# Identity-Based Encryption Technique with Ranking for Multi-Keywords Search in Cloud (IDERMKS).

1. INTRODUCTIONAs the volume of data keep increasing tremendously, the challenge is not only on how to store and manage these data but also on how to effectively and efficiently analyze these data to gain knowledge in making smart decision [1] [2] [3] and also on the security and privacy issues of these data.

Cloud computing has recently emerged as the promising technology for handling big data. Cloud computing is a network-based environment that centers on sharing computations or resources. Today, Cloud computing is one of the most thrilling technologies due to its ability to decrease costs relating to computing while increasing flexibility and scalability for computer processes. It is a revolutionary technology that has influenced how computer hardware and software are designed [4]. It provide enormous benefit like on demand and decreased cost of usage, easy access, quick deployment, flexibility, resource management etc. These benefits have accounted for different data owners outsourcing voluminous data onto the cloud. IT organizations have articulated concern about critical security issues that exist with the widespread implementation of cloud computing.

Security in cloud has been one of the most argued-about issues in the field of cloud computing. The risks of compromised security and privacy may be lesser when the data were to be stored on individual machines instead of in cloud. This has made privacy preserving a very important concern in cloud computing. Although cloud service providers are able to deliver highly available storage and massively parallel computing resources, the security and privacy of data stored on the cloud is a big challenge since the server might be curious or might illegally inspect and access user's sensitive data and also unauthorized users may be able to intercept people's data. Outsourcing data into the cloud is economically attractive for the cost and complexity of long-term large-scale data storage, however, it does not offer any guarantee on data integrity and availability. This problem may impede the successful deployment of the cloud architecture. Traditional cryptographic primitives for the purpose of data security protection cannot be directly adopted on data as users no longer possess the storage of their data physically. Thus, how to efficiently verify the correctness of outsourced cloud data without the local copy of data files becomes a big challenge for data storage security in Cloud Computing. Therefore, to fully guarantee the security of the data and save the cloud users' computation resources, it is of appropriately importance to enable public auditability for cloud data storage so that the users may resort to a Third Party Auditor (TPA), who has the expertise and capabilities to audit the outsourced data when needed[5].

For the outsourced data not to be leaked to external parties, exploiting data encryption before outsourcing is one way to alleviate this privacy concern. However, without an appropriate designed auditing protocol, encryption itself cannot prevent data from "flowing away" towards external parties during the auditing process. Encryption of the data helps in protecting data confidentiality of the user but searching for data also becomes a challenge. Secure search over encrypted data was first initiated by Song et al [6]. They proposed a cryptographic primitive concept called searchable encryption which enables users to perform a keyword-based search on an encrypted data, just as on plaintext data.

Wang et al [7] were the first to define secure search over encrypted cloud data, however, further development has been made by [8], [9], [10], [11] which incur high storage and computational cost. These researches and more, seeks to reduce computation and storage cost and also enrich the category of search functions such as fuzzy keyword search, secure ranked multi-keyword search and similarity based search but they are limited to single-owner model.

In 1984, Shamir [12] designed a public key encryption scheme in which the public key can be an arbitrary string. Shamir proposed the idea of identity-based cryptography in such a scheme there are four algorithms: Setup, Extract, Encrypt and Decrypt. The notion of identity based cryptosystem is that the public key can be an arbitrary string. Shamir's original motivation for identity-based encryption was to simplify certificate management in e-mail systems. Since the problem was posed in 1984 there have been several proposals for IBE schemes [13][14][15][16]. Boneh and Franklin in 2001, designed the first practical identity-based cryptosystem [17]. Zhang et al [18] defined a multi-owner model for privacy preserving keyword search over encrypted cloud data, however, their scheme did not use an identity based encryption scheme. A similar system was proposed in [19] of which a cryptographic techniques, query, response randomization and ranking capability was used. However, IDE was not used in their system and the approach used in the scheme formulation is quite different from our scheme.

The literatures reviewed above elaborate various concepts on cloud computing, security in cloud, auditable cloud data, searchable encryption, ranked multi-keyword search, privacy preserving, and identity-based cryptography. However, none of these literatures considered combining these concepts.

In our work, we propose an Identity-based technique with ranking for multi-keywords search (IDERMKS) in cloud. The framework of our scheme and its security requirements are defined. We prove that our proposed scheme satisfies the ciphertext and trapdoor indistinguishability in the random oracle [20][21]. Finally, we demonstrate the advantage of our IDERMKS scheme by comparing with previous PEKS schemes.

This paper is organized as follows: Section II outlines the system model, threat model, design goals and the architectural design, Section III defines the preliminaries, Section IV outlines the algorithms and the security requirements of our scheme. We present the proposed IDERMKS scheme and its security analysis in Section V. Section VI, we briefly outline the privacy preserving ranking method which we adopted from [18], Section VII gives the performance analysis of our scheme and we conclude in section VIII.

2. SYSTEM AND THREAT MODEL, DESIGN GOALS AND ARCHITECTURAL DESIGN

In this section, we describe the system model, threat model, design goals, and gives the architectural design of our proposed scheme.

2.1 System Model

In the proposed IDERMKS scheme, we have three entities namely; data owner, data user and cloud server. The data user has a collection of files that have to be outsourced to the cloud. Before outsourcing, these files need to be encrypted in order to ensure the confidentiality of the files. To enable efficient and adequate search operation on these files, the data owner builds a secure searchable index on keywords sets extracted from the files. The data owner then outsource the encrypted files together with their indexes to the cloud server. When a data user wants to search keywords over these encrypted files, the data user submit the hashed keywords to the data owner which are then used by the data owner to create a trapdoor for the data user. Upon receiving the trapdoor, the data user submit it to the cloud. The cloud server then searches through the encrypted indexes of each data owner and return the corresponding set of files for the indexes that matches and have the same pattern as that of the trapdoor. The cloud then rank these files based on their relevance score and return the top-k important files to the data user. The data user can then obtains a decryption key from the data owner to decrypt the file.

2.2 Threat Model

In the threat model, we assume that the cloud server is honest but curious. The server follows our protocol but keen to know the content of the encrypted files, keywords and the relevance scores which is the same as in [18]. The data owner and data users who are authenticated by PKI are assumed to be trusted.

2.3 Design Goals

In order to ensure the privacy preserving of the ranked multi-keyword search, the proposed IDERMKS scheme should satisfy some of the security and performance goals in [18]. The proposed IDERMKS should;

a) Enable multi-keyword search over encrypted files from different data owners.

b) Not hinder new data owners from entering the system, thus, providing data user scalability.

c) Ensure that authenticated data users perform correct and appropriate search

2.4 Architectural Design

Our proposed scheme enables a data user to perform a search over encrypted cloud data by making a multi-keyword search query which protect the system-wide privacy in cloud computing. The use of keyword relevance is an intermediate similarity semantic that has been selected among numerous semantics, in which a number of keywords from the search query is used. The appearance of keywords in the file will be used to measure the relevance of that file to the query. The ranking system used in our scheme proves to be very effective and efficient in implementing and returning the highly relevant files to terms submitted in the search query. Ranked based search can appropriately eliminate unnecessary network traffics by returning only the top relevant files which are of user's interest. Figure 1 shows the architecture of the proposed IDERMKS that outlines the various processes and entities of the scheme.

3. PRELIMINARIES

This section gives brief review of the various concepts of bilinear pairing, identity-based encryption and other related mathematical problems used in this paper.

3.1 Bilinear Pairing

Let ([G.sub.1], +) and ([G.sub.2],.) be cyclic groups of prime order q. Let [g.sub.1] and [g.sub.2] be the generator of [G.sub.1] and [G.sub.2] respectively. Let e be a bilinear map defined as e: [G.sub.1] X [G.sub.1] [right arrow] [G.sub.2] which satisfies the following conditions:

a) Bilinearity: If [for all]P, Q [member of] [G.sub.1] and a, b [member of] [Z.sub.q.sup.*], then e(aP, bQ) = e[(P, Q).sup.ab]

b) Non-degenerate: e(P,Q) [not equal to] 1

c) Computability: There is an efficient algorithm to compute e(P, Q), [for all]P, Q [member of] [G.sub.1]

3.2 Identity-Based Encryption

An identity-based scheme has four algorithms: Setup, Extract, Encrypt and Decrypt.

a) Setup: It takes a security parameter k and returns system parameters (params) and a master-key. The system parameters include a description of finite message space M and a description of a finite cipher text space C. Ideally, the system parameters will be publicly known, while the master-key will be known only to the Private Key Generator (PKG).

b) Extract: It take as input params, master-key and an arbitrary string ID which will be used as the public key ID [member of] [{1,0}.sup.*] and d is the corresponding private decryption key. The extract algorithm extract a private key from the given public key.

c) Encrypt: It take param, ID and M e M as input and returns C [member of] C.

d) Decrypt: It takes as input param, C [member of] C and and private key d and returns M [member of] M. These algorithm must satisfy the standard constraint such that if d is the private key generated by the Extract algorithm when given the public key ID, then [for all]M [member of] M: Encrypt(param, ID, M) = C and Decrypt(param, C, d) = M

3.3 Other mathematically related problems and assumptions. Here, we give two mathematical hard problems and also define their corresponding assumptions

a) Bilinear Diffie-Hellman (BDH) problem: The BDH problem in (e,[G.sub.1],[G.sub.2]) is defined as follows: Given P, aP, bP, cP [member of] [G.sub.1] for some a, b, c [for all] [Z.sub.q.sup.*], compute e[(P, P).sup.abc] [member of] [G.sub.2].

b) Computational Diffie-Hellman (CDH) problem: The CDH problem in (e,[G.sub.1],[G.sub.2]) is defined as follows: Given P, aP, bP [member of] [G.sub.1] for some a, b [member of] [Z.sub.q.sup.*], the CDH problem is to compute abP [member of] [G.sub.1]

Definition 1: Given [??]P, aP, bP, cP[??] [member of] [G.sub.1] for some a, b, c [member of] [Z.sub.q.sup.*], the BDH assumption is that, there is no probabilistic polynomial-time adversary A with a non-negligible probability that can compute e[(P, P).sup.abc] [member of] [G.sub.2]. The adversary A is said to have an advantage

AdvBDH(t) = Pr[A(P, aP, bP, cP) = e[(P, P).sup.abc] ] [greater than or equal to] [epsilon] within a running time t.

Definition 2: Given [??]P, aP, bP[??] [member of] [G.sub.1] for some a, b [member of] [Z.sub.q.sup.*], the CDH assumption is that, there is no probabilistic polynomial-time adversary A with a non-negligible probability that can compute abP [member of] [G.sub.1]. The adversary A is said to have an advantage AdvCDH(t) = Pr[A(P, aP, bP) = abP] [greater than or equal to] [epsilon] within a running time t.

4. ALGORITHMS AND SECURITY REQUIREMENTS

We define the algorithms and security requirements for the proposed IDERMKS by modifying the ones in [8] [17] [22].

4.1 Algorithms

An IDERMKS scheme consist of seven (7) algorithms, outlined as follows:

a) Setup algorithm: This is a probabilistic algorithm which takes as input, a security parameter l. It then output a master secret key and a master public key Ppub.

b) Key extract algorithm: This is a deterministic algorithm which takes a user's identity, a master secret key and system parameters as input and the output a user Secret key skID.

c) IDERMKS algorithm: This is a probabilistic algorithm which takes a public key of the data owner, a set of keywords in a document and system parameters as input and then output IDERMKS ciphertext which is the searchable index I. The data owner encrypts each document with Symmetric-Key Encryption method using different keys for each document. The data owner then encrypts the symmetric keys with a Public Key Encryption which has blinding capabilities and in our case an Identity-Based Encryption method will be used.

d) Trapdoor generation algorithm: This is a probabilistic algorithm which takes as input a multiple keywords, data user's secret key and system parameters and then generate the trapdoor Tw.

e) Test algorithm: This is a deterministic algorithm which takes as input the IDERMKS ciphertext I, trapdoor [T.sub.w] and system parameters. Upon receiving a query request [T.sub.w] from the user, the cloud server match the queried keywords from all keywords stored on it against the query request. It then extract all files that contain the query request to obtain a candidate file set.

f) Ranking algorithm: The cloud ranks the files in the candidate file set obtained after running the testing algorithm and find the top-k relevant file. An order and privacy preserving encoding scheme which encodes the relevance score to obtain the top-k search result will be adopted into our IDERMKS scheme.

g) Retrieval algorithm: This algorithm takes a blinded encrypted symmetric key as input and output a blinded symmetric key which was used to encrypt a document.

4.2 Security Requirements

An IDERMKS scheme must meet the following security requirements: (1) ciphertext indistinguishability and (2) trapdoor indistinquishability.

4.2.1 Ciphertext indistinguishability

We define a security for Indistinguishability of Ciphertext from Ciphertext of Chosen Keyword Attack (IND-CC-CKA) of the IDERMKS scheme. The Game interaction between an adversary A and a challenger C below defines the security of our scheme.

Game 1--

a) Setup: The Challenger C runs the setup algorithm to generate the master private key and a master public key [P.sub.pub]. C generate A's private key skIDA by running the key extract algorithm with the corresponding public key for the identity IDA. The C's master public key [P.sub.pub] and the system parameters are given to the adversary A. The challenger then keeps the master private key msk and gives A's private key skIDA to him.

b) Phase 1: The adversary A may make series of different queries to challenger C in an adaptive manner as follows:

* Key generation queries: The adversary gives an identity IDA to the Challenger. The challenger returns skIDA to A by running the key extract algorithm.

* Trapdoor queries: A makes this queries for keywords W, the challenger C returns the trapdoor Tw for keyword W to A by running the Trapdoor Generation algorithm.

c) Challenge: The adversary A sends ([ID.sub.c.sup.*], [W.sub.0.sup.*][W.sub.1.sup.*]) to the Challenger C where [W.sub.0.sup.*] and [W.sub.1.sup.*] are two challenged keywords. C chooses a random value b [member of] {0,1} and uses [W.sub.b.sup.*] to generate IDERMKS ciphertext [I.sup.*] by running the IDERMKS algorithm then send the result ciphertext to the adversary A.

d) Phase 2: Adversary A continues to make the key extract queries for any identity IDi and the trapdoor query for any keyword [W.sup.i] to the challenger C subject to the restriction that IDi [not equal to] ID[c.sup.*] and [W.sup.i][not equal to]{[W.sub.0.sup.*] or [W.sub.1.sup.*]}.

e) Guess: Finally, the adversary A output a guess of a value b e {0,1}. A wins the game if b = b from the game above, we define the advantage of the adversary A in breaking the ciphertext indistinguishability as the probability that the adversary A wins.

Definition 3: An IDERMKS scheme is said to have satisfied ciphertext indistinquishability against adaptive chosen plaintext attack if no adversary have a non-negligible advantage in Game 1.

4.2.2 Trapdoor indistinguishability

This means that an adversary is unable to distinguish the trapdoor for two challenged keywords chosen by him. A Game between a probabilistic polynomial time adversary A and a challenger C has been defined in Game 2 to represent the trapdoor indistinguishability.

Game 2--

a) Setup: This phase is same as defined in Game 1.

b) Phase 1: In this phase, the Adversary A makes a number of dissimilar of queries to the Challenger C in an adaptive manner. This phase is also same as one defined in Game1.

c) Challenge: The adversary A sends ([ID.sub.c.sup.*], [W.sub.0.sup.*][W.sub.1.sup.*]) to the Challenger C where [[W.sub.0.sup.*].sup.*] and [[W.sub.1.sup.*].sup.*] are two challenged keywords. C chooses a random value b [member of] {0,1} and uses [W.sub.b.sup.*] to compute a trapdoor T[W.sub.b.sup.*] by running the trapdoor algorithm. The restrictions are that IDi [not equal to] ID[c.sup.*] and [W.sup.i] [not equal to] {[W.sub.0.sup.*] or [W.sub.1.sup.*]}. The Challenger then sends the trapdoor T[W.sub.b.sup.*] to A

d) Phase 2: This phase is same as one defined in Game 1.

e) Guess: Finally, the adversary A output a guess of a value b' [member of] {0,1}. A wins the game if b' = b.

From the game 2 above, we define the advantage of the adversary A in breaking the trapdoor indistinguishability as the probability that the adversary A wins.

Definition 4: An IDERMKS scheme is said to have satisfied trapdoor indistinquishability against adaptive chosen plaintext attack if no adversary have a non-negligible advantage in Game 2.

5. THE PROPOSED IDERMKS SCHEME AND ITS SECURITY ANALYSIS

5.1 The Proposed IDERMKS Scheme

The proposed scheme consist of seven algorithms: System Setup, Key Extract, IDERMKS, Trapdoor Generation, Test, Ranking and Retrieval. Details of scheme is outlined below:

a) Setup: Given l as a security parameter, a trusted Private Key Generator (PKG) runs the setup algorithm as follows in order to generate a master private key and a master public key. Let [G.sub.1] be an additive cyclic group generated by P with a prime order of q and Let [G.sub.2] be a multiplicative cyclic group with same prime order q. The PKG generate a bilinear map e: [G.sub.1] X [G.sub.1] [right arrow] [G.sub.2]. The PKG randomly chooses a master secret key x [member of] [Z.sub.q.sup.*] and compute the system public key [P.sub.pub] as [P.sub.pub] = x.P. Let [H.sub.1], [H.sub.2] and [H.sub.3] be three cryptographic hash function such that [H.sub.1]: [{0,1}.sup.*] [right arrow] [G.sub.1.sup.*], [H.sub.2]: [{0,1}.sup.*] [right arrow] [G.sub.1.sup.*] and [H.sub.3]: [G.sub.2] [right arrow] [{0,1}.sup.n] where n is a fixed length depending on l. The algorithm then publishes the system's parameters {[G.sub.1], [G.sub.2], e, [H.sub.1], [H.sub.2], [H.sub.3], P, Ppub, n, params} and the master secret keys x is kept secret.

b) Key Extract: Given a user's identity [ID.sub.i] [member of] [{0,1}.sup.*], the PKG computes user's private key as skID, [left arrow] XQ[ID.sub.i] where [QD.sub.i] [LEFT ARROW] [H.sub.1]([ID.sub.i]).

c) IDERMKS: The data owner chooses a random value a [member of] [Z.sub.q.sup.*] and compute the searchable index on a set of keywords for a document as I = (U, V) where U = aQiD and V = [H.sub.3](k) and k is computed as k = e(U, Ppub)e([H.sub.2]([W.sub.i]), P).

d) Trapdoor Generation: Let W be the set of keywords that the data user want to search for. The data user compute t = [H.sub.2](W) and sends it to the data owner. The data owner then generate the trapdoor Tw = askiD + t and sends it to the data user.

e) Test: Given the IDERMKS ciphertext index I and the trapdoor[T.sub.w], the cloud server runs the test algorithm to check if [H.sub.3](e([T.sub.w], P) = V then it will return file and it ID.

f) Ranking: We adopt a privacy preserving ranked search scheme implement in [18] into our IDERMKS scheme to facilitate the ranking of the candidate file to determine which file is more relevant to a certain keyword according to the encoded relevance scores. In this ranking scheme, the cloud server makes a comparison of the encoded relevant scores of the files without knowing the actual contents of the files.

g) Retrieval: Given the Identity Based Encryption of the symmetric key C = ([C.sub.i], [C.sub.2]) where [C.sub.i] = rP and C2 = [sk.(e(Ppub, QID).sup.r]) where r [member of] [Z.sub.q.sup.*] used to encrypt the document. The data user blind C by computing C' = [(C).sup.[alpha]] = ([C.sub.1.sup.[alpha]], [C.sub.2.sup.[alpha]]) and sends it to the data owner. The data owner returns [sk.sup.[alpha]] = [C.sub.2.sup.[alpha]].e[(r[alpha]Ppub, QiD).sup.-1] to the data user. The secret key can be deduced thereafter and be used to decrypt the encrypted document [23].

5.2 Security Analysis of the Proposed IDERMKS Scheme

We prove our IDERMKS system is a non-interactive searchable encryption scheme that is semantically secure in a random oracle model [20][21][22]. The proof of security relies on the difficulty of the BDH and CDH problem.

5.2.1 Ciphertext Indistinguishability

We demonstrate that the proposed IDERMKS scheme satisfies ciphertext indistinguishability under an adaptive chosen plaintext attack. For the security of our scheme to be simplified, we prove a lemma in which the adversary is assumed to be an outside attacker.

Lemma 5.2.1 We assume that is an adversary A with a non-negligible probability e1 can break the ciphertext indistinguishability of the proposed IDERMKS scheme in the random oracle under an adaptive chosen plaintext attack, then there exist a challenger C with a non-negligible probability [mathematical expression not reproducible] who can solve the BDH problem where qR and q[H.sub.3] represents the maximum numbers of making key extract and [H.sub.3] queries respectively.

Proof: The Challenger C is given an input of the Bilinear Diffie Hellman (BDH) parameters as {q,[G.sub.1],[G.sub.2], e} produce by G and a random instances (P, aP, bP, cP) = (P, [P.sub.1], [P.sub.2], [P.sub.3]) of the BDH problem for these parameters, that is P is random in [G.sub.1.sup.*] and a, b, c e [Z.sub.q.sup.*] where q is the order of [G.sub.1] and [G.sub.2]. Let D = e[(P, P).sup.abc] [member of] [G.sub.2] be the solution to the BDH problem. The Challenger C find D by interacting with the adversary A as follows:

a) Setup: The challenger C runs the setup algorithm to generate the public parameters {q, [G.sub.1], [G.sub.2], e, P, [P.sub.pub], [H.sub.1], [H.sub.2], [H.sub.3]} by setting QD = [P.sub.2] and Ppub = [P.sub.3]. [H.sub.1], [H.sub.2] and [H.sub.3] are random oracles controlled by C. The challenger C gives the public parameters to A. The challenger C generate A's private key skIDA by running the Key Extract algorithm with the public key begin IDA as skIDA = aQD = abP. The challenger keeps the secret key msk and A's private key skIDA is given to him.

b) [H.sub.1] queries: The adversary A can query the random oracle [H.sub.1] at any time in an adaptive manner. To respond to these queries, the Challenger C make a list of tuples < [ID.sub.i], Q.sub.i], [b.sub.i], coini >, called the [H.sub.1.sup.LIST] Until the adversary A makes queries to the oracle, [H.sub.1.sup.LIST] is initially empty. When the adversary A queries the oracle with IDi the Challenger C responds as follows:

i. If query IDi already exist on the [H.sub.1.sup.LIST] in the tuple < [ID.sub.i], [Q.sub.i], [b.sub.i], coin > then C responds with [H.sub.1]([ID.sub.i]) = [Q.sub.i] [member of] [G.sub.i.sup*]

ii. Otherwise C generates a random coin [member of] {0,1} so that Pr[coin.sub.i] = 0]= [delta] for some [delta] that will be determined later.

iii. The challenger C picks a random value [b.sub.i] [member of] [Z.sub.q.sup.*]. If [coin.sub.i] = 0, compute [Q.sub.i] = [b.sub.i]P [member of] [G.sub.1.sup.*]. If [coin.sub.i] = 1, compute [Q.sub.i] = [b.sub.i]QID [member of] [G.sub.1.sup.*]

iv. The challenger C adds the tuple < [ID.sub.i], [Q.sub.i], [b.sub.i], coin > to [H.sub.1.sup.LIST] and respond to A with [H.sub.1]([ID.sub.i]) = [Q.sub.i]. Note that [Q.sub.i] is uniform in [G.sub.1]* and is independent of A's current view as required.

c) [H.sub.2] queries: At any time, the adversary A can query the random oracle [H.sub.2]. To respond to the queries, the Challenger C maintains a list of tuple < [ID.sub.i], [W.sub.i], Q[W.sub.i], [x.sub.i] > called the [H.sub.2.sup.LIST]. The list is initially empty until the adversary A makes a query. When the adversary A queries the oracle [H.sub.2] for < [ID.sub.i], [W.sub.i] >, The challenger C respond as follows:

i. If < [ID.sub.i], [W.sub.i] > appears in the list [H.sub.2.sup.LIST], C respond with [H.sub.2]([ID.sub.i], [W.sub.i]) = Q[W.sub.i]

ii. Otherwise, the challenger C randomly select a value [x.sub.i] [member of] [Z.sub.q.sup.*] and compute Q[W.sub.i] = [H.sub.2]([ID.sub.i], [W.sub.i]) = [x.sub.i].P. Finally, C adds the tuple < [ID.sub.i], [W.sub.i], Q[W.sub.i], [x.sub.i] > in the list [H.sub.2.sup.LIST] and respond to the adversary A with [H.sub.2]([ID.sub.i], [W.sub.i]) = Q[W.sub.i]

d) [H.sub.3] queries: The adversary A can query the random oracle [H.sub.3] at any time. To respond to the query, the challenger C maintains a list of tuple < [m.sub.i], [n.sub.i] > called [H.sub.3.sup.LIST] as described as follows: Until the adversary A queries the oracle [H.sub.3] for [m.sub.i], the list [H.sub.3.sup.LIST] is initially empty. When the adversary queries the oracle [H.sub.3], the challenger C responds as follows:

i. If [m.sub.i] appear in the list [H.sub.3.sup.LIST], the challenger C responds with [H.sub.3]([m.sub.i]) = [n.sub.i]

ii. Otherwise, the challenger C randomly select a value [n.sub.i] [member of] [{0,1}.sup.n] and set [H.sub.3]([m.sub.i]) = [n.sub.i]. Finally, C add the pair < [m.sub.i], [n.sub.i] > to the [H.sub.3.sup.LIST] and responds to the adversary A with [H.sub.3]([m.sub.i]) = [n.sub.i].

e) Phase 1: Let [ID.sub.i] be a private key extraction query issued by A. The Challenger C responds as follows:

i. C runs the algorithm for responding to [H.sub.1]-Queries to obtain a [Q.sub.i] [member of] [G.sub.1.sup.*] such that [H.sub.1]([ID.sub.i]) = [Q.sub.i] and let < [ID.sub.i], [Q.sub.i], [b.sub.i], [coin.sub.i] > be the corresponding tuple on the [H.sub.3.sup.LIST]. If [coin.sub.i] = 1, then C reports failure and abort. The attack on IDERMKS failed.

ii. If coin = 0, C gets Qi = biP and Define sk[ID.sub.i], = [b.sub.i]Ppub [member of] [G.sub.1.sup.LIST]. Observe that skID, = c[Q.sub.i] and therefore sk[ID.sub.i], is the private key associated to public key [ID.sub.i]. The Challenger then sends sk[ID.sub.i] to A.

f) Trapdoor queries: When the adversary A issues a query for the trapdoor for keyword < IDi, wi >, the Challenger C then access the corresponding tuples < IDi, [Q.sub.i], [b.sub.i], [coin.sub.i] > and < [ID.sub.i], [W.sub.i], Q[W.sub.i], [x.sub.i] > in the [H.sub.1.sup.LIST] and [H.sub.2.sup.LIST] respectively and computes [T.sub.wi], = [x.sub.i]skID +t where [x.sub.i] [member of] [Z.sub.q.sup.*] and t = [H.sub.2]([W.sub.i]) and returns [T.sub.wi] to the adversary A.

g) Challenge: The adversary A sends ([ID.sub.c.sup.*], [W.sub.0.sup.*][W.sub.1.sup.*]) to the challenger C where [W.sub.0.sup.*] and [W.sub.1.sup.*] are two challenged keywords. Upon receiving ([ID.sub.c.sup.*], [W.sub.0.sup.*][W.sub.1.sup.*]) from A, the challenger C chooses a random value b [member of] {0,1} and access the corresponding tuple < [ID.sub.c.sup.*], [W.sub.0.sup.*], [W.sub.1.sup.*], x* > in [H.sub.2.sup.LIST] to generate an IDERMKS ciphertext [I.sup.*] = ([U.sup.*], [V.sup.*]) = ([U.sup.*], R), where R [member of] [{0,1}.sup.n] is a random value. The restrictions are that the adversary A did not make any private key extraction for [ID.sub.c.sup.*] and A did not also make a trapdoor query for [W.sub.0.sup.*] and [W.sub.1.sup.*]. Finally, the challenger C sends [I.sup.*] to the adversary A.

h) Phase 2: A can continue to make the key extract queries adaptively for any identity IDC and the trapdoor query for any keyword W to the challenger C subject to the restriction that IDc [not equal to] ID[c.sup.*] and w [not equal to] {[W.sub.0.sup.*] or [W.sub.1.sup.*]}.

i) Guess: Finally, the adversary A output a guess of a value b' [member of] {0,1}. A wins the game if b' = b.

By the assumption, A with a non-negligible probability e1 can distinguish the IDERMKS ciphertext [I.sup.*] under an adaptive chosen plaintext attack. Now, the challenger C picks a tuple < [m.sup.*], [n.sup.*] > in the [H.sub.3.sup.LIST] and outputs [v.sup.*] = [m.sup.*] / e([H.sub.2]([W.sub.b]), P) as the solution for the BDH instance (P, aP, bP, cP) for a,b,c e [Z.sub.q.sup.*]. In the following, we demonstrate that the output [m.sup.*]/e([H.sub.2](wb),P) is equal to e[(P, P).sup.abc] where U = aQID and [m.sup.*] = (e(U, Ppub)e([H.sub.2](wb),P)). The challenger C accesses the corresponding tuple < [ID.sub.c.sup.*], [W.sub.b.sup.*], Q[W.sub.b.sup.*], [W.sub.x.sup.*] > in the list [H.sub.3.sup.LIST] and computes

[m.sub.i.sup.*] / e([H.sub.3]([W.sub.b]), P) = e(U, [P.sub.pub]).e([H.sub.3]([W.sub.b]), P) / e([H.sub.3]([W.sub.b]), P) = (e(aQID, Ppub).e([H.sub.3]([W.sub.b]), P) / e([H.sub.3]([W.sub.b]), P) = (e(aQID, cP) = e(abP, cP) = e[(P, P).sup.abc]

By adopting the similar technique used in [22][24], we compute the probability of our IDERMKS scheme by discussing the probability that the challenger C does not abort during the simulation. Suppose the adversary A makes qR queries to the key extract query. In such an instance, the probability that the challenger C does not abort in phase 1 or 2 is [([delta]).sup.qR] and the probability that the challenger C does not abort during the challenged step is (1 - [delta]). Therefore, the probability that the challenger C does not abort during the simulation is [([delta]).sup.qR] x [(1 - [delta]).sup.qR]. This value can be maximize at [mathematical expression not reproducible]. By using [[delta].sub.abt], the probability that the challenger C does not abort is at least [mathematical expression not reproducible]. We note that the probability analysis uses the same techniques as Coron's analysis of the Full Domain Hash in [25]. The Challenger C outputs the correct D with the probability at least [mathematical expression not reproducible] [25] where qH3 denotes the total number of making [H.sub.3] queries. Therefore, the Challenger C with a probability [mathematical expression not reproducible] can solve the BDH problem. This contradict to the BDH assumption.

By Lemma 6.1, we obtain the following theorem.

Theorem 5.2.1: The proposed IDERMKS scheme satisfies the ciphertext indistinquishability against an adaptive chosen plaintext attack under the BDH assumption in the random oracle.

5.2.2 Trapdoor Indistinguishability

Lemma 5.2.2 In the random oracle model, we assume that if there is an adversary A with a non-negligible advantage that can break the trapdoor indistinguishability of the proposed IDERMKS scheme under the adaptive chosen keyword attack, then there exist a challenger C with a non-negligible advantage who can solve the computational CBH problem.

Proof: We assume that the challenger C receives a CDH instance of (P, aP, bP) for a, b [member of] [Z.sub.q.sup.*] where q is the order of [G.sub.1] and [G.sub.2]. By interacting with the adversary A, the challenger C will return the CDH solution abP in Game 2.

a) Setup: The Challenger C runs the setup algorithm to generate public parameters {q, [G.sub.1], [G.sub.2], e, P, Ppub, [H.sub.1], [H.sub.2], [H.sub.3]} where QID = bP Ppub = aP. [H.sub.1], [H.sub.2] and [H.sub.3] are random oracles controlled by C. The challenger C gives the public parameters to A. The Challenger C generate A's private key skIDA by running the Key Extract algorithm with the public key begin IDA as skIDA = XQID. The challenger C keeps the secret key msk and A' s private key skIDA is given to him.

b) [H.sub.1] queries: The adversary A can query the random oracle [H.sub.1] at any time in an adaptive manner. To respond to these queries, the Challenger C make a list of tuples < [ID.sub.i], [Q.sub.i], [b.sub.i], [coin.sub.i] >, called the [H.sub.1.sup.LIST] Until the adversary A makes queries to the oracle, [H.sub.3.sup.LIST] is initially empty. When the adversary A queries the oracle with [ID.sub.i] the Challenger C responds as follows:

i. If query IDi already exist on the [H.sub.3.sup.LIST] in the tuple < [ID.sub.i], [Q.sub.i], [b.sub.i], [coin.sub.i] > then C responds with [H.sub.1]([ID.sub.i]) = [Q.sub.i] [member of] [G.sub.1.sup.*]

ii. Otherwise C generates a random coin [member of] {0,1} so that Pr[coin.sub.i] = 0] = [delta] for some [delta] that will be determined later.

iii. The challenger C picks a random value [b.sub.i] [member of] [Z.sub.q.sup.*]. If [coin.sub.i] = 0, compute [Q.sub.i] = [b.sub.i]P [member of] [G.sub.1.sup.*]. If [coin.sub.i] = 1, compute Qi = [b.sub.i]QID [member of] [G.sub.1.sup.*] iv. The challenger C adds the tuple < [ID.sub.i], [Q.sub.i], [b.sub.i], [coin.sub.i] > to [H.sub.1.sup.LIST] and respond to A with [H.sub.1]([ID.sub.i]) = [Q.sub.i]. Note that [Q.sub.i] is uniform in [G.sub.1.sup.*] and is independent of A's current view as required.

c) [H.sub.2] queries: At any time, the adversary A can query the random oracle [H.sub.2]. To respond to the queries, the Challenger C maintains a list of tuple < [ID.sub.i], [W.sub.i], Q[W.sub.i], [x.sub.i] > called the [H.sub.2.sup.LIST]. The list is initially empty until the adversary A makes a query. When the adversary A queries the oracle [H.sub.2] for < [ID.sub.i], [W.sub.i] >, The Challenger C respond as follows:

i. If < [ID.sub.i], [W.sub.i] > appears in the list [H.sub.2.sup.LIST], C respond with [H.sub.2]([ID.sub.i], [W.sub.i]) = Qwi

ii. Otherwise, the challenger C randomly select a value x, [member of] [Z.sub.q.sup.*] and compute Qw, = [H.sub.2]([ID.sub.i], [W.sub.i]) = [x.sub.i].P. Finally, C adds the tuple < IDi, wi, Q[W.sub.i], [x.sub.i] > in the list [H.sub.3.sup.LIST] and respond to the adversary A with [H.sub.2](IDi, wi) = Q[W.sub.i]

d) [H.sub.3] queries: The adversary A can query the random oracle [H.sub.3] at any time. To respond to the query, the challenger C maintains a list of tuple < [m.sub.t], [n.sub.i] > called [H.sub.3.sup.LIST] as described as follows: Until the adversary A queries the oracle [H.sub.3] for [m.sub.i], the list [H.sub.3.sup.LIST] is initially empty. When the adversary queries the oracle [H.sub.3], the challenger C responds as follows:

i. If mi appear in the list [H.sub.3.sup.LIST], the challenger C responds with H 3(mi) = ni

ii. Otherwise, the challenger C randomly select a value ni [member of] [{0,1}.sup.n] and set H3(mt) = m. Finally, C add the pair < mi, ni > to the [H.sub.3.sup.LIST] and responds to the adversary A with H3(mi) = ni.

e) Phase 1: Let ID, be a private key extraction query issued by A. The Challenger C responds as follows:

i. C runs the algorithm above for responding to [H.sub.1]-Queries to obtain a [Q.sub.i] [member of] [G.sub.1.sup.*] such that [H.sub.1]([ID.sub.i]) = [Q.sub.i] and let < IDi, Qi, bi, [coin.sub.i] > be the corresponding tuple on the [H.sub.1.sup.LIST]. If coini =1, then C reports failure and abort. The attack on IDERMKS failed.

ii. If coini = 0, C gets Qi = biP and Define skIDi = biPpub [member of] [G.sub.1]. Observe that skIDi = cQi and therefore skm is the private key associated to public key IDi. The Challenger then sends skIDi to A.

f) Trapdoor queries: When the adversary A issues a query for the trapdoor for keyword < IDi, wt >, the Challenger C then access the corresponding tuples < IDi, Qi, bi, coin, > and < IDi, wi, Qw,, x, > in the [H.sub.1.sup.LIST] and [H.sub.2.sup.LIST] respectively and computes Twi = x,skIDi +1 where x, e Zq and t = [H.sub.2](w,) and returns Tw, to the adversary A.

g) Challenge: The adversary A sends ([ID.sub.c.sup.*], [W.sub.0.sup.*][W.sub.1.sup.*])to the challenger C where where W [0.sup.*] and W [1.sup.*] are two challenged keywords with the restrictions that IDi [not equal to] [ID.sub.c.sup.*] and [W.sup.i] [not equal to] {[W.sub.0.sup.*] or [W.sub.1.sup.*]}. The Challenger C then access the corresponding tuples < IDi, Qi, bi, coini > and < IDi, wi, Qw,, xi > in the [H.sub.1.sup.LIST] and [H.sub.2.sup.LIST] respectively and computes [T.sub.wb.sup.*] = xskm +t where x, [member of] [Z.sub.q.sup.*] and t = [H.sub.2](w,) and finally C sends the trapdoor [T.sub.wb.sup.*] to the adversary A.

h) Phase 2: A can continue to make the key extract queries adaptively for any identity IDC and the trapdoor query for any keyword W to the challenger C subject to the restriction that IDc [not equal to] [ID.sub.c.sup.*] and w [not equal to] {[W.sub.0.sup.*] or [W.sub.1.sup.*]}. i) Guess: Finally, the adversary A output a guess of a value b' [member of] {0,1}.

By assumption, the adversary A with a non-negligible advantage can distinguish the trapdoor [T.sub.wb.sup.*] under the adaptive chosen keyword attack. Meaning the trapdoor [T.sub.wb.sup.*] satisfy the equation e([T.sub.wb.sup.*], P) = V where V' = e(U, Ppub).e ([H.sub.2](wb), P) and let U = xQID. We can then obtain:

e([T.sub.wb.sup.*], P) = e(U, Ppub).e([H.sub.2](wb), P) e([T.sub.wb.sup.*], P) = e(XQID, aP).e([H.sub.2](wb), P) e([T.sub.wb.sup.*], P) = e[(abP, P).sup.x]. e([H.sub.2](wb), P) e([T.sub.wb.sup.*], P)/ e([H.sub.2](wb), P) = e[(abP, P).sup.x] e([T.sub.wb.sup.*] - [H.sub.2][(wb), P).sup.x-1] = e(abP, P)

Which implies that e(abP, P) = e([T.sub.wb.sup.*] - [H.sub.2][(wb), P).sup.x-1]. Hence the Challenger can obtain abP = [x.sup.-l] ([T.sub.wb.sup.*] - [H.sub.2](w*)), which contradicts the CDH assumption.

Theorem 5.2.2 The proposed IDERMKS scheme satisfies the trapdoor indistinquishability against an adaptive chosen plaintext attack under the CDH assumption in the random oracle.

6 PRIVACY PRESERVING RANKED SEARCH

Our scheme adopts the privacy preserving ranked search used in [25]. It is important to note that, after the retrieval of all the candidate files, the cloud server cannot return all the undifferential files to the data user due to (1) acquisition of excessive communication cost and overhead for the system if the cloud server decides to return all the candidate files.(2) the data users may only be concerned with the top-k relevant files that correspond to their queries. The scheme in [25] illustration an additive order preserving and privacy preserving encoding scheme. It then uses the encoded relevance score to obtain the top-k search result.

7 PERFORMANCE ANALYSIS AND COMPARISON

In this section, we analyze the performance of the proposed IDERMKS scheme. Comparison will be made between the IDERMKS scheme and previously proposed PEKS schemes. In order to make it convenient in evaluating the computational cost of our IDERMKS scheme, our concentration will be on some time-consuming operations and by adopting similar strategy used in [22] and we define the time consuming operation as follows:

* TGe: The execution time of a bilinear map operation e: [G.sub.1] x [G.sub.1] [right arrow] [G.sub.2]

* TGmul : The execution time for scalar multiplication operation in [G.sub.1]

* TGH : The execution time for map-to-point hash function, thus [H.sub.1], [H.sub.2], [H.sub.3]: [{0,1}.sup.*] [right arrow] [G.sub.1]

* [T.sub.inv] : The execution time of a modular inverse operation in [Z.sub.q]

The most time consuming operation is the time for executing a bilinear map operation TGe as compared to the other operations stated above. In [26][27], the performance simulation results show that [TG.sub.e] [approximately equal to] 2.5TGmul. We therefore analyze the performance for our IDERMKS scheme for each phase. In the IDERMKS ciphertext generation phase, it required 2TGe + TGmul + (n + 1)TGH to generate an IDERMKS ciphertext, where n represents the total number of keywords. In the trapdoor generation phase, TGmul + TGH is required to generate a trapdoor [T.sub.w].

Table 1 list the comparison between our IDERMKS scheme and the previously proposed dPEKS schemes [22][28][29][30] in terms of public key setting and performance. From the table in [22], it is easy to realize that the two schemes in [28][30] do not support conjunctive keywords since they require nTGe + (n + 1)TGmul + nTGH in the generation of their ciphertexts. We categorically state that our scheme is more efficient in the ciphertext generation when n is sufficiently large. Furthermore, our scheme is more efficient in the trapdoor generation phase as compared to other dPEKS schemes. Also, our scheme is based on ID-system which eliminates the load of certificate management associated with the other schemes which are based on pairing-based public key system. The scheme in [22] is also based on ID-system but our IDERMKS is more efficient in terms of ciphertext and trapdoor generation.

8 CONCLUSION

In this paper we proposed an IDERMKS scheme which supports conjunctive keywords. We defined the framework and the security requirements for our IDERMKS scheme and when we compared our scheme to previous dPEKS scheme, the performance of our scheme is more efficient in both the ciphertext and trapdoor generation phase. Our ID-based system also has an advantage of eliminating certificate management associated with PKI. We also demonstrated that our IDERMKS scheme possesses the ciphertext indistinquishability and trapdoor indistinguishability under BDH and CDH assumptions respectively. Our future work will seek to implement our IDERMKS scheme in a hybrid cloud and also to investigate our IDERMKS scheme without random oracle with a pairing free algorithm to determine how feasible it will be in cloud.

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Regina Esi Turkson, School of Computer Science and Engineering, University of Electronic Science and Technology of China, Email: regina_turkson@yahoo.com or rturkson@ucc.edu.gh

Yongjian Liao, School of Information and Software Engineering, University of Electronic Science and Technology of China, Email: liaoyj@uestc.edu.cn

Edward Yeallakuor Baagyere, School of Information and Software Engineering, University of Electronic Science and Technology of China, Email: ybaagyere@uds.edu.gh

TABLE 1: Comparison between our IDERMKS and previously proposed dPEKS schemes Scheme of Hwang and Lee [29] Public Key Setting Pairing-based Certificate Management Required Computational cost for ciphertext (2n + 2)T[G.sub.mul] generation (conjunctive n keywords) +2nT[G.sub.H] Computational cost for ciphertext 3T[G.sub.mul] generation (1 keywords) +2T[G.sub.H] Computational cost for trapdoor 3T[G.sub.mul]+ 2T[G.sub.H] generation +[T.sub.inv] Computational cost for test 3T[G.sub.e] +T[G.sub.H] Scheme of Rhee et al. [30] Public Key Setting Pairing-based Certificate Management Required Computational cost for ciphertext (2n + 2)T[G.sub.mul] generation (conjunctive n keywords) +2nT[G.sub.H] Computational cost for ciphertext T[G.sub.e] + 2T[G.sub.mul] generation (1 keywords) +T[G.sub.H] Computational cost for trapdoor 3T[G.sub.mul] + 2T[G.sub.H] generation +[T.sub.inv] Computational cost for test T[G.sub.e] + 2T[G.sub.mul] +T[G.sub.H] Scheme of Hu and Liu [28] Public Key Setting Pairing-based Certificate Management Required Computational cost for ciphertext (2n + 2)T[G.sub.mul] generation (conjunctive n keywords) +2nT[G.sub.H] Computational cost for ciphertext T[G.sub.e] + 2T[G.sub.mul] generation (1 keywords) +T[G.sub.H] Computational cost for trapdoor 3T[G.sub.mul] + 2T[G.sub.H] generation +[T.sub.inv] Computational cost for test T[G.sub.e] + 2T[G.sub.mul] +[T.sub.inv] Scheme of Wu et al. [22] Public Key Setting ID-based Certificate Management Not Required Computational cost for ciphertext T[G.sub.e] + (n + 2)TGmu generation (conjunctive n keywords) +(n + 2)T[G.sub.H] Computational cost for ciphertext T[G.sub.e] + 3T[G.sub.mul] generation (1 keywords) +3T[G.sub.H] Computational cost for trapdoor 2T[G.sub.mul] + T[G.sub.H] generation Computational cost for test 2T[G.sub.e] + T[G.sub.mul] +[T.sub.inv] 314 Our IDERMKS Public Key Setting ID-based Certificate Management Not Required Computational cost for ciphertext 2T[G.sub.e] + T[G.sub.mul] generation (conjunctive n keywords) +(n + 1)T[G.sub.H] Computational cost for ciphertext 2T[G.sub.e] + T[G.sub.mul] generation (1 keywords) +2T[G.sub.H] Computational cost for trapdoor T[G.sub.mul] + T[G.sub.H] generation Computational cost for test T[G.sub.e] + 2T[G.sub.mul]

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Author: | Turkson, Regina Esi; Liao, Yongjian; Baagyere, Edward Yeallakuor |
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Publication: | International Journal of Cyber-Security and Digital Forensics |

Article Type: | Report |

Date: | Sep 1, 2018 |

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