Journal of Cloud Computing

Advances, Systems and Applications

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An efficient and traceable KP-ABS scheme with untrusted attribute authority in cloud computing

Journal of Cloud ComputingAdvances, Systems and Applications20165:2

https://doi.org/10.1186/s13677-016-0052-1

Received: 17 November 2015

Accepted: 6 March 2016

Published: 12 March 2016

Abstract

ABE has been widely applied for secure data protection in cloud computing. In ABE, user’s private keys are generated by attribute authority, thus, attribute authority has the ultimate privileges in the system and can impersonate any users to forge valid signatures. Once the attribute authority become dishonest or be invaded in cloud systems, the system’s security will be at risk. To better solve the problem mentioned above, in this paper, we propose a key-policy attribute based signature scheme with untrusted authority and traceability (KP-ABS-UT). In our scheme, the signer’s private key is composed by two components: one part is distributed by attribute authority and the other part is chosen privately by the signer’s self. Thus attribute authority cannot forge any signatures which should be signed by legal users. Besides, our scheme introduces an entity “tracer”, which can trace the identity of signer when necessary. By security analysis and efficiency comparison, we prove our KP-ABS-UT scheme meets the requirements of unforgeability as well as lower computation cost.

Keywords

Access structure Untrusted authority Traceability

Introduction

With the various information resources increasing rapidly in the cloud, users are faced with urgent problems like how to make data sharing among resources efficiently and securely. On this occasion, Sahai proposed a new cryptographic primitive named “ABE” (attribute based encryption) [1]. In ABE mechanism [14], a user’s identity is described by several attributes rather than a single string. A data receiver is capable of getting access to the data when the attributes he possesses match with the structure made by the data owner. KP-ABE [2, 3] is a typical class of ABE.As is shown in Fig. 1, a user’s private key corresponds to an access tree structure. Each leaf node stands for the attribute a user owns; the non-leaf node describes the access policy of these attributes. The ciphertext corresponds to an attribute set such as {Attr1, Attr4}. Due to its capability of providing flexible access control for data sharing between users, KP-ABE is gradually becoming an effective tool for secure data sharing in complex networks.
Fig. 1

Access structure for KP-ABE

After the proposal of ABE, ABS (attribute based signature) has been developed as a primitive to solve the data authentication problem in attribute based cryptosystem. ABS was originally proposed by Maji et al. in [5]. In ABS mechanism [69], a user is capable of signing a message using the private key component of the attribute set he owns. A receiver validates the signature by utilizing system public parameters. The signature can be verified to an attribute set or access structure which the signer possesses. Since the advent of the notion, many ABS schemes have been proposed. However, most of the existing ABS schemes have one thing in common, that is the attribute authority has the ultimate privileges in the whole system. In order to keep the whole system running safely, attribute authority must be honest and highly protected. In the open network systems, attribute authority are vulnerable to external as well as internal threat. Once being invaded, it can impersonate any users to forge legal signatures, which will threaten the whole security of the system. What’s more, anonymity is an important feature of common attribute based signature mechanism. However, some malicious users may take advantage of this feature to release illegal information without taking responsibility.

To better solve the problems discussed above, in this paper, we construct a KP-ABS-UT (key-policy attribute based signature with untrusted authority and traceability) scheme, which has the following merits: (1) the signature is unforgeable, even attribute authority cannot impersonate any users to forge legal signatures; (2) the overall computation cost of the whole process of signing and verifying is reduced sharply; (3) the signer’s identity can be traced by the system administer if necessary.

The rest sections are arranged as follows:

In section 2, we give the literature review and fundamental mathematical preliminaries and notions. The security model and detailed algorithms of our KP-ABS-UT scheme are constructed in section 3 and section 4. Section 5 gives the security proof and performance comparison of our scheme. At last, the conclusion of this paper and prospects on future directions are made in section 6.

Related works and preliminaries

Related works

The notion of ABS was first proposed by Maji in [5]. Since then, many ABS schemes have been proposed by researchers worldwide. Existing literatures of ABS can be classified to three types in terms of the data access structure :(1)ABS using threshold structure [1012]; (2) ABS using LSSS access structure [1315];(3) ABS using access tree [16], which has been illustrated in Fig. 1. Besides access structure, different ABS schemes have different advantages and performances. A. Escala et al. in [13] proposed a user-revocable attribute based signature. If a user drops some of the attributes, his signing privilege can be exactly withdrawn. Their scheme also achieves adaptively security in the standard model. S. Shahandashti et al. in [10] construct a threshold ABS and applied it to credential systems. Their scheme enables a signature holder to prove possession of signatures by revealing only the relevant attributes of the signer, hence providing signer attribute privacy for the signature holder. S. Kumar et.al in [11] proposed an ABS which is equipped with multiple threshold access structure. Their scheme is efficient as well as stretchable. Tatsuaki et al. in [16] proposed a decentralized multi-authority ABS. In their scheme, due to the introducing of multiple authorities, thus no central CA is needed. However, this also brings about other problems such as parameter synchronization, time synchronization, etc. Besides, the efficiency of their scheme can be further improved. S.L Ding et al. proposed a traceable ABS in [14], which allows PKG and the issuing authority join together to trace the identity of a malicious signer. However, since the trace algorithm needs frequent participation of PKG, it may be exposed to more internal and external security risks. Li et al. in [15] proposed a novel ABS with hidden attribute property. In their scheme, anonymous user revocation is achieved. Liu et al. in [17] proposed an attribute based multisignature scheme, which allows a number of users to participate to authenticate a message with only one signature. Their scheme is shown to be secure and is more appropriate to be applied for wireless communications. Chen et al. in [9] proposed a new paradigm named “Outsourced ABS”. The computation load in their scheme is reduced sharply by delegating most of the computation work to a semi-trusted server, thus this will relieve the terminal devices from heavy computation burden. The high efficiency and security level makes their scheme an excellent method for providing data authentication in cloud computing. Xu et al. in [18] propose an ABS scheme with dynamic user revocation. Their scheme has superior performance with regard to scalability, which is able to be applied to massive data storage environments.

The above schemes have laid solid foundations for filling the gap between theoretical proposal and practical application of ABS. However, in the above schemes, the attribute authority has the ultimate privilege in the system. Even worse, it can forge as any legal users and generates a valid signature. Once the attribute authority become dishonest, the system will be exposed to potential risks. Although ABS with multiple attribute authorities has been proposed in [16], if these attribute authorities collude with each other to obtain a user’s private key, a legal signature can still be successfully forged. Thus, it is of significance to cut down the privilege of the attribute authority in ABS systems.

Bilinear pairings

Denote G 1 and G 2 to be cyclic groups of prime order q. Let g be a generator of G 1. A bilinear pairing ê : G 1 × G 1 → G 2 has three features [19]:
  • Bilinearity: For a, bZ q , ê(g a , g b ) = ê(g, g) ab .

  • Non-degeneracy: There exists P, QG 1 which satisfy ê(g, g) ≠ 1.

  • Computability: The value of ê(u, v) can be calculated for any u, vG 1.

Hardness assumption

Decision Bilinear Diffie-Hellman problem:

Picks random numbers \( a,b,c,z\in {Z}_q^{*} \) , assuming that the value of (g, g a , g b , g c , z) are given, no probabilistic polynomial-time algorithm can distinguish the tuples (A = g a , B = g b , C = g c , ê(g, g) abc ) and (A = g a , B = g b , C = g c , ê(g, g) z ) with a non-negligible probability.

Lagrange Interpolation function

For a polynomial p(x) in Z q of order d − 1 and a set S in Z q with the size |S| = d is computed by \( p(x) = {\displaystyle {\sum}_{i\in S}p(i){\prod}_{j\in S,j\ne i}\frac{x-j}{i-j}} \).

Our model and assumptions

Formal model of our scheme

In order to clearly describe our KP-ABS-UT, we define some notations listed in Table 1.
Table 1

Notations and their corresponding meanings

Notations

Meanings

G 1, G 2

Groups

ê

Bilinear paring operation

H 1

Hash function

PK

System public parameters

MK

System master key

AA

Attribute authority

U

Global attribute set

A i

Attribute i

D i

Private key of attribute i

M

Plaintext

T i

Access structure

x

Node in the access tree

Δ i,S(x)

Lagrange interpolation function

q x

Polynomial defined by AA

p x

Polynomial defined by signer

v

Signature to be verified

id l

Identity of user l

The model of our system is shown in Fig. 2. Our model consists of 4 entities: “AA” (Attribute authority), “tracer”, “signer” and “receiver”. AA provides system’s public parameters, generates part of user’s private key and distributes them to users. After receiving the original key generated by AA, signer re-generates his own private key by adding a secret component. Then signer signs the plaintext using the private key generated by himself and uploads the signature to cloud center. Cloud center provides secure data storage and responses the access request made by data receiver. Receiver verifies if the signature is a valid one. Tracer is responsible for revealing the signer’s identity when necessary.
Fig. 2

The system model of KP-ABS-UT in cloud computing

Formulized definitions of algorithms in KP-ABS-UT:
  • Setup{1 λ } → {PK, MK}:This algorithm is operated by AA. Take in a security parameter, it generates PK and MK. PK is shared by users while MK is kept private by AA.

  • \( Private\ key\ generation\left\{PK,MK,{T}_k,s\right\}\mathit{\to}\left\{{D}_{i,{T}_k,i{d}_l}\right\} \): The private key generation algorithm is an interaction between AA and a user with an access structure T k . On input PK, MK, T k , AA firstly returns \( {D}_{i,{T}_k} \) to be the initial attribute private key for T k . Then each signer embeds a secret component into \( {D}_{i,{T}_k} \) to re-generate his own private key \( {D}_{i,{T}_k,i{d}_l} \).Signer sends certain information referring to the secret component to tracer. Tracer assigns a unique identity number to the signer and stores the relationship of each signer with his identity.

  • \( Sign\left\{{D}_{i,{T}_k,i{d}_l},PK,M,{\sigma}_{i{d}_l}\right\}\mathit{\to}\left\{v\right\} \): This algorithm is run by a signer which the systems public parameter PK, a plaintext M, signer’s private key \( {D}_{i,{T}_k,i{d}_l} \) and a mathematical constant \( {\sigma}_{i{d}_l} \) which can reveal signer’s identity as input. Then the algorithm outputs a signature v for the plaintext M.

  • Verify{PK, v, M} → {θ}: This algorithm is run by the receiver. On input PK and the plaintext M with the signature, it outputs a value θ. If θ = 1 then the signature is a valid one, if θ = 0 then the signature is invalid.

  • Trace{MK, v} → {id l }: This algorithm is run by the tracer. On input the signature v,the tracer can pinpoint the exact identity of a signer.

The essential unforgeability of our KP-ABS-UT

Definition 1

Our KP-ABS-UT has the existential unforgeability if no Adversary has non-negligible advantage in the following game played by a Challenger and an Adversary.

Setup : Challenger runs Setup procedure to obtain the system parameters and sends PK to Adversary.

Sign queries : Adversary chooses an access control structure T i ,a plaintext M. Challenger calculates the attribute private key \( {D}_{i,{T}_k,i{d}_l} \) and runs the Sign procedure to calculate the signature \( v= Sign\left(PK,M,{D}_{i,{T}_k,i{d}_l}\right) \).After then, Challenger sends the signature v to Adversary.

Challenge : Adversary chooses a plaintext M*,a challenging access structure T c  , and calculates the signature v*.

Challenger verifies the signature by running the procedure Verify(PK, v*) and outputs a value θ.

Adversary wins if the output of Verify algorithm is valid.

Denote Adv(A) = |Pr[θ = 1]| to be advantage of Adversary.

Our construction to KP-ABS-UT scheme

Concrete algorithms:

Setup: Let G 1 and G 2 be two cyclic groups of prime order p, while g is the generator of G 1. Let ê : G 1 × G 1 → G 2 be a bilinear paring. Define H 1 : {0, 1}* → Z p .AA picks \( {t}_i\in {Z}_p^{*} \) for each A i in the system. Besides, AA picks another secret number \( y\in {Z}_p^{*} \) and calculates ê(g, g) y .Set MK to be {t i , y} while PK to be \( \left\{{G}_1,{G}_2, p,g,{H}_1,\widehat{e}{\left(g,g\right)}^y,{g}^{t_i}\right\} \).

Private key generation: AA randomly chooses a polynomial q x for each node x in the access tree and sets the degree d x of q x to be one less than the threshold value k x of that node (d x  = k x  − 1). For any other node (except root node) in the access tree, let q x (0) = q parent(x) index(x). For the root node AA sets q root (0) = y. To avoid the signature forgery which can be made by AA, the signer also chooses a polynomial p x likewise and picks a secret number s. For the root node AA sets q root (0) = y while the signer sets p root (0) = s. Once the polynomials have been decided, the private key is of each leaf node x in the access tree can be denoted by \( {D}_{i,{T}_k,i{d}_l}=\left\{{g}^{\frac{q_x(0).{p}_x(0)}{t_i}},i\in {T}_k\right\} \). Signer sends the value of g s to tracer. For each signer, tracer chooses a global unique identifier \( i{d}_l\in {Z}_p^{*} \) to describe his identity (Without loss of generality, the signer’s identity is denoted by id l ). Tracer calculates \( {\sigma}_{i{d}_l}={\left({g}^s\right)}^{i{d}_l} \) and sends \( {\sigma}_{i{d}_l} \) back to the signer. Meanwhile, it stores a list recording the relationship between singer’s unique identifier and \( {\sigma}_{i{d}_l} \).

Sign: For a plaintext M, the signer picks a random number \( r\in {Z}_p^{*} \) and calculates:
$$ \begin{array}{l}{C}_0=\widehat{e}{\left(g,g\right)}^{yrs},{C}_1=\widehat{e}{\left(g,g\right)}^{ys}\\ {}{C}_2 = \left\{{D_{i,{T}_k,i{d}_l}}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\right\}=\left\{{\left({g}^{\frac{q_x(0).{p}_x(0)}{t_i}}\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\right\}\end{array} $$
(1)

The signature can be denoted by \( v=\left\{{C}_1,{C}_2,{C}_3,M,{\sigma}_{i{d}_l},{g}^s,{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s\right\} \). The signer sends v to the receiver.

Verify: After receiving the signature v = {C 1, C 2, C 3, M, σ id , g s , H 1(M||σ id ) s } from the signer, receiver firstly calculates: ê(H 1(M||σ id ) s , g) = ê(H 1(M||σ id ), g s ).

If the equation is set up, then calculates \( VerityNode\left(x,PK,v\right)=\widehat{e}\left({g}^{t_i},{C}_2\right) \).

If x is a leaf node, then the calculation process is as follows:
$$ VerityNode\left(x,PK,v\right)=\left\{\begin{array}{c}\hfill \widehat{e}\left({g}^{t_i},{C}_2\right)=\widehat{e}\left({g}^{t_i},{\left({g}^{\frac{q_x(0).{p}_x(0)}{t_i}}\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\right)\hfill \\ {}\hfill =\widehat{e}{\left({g}^{t_i},{g}^{\frac{q_x(0).{p}_x(0)}{t_i}}\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\hfill \\ {}\hfill \kern2.25em =\widehat{e}{\left({g}^{p_x(0)},{g}^{q_x(0)}\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s} \cdot \widehat{e}\left({g}^{r.{p}_x(0)},{g}^{q_x(0)}\right)\hfill \\ {}\hfill \perp, otherwise\hfill \end{array}\right. $$
(2)

All the value calculated from VerityNode(x, PK, v) will be stored as F z . For any F z  ≠ 0, the algorithm calculates F root (the value of root node) using Lagrange interpolation method.

If x is a non-leaf node, z is the child node of x,then the value of VerityNode(x, PK, v) is calculated as follows:

Let i = index(z), S x , = {index(z) : zS x }
$$ \begin{array}{l}{F}_x={\displaystyle \prod_{z\in {S}_x}}{F_z}^{\varDelta_{i,Sx\hbox{'}\;(0)}}\\ {}={\displaystyle \prod_{z\in {S}_x}}\widehat{e}{\left({\left({g}^{p_z(0)}\right)}^{\varDelta_{i,Sx\hbox{'}\;(0)}},{g}^{q_z{(0)}^{\varDelta_{i,Sx\hbox{'}\;(0)}}}\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\\ {}={\displaystyle \prod_{z\in {S}_x}}\widehat{e}{\left({\left({g}^{p_{parent}{(z)}^{\left( index(z)\right)}}\right)}^{\varDelta_{i,Sx\hbox{'}\;(0)}},\left({g}^{q_{parent}{(z)}^{{\left( index(z)\right)}^{\kern0.5em {\varDelta}_{i,Sx\hbox{'}(0)}}}}\right)\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\\ {}={\displaystyle \prod_{z\in {S}_x}}\widehat{e}{\left({\left({g}^{p_x(i)}\right)}^{\varDelta_{i,Sx\hbox{'}\;(0)}},\left({g^{q_x(i)}}^{\;{\varDelta}_{i,sx\hbox{'}\;(0)}}\right)\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\\ {}=\widehat{e}{\left({g}^{p_x(0)},{g}^{q_x(0)}\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s}\cdot \widehat{e}\left({g}^{r.{p}_x(0)},{g}^{q_x(0)}\right)\end{array} $$
(3)

Then, the algorithm calculates F root (the value of root node) by recursive function.

Finally, the algorithm verifies if:
$$ {F}_{root}={C_1}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s}\cdot {C}_0 $$
(4)

If the equation is set up, then the signature is a legal one and the signer’s attributes satisfies the claimed structure.

Trace: The tracer searches the list recording the relationship between \( i{d}_l-{\sigma}_{i{d}_l} \), then the identity of the signer can be pinpointed.

Security proof and performance evaluation

Correctness proof

To calculate the value of F root , it is essential to obtain the value of leaf nodes which satisfy the access structure.

Since the value of \( VerityNode\left(x,PK,v\right)=\widehat{e}{\left({g}^{p_x(0)},{g}^{q_x(0)}\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}, \) whether x is a leaf node or non-leaf node, consequently, the value of root node F root can be calculated by:
$$ \begin{array}{l}{F}_{root}=\widehat{e}{\left({g}^{p_{root}(0)},{g}^{q_{root}(0)}\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\\ {}{p}_{root}(0)=s,{q}_{root}(0)=y\end{array} $$
(5)
Consequently, if the signature is valid then the equation is set up:
$$ \begin{array}{l}{F}_{root}\kern0.5em =\widehat{e}{\left({g}^s,{g}^y\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\\ {}={\left(\widehat{e}{\left(g,g\right)}^{ys}\right)}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\\ {}={C}_0\cdot {C_1}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s}\end{array} $$
(6)

Unforgeability

Theorem 1

If our scheme can be broken by an Adversary then it can be constructed that a simulator with a non- negligible advantage solves the DBDH problem successfully.

Proof

In the challenge game, if an Adversary can break our scheme with advantage ε, then a simulator be constructed to solve the DBDH problem with an advantage of \( \frac{\varepsilon }{2} \).

The construction process of the simulator is as follows:

Phase 1 Setup:

Challenger sets the parameters as follows:

Defines a global attribute set U = {1, 2, … n}.

Denotes id l to be the unique identifier of Adversary.

Defines G 1 and G 2 be two cyclic groups of prime order p. The generator of G 1 is denoted by g.

Defines a bilinear paring ê : G 1 × G 1 → G 2.

Picks μ {0, 1}, a, b, c, zZ p .
$$ \left\{\begin{array}{c}\hfill \left(A,B,C,Z\right)=\left({g}^a,{g}^b,{g}^c,\widehat{e}{\left(g,g\right)}^{abc}\right)\kern1.5em if\ \mu =0\hfill \\ {}\hfill \left(A,B,C,Z\right)=\left({g}^a,{g}^b,{g}^c,\widehat{e}{\left(g,g\right)}^z\right)\kern2.5em if\ \mu =1\hfill \end{array}\right. $$
(Let)

The aim of simulator is to output a value μ* as a guess of μ.

The simulator runs Adversary as sub-program and plays the role of Challenger.

Adversary defines attribute set γ.

Simulator randomly chooses r i , β i Z p and sets PK to be \( Y=\widehat{e}\left({g}^a,{g}^b\right)=\widehat{e}{\left(g,g\right)}^{ab},{T}_i=\left\{\begin{array}{c}\hfill {g}^{r_i},\kern0.75em if\ i\in \gamma \hfill \\ {}\hfill {g}^{b{\beta}_i}, if\ i\notin \gamma \hfill \end{array}\right. \)

Phase 2 Queries :

Private key generation queries:When Adversary asks a Private key generation query for access structure T k , simulator responds as follows:

Chooses two polynomial q x , p x for each node x in the access structure T k and sets the degree d x of q x , p x to be one less than the threshold value k x of that node (d x  = k x  − 1). Besides, simulator sets q root (0) = y, p root (0) = s. For any other node (except root node) in the access tree, let q x (0) = q parent(x) index(x). Then simulator calculates the private key and sends \( {D}_{i,{T}_k,i{d}_l} \) back to Adversary.The format of \( {D}_{i,{T}_k,i{d}_l} \) is as follows:
$$ {D}_{i,{T}_k,i{d}_l}=\left\{\begin{array}{c}\hfill {g}^{\frac{q_x(0){p}_x(0)}{t_i}}={g}^{\frac{q_x(0){p}_x(0)}{r_i}}, if\ att(x)\in \gamma \hfill \\ {}\hfill {g}^{\frac{q_x(0){p}_x(0)}{t_i}}={g}^{\frac{q_x(0){p}_x(0)}{b{\beta}_i}}, if\ att(x)\notin \gamma \hfill \end{array}\right. $$

Sign queries: When Adversary asks a Sign query for access structure T k and a plaintext M, simulator responds as follows:

Firstly, simulator calculates the private key \( {D}_{i,{T}_k,{\sigma}_{i{d}_l}} \).

Obtaining the private key of access structure T k by running

Randomly chooses \( r\in {Z}_p^{*} \), let \( {\sigma}_{i{d}_l}={\left({g}^s\right)}^{i{d}_l} \), then calculates:
$$ {C}_0=\widehat{e}{\left(g,g\right)}^{abrs},{C}_1=\widehat{e}{\left(g,g\right)}^{abs},\ {C}_2 = \left\{{D_{i,{T}_k,i{d}_l}}^{H_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^s+r}\right\} $$
(7)

Then signer outputs \( v=\left\{{C}_1,{C}_2,{C}_3,M,{\sigma}_{\sigma_{i{d}_l}},{g}^s,{H}_1{\left(M\left|\right|{\sigma}_{\sigma_{i{d}_l}}\right)}^s\right\} \) and sends it to Adversary.

It can be seen that the simulator is consistent with our scheme.

Phase 3:Challenge:

Adversary outputs a new signature v l with a challenging access structure T l and plaintext M.

Adversary cannot make Sign queries about T l and v l is not gained by a previous Sign query.

Adversary forges the signature as the following process:

Chooses a polynomial p lx .Sets p lroot (0) = s l . Picks r l Z p , returns \( {g}^{s_l} \) to simulator.

Simulator chooses sends \( {\sigma}_{\sigma_{i{d}_l}}={\left({g}^{s_l}\right)}^{id} \) back to Adversary.

Adversary forges the private key \( {D_{i,{T}_l,{\sigma}_{i{d}_l}}}^{*} \) of the access structure T l .
$$ {D_{i,{T}_l,{\sigma}_{i{d}_l}}}^{*}=\left\{\begin{array}{c}\hfill {g}^{\frac{{q_x}^{*}(0){p_l}_x(0)}{t_i}}={g}^{\frac{{q_x}^{*}(0){p_l}_x(0)}{r_i}}, if\ att(x)\in \gamma \hfill \\ {}\hfill {g}^{\frac{{q_x}^{*}(0){p_l}_x(0)}{t_i}}={g}^{\frac{{q_x}^{*}(0){p_l}_x(0)}{b{\beta}_i}}, if\ att(x)\notin \gamma \hfill \end{array}\right. $$
(8)
Adversary calculates \( {v}_l=\left\{{C}_0,{C}_1,{C}_2,M,{g}^{s_l}, {\sigma}_{\sigma_{i{d}_l}},{H}_1{\left(M\left|\right|{\sigma}_{\sigma_{i{d}_l}}\right)}^{s_l}\right\} \):
$$ \begin{array}{l}{C}_0=\widehat{e}{\left(g,g\right)}^{ab{s}_l},{C}_1=\widehat{e}{\left(g,g\right)}^{ab.{r}_l{s}_l}\\ {}{C}_2 = {D_{i,{T}_l,{\sigma}_{i{d}_l}}}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}=\left\{\begin{array}{c}\hfill {\left({g}^{\frac{{q_x}^{*}(0){p_l}_x(0)}{r_i}}\right)}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}, if\ att(x)\in \gamma \hfill \\ {}\hfill {\left({g}^{\frac{{q_x}^{*}(0){p_l}_x(0)}{b{\beta}_i}}\right)}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}, if\ att(x)\notin \gamma \hfill \end{array}\right.\end{array} $$
(9)
$$ Verify: $$
Simulator firstly calculates \( VerityNode\left(x,PK,{v}_l\right)=\widehat{e}\left({g}^{r_i},{C}_2\right) \).
$$ \begin{array}{l} VerityNode\left(x,PK,{v}_l\right)=\left\{\begin{array}{c}\hfill \widehat{e}\left({g}^{r_i},{\left({g}^{\frac{{q_x}^{*}(0){p_l}_x(0)}{r_i}}\right)}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}\right), if\ att(x)\in \gamma \hfill \\ {}\hfill \widehat{e}\left({g}^{b{\beta}_i},{\left({g}^{\frac{{q_x}^{*}(0){p_l}_x(0)}{b{\beta}_i}}\right)}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}\right), if\ att(x)\notin \gamma \hfill \end{array}\right.\\ {}=\widehat{e}{\left({g}^{{p_l}_x(0)},{g}^{{q_x}^{*}(0)}\right)}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}\end{array} $$
(10)
By recalling the recursive function, the value of root node F root can be calculated by:
$$ \begin{array}{l}{F}_{root}=\widehat{e}{\left({g}^{{p_l}_{root}(0)},{g}^{{q_{root}}^{*}(0)}\right)}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}\\ {}=\widehat{e}{\left({g}^{s_l},{g}^{{q_{root}}^{*}(0)}\right)}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}\\ {}={\left(\widehat{e}{\left(g,g\right)}^{s_l{q_{root}}^{*}(0)}\right)}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}\kern1.25em \end{array} $$
(11)
According to the setting in Setup phase, let s l  = c, \( {r}_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}=f, \) then
$$ \begin{array}{l}{F}_{root}={\left(\widehat{e}{\left(g,g\right)}^{s_l\cdot {q_{root}}^{*}(0)}\right)}^{r_l+{H}_1{\left(M\left|\right|{\sigma}_{i{d}_l}\right)}^{s_l}}\\ {}=\left\{\begin{array}{c}\hfill {\left(\widehat{e}{\left(g,g\right)}^{abc}\right)}^f, if\ \mu =0\ \hfill \\ {}\hfill {\left(\widehat{e}{\left(g,g\right)}^z\right)}^f, if\ \mu =1\hfill \end{array}\right.\end{array} $$
(12)

Then we will discuss the advantage of simulator in breaking the DBDH assumption.

When μ = 0, Adversary forges the signature successfully. According to our assumption, the probability of this incident is \( \frac{1}{2}+\varepsilon \). When Adversary successfully forges the signature, simulator guesses u* = 0 and simulator has a \( \frac{1}{2}+\varepsilon \) probability of making the correct guess of u*. The probability of simulator making the correct judgment can be denoted by:
$$ Pr\left({u}^{*}=u\Big|\mu =0\right)=\frac{1}{2}+\varepsilon $$
(13)
When μ = 1, Adversary fails to forge the signature. The result of output is rejected symbol . Under this condition simulator guesses μ* randomly. The probability of simulator making the correct judgment can be denoted by:
$$ Pr\left({u}^{*}=u\Big|\kern0.1em \mu =1\right)=\frac{1}{2} $$
(14)
As is mentioned above, the advantage of simulator is:
$$ \begin{array}{l}\frac{1}{2} Pr\left({u}^{*}=u\Big|\kern0.1em \mu =0\right)+\frac{1}{2} Pr\left({u}^{*}=u\Big|\kern0.1em \mu =1\right)-\frac{1}{2}\\ {}=\frac{1}{2}\left(\frac{1}{2}+\varepsilon \right)+\frac{1}{2}\times \frac{1}{2}-\frac{1}{2}\\ {}=\frac{\varepsilon }{2}\end{array} $$
(15)

Unforgeability with untrusted authorities and traceability

In attribute based signature mechanism, users’ attribute private keys are generated and distributed by AA. Under this circumstance, AA is capable of forging any signatures without being detected. To keep the whole system running safely, AA must be honest and highly protected from being invaded. However, in the complex network systems, there are variety risks which may result in the fault of AA. Consequently, for the sake of safety, AA’s rights in the system should be reduced to some extent. In our KP-ABS-UT, the private key used for signature is generated by joint efforts of both AA and user, so it is computational infeasible for AA to forge any signatures which should be signed by legal users. Besides, since different users randomly choose different secret numbers to embed into their initial private keys distributed from AA, it is computational infeasible for a user to personate any other users to forge a legal signature in the system.

Anonymity is another important feature of common attribute based signature mechanism. However, some malicious users may take advantage of this feature to send out illegal information without taking responsibility. To prevent such incidents happening, in our KP-ABS-UT scheme, the signer’s identity can be revealed with the introducing of “tracer”. When data receivers receive illegal information from a malicious signer, they can send the signature v to tracer. Since tracer has stored \( {\sigma}_{i{d}_l}={\left({g}^s\right)}^{i{d}_l} \) with user’s identity, the signer’s identity can be located by searching the \( i{d}_l-{\sigma}_{i{d}_l} \) list. Then tracer calculates \( {\sigma_{i{d}_l}}^{-i{d}_l}={\left({\left({g}^s\right)}^{i{d}_l}\right)}^{-i{d}_l}={g}^s \). Since s is the secret number used in the Sign algorithm and g s has been sent to tracer, a signer cannot deny the message he signed before.

Efficiency and comparison analysis

In our KP-ABS-UT scheme, assume “n” to be the amount of attributes which a signer owns. The Sign algorithm needs (n + 3) times of exponential operation and 1 hash operation. The computation of Sign algorithm is low since it does not need any paring operation. The Verify algorithm needs (n + 1) times of paring operation and 1 hash operation. Since the computation cost of paring operation is much more than any other operation, in this paper, we mainly compare the number of paring operation with other ABS schemes [9, 14, 18] with respect to efficiency. The efficiency and performance comparison results are listed below in Table 2.
Table 2

Efficiency and performance comparison

Scheme

Access method

Traceability

With untrusted AA

Computation cost

[14]

LSSS

Yes

No

(n + 3) paring

[9]

Threshold

No

No

4 paring

[18]

Threshold

No

No

(n + 4) paring

Our scheme

Access tree

Yes

Yes

(n + 1) paring

From the result we can see that the computation cost is lower in our KP-ABS-UT since the number of bilinear paring operation is reduced. Thus our plan has a higher efficiency. Besides, our KP-ABS-UT uses access tree as control structure, each signer’s private key corresponds a certain structure. It is computational infeasible for different users to collude their private keys with each other to forge a legal signature. With the introducing of user’s secret key component and the entity “tracer”, the signature cannot be forged by AA and can be traced to identity if a malicious user releases information illegally. The overall comparison shows our KP-ABS-UT is of better performance and is more appropriate for secure data verification in open network systems such as cloud computing, etc.

Conclusion

In this paper, we propose a KP-ABS with untrusted authorities and traceability, which is also of high efficiency. In our scheme, the signer’s private key is composed by two components: one part is distributed by attribute authority and the other part is chosen privately by the signer’s self. The signature cannot be forged by any other users including attribute authority. What’s more, the identity of signer can be traced. Our scheme is proved to be secure and of better performance with respect to efficiency.

Our future work should focus on the attribute revocation and key refreshing in our KP-ABS scheme. Once key exposure happens, although the system can trace the traitor, however, user’s private keys still need to be refreshed for the safe of privacy. Consequently, research on the attribute based refreshing mechanisms should be taken into our future research direction. Besides, outsourcing ABS [9] with untrusted attribute authorities also merits attention.

Declarations

Acknowledgement

This research is supported by the National Natural Science Foundation of China (60973140, 61170276, 61373135).

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Key Lab of Broadband Wireless Communication and Sensor Network Technology, Ministry Education, Nanjing University of Posts and Telecommunications

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© Hong and Sun. 2016