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Identitybased remote data checking with a designated verifier
Journal of Cloud Computing volume 11, Article number: 7 (2022)
Abstract
Traditional remote data possession auditing mechanisms are usually divided into two types: private auditing and public auditing. Informally, private auditing only allows the original data owner to check the integrity of its outsourced data, whereas anyone is allowed to perform the checking task in public auditing. However, in many practical applications, the data owner expects some designated verifier (instead of others) to audit its data file, which cannot be covered by the existing private or public auditing protocols. Thus, in a recent work, Yan et al. proposed a new auditing technique with a designated verifier [IEEE Systems Journal, 12(4): 17881797, 2020]. Nevertheless, we note that this protocol suffers from complicated management of certificates and hence heavily relies on public key infrastructure. To overcome this shortcoming, in this paper, we propose an identitybased auditing protocol with a designated verifier, which not only avoids the introduction of certificates, but also has the desired property of only allowing specific verifier to audit. Its security is based on the classical computational DiffieHellman and Weil DiffieHellman assumptions. Finally, performance analysis shows that our proposed protocol is very efficient and suitable for some reallife applications.
Introduction
In recent years, cloud storage has become an attractive technique for users or data owners (DOs) to store their data since it may have much lower prices than the cost to maintain them on personal devices. When enjoying the services provided by cloud companies, the DOs can freely access and conveniently share these outsourced data on different devices and locations. However, they also completely lose the control of its data after uploading them to the cloud service provider (CSP). Meanwhile, for kinds of internal or external reasons, it is extremely possible to lose user’s data for the CSP [1]. For example, the CSP may deliberately delete user’s rarely used data to save its own space and thus attract more clients to gain more economic benefits. In addition, the cloud companies may also be attacked by hackers and hence “passively” lose user’s data [2].
Under this situation, the DOs naturally worry about the integrity of their outsourced data and urgently hope that there exists a mechanism to help them to audit or check data’s integrity. If it is broken, then they should obtain compensations. As a result, remote data possession checking (RDPC) mechanisms, like provable data possession (PDP) or proof of retrievability (PoR) were developed and broadly studied in the last decades [3, 4].
According to the manners of auditing, RDPC mechanisms are divided into two types: private auditing (see [5]) and public auditing (see [6]). In private auditing, the auditor is just the DO itself. But in public auditing, anyone is allowed to perform the task of auditing. A popular way is to outsource the auditing task to some thirdparty auditor (TPA), who may have more professional knowledge on auditing and more computing power. Therefore, the public auditing is very popular in practical applications.
However, in some situations, the DO hopes or needs to limit the verifier’s identity, and just allows a designated auditor to check its data’s integrity. For example, a financial staff in a company outsourced some sensitive data, such as employees’ salaries, to some CSP. Then the chief financial officer is just the one that is allowed to check the integrity of these data. Another example is that some TPA company gives its clients a lower discount if the DO only chooses this TPA company (instead of others) as the auditor of its stored data. Therefore, in these practical scenarios, how to design RDPC mechanisms with a designated verifier is an interesting but challenging problem.
In [7], Yan et al. recently designed an efficient designatedverifier PDP (DVPDP, for short) protocol, where the DO specifies a designated verifier to audit the integrity of its data. Meanwhile, they also analyzed its security in the random oracle model.
However, it is well known that most of the existing public (DV)PDP protocols are based on the management of certificates and heavily relies on public key infrastructure (PKI) technique [8]. The reason lies in that other entities in the protocol need to securely obtain and use the public keys of some participants, but how to fix the relationship between the received public key and its true identity is hard to handle. In other words, the genuineness of user’s public key is hard to guarantee. The traditional method is issuing certificates for these users by a trusted certificate authority (CA) center. However, the complex management procedures on certificates, including generation, storing, delivery, updating, and revocation etc., additionally increase communication/computational costs, which are timeconsuming and expensive, and thus greatly reduce the efficiencies of the designed PDP protocols. In addition, PKI’s security cannot be completely guaranteed, especially when CA center is controlled by some malicious hacker. For instance, after discovering more than 500 fake certificates, web browser vendors were forced to blacklist all certificates issued by DigiNotar, a Dutch CA, in 2011 [9].
Motivation. We note that the identitybased (IB) cryptography does not suffer this problem [10]. More specifically, in identitybased primitives, user’s public key is set as its name, email address, or identitycard number and so on, which does not need the certificates to authenticate its validity. Then a trusted key generation center (KGC) generates secret keys for different users corresponding to these identities. When all users have their secret keys issued by the same KGC, individual public keys become obsolete, which removes the need for explicit certification and all necessary costs. These features make the identitybased paradigm particularly appealing for use in conjunction with PDP protocols. This is also the reason why many proposed PDP protocols were twisted into IBPDP ones [11, 12]. Hence, it becomes an interesting problem to introduce the identitybased technique to IBPDP protocol.
Our Contributions. In this paper, we consider the new notion of IBPDP protocol with a designated verifier (IBDVPDP, for short), which can be seen as an improvement on DVPDP proposed by Yan et al. in [8]. The main contributions of this paper can be summarized as follows.

1)
For the first time, we propose the notion of IBDVPDP protocol, which provides the integrity checking of user’s stored data under the constraint of a designated verifier. In our proposed protocol, anyone knows the cloud user’s identity but only the designated verifier is allowed to perform the auditing task on behalf of the data owner, which may find special applications in practical scenarios.

2)
Second, we define the security model of IBDVPDP protocol, which not only satisfies the security definition of previous IBPDP protocol, but also describes the security against undesignated verifier. More precisely, we first consider the security against malicious CSP, which is the essential security for PDP protocol. Then in order to guarantee the DV’s interests, we discuss the security against any unauthorized verifier.

3)
Then, we propose a concrete construction of IBDVPDP protocol and prove its security within our model based on the computational DiffieHellman (CDH) and Weil DiffieHellman (WDH) assumptions. Meanwhile, we emphasize that this security relies on the classical random oracle model.

4)
Finally, we also analyze the performances of our proposed protocol, including the communication overheads, computational/storage costs (theoretical analysis), and experimental results, which shows that our protocol is very efficient and suitable for practical applications.
Related Works. In 2007, Juels et al. first proposed the notion of PoR to model the security of outsourced data, where the errorcorrecting code was combined with the spotchecking of data to verify the integrity [13]. However, this scheme does not support public checking and only allows limited number of checking. At the same time, Ateniese et al. presented a similar technique named PDP, which is based on the RSAhomomorphic authenticators [14]. It allows public auditing and unlimited number of verifications. Later, Dodis, Vadhan and Wichs constructed PoR schemes via hardness amplification [15]. Note that the notion PoR is obviously stronger than PDP. Therefore, constructing a secure PDP protocol is easier than instantiating a PoR scheme. In 2017, Zhang et al. presented a general framework to design secure PDP protocols using homomorphic encryption schemes [2].
In 2016, Chen et al. first revealed an intrinsic relationship between network coding (see [16–18]) and secure cloud storage protocol, although both of them seem to be very different in their nature and were studied independently [19]. But the reverse direction does not hold in general unless imposing additional constraints. In recent work [20], Chang et al. defined a new notion of admissible PoR and proved that it is also possible to design secure network coding from admissible PoR protocols.
Consider that most of the previous PDP or PoR protocols heavily reply on PKI technique and must deal with complicated management of public key certificates. As a result, many identitybased PDP or PoR protocols were also proposed in past years,which borrow the identitybased idea in public key cryptography [10].More concisely, Wang et al. designed IBPDP protocol in public clouds [21], in which they did not consider the problem of data’s privacy preserving. Then, in 2017, Yu et al. constructed a secure IBPDP protocol with perfect data privacy preserving to improve Wang et al.’s result [9]. Moreover, Xue et al. designed identitybased public auditing protocol against malicious auditors via the new technique of blockchain [12].
In order to decrease the system complexity, later, Li et al. also proposed a new identitybased data storage protocol, which uses homomorphic verifiable tag to achieve it [8]. In addition, in multicopymulticloud case, they also considered the efficient construction of IBPDP protocol [22]. Recently, Chang et al. revisited this protocol and proposed some improvements on its performances [23].
In addition, Zhang et al. constructed an IBPDP protocol, which is secure against keyexposure attack and based on Lattices [24]. Later, Mary and Rhymend proposed a stronger keyexposure model and gave some suggestions on how to construct this kind of IBPDP protocol [25].
Note that the previous IBPDP protocols did not consider the designatedverifier case. Hence, our proposed IBDVPDP protocol is the first one and will find applications in practical scenarios.
Organizations. The organizations of this paper are as follows. In “Preliminaries” section, we introduce some basic notations and notions. In “Our proposed construction and its security” section, the proposed protocol and its security analysis are given. In “Performances analysis” section, we analyze the performances and compare them with three other related PDP protocols. Finally, conclusions are presented in “Conclusions” section.
Preliminaries
Basic notations
Now, we first present the basic notations used in this paper. More precisely, we use λ or 1^{λ} to denote the security parameter. PPT is the abbreviation of “Probabilistic Polynomial Time”. For a natural number n, the symbol [n] means the set of {1,2,⋯,n}. For an algorithm A, the symbol ^{′′}y←A(x_{1},x_{2},⋯)^{′′} denotes that it takes x_{1},x_{2},⋯ as inputs, and will output y. For a group \(\mathbb G, \mathbb G\) denotes the bit length of each element in it. A function f(λ) is negligible if, for any c>0, there exists a \(\lambda _{0}\in \mathbb Z\) such that for any λ>λ_{0}, it always holds that f(λ)<λ^{−c}. Other symbols and their corresponding definitions are listed in Table 1.
Blinear map
Consider two cyclic groups \(\mathbb G_{1}, \mathbb G_{2}\) with the same prime order q. A map \(e:\mathbb G_{1}\times \mathbb G_{1}\rightarrow \mathbb G_{2}\) is called a bilinear map if the following conditions hold.

NonDegeneracy. For any generators g_{1},g_{2} of \(\mathbb G_{1}\), it holds that \(e(g_{1},g_{2})\neq 1_{\mathbb G_{2}}\), in which \(1_{\mathbb G_{2}}\) is the identity element of \(\mathbb G_{2}.\)

Computability. It is efficient to compute e(g_{1},g_{2}) for any \(g_{1},g_{2}\in \mathbb G_{1}.\)

Bilinearity. For any \(a,b\in \mathbb Z_{q}, g_{1},g_{2}\in \mathbb G_{1}\), it holds that
$$e(g_{1}^{a},g_{2}^{b})=e(g_{1},g_{2})^{ab}.$$
CDH assumption
Define \(\mathbb G_{1}\) as a cyclic group with prime order q and generator g. Randomly choose a,b from \(\mathbb Z_{q}\) and compute g^{a},g^{b}. The CDH assumption means that, given the tuple (g,g^{a},g^{b}), any PPT algorithm cannot compute and output the element \(g^{ab}\in \mathbb G_{1}.\)
Weil DiffieHellman Assumption
Here, we recall the Weil DiffieHellman (WDH) assumption in [26]. Let \(e:\mathbb G_{1}\times \mathbb G_{1}\rightarrow \mathbb G_{2}\) be a bilinear map and g be a generator of \(\mathbb G_{1}\). Randomly choose a,b,c from \(\mathbb Z_{q}\) and compute g^{a},g^{b},g^{c}. The WDH assumption refers to that, given the tuple (g,g^{a},g^{b},g^{c}), there is no efficient algorithm to compute \(e(g,g)^{abc}\in \mathbb G_{2}\).
System model
In this subsection, we formally introduce the system model of IBDVPDP protocol. A graphical description can be found in Fig. 1. In particular, the whole system includes four entities: KGC, DO, DV, and CSP.

KGC: Given the identity of DO or DV, this entity will issue secret key for it.

DO: This entity is just the data owner, who intends to outsource its original data file to some CSP.

DV: This is an entity which is designated by DO and will check the integrity of DO’s stored file.

CSP: This entity has great storage spaces and provides storing services for lots of DOs.
Then, an IBDVPDP protocol consists of the following seven PPT algorithms.

(params,msk)←Setup(1^{λ}). This algorithm initializes the system and generates the public system parameter params as well as the master secret key msk.

sk_{ID}←Extract(msk,ID). This algorithm generates secret keys for users (including the DV) with different identities ID. Now, the DO (resp. DV) can obtain its secret key sk_{O} (resp. sk_{V}) by inputting its identity ID_{O} (resp. ID_{V}) into this algorithm.

trd←TrapdoorGen((sk_{O},ID_{V}) or (sk_{V},ID_{O})). This is a trapdoorgeneration algorithm, which takes the pair (sk_{O},ID_{V}) or (sk_{V},ID_{O}) as input, and computes a common private trapdoor trd between the DO and DV.

T←TagGen(sk_{O},F). This algorithm is run by DO to generate an authenticated file T for the original data file F. Then T is transmitted to a CSP for storing.

chal←Challenge. This is an algorithm run by the DV (in the auditing phase) to generate a challenge message chal, which will be sent to the CSP.

Γ←ProofGen(chal,T). When receiving the challenge message chal from the DV, the CSP computes a proof Γ based on the stored T.

b←Verify(trd,chal,Γ). This is the final checking algorithm of the DV, which will output a bit b based on the challenge message chal and the returned proof Γ. If b=1, then the DO’s stored data file is intact. Otherwise, its integrity is broken.
When combining the entities with the above algorithms, the system will work as follows. The KGC first runs Setup to obtain params and msk, and broadcasts params to other entities. Given the identity ID, the KGC runs sk_{ID}←Extract(msk,ID) and returns sk_{ID} to the entity with ID. After receiving the secret key sk_{O} and DV’s identity ID_{V}, the DO computes the trapdoor trd, which can also be computed by the DV, by running TrapdoorGen. Then the DO continues to generate the authenticated file T for the original data file F by performing TagGen(sk_{O},F), and transmits T to CSP for storing. In the auditing phase, the DV runs Challenge to get chal and sends it to CSP, who will compute Γ based on chal and the stored T, and then returns it to the DV. Finally, the DV checks the validity of the returned Γ by running Verify.
Security model
In this subsection, we define the security model of an IBDVPDP protocol. Before describing its definition, we would like to give some underlying intuition. Concretely, here, we need to consider two kinds of attacks. One is to model the misbehavior of malicious CSP. That is, if some stored data blocks are lost by CSP, then the designated verifier can detect it with extremely high probability. Another one is to model the security of DV, which also means that checking whether an undesignated verifier is able to perform the task of auditing or not. For a secure IBDVPDP protocol, it should satisfy that any undesignated verifier cannot obtain the corresponding trapdoor and thus cannot perform the task of auditing.
Now, we formally give the definition of the former one, which is modeled by the following security game played between a challenger CH^{I} and an adversary \(\mathcal {CSP}\).

Initialization. The challenger first runs Setup(1^{λ}) to get the public system parameter params and the master secret key msk. Give params to \(\mathcal {CSP}\).

Queries. The adversary is allowed to adaptively make the following queries.

\(\texttt {SecretKeyExtract}. \mathcal {CSP}\) submits an identity ID to CH^{I}, who will run sk_{ID}←Extract(msk,ID) and return sk_{ID} to it.

AuthenticatedFileQuery. The adversary \(\mathcal {CSP}\) submits an identity ID and a data file F to CH^{I}, who runs
$$T\leftarrow\texttt{TagGen}(sk_{ID},F)$$and returns it to \(\mathcal {CSP}.\)

AuditingQuery. This kind of query is based on the queried T in the previous step. More precisely, the challenger first runs chal←Challenge and gives chal to \(\mathcal {CSP}\), who computes and returns a proof Γ by running ProofGen(chal,T). When receiving Γ,CH^{I} obtains the trapdoor trd by running the algorithm TrapdoorGen(sk_{O},ID_{V}), and continues to run
$$b\leftarrow\texttt{Verify} (trd,chal,\Gamma).$$Finally, the bit b is transmitted to \(\mathcal {CSP}.\)


Final Phase. In this phase, the challenger CH^{I} submits a challenge message chal^{∗} to \(\mathcal {CSP}\) in order to check the integrity of some data file F queried in AuthenticatedFileQuery, who finally returns a proof Γ^{∗}.
If
and the following conditions hold,

1)
The identities ID_{O} and ID_{V} are not queried to the oracle SecretKeyExtract, and

2)
The returned proof Γ^{∗} does not equal to the correct one Γ, which would be honestly computed in ProofGen(chal^{∗},T),
then we call the IBDVPDP protocol is secure against any PPT malicious CSP.
Next, we consider the security on the trapdoor of the DV. In order to clearly introduce the security model on trapdoor, we would like to give some underlying intuition. More concretely, for an adversary, it wants to find the correct trapdoor after seeing the transmissions between the entities, including the initialization, queries and so on. Here, we still introduce the following security game played by a challenger CH^{II} and an adversary \(\mathcal A.\)

These phases “Initialization” and “Queries” are the same as the ones appeared in the above game except that CH^{I} and \(\mathcal {CSP}\) are replaced by CH^{II} and \(\mathcal A\), respectively.

Final Phase. In final, the adversary outputs a trapdoor trd^{∗}. Naturally, the restriction is that \(\mathcal A\) is not allowed to query ID_{O} and ID_{V} to SecretKeyExtract oracle.
If for any PPT adversary \(\mathcal A\), the probability that its output trd^{∗} equals to the trapdoor trd of the DV is negligible, then we call the IBDVPDP protocol is secure for the DV. In other words, this security describes that any undesignated entity is “hard” to obtain the trapdoor and hence is not able to perform the task of auditing.
Our proposed construction and its security
Now, we formally introduce the construction of our proposed IBDVPDP protocol as follows.

(params,msk)←Setup(1^{λ}): This is an algorithm that generates the system parameters params and KGC’s master secret key msk when given the security parameter 1^{λ}. Concretely, it chooses two cyclic groups \(\mathbb G_{1}, \mathbb G_{2}\) with the same prime order q (q≥λ), and e a bilinear map from \(\mathbb G_{1}\times \mathbb G_{1}\) to \(\mathbb G_{2}\). Assume that g is a generator of \(\mathbb G_{1}\), and H_{1},H_{2} are two hash functions from {0,1}^{∗} to \(\mathbb G_{1}\). Moreover, define \(\phi :\mathbb Z_{q}^{*}\times \{1,2,\cdots,n\}\rightarrow \mathbb Z_{q}^{*}, \pi :\mathbb Z_{q}^{*}\times \{1,2,\cdots,n\}\rightarrow \{1,2,\cdots, n\}\) as pseudorandom function and pseudorandom permutation, respectively. Then randomly choose x from \(\mathbb Z_{q}^{*}\) and compute P_{0}=g^{x}. Finally, set
$$params=(\mathbb G_{1},\mathbb G_{2},e,q,g,H_{1},H_{2},P_{0},\phi,\pi)$$and msk=x.

sk_{ID}←Extract(msk,ID): This is a secretkeygeneration algorithm for a user with identity ID. In particular, for the inputs of KGC’s master secret key msk and user’s identity ID, it computes sk_{ID}=H_{1}(ID)^{x} and outputs it as the generated secret key (for ID).
In fact, by running this algorithm, the DO with identity ID_{O} and the DV with ID_{V} can obtain their secret keys sk_{O} and sk_{V}, respectively.

trd←Trapdoor((sk_{O},ID_{V}) or (sk_{V},ID_{O})): If the input is (sk_{O},ID_{V}), then this algorithm computes the trapdoor trd as follows.
$$trd:=e\left(sk_{O},H_{1}(ID_{V})\right) =e\left(H_{1}(ID_{O}),H_{1}(ID_{V})\right)^{x}.$$Similarly, if the input is (sk_{V},ID_{O}), then this algorithm computes the trapdoor trd as follows.
$$trd=e\left(sk_{V},H_{1}(ID_{O})\right) =e\left(H_{1}(ID_{O}),H_{1}(ID_{V})\right)^{x}.$$ 
T←TagGen(sk_{O},F): This is the taggeneration algorithm for the original data file F and run by DO, who has the secret key sk_{O}. Specifically, first randomly choose a filename Fid from {0,1}^{λ} (for F). Parse F into n blocks {m_{1},m_{2},⋯,m_{n}}, and each block m_{i} has s samelength segments \((m_{i,1},m_{i,2},\cdots,m_{i,s})\in \left (\mathbb Z_{q}\right)^{s}\). That is,
$$ \begin{aligned} F&=\{m_{1},\cdots,m_{n}\} \\ &=\{(m_{1,1},m_{1,2},\cdots,m_{1,s}),\cdots, (m_{n,1},m_{n,2},\cdots,m_{n,s})\}. \end{aligned} $$Then randomly choose u_{1},u_{2},⋯,u_{s} from \(\mathbb G_{1}\), t from \(\mathbb Z_{q}^{*}\), and compute
$$\begin{array}{*{20}l} T_{i}=sk_{O}\cdot\left(H_{2}\left(trdFidi\right)\cdot \prod_{j=1}^{s} u_{j}^{m_{i,j}}\right)^{t}, \end{array} $$(1)for each i∈[n]. Define R=g^{t} and select an identitybased signature (IBS) algorithm IBS.Sig to compute
$$T_{Fid}=\texttt{IBS.Sig}(u_{1}u_{2}\cdots u_{s}RFid).$$Now, define
$$T=\left(F,u_{1},u_{2},\cdots,u_{s},R,Fid,T_{Fid}, T_{1},T_{2},\cdots,T_{n}\right)$$and output it, which will be transmitted to CSP for storing.

chal←Challenge: This is the challengemessagegeneration algorithm and run by the DV. In particular, in order to check the integrity of user’s data, the designated verifier randomly chooses ℓ from [n], which denotes the number of challenged blocks, and k_{1},k_{2} from \(\mathbb Z_{q}^{*}\). Output the challenge message chal=(ℓ,k_{1},k_{2}), which will be sent to CSP.

Γ←ProofGen(chal,T): This algorithm will generate the returned proof Γ and hence run by the CSP. Specifically, when receiving the challenge message chal from the verifier, it first computes
$$\begin{array}{*{20}l} a_{i}=\phi(k_{1},i)\in\mathbb Z_{q}^{*},\,c_{i}=\pi(k_{2},i)\in [n] \end{array} $$(2)for 1≤i≤ℓ. Then calculate
$$\sigma=\prod_{i=1}^{\ell} T_{c_{i}}^{a_{i}},\text{ and }M=\sum_{i=1}^{\ell} a_{i}\cdot m_{c_{i}}\in \left(\mathbb Z_{q}\right)^{s}.$$Finally, return
$$\Gamma=\left(u_{1},u_{2},\cdots,u_{s},R,Fid,T_{Fid},\sigma, M\right)$$to the DV.

1/0←Verify(trd,chal,Γ): After receiving the returned proof Γ from CSP, the DV first checks if T_{Fid} is a valid signature on the message (u_{1}u_{2}⋯u_{s}RFid). If it isn’t, refuse this proof and output 0. Otherwise, compute a_{i},c_{i} from k_{1},k_{2} as in (2). Then check if
$$ \begin{aligned} e\left(\sigma,g\right)&=e\left(H_{1}(ID_{O}),P_{0}\right)^{\sum_{i=1}^{\ell} a_{i}}\cdot \\ &e\left(\prod_{i=1}^{\ell} \left(H_{2}\left(trdFidc_{i}\right)^{a_{i}},R\right)\right)\cdot e\left(\prod_{j=1}^{s}u_{j}^{M_{j}},R\right), \end{aligned} $$(3)in which M_{j} is the jth component of M. If it is, output 1. Otherwise, output 0.
The correctness of (3) can be verified as follows.
Brief review of yan et al.’s dVPDP protocol
In order to clearly express our improvement on Yan et al.’s DVPDP protocol, here, we briefly review their protocol. More precisely, the publicprivate key pairs of DO and DV are \((x,X=g^{x})\in \mathbb Z_{q}^{*}\times \mathbb G_{1}\) and \((y,Y=g^{y})\in \mathbb Z_{q}^{*}\times \mathbb G_{1}\), respectively. Then the DO (resp. DV) can calculate the trapdoor α=H_{1}(Y^{x})∈{0,1}^{k} (resp. α=H_{1}(X^{y})). For each data block \(m_{i}\in \mathbb Z_{q}^{*}\), compute its tag
where F_{id} is the unique identification of the file, and \(u\in \mathbb G_{1}\) is public. The computations of challenge message chal and returned proof Γ are the same as the ones in our construction. In the final verification algorithm, the DV computes and checks if
If it is, then output 1. Otherwise, output 0.
Security analysis
In this subsection, we analyze the security of the IBDVPDP protocol presented in the above section.
Theorem 1
If the CDH assumption holds in \(\mathbb G_{1}\), and H_{1},H_{2} are seen as random oracles, then the above IBDVPDP protocol is secure against any malicious CSP in the random oracle.
Before presenting the detailed proof, we would like to introduce the underlying reasons why the security of IBDVPDP can be reduced to the CDH assumption. In particular, given the tuple \((g,g^{a},g^{b})\in \mathbb G_{1}^{3}\), the algorithm \(\mathcal B\), who tries to solve the CDH problem, can set P_{0}=g^{a} as the master public key since the malicious CSP is not allowed to query the master secret key in the whole query process. The extracted secret key for each user can also be simulated by randomly masking the group element P_{0}. The trickiest step is the simulation of taggeneration for each data block, which can be handled by embedding \(\mathcal B\)’s another group element g^{b}. Then the auditing query is natural and easy to deal with.
Proof. Let \(\mathcal {CSP}\) be a PPT CSP, who attacks on the proposed IBDVPDP protocol. Consider \(\mathcal B\) as an attacker on the CDH assumption. That is, given the tuple \((g,g^{a},g^{b})\in \mathbb G_{1}^{3}\), its target is to output the element g^{ab} for the unknown \(a,b\in \mathbb Z_{q}.\) Now, \(\mathcal B\) simulates the environment for \(\mathcal {CSP}\) and wants to obtain its answer by using \(\mathcal {CSP}\) as a subroutine.

Initialization.\(\mathcal B\) first generates params as in Setup except for the item P_{0}. More precisely, it sets P_{0}=g^{a} and implicitly defines the master secret key x as the unknown a. Give params to \(\mathcal {CSP}\). In addition, \(\mathcal B\) also initializes an empty list L used to store the queryanswer pairs.

Queries. The adversary \(\mathcal {CSP}\) is allowed to adaptively make the following queries.

HashQueries. For the query ID to the H_{1}oracle, \(\mathcal B\) checks if (ID,H_{1}(ID)) is in the list L. If it is, return H_{1}(ID) to \(\mathcal {CSP}\). Otherwise randomly choose \(\tau \in \mathbb Z_{q}\) and compute H_{1}(ID)=g^{τ}. Return H_{1}(ID) to \(\mathcal {CSP}\) and store (ID,H_{1}(ID)) into L. In addition, \(\mathcal B\) also defines H_{1}(ID_{O}) as g^{b}.

\(\texttt {SecretKeyExtract}. \mathcal {CSP}\) submits an identity ID to \(\mathcal B\), who computes
$$sk_{ID}=(g^{a})^{\tau}=g^{a\tau}=(H_{1}(ID))^{a}.$$Here, τ is the random chosen element when answering the query to H_{1}oracle on ID. Since \(\mathcal {CSP}\) is not allowed to query the secret key of \(ID_{O}, \mathcal B\) implicitly sets sk_{O}=(g^{b})^{a}=g^{ab}.

AuthenticatedFileQuery. The adversary \(\mathcal {CSP}\) submits an identity ID and a data file F to \(\mathcal B\). If ID≠ID_{O}, then \(\mathcal B\) normally answers this query because it has the true secret key sk_{ID}. Else if ID=ID_{O}, then \(\mathcal B\) will simulate as follows. First, compute the trapdoor trd by running Trapdoor(sk_{V},ID_{O}). Then randomly choose Fid as the filename of F and parse F into n blocks
$$\begin{aligned} F&=\{m_{1},\cdots,m_{n}\} \\ &=\{(m_{1,1},\cdots,m_{1,s}),\cdots, (m_{n,1},\cdots,m_{n,s})\}. \end{aligned} $$Randomly choose x_{1},⋯,x_{s},r from \(\mathbb Z_{q}\) and compute
$$u_{1}=(g^{b})^{x_{1}},\cdots,u_{s}=(g^{b})^{x_{s}},R=(g^{a})^{r}.$$That is,
$$\prod_{j=1}^{s} u_{j}^{m_{i,j}}=\prod_{j=1}^{s}\left(g^{b}\right)^{x_{j}m_{i,j}} =\left(g^{b}\right)^{\sum_{j=1}^{s}x_{j}m_{i,j}},$$and implicitly set t=ar. Moreover, define
$$H_{2}(trdFidi)=\left(g^{b}\right)^{\alpha_{i}}\cdot g^{\beta_{i}},$$where α_{i} satisfies
$$1+r\alpha_{i}+r\sum_{j=1}^{s}{x_{j}m_{i,j}}=0.$$Here, we know that, for 1≤i≤n, it holds that
$$\begin{array}{*{20}l} T_{i}&=sk_{O}\cdot\left(H_{2}\left(trdFidi\right)\cdot \prod_{j=1}^{s} u_{j}^{m_{i,j}}\right)^{t}\\ &=g^{ab}\cdot\left(\left(g^{b}\right)^{\alpha_{i}} g^{\beta_{i}}\cdot \left(g^{b}\right)^{\sum_{j=1}^{s}x_{j}m_{i,j}} \right)^{ar}\\ &=g^{ab}\left(g^{(ab)(r\alpha_{i}+r\sum x_{j}m_{i,j})}\cdot \left(g^{a}\right)^{r\beta_{i}}\right)\\ &=\left(g^{ab}\right)^{1+r\alpha_{i}+r\sum{x_{j}m_{i,j}}}\cdot \left(g^{a}\right)^{r\beta_{i}}\\ &=\left(g^{a}\right)^{r\beta_{i}}. \end{array} $$Finally, generate T_{Fid} by signing the item
$$u_{1}u_{2}\cdotsu_{s}RFid,$$and return
$$T=\left(F,u_{1},u_{2},\cdots,u_{s},R,Fid,T_{Fid}, T_{1},T_{2},\cdots,T_{n}\right)$$to \(\mathcal {CSP}.\)

AuditingQuery. In this step, \(\mathcal B\) simulates the DV and sends a challenge message chal to \(\mathcal {CSP}\), who computes and returns a proof Γ. Then \(\mathcal B\) checks the validity of Γ by running
$$b\leftarrow\texttt{Verify} (trd,chal,\Gamma),$$and gives the bit b to \(\mathcal {CSP}.\)


Final Phase. In this phase, the algorithm \(\mathcal B\) submits a challenge message chal^{∗}=(ℓ,k_{1},k_{2}) to \(\mathcal {CSP},\) who returns a forged proof
$$\Gamma^{*}=\left(u_{1},u_{2},\cdots,u_{s},R,Fid,T_{Fid},\sigma^{*}, M^{*}\right).$$
Let
be the correct proof, which would be honestly computed in ProofGen(chal^{∗},T). Hence, (σ^{∗},M^{∗})≠(σ,M).
Since both of the proofs Γ and Γ^{∗} can pass the verification of (3), it holds that
and
Divide (4) by (5) and we can get
Because r,x_{1},x_{2},⋯,x_{s} are random and at least one \((M_{j}M^{*}_{j})\) does not equal to 0, we know that \(\sum _{j=1}^{s} rx_{j}(M_{j}M^{*}_{j})=0\) occurs with probability 1/q, which is negligible. Therefore, \(\mathcal B\) can obtain the answer g^{ab} by computing
This ends the proof of Theorem 1.
As for the security of the DV, we have the following:
Theorem 2
If the WDH assumption holds for the bilinear map \(e:\mathbb G_{1}\times \mathbb G_{1}\rightarrow \mathbb G_{2}\), then our proposed IBDVPDP protocol is secure for the designated verifier.
Now, we first explain the underlying intuition of this theorem’s proof. Given the tuple \((g,g^{a},g^{b},g^{c})\in \mathbb G_{1}^{4}\), the algorithm \(\mathcal B'\) attacking on the WDH assumption can naturally set g^{a} as the master public key P_{0}, and g^{b},g^{c} as the randomoracle queries to H_{1} for the identities ID_{O} and ID_{V}, respectively. Then the finding of trapdoor trd=e(g,g)^{abc} for undesignated verifier is just the WDHsolution of \(\mathcal B'\). This is also the reason why WDH assumption is used in this paper.
Proof. Assume that \(\mathcal A\) is an adversary who wants to find out the trapdoor of the DV. Then we will construct another algorithm \(\mathcal B'\) attacking on the WDH assumption. In particular, given the tuple \((g,g^{a},g^{b},g^{c})\in \mathbb G_{1}^{4}, \mathcal B'\) intends to compute and output the element \(e(g,g)^{abc}\in \mathbb G_{2}.\) Hence, it simulates the environment for \(\mathcal A\) and invokes \(\mathcal A\) as a subroutine.
The simulation of \(\mathcal B'\) is similar to that of the algorithm \(\mathcal B\) constructed in Theorem 1. That is, \(\mathcal B'\) also sets P_{0}=g^{a},H_{1}(ID_{O})=g^{b}. Moreover, it defines H_{1}(ID_{V})=g^{c}. In this way, \(\mathcal B'\) can simulate all the queries from \(\mathcal A\) as in the above theorem. Finally, when \(\mathcal A\) outputting trd^{∗} as a guess of the trapdoor of DV, \(\mathcal B'\) also outputs it as its answer to the WDH assumption.
From the simulation of \(\mathcal B'\), we know that
Therefore, if \(\mathcal A\) can correctly find the trapdoor trd, then \(\mathcal B'\) is also able to solve the WDH assumption. From the hardness of WDH assumption, we know that our proposed IBDVPDP protocol is secure for the DV. This also ends the proof of Theorem 2.
Further improvement
Note that, in our proposed protocol, the authenticated file T has the form of
In order to further reduce the communication cost, we suggest the following improvements. Define a new hash function \(H:\{0,1\}^{*}\rightarrow \mathbb G_{1}\). Then generate
and define T_{Fid} as the IBS of (RFid). In the security proof, the hash function H is still viewed as a random oracle, which provides randomness for u_{1},u_{2},⋯,u_{s}. In this case, the communications of u_{1},u_{2},⋯,u_{s} can be reduced from T. Similarly, they can also be reduced from the returned proof Γ because the DV can compute them online from Fid.
Error detection probability
Since our proposed IBDVPDP protocol adopts the random sampling method to detect the corruption of user’s data, we now discuss its error detection probability for the DV. Specifically, in our protocol, the DV chooses ℓ blocks in each challenge. Assume that d blocks are corrupted by CSP and define X as a random variable, which describes the number of challenged blocks matching the corrupted ones. Moreover, P_{X} denotes the probability that CSP’s misbehavior is detected. Hence, we have
Obviously, the more challenged blocks, the higher error detection probability. If 5,000 blocks out of 1,000,000 ones are tampered, then the error detection probability P_{X} is greater than 80% when challenging 321 blocks (for the DV). Similarly, if 10,000 blocks are corrupted, then randomly choosing only 300 blocks will realize that the error detection probability P_{X} is at least 95%.
Performances analysis
In this section, we evaluate the performances of our proposed IBDVPDP protocol from communication overhead, storage and computational costs as well as the experimental result. In order to present the practicality of our protocol, we compare it with two other IBPDP protocols and a certificateless PDP protocol with privacypreserving property, which were designed in [8] [9], and [4], respectively.
Communication overhead
The communication contents for an IBPDP protocol include the parts of transmitting the authenticated file T from data owner to CSP, a challenge message chal from the DV to CSP, and the returned proof Γ from CSP to the DV. Now, we respectively denoted by DOtoCSP, DVtoCSP, and CSPtoDV the corresponding communication overheads.
Recall that, in our protocol, the DO will send the authenticated file
to CSP. Thus, the communication overhead (i.e. DOtoCSP) for our protocol equals to
Similarly, we can evaluate the communication overheads from DO to CSP for the three protocols in [8], [9] and [4] as
and
respectively.
The challenge message chal in our protocol is
which has the length of \(2\cdot n+\mathbb Z_{q}^{*}\). The challengemessagegeneration algorithm in [8] is the same as that of our protocol and thus DVtoCSP for this protocol is also \(2\cdot n+\mathbb Z_{q}^{*}\). In addition, in [9], the challenge message is
in which, c_{1},c_{2} are in \(\mathbb G_{1}\) and \(\mathbb G_{2}\), respectively, \(i\in [n], v_{i}\in \mathbb Z_{q}^{*}\), and pf is a proof of knowledge. Therefore, the communication overhead DVtoCSP for this protocol equals to
We also know that the challenge message of Ji et al.’s protocol has length of
Finally, in our protocol, the returned proof Γ has the form of
which has the length of
Similarly, we are able to compute the communication overheads from CSP to DV for [8], [9], and [4] as
and
respectively.
The total comparisons on the communication overheads are listed in Table 2.
Storage cost
Now, we analyze the storage costs of the chosen protocols. First, in our protocol, there are four entities: KGC, DO, DV, and CSP. In the running of this protocol, the KGC will store its own master secret key \(msk=x\in \mathbb Z_{q}\), which has length of \(\mathbb Z_{q}\). For DO, after outsourcing its original authenticated data file to CSP, it only needs to store its private key \(sk_{ID}=H_{1}(ID)^{x}\in \mathbb G_{1}\), which has length of \(\mathbb G_{1}\). For CSP, it will store the transmitted data file T (from DO), which has length of
Finally, the DV needs to store its own private key \(sk_{V}\in \mathbb G_{1}\) and the challenged message \(chal=(\ell, k_{1},k_{2})\in [n]\times \mathbb Z_{q}^{*}\times \mathbb Z_{q}^{*}\), which have lengths of \(\mathbb G_{1}\) and \(n+2\mathbb Z_{q}^{*}\), respectively. Note that the storage cost of CSP just equals to the communication cost from DO to CSP, which has be discussed in the previous subsection. Hence, we do not consider it in the following parts.
Similarly, we can evaluate the storage costs of the involved entities for other protocols, and present the comparisons in Table 3.
Computational cost (Theoretical analysis)
Now, we analyze the computational cost of our protocol, which mainly consists of the computations of authenticated data file T (for DO), returned proof Γ (for the CSP), and the verification of Γ. The computational cost mainly relies on the expensive operations like pairing, multiplication, and exponentiation, since other operations such as the hash function or addition on group only have negligible costs. For clarity of the theoretical analysis, we denote by T_{p},T_{mul}, and T_{exp} the computational costs for pairing, multiplication and exponentiation (on group \(\mathbb G_{1}\)), respectively.
First, in our protocol, the generation of the authenticated data file T will need to compute n tags
u_{1},u_{2},⋯,u_{s},R, and T_{Fid}=IBS.Sig(RFid). Hence, the computation cost is
where T_{IBS} denotes the timeconsumption of generating a signature in an IBS scheme. If we set s=1 as in [8], then the cost is
Similarly, we can evaluate the computational costs for the authenticated data file T in [8], [9], and [4] as
where T_{S} is the timeconsumption for a signature algorithm (like BLSsignature),
and
respectively.
Next, we consider the computational cost of generating the proof Γ. In our protocol, Γ is generated by computing
which costs
However, the timeconsumptions of Γ in [8], [9], and [4] are
and
respectively.
Finally, the computational costs of verifying Γ for the three protocols are analyzed as follows. In our protocol, the DV will first verify the validity of T_{Fid}, whose timeconsumption is denoted by T_{Ver}, and then check the equality (3). The total computational cost is
If s=1, then it is
Similarly, we can evaluate the verifications of other protocols and calculate their computational costs as
where \(T^{\prime }_{Ver}\) denotes the timeconsumption for the verification of a signature scheme,
and
respectively.
In addition, we remark that our protocol is the first one for a designated verifier but the other two protocols are not. The total comparisons on the computational costs are listed in Table 4.
Experimental results
In order to further evaluate the performance of our proposed protocol, we implement it within the framework of “Charm” [27]. In particular, the 512bit SS elliptic curve from pairingbased cryptography (PBC) library is set as the basis of our experiments [28]. The DO and DV are simulated by a Huawei MateBook with the configuration of Intel Core i56200U CPU @2.3GHz and 16GB RAM. Then the CSP is simulated by a Huawei FusionServer 2288H V5 with the configuration of Xeor Bronze 3106@1.7GHz, 16GB RAM.
Since, in the three protocols, standard signature or IBS scheme is needed, we choose the BLSsignature [29] and the IBS scheme in [30] or [4] as building blocks to implement them. Now, we choose a data file with size of 500 MB, which is parsed as 100, 200, 300, 400, and 500 blocks. Then the timeconsumptions for the generations of authenticated files in the three protocols are listed in Fig. 2. For each authenticated file, we set the numbers of challenged blocks as 30, 60, 90, 120, and 150. Then the timeconsumptions for the generation of returned proof and the verification for DV are presented in Fig 3 and 4, respectively.
From the comparisons of experimental results, we can see that our proposed protocol is competitive especially for the generation of the returned proof. However, we still explain that our protocol is the first one for the DV but the other ones are not. Hence, our protocol does not obviously reduce IBPDP’s efficiency, and can be naturally used in future reallife applications.
Conclusions
In many practical scenarios, the data owner only hopes some designated verifier to perform the auditing task, which is not covered by private or public auditing model. Yan et al. recently designed an auditing protocol with a designated verifier to resolve this problem. But their protocol naturally suffers from complicated management of certificates and heavy dependence on PKI. In this paper, for the first time, we introduce the identitybased remote data checking with a designated verifier. Compared with the existing PDP protocol with a designated verifier, our protocol avoids the introduction of PKI and management of public key certificates. Moreover, we also give the security model and prove that our protocol is provable secure based on the CDH and WDH assumptions in the random oracle model. The final analysis on performance shows that this protocol is also efficient and can be used in future reallife applications.
Future works. The first interesting problem is how to design more efficient IBDVPDP protocol so that it can find applications in practical scenarios. In addition, note that the security of our proposed IBDVPDP protocol relies on the ideal random oracle, and thus the second future work is finding and designing an efficient IBDVPDP protocol, which is provable secure in the standard model.
Availability of data and materials
The data and materials are available from the corresponding author on reasonable request.
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Acknowledgments
The authors would like to thank anonymous referees for their valuable suggestions and comments.
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This work is supported in part by National Natural Science Foundation of China (No. 61872284), and in part by Foundation of SKLOIS (No. 2021MS04).
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Yanyan Ji and Bilin Shao gave the main idea of this paper. The other three authors had worked equally during all this paper’s stages. The authors read and approved the final manuscript.
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Ji, Y., Shao, B., Chang, J. et al. Identitybased remote data checking with a designated verifier. J Cloud Comp 11, 7 (2022). https://doi.org/10.1186/s13677022002795
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DOI: https://doi.org/10.1186/s13677022002795