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Privacy-assured boolean adjacent vertex search over encrypted graph data in cloud computing.

1. Introduction

Data outsourcing is an important application in cloud computing, and it can reduce the overhead and expenses. Hence data owners could outsource a large number of graph data to the cloud for cost savings. A graph is particularly useful in many fields, such as network topological graph [1], collaboration network [2], and social network graph [3]. In cloud computing , security is a big concern [4] [5]. Due to the cloud server being not fully trusted, the graph data are usually encrypted before outsourcing to the cloud server for preventing the information leak. However, the encrypted graph data can put operations to great trouble, and this brings query users great inconvenience. Therefore, achieving the boolean adjacent vertex search over encrypted graph data in cloud computing is a very useful operation.

The adjacent vertex search which searches for all adjacent vertices of a query vertex is a very common operation in the graph [6][7][8], many other operations such as subgraph mining [9], community finding [10], and outlier detection [11], can be built based on the adjacent vertex search. The boolean adjacent vertex search over a graph is a more general operation, which searches for all adjacent vertices by a boolean expression. There are disjunction, conjunction and negation three basic operations in boolean expression [12]. For example, in a scientific collaboration network, each vertex represents a scientific researcher. We could search for the scientific collaborators through the adjacent vertex queries. We use ([t.sub.1], [t.sub.2], ..., [t.sub.x]} to represent x scientific researchers. The disjunctive operation allows a query user to search for the neighbor vertices which are adjacent to [t.sub.1] or [t.sub.2] ... or [t.sub.x]. The conjunctive operation allows a query user to search for the neighbor vertices which are adjacent to "[t.sub.1] and [t.sub.2] ... and [t.sub.x]". The negative operation allows a query user to search for the neighbor vertices which are not adjacent to particular vertices. Thus, through such boolean search, a query user can perform a more general adjacent vertex search. To achieve this query, we adopt boolean expression algorithms in mathematics [12]. However, it is a challenging task to directly conduct the boolean search over encrypted graph data. Considering the "pay-for-use" billing rule in the cloud, it is particularly uneconomical to download all the graph data and decrypt locally. Therefore, it is very meaningful to perform the boolean adjacent vertex search over encrypted graph data in cloud computing.

For searching encrypted data in the cloud, searchable encryption [13][14][15][16][17] is a very useful technique. The cloud server can securely perform search operation over encrypted data through a query token. Currently, searchable encryption is a hot research direction, and there have been a lot of literatures [18][19][20][21][22] in this field. Moataz et al. [18] presented an efficient searchable encryption scheme to perform boolean expression query, and proved the scheme secure through a simulation based formal proof. The problems of dynamic searchable encryption were studied in these literatures [19][20][22]. Cash et al. studied highly-scalable searchable encryption scheme with support for boolean queries in the literature [21]. But all the searchable encryption methods cannot be directly used to conduct boolean adjacent vertex search over encrypted graph data. Meanwhile, many studies of privacy-preserving graph queries [23][24][25][26] have already appeared in recent years. The problem of secure subgraph query was studied in the literatures [23][26]. Chase et al. [24] proposed the notion of structured encryption and the application of controlled disclosure. The problem of privacy-assured substructure similarity query over encrypted graph-structured data was investigated in the literature [25]. However, the problem of boolean adjacent vertex search over encrypted graph data has not been solved.

To solve the problem, we propose a solution to perform secure boolean adjacent vertex search over encrypted graph data in cloud computing (BASG). Our BASG scheme can achieve a boolean adjacent vertex query through an index, and insure the query security. We first construct graph vertices' normalized orthogonal keyword set and adjacent vector information, and then build a secure index. Then a query user can send the boolean query token to the cloud server, and the cloud server performs computing and search operation in the index by the query token. After the retrieving, the cloud server returns the boolean search results to the query user. The experiment results and security analysis show that our scheme is efficient and provably secure.

The contribution of this paper can be summarized as follows.

(1) We propose a scheme to solve the problem of the boolean adjacent vertex search over encrypted graph data in cloud computing.

(2) We formally analyze the security of our scheme, and it ensures the privacy of the boolean query tokens and the search results.

(3) The experiment results demonstrate the efficiency of our scheme.

The rest of this paper is organized as follows. Section 2 summarizes the related work. Section 3 introduces the basic knowledge of the paper. Section 4 provides the scheme introduction and a description of BASG. Section 5 analyzes the security of our scheme. Section 6 evaluates our scheme from the experiments. Finally, section 7 concludes the paper.

2. Related Work

There are two kinds of encryption search models: the symmetric encryption search model and the asymmetrical encryption search model [13][15][16][17]. Generally speaking, the symmetric encryption search is far more efficient than the asymmetrical encryption search, thus we use the symmetric encryption search model in our scheme designing.

The searchable encryption [13][14][15][16][17] has been widely studied as a cryptographic primitive and is very effective for achieving search on encrypted data. Song et al. [13] first proposed the notion of searchable symmetric encryption which took advantage of the stream cipher to encrypt all the words in each file. Goh [14] first introduced the notion of the secure index, which made use of a bloom filter to build an index for each file. The problem of searching on data that was encrypted using a public key system was studied in the literature [15]. Chang et al. [16] presented a definition based on simulation, and it aimed to insure the privacy of the index and encrypted query word. Curtmola et al. built non-adaptive and adaptive searchable symmetric encryption schemes respectively in the literature [17]. Recently some extended searchable encryption methods have been put forward in the literatures [18][19][20][21][22]. An efficient scheme was presented in the literature [18] to focuse on the enhancement of searchable encryption, and the scheme supported any boolean expression query over encrypted data. The construction of boolean symmetric searchable encryption in the paper is mainly based on the orthogonalization of the keyword field according to the Gram-Schmidt process. In addition, the security and the efficiency were formally analyzed. The dynamic searchable symmetric encryption scheme was proposed in the literature [19]. The dynamic symmetric searchable encryption schemes in very-large databases was designed and implemented in the literature [20]. Cash et al. [21] proposed the highly-scalable searchable symmetric encryption supporting boolean queries. A new dynamic searchable symmetric encryption scheme that achieved the highest privacy and low information leakage was developed in the literature [22]. But all the schemes cannot be directly used to perform boolean adjacent vertex search.

In recent years, some studies on privacy-preserving graph queries have appeared [23][24][25][26]. Cao et al. [23] defined and solved the problem of privacy-preserving query over encrypted graph-structured data in cloud computing. To improve efficiency, the principle of "filtering-and-verification" was utilized to prune as many negative data graphs as possible before verification, and a feature-based index was pre-built to provide feature-related information for every encrypted data graph. Then, a secure inner product computation was proposed to meet the challenge of supporting graph semantics, and a variety of privacy requirements were achieved. The notion of structured encryption and the application of controlled disclosure were introduced in the literature [24]. In this paper, the schemes were constructed for encrypting graph data queries, but did not perform more complex queries such as boolean queries and similarity queries on encrypted graph data. Zhang et al. [25] constructed a framework and a series of algorithms to solve the problem of privacy-assured substructure similarity query over encrypted graph-structured data based on the privacy homomorphism and obscuration methods. Fan et al. proposed a private subgraph query approach over outsourced graph databases in the literature [26]. One idea of this paper was to determine the minimized candidate subgraphs that contained at least a candidate mapping, and then candidate mappings were enumerated from those subgraphs instead of the original graph. In the paper, a subgraph cache was proposed to prune the candidate matchings that were enumerated, and a robust encoding scheme and its verification method were presented for the users to determine the proper encodings for their queries without leaking query structures. However, all the schemes cannot solve the problem of boolean adjacent vertex search over encrypted graph data. In our paper, we construct a complete scheme based on the searchable encryption and the Gram-Schmidt algorithm, and formally analyze the query security. We build the adjacent vector information of all the graph vertices to generate a secure index, and the cloud server can perform computing and search operation by the index and encrypted query tokens. Through our mechanism, the more complex boolean adjacent vertex search could be achieved.

3. Preliminaries

3.1 Cryptographic Basics

Goldwasser and Micali introduced the notion of semantic security in the literature [27]. Informally, a system is semantically secure if whatever an adversary can compute about the plaintext given the ciphertext, he can also compute without the ciphertext [27]. In cloud computing, we need to consider the efficiency and cost, and symmetric encryption is more efficient than asymmetric encryption [15][17][28][29][30][31]. Therefore, we use symmetric encryption in this paper. A semantically secure symmetric encryption scheme is a set (Kge, Enc, Dec) consisting of three polynomial time algorithms [32]. Kge is a secret key generating algorithm which takes a security parameter as input, and returns a secret key. Enc is an encryption algorithm that takes a secret key and the plaintext as inputs, and returns the ciphertext. Dec is a decryption algorithm that takes the ciphertext and a secret key as inputs, and returns the decrypted content.

Generally speaking, a pseudorandom function is the function that cannot be distinguished from a truly random function by any efficient procedure that can get the values of the function at arguments of its choice [32]. If the pseudorandom function is an injective function, the function is a permutation, and the pseudorandom permutation and truly random permutation are indistinguishable in polynomial time [32].

3.2 Notations and Formulas

For simplifying the following description, let t [right arrow] T denote that an element t is selected from the set T, and t [??] T denote that an element t is selected from the set T uniformly and randomly [17][32]. The main notations used in this paper are listed in Table 1.

There are three basic boolean query operations: the disjunction, the conjunction and the negation. To build our search scheme, similar to the literature [18], we adopt the Gram-Schmidt algorithm [12][18] to achieve the boolean adjacent vertex search. In linear algebra, every spanning set of a vector space must contain at least as many elements as any linearly independent set of vectors from the vector space [12].

Theorem 1. Gram-Schmidt algorithm [12]. Let ([[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n]} be a linearly independent vectors set of a pre-Hilbert space, [theta]([[alpha].sub.i], [[alpha].sub.j]) denotes the scalar product of at and [[alpha].sub.j], the Gram-Schmidt process works as follows:

[[beta].sub.1] = [[alpha].sub.1];

[[beta].sub.k] = [[alpha].sub.k] [k-1.summation over (i=1)] [theta]([[alpha].sub.k], [[beta].sub.i])/ [theta]([[beta].sub.i], [[beta].sub.i]) [[beta].sub.i], k = 2,...,n;

[[eta].sup.k] = [[beta].sub.k]/[square root of [theta]([[beta].sub.k], [[beta].sub.k])], k = 1,..., n.

Thus, we obtain the normalized orthogonal set ([[eta].sub.1], [[eta].sub.2], ..., [[eta].sub.n]} of vectors set ([[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n]}.

4. BASG Construction

4.1 Scheme Overview

In the cloud, the outsourcing system model illustrated in Fig. 1 mainly consists of three different entities: the cloud server, the data owner, and the query users. The cloud server is "honest-but-curious", that is, it honestly follows the protocol specification, but can curiously infer and analyze the contents on the cloud server and try to learn the additional information. The data owner can outsource the encrypted graph data and the index to the cloud server. When searching, the query user sends the search tokens to the cloud server, and the cloud server can perform retrieval through the encrypted search tokens and the secure index and return the search results to the query user. For protecting the privacy, our scheme follows the security definition of searchable encryption [17].

Our BASG scheme can perform the boolean adjacent vertex search over encrypted graph data in cloud computing, and should achieve the following design goals.

(1) Secure boolean adjacent vertex search functionality. The user can correctly search all the query results by a boolean token.

(2) Security. The cloud server cannot learn the contents of the query tokens and the encrypted search results.

(3) Efficiency. As little as possible computation and storage cost is spent in the search.

Some data structures such as the index, the array, and the query table are used in our BASG scheme. The data owner stores the adjacent information in the index, the query user sends the encrypted boolean search tokens to the cloud server, and the cloud server can return the search results to the query user after the retrieval. To guarantee the privacy of the stored contents and query tokens in the cloud, we must assure the security and correctness in our scheme implementation.

To attain the goal, we carry out three steps to achieve the boolean adjacent vertex search scheme. Firstly, we need to build normalized orthogonal keyword set from the graph vertex set, and construct the adjacent vertex label set of the graph vertex. Secondly, we build the adjacent vertex vector information of all the graph vertices, and generate a secure index. Finally, we construct the boolean query tokens according to the query requirements. Afterwards, the query user sends the search tokens to the cloud server, and the cloud server performs the retrieval by the query tokens along with the index and returns the query results to the user.

Consistent with most of the searchable symmetric encryption (SSE) methods [17][18] [30], we assume that the adversary (i.e., the cloud server) can adopt the adaptive attack model and the data owner has the mutual request authentication and search control mechanisms e.g., broadcast encryption [17] with the query users.

Fig. 2. Scheme designing

KeyGen(l):

Generate random secret key K [left arrow] [(0,1).sup.l].

BuildOrthoKeyword(A, K):

1. The set of all the vertices in graph G is A = {[a.sub.1],
[[alpha].sub.2], ..., [a.sub.n]}. To prevent the cloud server to
learn the number of adjacent vertices, m disturbing values
{[a.sub.n+1], [a.sub.n+2], ... , [a.sub.n+m]} are added into the
set A, and get the new vertices set A = {[[alpha].sub.2],
[[alpha].sub.2], ..., [a.sub.n+m]}, where m is the maximum degree
of the graph G.

2. Each element in the set A is transformed by pseudorandom
permutation, and get the set V = {[v.sub.1], [v.sub.2], ...,
[v.sub.n+m]}, that is [v.sub.i] = F(K, [a.sub.i]), and [v.sub.i]
[member of] {0,1}*, where 1 [less than or equal to] i [less than or
equal to] n+m.

3. Build normalized orthonormal set W = {[w.sub.1], [w.sub.2], ...,
[w.sub.n+m]} from the set V through theorem 1, building process is
as follows.

(1) Define temporary variables [u.sub.1], [u.sub.2], ...,
[u.sub.n+m], get the following results:

[u.sub.1] = [v.sub.1];

[u.sub.k] = [v.sub.k] - [k-1.summation over (i=1)]
[theta]([v.sub.k], [u.sub.i])/[theta]([v.sub.i], [u.sub.i])
[u.sub.i], k = 2, ..., n+m.

(2) Compute and get [w.sub.k] = [u.sub.k]/[square root of
([u.sub.k], [u.sub.k])], k = 1, ..., n+m.

BuildIndex(W, K):

1. Let T = [1, n] denote all integers from 1 to n, let [J.sub.i]
[subset or equal to] T denote the label of adjacent vertices of
[v.sub.i], where 1 [less than or equal to] i [less than or equal
to] n. If [absolute value of [J.sub.i]] < m, m - [absolute value of
[J.sub.i]] integers are uniformly at random from the set {n+1, ...
, n+m}, and get a set [P.sub.i], [P.sub.i] [subset] {n+1, ... ,
n+m}.

2. Compute adjacent vertex vector of each graph vertex [v.sub.i],
that is [N.sub.i] = [summation over (k[member
of][J.sub.i][union][P.sub.i])][w.sub.k], where 1 [less than or
equal to] i [less than or equal to] n. Thus, get the index table I
= {[N.sub.1], [N.sub.2], ..., [N.sub.n]}.

TokenGen(A, {#}, K):

1. Let M ={[a.sub.1], [a.sub.2], ..., [a.sub.n]} denote a subset of
set A, an arbitrary boolean expression on M can be expressed as
B(M) = [a'.sub.1]# [a.sub.2] # *** # [a'.sub.r] where # is a binary
logical operator, # [member of] {[and], [disjunction]}, [a'.sub.i]
[member of] {[a.sub.i], [[??].sub.i], [PHI]}, [[??].sub.i] denotes
the negation of [a.sub.i], [PHI] represents null value. For a
conjunctive expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] code [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where Y is the set of vertex labels in the conjunctive expression,
and |Y| strictly positive integers [{[x.sub.j]}.sub.1[less than or
equal to]j[less than or equal to][absolute value of Y]]. are picked
at random.

2. All boolean expressions can be expressed in disjunctive normal
form, thus, the expression B(M) can be transformed into
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] We introduce a
new vertex value [lambda] which is the adjacent vertex of each
graph vertex by default. After the Gram-Schmidt process, we have
one normalized orthogonal vector y transforming from [lambda]. For
each r e [1,s], We define: [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] where [[rho].sub.r] is a random positive
integer, [[mu].sub.r,i] is a positive integer selected randomly,
and [[delta].sub.r,i] is a random negative integer. [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION
NOT REPRODUCIBLE IN ASCII] represent respectively the labels of
positive and negative keywords in the boolean query expression. To
avoid the particular case where there are only negative vectors in
the conjunction [C.sub.r], then we introduce a vector y in our
definition. Otherwise, [Q.sub.r] would have to be divided by 0.

3. After the expression transformation and coding, get the query
token [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Search(Q, I):

1. Perform search operation:

For 1 [less than or equal to] i [less than or equal to] n:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an empty
set, then [theta]([Q.sub.r], [N.sub.i]) = 1, the graph vertex
[a.sub.i] matches the query token, and we add [v.sub.i] to the
search results set R.

2. The search results set R is returned to the query user.


4.2 Scheme Designing

Our main work in our scheme is to build the secure index and the boolean query tokens. Then the cloud server can perform the search and computing. The several algorithms used in our scheme are illustrated as follows.

* KeyGen([1.sup.l]): Takes a secure parameter l as input and outputs a symmetric key K.

* BuildOrthoKeyword(A,K): Takes the vertex set A in the graph data G and the symmetric key K as inputs, outputs a normalized orthogonal keywords set W.

* BuildIndex(W,K): Takes the normalized orthogonal keywords set Wand the symmetric key K as inputs, outputs a secure index I.

* TokenGen(W,{ #},K): Takes the normalized orthogonal keywords set W, binary logical operator set {#} and the symmetric key K as inputs, outputs a query token Q.

* Search(Q, I): Takes the query token Q and the index I as inputs, outputs the query results set R.

The pseudorandom permutation [32], the symmetric encryption algorithm [32][33][34][35], and the Gram-Schmidt algorithm [12][18] are used in our scheme. Let l be security parameter that can be used in the key generating algorithm. Our proposed boolean adjacent vertex search scheme is described in Fig. 2. The building process of our scheme is as follows.

(1) Normalized Orthogonal Keyword Construction

For the outsourcing graph G, A = {[a.sub.1], ... , [a.sub.n]} ([a.sub.i] is a positive integer, and 1 [less than or equal to] i [less than or equal to] n) is the vertices set. To hide the number of adjacent vertices for each vertex, we add m disturbing integer values {[a.sub.n+1], ... , [a.sub.n+m]} (m is the maximum degree of the graph), and get the changed vertices set A = {[a.sub.1], ... , [a.sub.n+m]}. The data owner needs to build the normalized orthogonal set of the graph vertices by BuildOrthoKeyword algorithm. The data owner first transforms each element [a.sub.i] (1 [less than or equal to] i [less than or equal to] n+m) via the pseudorandom permutation f(K, *) (K is a secret key), and get the integer value [v.sub.i] = (K, [a.sub.i]), that is [v.sub.i] [member of] {0, 1}*. The transformed set is V = {[v.sub.1], ... , [v.sub.n+m]}. Then, the data owner transforms the set V into the normalized orthogonal set by theorem 1, and the transformed vectors set is W = {[w.sub.1], ... , [w.sub.n+m]}.

Example 1. Fig. 3 shows a scientific collaboration graph, each vertex represents a scientific researcher, and each edge represents the cooperative relationship between two researchers. The graph has 5 vertices and 8 edges, and the vertices set is A = {[a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4], [a.sub.5]}, The maximum degree of the graph is 4. we add 4 disturbing integer values {[a.sub.6], [a.sub.7], [a.sub.8], [a.sub.9]} in the set A and get the changed vertices set A = {[a.sub.1], ... , [a.sub.9]}. We perform the calculation by the pseudorandom function v, = f(K, [a.sub.i]) (1 [less than or equal to] i [less than or equal to] 9), and the resulting values are denoted as a set V=

{[v.sub.1], ... , [v.sub.9]}. The elements in the set V are transformed by theorem 1, and we get the normalized orthogonal vectors set W = {[w.sub.1], ... , [w.sub.9]}.

(2) Building Index

To achieve querying on the encrypted cloud server, the data owner needs to build the secure index by BuildIndex algorithm. And the cloud server can accomplish the boolean adjacent vertex search by means of the secure index. In the first place, the data owner builds the adjacent index of each graph vertex, that is, for each graph vertex [a.sub.i] (1 [greater than or equal to] i [less than or equal to] n), its all adjacent vertices label is denoted as the set [J.sub.i]. The elements among the set [J.sub.i] which is the subset of the set T = [1, n] (T is the set of positive integers smaller than n) are positive integers. If [absolute value of J] < m, m - [absolute value of [J.sub.i]] integers are randomly chosen from the set {n+1, .... , n+m}, and these integers form a set [P.sub.i]. In this way, when computing the adjacent vertex vector [N.sub.i], it can pad [absolute value of [P.sub.i]] disturbing adjacent vertices. Afterwards, the data owner computes the adjacent vertex vector of each graph vertex a, and get the result [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [D.sub.i] = [J.sub.I] [union] [P.sub.i]. After all the computations, we have generated the index I = {[N.sub.1], ..., [N.sub.n]}. In the end, the data owner stores the encrypted index on the cloud server.

Example 2. As mentioned above, the vertices set in Fig. 3 is A = {[a.sub.1], ..., [a.sub.9]}, and the normalized orthogonal vectors set of which is W = {[w.sub.1], ... , [w.sub.9]}. we definite the label set of the vertices {[a.sub.1], ... , [a.sub.9]} as {1, ... , 9}. All the adjacent label sets of graph vertices are as follows: [D.sub.1] = {2, 3, 5} [union] {7} = {2, 3, 5, 7}, [D.sub.2] = {1, 3, 4, 5}, [D.sup.3] = {1, 2, 4, 5}, [D.sub.4] = {2, 3} [union] {6, 8} = {2, 3, 6, 8}, [D.sub.5] = {1, 2, 3} [union] {8} = {1, 2, 3, 8}, where {7}, {6, 8} and {8} are subsets that are randomly selected from the set {6, 7, 8, 9}. The adjacent vector of each graph vertex is computed as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and then the search index I = {[N.sub.1], [N.sub.2], [N.sub.3], [N.sub.4], [N.sub.5]} is generated.

(3) Building Query Tokens and Performing Search Operation

In the boolean adjacent vertex search, we need to build the boolean expression query tokens according to the requirements. There are usually three kinds of operations: the disjunction operation ([disjunction]), the conjunction operation ([conjunction]), and the negation operation [(.sup.~]), where the disjunction and conjunction are binary operations. Let M = {[a.sub.1], ... , [a.sub.t]} be an integer set of keywords, and M is the subset of the set A. A boolean query expression on M is expressed as B(M) = [a'.sub.1] # [a'.sub.2] #...# [a'.sub.t] , where # is a binary operator, and # [member of] {[disjunction],[and]}. And [a'.sub.i] comes from the set {[a.sub.i], [[??].sub.i], [empty set]}, where [[??].sub.i] denotes the negation of [a'.sub.i], [empty set] represents null value, and 1 [less than or equal to] i [less than or equal to] t .

For a conjunctive expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [a.sub.j], is a positive keyword, we define [18]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where K and Y are respectively the secret key and the set of vertex labels in the conjunctive expression, and [absolute value of Y] strictly positive integers [{[X.sub.j]}.sub.1[less than or equal to]j[less than or equal to][absolute value of Y]] are picked at random. The query user sends [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to the cloud server. For each adjacent vertex vector [N.sub.i](1 [less than or equal to] i [less than or equal to] n), the cloud server performs the following computing [18]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If Y [intersection] [D.sub.i] = Y then: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [N.sub.i] matches the conjunctive expression. Else no adjacent vertex vector matches the conjunctive expression.

The disjunctive (conjunctive) expression consisting only of a finite number of words is called simple disjunctive (conjunctive) expression. The disjunctive (conjunctive) expression consisting only of a finite number of simple conjunctive (disjunctive) expressions is called disjunctive (conjunctive) normal form. Any propositional formulae has an equivalent disjunctive normal form and an equivalent conjunctive normal form. In boolean logic, a disjunctive normal form is a standardization of a logical formula which is a disjunction of conjunctive clauses [12]. In our paper, we only consider our boolean query expression as disjunctive normal form: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and s is is the number of conjunction expressions.

The correctness analysis of search operation:

We now consider each conjunction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in boolean query expression B(M). We split [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into two partitions such that: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represent respectively the labels of positive and negative keywords in the boolean query expression [18]. In our scheme, we introduce a new vertex which is the adjacent vertex of each graph vertex by default. To this extent, we add an arbitrary value [lambda] in the vertices set. After the Gram-Schmidt process, we have one normalized orthogonal vector y transforming from X. For each r [member of] [1,s], we pick at random [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] positive integers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] negative integer [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and one random positive integer pr. we define [18]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The reason why we have introduced a vector y is for the case where there are only negative vectors in the conjunction [C.sub.r]. Otherwise, [Q.sub.r] would have to be divided by 0. Thus we get the query token:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When the user sends the query token Q to the cloud server, for each adjacent vertex vector [N.sub.i], the following computations can be performed:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, the results of [theta](Q, [N.sub.i]) are interpreted as follows:

* If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an empty set, then [theta]([Q.sub.r], [N.sub.i]) 1, the graph vertex [a.sub.i] matches the query token, and we add vi to the search results set R.

* If [for all]r [member of] [1, s], [theta]([Q.sub.r], [N.sub.i]) [not equal to] 1, no graph vertex matches the query token.

After the retrieval, the cloud server can return the search results set R to the query user.

5. Security Analysis

In this section, we analyze the security of our proposed scheme. We first introduce some notions used in [17] and adapt them for our BASG scheme.

History: A history is an interaction between the user and the cloud server, including a graph G and a set of q boolean expression queries {B([M.sub.1]),..., B([M.sub.q])}, denoted as [H.sub.q] = (G, B([M.sub.1]),..., B([M.sub.q])). The partial history is denoted as [H.sup.t.sub.q] = (G, B([M.sub.1]),., B([M.sub.t])), where t [less than or equal to] q.

View: Let [H.sub.q]= (G, B([M.sub.1]), ..., B([M.sub.q])) is a history

over q queries, the cloud server can only see an encrypted version of the history, let [G.sub.E] = Enc(K,G), and B([M.sub.i]) is encoded as [Q.sub.i] (1 [less than or equal to] i [less than or equal to] q), a view is denoted as [V.sub.k]([H.sub.q]) =([G.sub.E], I, [Q.sub.1],..., [Q.sub.q]). The partial view is [V.sup.t.sub.k]([H.sub.q]) = ([G.sub.E], I, [Q.sub.1],..., [Q.sub.t]), where t [less than or equal to] q.

Access Pattern: Let [H.sub.q] = (G, B([M.sub.1]), ..., B([M.sub.q])) is a history over q queries, the access pattern is the tuple [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the returned search results matching to the query token [Q.sub.i].

Search Pattern: Let [H.sub.q]= (G, B([M.sub.1]), ..., B([M.sub.q])) is a history over q queries, the search pattern is a binary symmetric matrix [[PI].sub.q] , such that [[PI].sub.q] [i,j] =1 if B([M.sub.i]) = B([M.sub.j]), and [[PI].sub.q] [i,j]= 0 otherwise, for 1 [less than or equal to] i, j [less than or equal to] q.

Trace: Let [H.sub.q]= (G, B([M.sub.1]),..., B([M.sub.q])) is a history over q queries, the trace is the sequence [T.sub.r]([H.sub.q])=([absolute value of G], R([H.sub.q]), [[PI].sub.q]). Where [absolute value of G] is the total size of the graph G, R([H.sub.q]) and [[PI].sub.q] are access pattern and search pattern of history [H.sub.q] respectively. The trace of partial history is denoted as [T.sub.r]([H.sup.t.sub.q]) = ([absolute value of G], R([H.sup.t.sub.q]), [[PI].sub.t]), where t [less than or equal to] q.

In our scheme, the security goal is to protect user privacy by preventing the cloud server from learning information of the graph data, the index and the query tokens. Here we adapt the simulation-based security model of [17], and prove our scheme meets the adaptive semantic security. The adaptive attack model considers the adversary (i.e., the cloud server) can choose the search request based on the search tokens and search outcomes of previous searches [17].

For the security guarantee, we follow the security definition used in the previous SSE schemes [17]. And our security strength is captured by the statement that given two histories with the identical trace, the cloud server cannot distinguish the views of the two histories [16][17]. In other words, the cloud server is not able to learn additional knowledge beyond the contents we are ready to leak (i.e., the trace) and thus our scheme is secure. The security theorem for our boolean adjacent vertex search scheme is stated below.

Theorem 2. Our boolean adjacent vertex search scheme meets adaptive semantic security.

Proof We are going to describe a polynomial-size simulator S. For all q [member of] N, given only the trace of a partial history, the randomly chosen key K, given trace [T.sub.r]([H.sup.t.sub.q]) for all 0 [less than or equal to] t [less than or equal to] q, the simulator S can generate a view ([V.sup.t.sub.q]) such that ([V.sup.t.sub.q]) is computationally indistinguishable from the adversary's view [V.sup.t.sub.q]([H.sub.q]).

For t = 0, the simulator S constructs the index I* = {[N.sup.*.sub.1], ..., [N.sup.*.sub.n]} on the partial history [T.sub.r]([H.sup.0.sub.q]), where [N.sup.*.sub.i] [??] [{0,1}.sup.m] (m represents the modulus of vector of graph vertex) for all 1 [less than or equal to] i [less than or equal to] n. And the simulator generates at random [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The simulator S keeps a copy of I* to be able to simulate future partial views for 1 [less than or equal to] t [less than or equal to] q. It is obvious that I* is indistinguishable from I, and the encrypted graph [G.sup.*.sub.E] in ([V.sup.t.sub.q])* is indistinguishable from GE in [V.sup.t.sub.k]([H.sub.q]), otherwise one could either distinguish between the output of the pseudorandom permutation and a random string of the same size, or between the output of a semantically secure symmetric encryption scheme and a random string of the same size. Thus, (Vq0)* is indistinguishable from [V.sup.0.sub.k][H.sub.q]).

For 1 [less than or equal to] t [less than or equal to] q, [T.sub.r] ([H.sub.q]) contains the search pattern matrix [[PI].sub.t] for t queries. The simulator S will construct the query tokens ([Q.sup.*.sub.1],..., [Q.sup.*.sub.t]) included in ([V.sup.t.sub.q])*. In the query tokens construction, the simulator S reuses the tokens (Q*i,..., Q*t-1) that were included in (Vqi-1)*, and we assume that the simulator S remembers ([V.sup.t-1.sub.q])* and can reuse the query information in it, alternatively, S can reconstruct these query tokens from [T.sub.r]([H.sup.t-1.sub.q]).

To construct [Q.sup.*.sub.t], the simulator S first checks if [H.sup.t-1.sub.q] contains B(Mt) by checking if [[PI].sub.t] [t,j] = 1 for any 1 [less than or equal to] j [less than or equal to] t - 1. If [H.sup.t-1.sub.q] cannot contain B([M.sub.t]), Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the vertex set of B([M.sub.t]), for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] builds an association between each entry of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a value of generated I* at the previous phase such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are pairwise distinct. The simulator S creates [Q.sup.*.sub.t] [??] [{0,1}.sup.m] x [{0,1}.sup.y] (y represents the number of disjunctions of the simulator's guess) such that for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] there exists at least one 1 [less than or equal to] x [less than or equal to] y such that the inner product [theta]([Q.sup.*.sub.t,x], [N.sup.*.sub.i]) = 1. Otherwise, if [H.sup.t-1.sub.q] contains B([M.sub.t]), then the simulator S retrieves the query information associated with B([M.sub.t]) and assigns it to [Q.sup.*.sub.t]. This makes sure that if [H.sup.t.sub.q] contains repeated boolean expression queries, then the query information included in ([V.sup.t.sub.q])* are identical.

It's easy to see that the query tokens ([Q.sup.*.sub.1],..., [Q.sup.*.sub.t]) in ([V.sup.t.sub.q])* are indistinguishable from the query tokens ([Q.sub.1],..., [Q.sub.t]) in [V.sup.t.sub.k]([H.sub.q]), otherwise one could distinguish between the output of the pseudorandom permutation and a random string of the same size. For all 0 [less than or equal to] t [less than or equal to] q, we claim that there is no probabilistic polynomial-size adversary can distinguish between ([V.sup.t.sub.q])* and [V.sup.t.sub.k]([H.sub.q]). Thus, we have proven the theorem.

6. Experimental Evaluations

In this section, we demonstrate a thorough experiment evaluation of our scheme on the Enron email network graph [36][37]. Our experiment is using C programming language on both a local machine and a cloud server. The local machine has an Intel Core 2 CPU running at 2.4 GHz, and 4GB of RAM. The cloud server is equipped with 4 CPU cores (2 x 2.8 GHz) and 8GB of RAM. In our experiment the dominant factor affecting the performance is the search phase as it is performed by the cloud server for the query users. The performance of the index construction, query tokens generation, and decryption actions in our scheme are evaluated on the client side.

In our experiment evaluation, we consider two cases about the graph data picked randomly from real network graph set, one contains more edges (shorter form GME), the other contains less edges (shorter form GLE) which have about half of the number of the former's edges. To facilitate comparison, we take the same boolean query in all these searches, and do not take into account the query construction time or the network communication overheads.

6.1 Building Index

In our index construction scheme, we first need to build the normalized orthogonal vectors of graph vertices by Gram-Schmidt algorithm. And then we construct the adjacency vector of each graph vertex and the secure index of our scheme.

The results of normalized orthogonal vectors construction can be seen in Fig. 4. The T-axis of Fig. 4 represents the runtime of normalized orthogonal vectors construction. The X-axis of Fig. 4 shows the number of vertices. As can be seen in the figure, the normalized orthogonal vectors construction time is almost linear in the number of vertices in the graph.

The index construction process includes orthogonal vectors building and secure index constructing, and the orthogonal vectors building is related to the number of vertices. The index construction results are plotted in Fig. 5, where the 7-axis shows the runtime of index construction, and the X-axis represents the number of vertices in the graph. As can be seen in Fig. 5, the index construction time varies almost linearly with the number of vertices in the graph. Generally, the graph that contains more edges should possess more adjacent vertices, and the index construction time of GME is more than that of GLE. Similarly, as shown in Fig. 6, the size of index construction is nearly linear to the number of vertices, and the index size of GLE is less than that of GME.

6.2 Performing Query

In the query operation, the cloud server executes the retrieving process by the encrypted query tokens and the search index. The query process includes the retrieving on server side and the decryption of search results. The retrieving results are plotted in Fig. 7, where the 7-axis represents the query time, and the X-axis shows the number of vertices in the graph.

As shown in Fig. 7, the query time is almost linear to the number of vertices in the graph, and the query of GLE has the higher efficiency than that of GME. This shows that the query will cost more time with the increasing number of edges in the graph under the same vertices.

After retrieving, the cloud server returns the encrypted search results to the query user, and the user will decrypt the results using the keys. The decryption process is illustrated in Fig. 8, where the 7-axis shows decryption time, and the X-axis represents the number of vertices in the graph. Fig. 8 shows the time cost to decrypt the search results. The decryption operation is mainly determined by the size of the search results, and the decryption process is closely related to the number of vertices and edges of the graph.

As can be seen from Fig. 5 and 6, the index construction cost is acceptable given that these computations and operations are performed on the client side. Considering much more complex queries can be handled, the search cost on the server side would be acceptable.

7. Conclusion

In this paper, we propose a solution to achieve the secure boolean adjacent vertex search over encrypted graph data in cloud computing. As the cloud server cannot be fully trusted, to accomplish the boolean adjacent vertex search, we make use of searchable encryption and the Gram-Schmidt algorithm to construct our scheme. With the implementation of our mechanism, the query users can achieve the more complex boolean adjacent vertex search. We formally prove the security in the paper. The experiment results with a real graph data set demonstrate efficiency of our scheme.

As our future work, we aim to build a dynamic boolean adjacent vertex search scheme which could support the operations of dynamic graph and meet the requirement of scalability. Another interesting direction is to build an efficient scheme which meets key management and update in an asymmetric setting.

http://dx.doi.org/10.3837/tiis.2016.10.029

Acknowledgement

The authors gratefully acknowledge the reviewers' comments and helpful suggestions.

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Hong Zhu received her B.S. degree, M.S. degree and Ph.D. degree from Huazhong University of Science and Technology in 1987, 1990 and 2001, respectively. Currently, she is a professor and a Ph.D. supervisor in School of Computer Science and Technology, Huazhong University of Science and Technology in China. Her main research areas are big data management technology, big data security technology, database theory and implementation technology, and database security technology.

Bin Wu received his B.S. degree from Shandong Normal University in 2003 and the M.S. degree from Dalian University of Technology in 2009. Currently he is a Ph.D. student of School of Computer Science and Technology in Huazhong University of Science and Technology. His research interests include database security, network security, and cloud security.

Meiyi Xie received her B.S. degree, M.S. degree and Ph.D. degree from Huazhong University of Science and Technology in 1996, 1999 and 2009, respectively. Currently, she is a lecturer in School of Computer Science and Technology, Huazhong University of Science and Technology in China. Her main research areas are modern database theory and technology, database security, and cloud security.

Zongmin Cui received the B.E. degree from Southeast University in 2002 and the M.S. degree from Huazhong University of Science and Technology in 2006. He received the Ph.D. Degree from Huazhong University of Science and Technology in 2014. He is currently an associate professor with the School of Information Science and Technology, Jiujiang University. His research interests include cloud security, data query and publish/subscribe system.

Hong Zhu (1), Bin Wu (1), Meiyi Xie (1) and Zongmin Cui (2)

(1) School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan, China

(2) School of Information Science and Technology, Jiujiang University, Jiujiang, China [e-mail: wubincs@gmail.com]

* Corresponding author: Bin Wu

Received December 8, 2015; revised March 29, 2016; revised May 21, 2016; accepted August 13, 2016; published October 31, 2016

Table 1. Summary of notations

Notations                     Denotations

G                          A graph data set

I                           A secure index

A                A set of all vertices in the graph G

W               A set of normalized orthogonal keywords
                   from all vertices in the graph G

n                The number of vertices in the graph G

m                  The maximum degree of the graph G

Q                            A query token

R                       A set of search results

1(key, *)           A pseudorandom permutation used
                        in our scheme designing

Enc(key, *)    A semantic security symmetric encryption
                 function used in our scheme designing

Dec(key, *)    A semantic security symmetric decryption
                 function used in our scheme designing
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