
 In a desired Multi-Authority CP-ABE (MA-CP-ABE) system,
different domains of attributes are managed by different
authorities. An encryptor can encrypt messages with any
access policy over the entire attribute universe.
7
History: Existing CP-ABE Schemes
 Goyal et al. [GPSW06]: CP-ABE notion.
 Bethencourt, Sahai and Waters [BSW07] : The first CP-ABE
scheme.
 Cheung and Newport [CN07]
are proposed to
 Goyal et al. [GJPS08]
achieve
better
and
 Waters [Waters08/11]
better expressiveness,
 Lewko et al.[LOSTW10]
 Okamoto and Takashima[OT10] efficiency and security.
[Waters08/11] and [LOSTW10]: expressive (any monotone access structure);
efficient; and secure. The two constructions are very similar, and the
difference is that [Waters08/11] is on prime order group while [LOSTW10] is
on composite order group. [Waters08/11] is selectively secure and
[LOSTW10] is adaptively secure.
8
History: Existing MA-CP-ABE Schemes
 Müller et al. [MKE09]: One Central Authority (CA) and
Multiple Attribute Authorities (AAs).
• Selectively secure.
• Key Escrow: The CA can decrypt all ciphertexts.
 Lewko and Waters [LW11] : Multiple AAs
• The AAs operate independently from each other.
• Adaptively secure, in the random oracle model.
• Key Escrow: Each AA can decrypt the ciphertexts whose
policy can be satisfied by the AA’s attribute domain.
9
Motivation
Construct an MA-CP-ABE system
 Different attribute domains are managed by different
authorities.
 Expressiveness, efficiency and security are not weaker
than that of the single-authority CP-ABE in [LOSTW10]:
 Expressiveness: Support any monotone access structure over the
entire attribute universe;
 Efficiency: similar to that of [LOSTW10];
 Security: adaptively secure in the standard model.
 No authority can independently decrypt any ciphertext.
10
Our Results
We constructed a new MA-CP-ABE system.
 Multiple CAs and Multiple AAs.
 The CAs issue identity-related keys to users but do not involve in
any attribute-related operations.
 The AAs issue attribute-related keys to users.
 Each AA manages a different attribute domain, and operates
independently from other AAs.
 A party may easily join the system as an AA by registering itself
to the CAs and publishing its attribute-related parameters.
 The expressiveness, efficiency and security are
comparable to that of the single-authority CP-ABE
scheme in [LOSTW10].
 No authority can independently decrypt any ciphertext.
11
Our Results
LOSTW10
(SA-) CP-ABE
LW11
MA-CP-ABE
Ours
MA-CP-ABE
Standard Model
Multi-Authority
Prevent Decryption by
Individual Authority
Partially
Size of Ciphertext
 + 
 + 
 + 
Size of Secret key
 +
||
 + ( + )
  +
||
  +
 +
||
 ++
Pairing Computation
of Decryption
Size of Public key
: The number of CAs.
: The number of AAs.
12
The rest of this presentation…
1. Bilinear map and access structure
2. Our construction
3. Extensions
13
Background
 Bilinear map:
  = 1 2 3 where 1 , 2 and 3 are three distinct primes;
  and  are cyclic groups of order ;
 :  ×  →  is a map such that
 (1) Bilinear: ∀ , ℎ ∈ , ,  ∈  , e g a , hb = e g, h
ab ;
 (2) Non-Degenerate: ∃ ∈ , such that (, ) has order  in  .
 LSSS: Any monotone access structure can be realized by a Linear SecretShare Scheme (LSSS). An LSSS is a labeled matrix (, ), where  is a  × 
matrix over ∗ and  labels each row with a share holder. E.g.,
(2,2)
1 1 1 
1 1 2
 1 1 3 

(2,3)
D
1 2 0 
A
B
C
14
Our MA-CP-ABE Scheme: Idea
Start from the single authority CP-ABE of [LOSTW10]:
  ,  → , .
: , , ℎ,  ,   ,
 =   ∀ ∈  ; : , 3
 , ℎ ∈ 1  ,  ∈  are chosen randomly, 3 is a generator of 3 .
  , ,  →  = (, ,  ∀ ∈  ).
=
 ℎ 0 ,
=
 0′

,
 =   ∀  ∈ .
  ∈  ,  0 , 0′ ,  ∈ 3 are chosen randomly.
  , ,  ,  → .
 =  ⋅  , 
 , ′
= ℎ  ,  = ℎ  ⋅ 

−
, ′ =  ∀ ∈ {1,2 … , }.
 ,  ∈  are chosen randomly.
  , ,  → .
 ⋅
  ∈
 Constants { } satisfy
  ,  
′ ,  
 ′, 
∈  
= (1,0, … , 0).

= .
15
Our MA-CP-ABE Scheme: Idea
  , ,  →  = (, ,  ∀ ∈  ).
 =  ℎ 0 ,
 =  0′ ,

 =   ∀  ∈ .
  ∈  ,  0 , 0′ ,  ∈ 3 are chosen randomly.
Have no relation with attributes
Bind all attribute-related keys of a user
together;
Prevent collusion attack from different
users (Distinct random  for each user);

 = 
 =    =   /0′
′
=  
Ideas:
 Separate the single authority to one CA and multiple AAs
 CA is responsible for choosing  and generating    for users;
 When a user submits his  to an AA, the AA generates  by using
   .
Problem:  is submitted to AA by the user, so that two users (e.g., Bob and Tom)
can launch a collusion attack by submitting the same .
Solution: Use digit signature to bind  and the identity of a user together.
16
Our MA-CP-ABE Scheme: Idea
One-CA-Multi-AA
   → , .
: , , ℎ,  ,   ,
 =   ∀ ∈ , 3 , ; : , 
  ,  →  ,  .
 : , , ℎ,  ,   ,
 =   ∀ ∈  ;  :  ∀ ∈ 
   → (, ,  ∀ ∈  ).
 =  ℎ 0 ,
 =  0′ ,

 =   ∀  ∈ .
 = (, || )
  , ,  →  .
 =  ℎ 0 ,
 =  0′ ,
 =  
  ,  , ,  ,  → .
 =  ⋅  , 
 , ′
= ℎ  ,  = ℎ  ⋅ 

−
, ′ =  ∀ ∈ {1,2 … , }.
17
Our MA-CP-ABE Scheme: Idea
One-CA-Multi-AA
 Problem:
Multi-CA-Multi-AA
In the One-CA-Multi-AA system, the CA holds the value of , so that it can
decrypt all ciphertexts.
 Solution
Introduce multiple CAs: CA1, …, CAD . Each CAd chooses 
independently, and publishes  ,   to the public parameters.
In  algorithm,  =  ⋅  ,    .
Implicitly, we have set that  = 1 + 2 + ⋯ +  .
 Only when all CAs collude together, can they decrypt a ciphertext.
18

1
……

1
1
User 
 = {1 , 2 }
1 ∈ 1 , 2 ∈ 
……

Our MA-CP-ABE Scheme: Idea
Naive Multi-CA-Multi-AA
   → : , , ℎ, 3
   →  :  ,   ,  ;  :  , 
  ,  →  :  =   ∀ ∈  ;  :  ∀ ∈ 
  ,  → (, , , ).
, =  ℎ, 0 , , = , 0′ , , = ( ,  || d|| , )
  {, , , | = 1,2, … , },  → {,, }.


,, = ,
,, ,
 = 1,2, … 
  , ,  ,  → .
=⋅
 , 
  ,  ′
= ℎ  ,  = ℎ  ⋅ 

−
, ′ =  ∀ ∈ {1,2 … , }.
  , ,  → .
  ∈
=1  
  , , 

′ ,   ,,
 ′ , ,

=
=1  
1
 , 
  .
20
Our MA-CP-ABE Scheme: Idea
Naive Multi-CA-Multi-AA
Our MA-CP-ABE
 Problem:
 When an attacker corrupts a CA, collusion attack can be launched.
 E.g.,  = 2,  = 2. 1 ∈ 1 , 2 ∈ 2 .  = 1 ,  = {2 }. CA1 is corrupted by
Bob and Tom, while CA2 is still secure. In such a case, Bob and Tom should
not be able to decrypt a ciphertext with policy (1  2 ). However,
 Bob obtains ,2 , ,2 from CA2 ; then obtains 1,,2 from AA1 ;
 They set  ,1 = ,2 , and submit this  ,1 to AA2 . AA2 is cheated
and believes that this “ ,1 " is legal, because Bob and Tom control
CA1 so that they can generate the valid signature. Then AA2 generates
"2 ,,1 " by using this " ,1 ", which is actually "2 ,,2 " for
",2 “.
 For the ciphertext, they can reconstruct  ,  2 by using
,2 , ,2 , {1,,2 , 2,,2 }. --- COLLUSION ATTACK WORKS.
21
Our MA-CP-ABE Scheme: Idea
Our MA-CP-ABE
Naive Multi-CA-Multi-AA
 Solution: Each time CAd generates , = , ′, it must show the knowledge
of , to AAk . We addressed this by reusing the CP-ABE scheme of [LOSTW10].
1
When  visits  ,
 regards  as the
“attributes” of the user
User 
1 = { 1,1 , 2,1 , … , (, 1)}
2 = 1,2 , 2,2 , … , , 2
 = { 1,  , 2,  , … , (, )}
2
 registers , to  ;
 uses , as the
public key corresponding
to “attribute (k,d)”
, = ,
1,1 = 1,1
1,2 = 1,2
2,1 = 2,1
2,2 = 2,2
1
2
1,1 , 1,2
2,1 , 2,2
22
Our MA-CP-ABE Scheme: Idea
Our MA-CP-ABE
Naive Multi-CA-Multi-AA
 [LOSTW10] , ,  →  = (, ,  ∀ ∈  ) .
=
 ℎ 0 ,
=
 0′
,

 =   ∀  ∈ .
When  visits  ,  regards 
= { 1,  , 2,  , … (, )}
as the “attributes” of the user:
, takes the place of 
 [Ours] ,  → (, , , , Γ,,   = 1  ).
, =  ℎ, 0 , , = , 0′ ,
Γ,, = ,
,
( = 1  ),
, = ( , || || || Γ,,1 || … ||Γ,, ) .
 uses Γgid,d,k to show to  that the corresponding , is generated
honestly.
23
Conclusion
We constructed an MA-CP-ABE system, where
 Different domains of attributes are managed by different attribute
authorities, which operate independently from each other.
 No authority can independently decrypt any ciphertext.
LOSTW10
(SA-) CP-ABE
LW10
MA-CP-ABE
Ours
MA-CP-ABE
Standard Model
Multi-Authority
Prevent Decryption by
Individual Authority
Partially
Size of Ciphertext
 + 
 + 
 + 
Size of Secret key
 +
||
 + ( + )
  +
||
  +
 +
||
 ++
Pairing Computation
of Decryption
Size of Public key
24
Extensions
 Large attribute universe construction:
 The size of public key is linear in ||.
 It can be avoided by using the idea of interpolation.
 Improving performance and reliability of the
system:
 In this paper,  = 1 + 2 + ⋯ +  is used to
distribute  to  CAs. It is a (, )-threshold policy, so
that all CAs must remain active.
 In the full version of this paper, general , Δ -threshold
policy is used. Only when  CAs are involved, they can
decrypt a ciphetext. The system works as long as no
more than Δ − D CAs fail.
25
References
• [SW05] Sahai, A., Waters, B.: Fuzzy identity-based encryption.
EUROCRYPT 2005.
• [GPSW06] Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based
encryption for finegrained access control of encrypted data. ACM CCS
2006.
• [BSW07] Bethencourt, J., Sahai, A., Waters, B.: Ciphertext-policy attributebased encryption. IEEE Symposium on Security and Privacy, 2007
• [CN07] Cheung, L., Newport, C.C.: Provably secure ciphertext policy abe.
ACM CCS 2007
• [GJPS08] Goyal, V., Jain, A., Pandey, O., Sahai, A.: Bounded Ciphertext
Policy Attribute Based Encryption. ICALP 2008, Part II.
• [Waters08/11] Waters, B.: Ciphertext-policy attribute-based encryption:
An expressive, efficient, and provably secure realization. PKC 2011
• [LOSTW10]Lewko, A.B., Okamoto, T., Sahai, A., Takashima, K., Waters, B.:
Fully secure functional encryption: Attribute-based encryption and
(Hierarchical) inner product encryption. EUROCRYPT 2010.
26
Reference
• [OT10] Okamoto, T., Takashima, K. : Fully secure functional encryption
with general relations from the decisional linear assumption. CRYPTO
2010.
• [MKE09] M¨uller, S., Katzenbeisser, S., Eckert, C.: On multi-authority
ciphetext-policy attribute-based encryption. Bulletin of the Korean
Mathematical Society 2009.
• [LW11] Lewko, A., Waters, B.: Decentralizing attribute-based encryption.
EUROCRYPT 2011.
27
Thanks.
Q&A
28

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