
This paper addresses a mixed and free convective Casson nanofluid flowing on an oscillating inclined poured plate with sinusoidal heat transfers and slip boundaries. As base fluid water is supposed and the suspension of nanofluid is formulated with the combination of individual copper (Cu), titanium dioxide (TiO2) and aluminum oxide (Al2O3) as nanoparticles, the dimensionless governing equations are generalized based on Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional operators for developing a fractional form. Then, for the semi-analytical solution of the momentum and thermal profiles, the Laplace transformation is utilized. To discuss the influences of various pertinent parameters on governing equations, graphical tablecomparison of the Nusselt number and skin friction is also inspected at different times and numerical schemes. As a result, it has been concluded that both the momentum and energy profiles represent the more significant results for the AB-fractional model as related to the CF-fractional model solution. Furthermore, water-based titanium dioxide (TiO2) has a more progressive impact on the momentum as well as the thermal fields as compared to copper (Cu) and aluminum oxide (Al2O3) nanoparticles. The Casson fluid parameter represents the dual behavior for the momentum profile, initially momentum field decreases due to the Casson parameter but it then reverses its impact and the fluid flow moves more progressively.
Citation: Ali Raza, Umair Khan, Aurang Zaib, Wajaree Weera, Ahmed M. Galal. A comparative study for fractional simulations of Casson nanofluid flow with sinusoidal and slipping boundary conditions via a fractional approach[J]. AIMS Mathematics, 2022, 7(11): 19954-19974. doi: 10.3934/math.20221092
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This paper addresses a mixed and free convective Casson nanofluid flowing on an oscillating inclined poured plate with sinusoidal heat transfers and slip boundaries. As base fluid water is supposed and the suspension of nanofluid is formulated with the combination of individual copper (Cu), titanium dioxide (TiO2) and aluminum oxide (Al2O3) as nanoparticles, the dimensionless governing equations are generalized based on Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional operators for developing a fractional form. Then, for the semi-analytical solution of the momentum and thermal profiles, the Laplace transformation is utilized. To discuss the influences of various pertinent parameters on governing equations, graphical tablecomparison of the Nusselt number and skin friction is also inspected at different times and numerical schemes. As a result, it has been concluded that both the momentum and energy profiles represent the more significant results for the AB-fractional model as related to the CF-fractional model solution. Furthermore, water-based titanium dioxide (TiO2) has a more progressive impact on the momentum as well as the thermal fields as compared to copper (Cu) and aluminum oxide (Al2O3) nanoparticles. The Casson fluid parameter represents the dual behavior for the momentum profile, initially momentum field decreases due to the Casson parameter but it then reverses its impact and the fluid flow moves more progressively.
Nowadays, a large amount of data is generated every day. Securely storing and processing such a large amount of data is a significant challenge [1,2,3]. Cloud computing [4], artificial intelligence [5,6,7,8], and big data techniques [9] are promising ways to address this challenge. Among them, cloud storage is essential because it provides a basic way of storing such a vast amount of data. Cloud storage services are becoming increasingly popular, and many people have outsourced their data, including files, movies, and photos, to the cloud. On the one hand, this service is very convenient for users since they do not need to maintain their data locally. Furthermore, they can access it via their mobile devices at any time and from anywhere. For some users, perhaps the most valuable aspect of cloud storage is the assurance that their outsourced data is almost never lost. On the other hand, the security of these outsourced files, movies, and photos cannot be guaranteed by the cloud storage service providers themselves. We need mechanisms to ensure their security, such as encryption, secure search, deduplication, and auditing techniques.
Due to limited local storage, many data owners want to outsource their files to a cloud server, such as movies, pictures, or music. Before outsourcing the file to the cloud server, the data owners encrypt their files using convergent encryption. Then they outsource the encrypted files to the cloud server. If many data owners (more than the predefined threshold) outsource the same encrypted files to the cloud server, these encrypted files will be the same and denoted as "popular." They will then be deduplicated, and the cloud server will only store one copy for all the data owners. However, if only a few data owners (less than the predefined threshold) outsource the encrypted files to the cloud server, these encrypted files will be denoted as "unpopular, " and they will not be deduplicated. In this way, the cloud server can save on storage costs. The technique of deduplication has been used by many cloud service providers, such as Amazon.
In this paper, we focus on a recently proposed scheme for cloud storage auditing [10] with deduplication. It supports different security levels and first introduces the concept of different security levels in this context. In this scheme, the outsourced data is categorized as popular or unpopular. If many data users have outsourced the same file to the cloud, this file can be considered popular. Otherwise, if the outsourced file has not been outsourced by many users, it can be categorized as unpopular. For popular files, Hou et al. suggest using convergent encryption to encrypt them, which is better for deduplication. In this way, cloud service providers can greatly reduce storage space. But for unpopular files, they suggest using probabilistic encryption to achieve semantic security, which is more secure than convergent encryption. Due to the unpopularity of these files, deduplication is no longer necessary. Generally speaking, this proposal is very interesting and valuable. However, we will show that it is not as secure as claimed, as there are some flaws in the scheme that invalidate its protocol's security. We also propose an improved scheme to achieve the security goal.
Our contribution can be summarized is two-fold. First, we focus on and demonstrate that the scheme proposed in [10] is not secure. Although it is the first relevant work on introducing data popularity to cloud auditing, this scheme is not entirely secure. We also analyze why their scheme has this security flaw and show how to avoid it. Then, we present an improved scheme building on the ideas proposed in [10], and provide a thorough analysis of its security. Our scheme addresses the security flaw present in the original proposal, and we explain in detail why it is more resistant to attacks.
In section 1, we provide the background, paper contribution, and organization. In section 2, we discuss related work. In section 3, we review the scheme proposed by Hou et al. In section 4, we present the attack. In section 5, we provide an improved proposal and briefly analyze its security. Finally, in section 6, we conclude the paper.
There is a large body of research in this context, and we have included the most relevant works to the one presented in this paper. This includes related work on encryption and deduplication techniques, as well as auditing schemes.
1. Encryption and deduplication are important techniques for ensuring the confidentiality and efficient management of outsourced data. While traditional encryption techniques, such as probabilistic public key encryption or symmetric encryption like AES, can achieve semantic security, they are not suitable for implementing functionality such as searching and deduplication. Therefore, novel encryption techniques have been developed specifically for use in cloud computing, including encryption with keyword search [11,12,13,14], encryption with access control [15,16], convergent encryption with deduplication [17,18], and others [19].
2. Auditing is an important way to ensure the integrity of the outsourced data. In 2007, Ateneo et al. [20] proposed the concept of provable data possession, which aims to allow the cloud servers to provide proof that they have stored the outsourced data well to the cloud users. Furthermore, the proof is very compact and the probability of cheating by the cloud servers is very low. This interesting primitive is actually a new auditing method for cloud storage. Since then, many cloud auditing schemes following this paradigm have been proposed, such as dynamic provable data possession [21], proof of retrievability [22,23], compact proof of retrievability [24], publicly verifiable auditing [25,26].
3. In 2016, Yu proposed a cloud data integrity checking scheme with an identity-based auditing mechanism from RSA [27]. Later, they proposed identity-based [28], attribute-based [29], and blockchain-based [30] cloud auditing schemes with different properties, and these are very interesting results in this field. Sometimes, the cloud service provider needs to use both auditing and deduplication techniques simultaneously. This way, the cloud service provider can reduce its costs when many users are outsourcing the same file, like popular movies, popular music, etc., and at the same time, the cloud users can ensure that their outsourced data, like files and photos, have not been lost or tampered with.
The system model of Hou et al.'s cloud auditing scheme with deduplication [10] is shown in Figure 1. There are three roles in the system: the data owners, the cloud server, and the TPA (third-party auditor). The system operates as follows:
1. To check the integrity of the outsourced files, the data owners, cloud server, and TPA (third party auditor) run the auditing scheme proposed by Hou et al. In this scheme, the data owners compute the authenticators for the data blocks of the files and outsource both the files and the authenticators to the cloud server. When the data owners want to check the integrity of the outsourced files, they delegate this task to the TPA. The TPA launches a challenge-proof game with the cloud server. First, the TPA sends a challenge to the cloud server requesting the server to return the aggregated data blocks and the corresponding aggregated authenticators as proof for the integrity of the outsourced file. Then, the cloud server returns the proof to the TPA, who checks the correctness of the proof using verification equations.
2. However, in the auditing process described above, the cloud server may be malicious. In an effort to reduce storage space, it may delete or modify some outsourced files without being detected by the data owners or the TPA. This means that the malicious cloud server has a strong incentive to delete the outsourced files. In the following section, we will demonstrate an attack on Hou et al's auditing scheme. In this attack, the malicious cloud server is able to forge the authenticator for any data block, which in turn invalidates their auditing scheme.
We will now review Hou et al.'s scheme [10]. The core data flow between the data owner and cloud storage server, between the IS and cloud storage server, and between the TPA and cloud storage server can be seen in Figures 2, 3, and 4.
Notations: Assume the file outsourced to the cloud by the data owner is F={m1,m2,⋯,mn}. Each mi={mi1,mi2,⋯,mis}, here 1≤i≤n. The file has its unique file identifier, it is signed with signature SSig to prevent the attackers to modify it. The user (data owner) keeps his secret key for generating SSig and publish the public key for signature.
1. Setup: With parameter k as the input,
(a) Running IG(1k) to generate G1 and G2, which are two cyclic multiplicative groups of large prime order p. There exists a e:G1×G1→G2 which is a bilinear pairing.
(b) We denote h:{0,1}∗→{0,1}∗ as an indexing function, ϕ:Z∗p×Z∗p→Z∗p and π:Z∗p×{1,2,⋯,n} as a PRF and a PRP, denote H1:{0,1}∗→G1, H2:G2→{0,1}l, H3:{0,1}∗→G1 as three cryptographic hash functions.
(c) Run εμ.Setup(k,n,t)→(pk,sk,S) where pk={p,G1,G2,e,H1,H2,H3,h,g,gpub}, n key shares {xi}n−1i=0 also generated.
2. Join: this algorithm is not directly related with the attack.
3. Upload: this algorithm is not directly related with the attack.
4. AuthGen: With a ciphertext of file C={c1,c2,⋯,cn} (specially C is Cϵμ or Cϵ) and a secret key ktag←Z∗p, the key v←gktag is computed and published by the user. For each ciphertext block ci(1≤i≤n), the authenticator Ti is generated by Ui and uploaded to the cloud.
(a) u1,u2,⋯,us are s generators of G1, which are chosen by Ui, r←Z∗p is also randomly chosen by Ui.
(b) Denote τ0=name||n||vr||u1||u2||⋯||us. Let ssk←Z∗p be signing key and Pssk←gssk the corresponding verification key. These are randomly generated by the user. The file tag is τ←τ0||SSigssk(τ0).
(c) For each data block the authenticator is computed by Ui as
Ti=(H3(name||i)⋅s∏j=1ucijj)ktag. |
(d)
{uktag(r(Cϵμ)i,1−(Cϵ)i,1)1,uktag(r(Cϵμ)i,2−(Cϵ)i,2)2,⋯,uktag(r(Cϵμ)i,s−(Cϵ)i,s)s} |
are computed by Ui and sent to IS.
(e) The file tag and {Ti}1≤i≤n are sent by Ui to the cloud.
5. PopulartityChange: For the popularity threshold t, the algorithm is executed whenever the users' number that are submitting the same index is higer than it. The file F is not needed to upload to cloud again by the user Ui. IS sends the set index to the cloud, and Ui sends it to the cloud. For all those users with file index in the set index, the storage cloud collects decryption shares of them. Then the ciphertext Cϵμ uploaded by these users can be decrypted by the storage cloud. Then, the ciphertext FC encrypted by the convergent encryption can be obtained by the storage cloud. Thus, as the ciphertext FC coincides with that for file F, the deduplication can be achieved. Finally, {uktag(r(Cϵμ)i,1−(Cϵ)i,1)1,uktag(r(Cϵμ)i,2−(Cϵ)i,2)2,⋯,uktag(r(Cϵμ)i,s−(Cϵ)i,s)s} are sent to the cloud by the IS. The new data block authenticator*
* In [10], T′i=Ti⋅∏sj=1uktag(rcϵμij−cϵij)j, but we think it should be T′i=Ti⋅∏sj=1uktag(r(Cϵμ)i,j−(Cϵ)i,j)j.
T′i=Ti⋅s∏j=1uktag(rcϵμij−cϵij)j |
for each user are created by the clouds.
6. ProofGen: With the {ci}1≤i≤n, {Ti}1≤i≤n as the input,
● The auditing challenge is generated by TPA as the following:
(a) The file tag gained by the TPA from the cloud and using the key Pssk it checks whether the correctness of signature on τ0. TPA rejects and halts if the signature is not correct.
(b) Otherwise, filename name, n, vr and {u1,u2,⋯,us} are recovered by the TPA. Then c, with 1≤c≤n is chosen by him.
(c) Parameters k1←Z∗p, k2←Z∗p are randomly selected by the TPA.
(d) The challenge chal=(c,k1,k2) is sent by the TPA to the cloud.
● The cloud yields lt=πk1(t) and at=ϕk2(t) wherein 1≤t≤c after receiving chal from the TPA. And then the proof T=∏ct=1Tatlt, ηj=∑ct=1at⋅clt,j,1≤j≤s is computed.
7. ProofVerify: With the proof P=(T,η) and the challenge massage chal=(c,k1,k2), TPA computes lt=πk1(t) together with at=ϕk2(t) wherein 1≤t≤c. Subsequently, the below verification equations are checked
e(T,g)=e(c∏t=1(H3(name||lt)ats∏j=1uηjj),v), |
e(T,g)=e(c∏t=1(H3(name||lt)ats∏j=1uηjj),vr). |
If one of them passed, the proof is valid.
The attack is executed according to the following steps:
1. The attacker can be the IS or the cloud. Note here the IS or the cloud can obtain
{uktag(r(Cϵμ)i,1−(Cϵ)i,1)1,uktag(r(Cϵμ)i,2−(Cϵ)i,2)2,⋯,uktag(r(Cϵμ)i,s−(Cϵ)i,s)s},1≤i≤n |
from Ui by running algorithm AuthGen. Concretely the IS or the cloud can obtain
{uktag(r(Cϵμ)1,1−(Cϵ)1,1)1,uktag(r(Cϵμ)1,2−(Cϵ)1,2)2,⋯,uktag(r(Cϵμ)1,s−(Cϵ)1,s)s},{uktag(r(Cϵμ)2,1−(Cϵ)2,1)1,uktag(r(Cϵμ)2,2−(Cϵ)2,2)2,⋯,uktag(r(Cϵμ)2,s−(Cϵ)2,s)s},{uktag(r(Cϵμ)3,1−(Cϵ)3,1)1,uktag(r(Cϵμ)3,2−(Cϵ)3,2)2,⋯,uktag(r(Cϵμ)3,s−(Cϵ)3,s)s},⋯⋯⋯,{uktag(r(Cϵμ)n,1−(Cϵ)n,1)1,uktag(r(Cϵμ)n,2−(Cϵ)n,2)2,⋯,uktag(r(Cϵμ)n,s−(Cϵ)n,s)s}. |
2. Let A1=urktag1, B1=uktag1, A2=urktag2, B2=uktag2, ⋯,⋯,⋯, As=urktags, Bs=uktags then
{uktag(r(Cϵμ)1,1−(Cϵ)1,1)1,uktag(r(Cϵμ)1,2−(Cϵ)1,2)2,⋯,uktag(r(Cϵμ)1,s−(Cϵ)1,s)s},{uktag(r(Cϵμ)2,1−(Cϵ)2,1)1,uktag(r(Cϵμ)2,2−(Cϵ)2,2)2,⋯,uktag(r(Cϵμ)2,s−(Cϵ)2,s)s},{uktag(r(Cϵμ)3,1−(Cϵ)3,1)1,uktag(r(Cϵμ)3,2−(Cϵ)3,2)2,⋯,uktag(r(Cϵμ)3,s−(Cϵ)3,s)s},⋯⋯⋯,{uktag(r(Cϵμ)n,1−(Cϵ)n,1)1,uktag(r(Cϵμ)n,2−(Cϵ)n,2)2,⋯,uktag(r(Cϵμ)n,s−(Cϵ)n,s)s} |
can be rewritten as
{A(Cϵμ)1,11B(Cϵ)1,11,A(Cϵμ)1,22B(Cϵ)1,22,⋯,A(Cϵμ)1,ssB(Cϵ)1,ss},{A(Cϵμ)2,11B(Cϵ)2,11,A(Cϵμ)2,22B(Cϵ)2,22,⋯,A(Cϵμ)2,ssB(Cϵ)2,ss},{A(Cϵμ)3,11B(Cϵ)3,11,A(Cϵμ)3,22B(Cϵ)3,22,⋯,A(Cϵμ)3,ssB(Cϵ)3,ss},⋯⋯⋯,{A(Cϵμ)n,11B(Cϵ)n,11,A(Cϵμ)n,22B(Cϵ)n,22,⋯,A(Cϵμ)n,ssB(Cϵ)n,ss}. |
3. With
{A(Cϵμ)1,11B(Cϵ)1,11,A(Cϵμ)1,22B(Cϵ)1,22,⋯,A(Cϵμ)1,ssB(Cϵ)1,ss},{A(Cϵμ)2,11B(Cϵ)2,11,A(Cϵμ)2,22B(Cϵ)2,22,⋯,A(Cϵμ)2,ssB(Cϵ)2,ss}, |
the attacker can compute A1, B1, ⋯,⋯,⋯, As, Bs as following. First let X1=A(Cϵμ)1,11B(Cϵ)1,11,X2=A(Cϵμ)1,22B(Cϵ)1,22,⋯,Xs=A(Cϵμ)1,ssB(Cϵ)1,ss, Y1=A(Cϵμ)2,11B(Cϵ)2,11,Y2=A(Cϵμ)2,22B(Cϵ)2,22,⋯,Ys=A(Cϵμ)2,ssB(Cϵ)2,ss then the above can be rewritten as
{X1,X2,⋯,Xs},{Y1,Y2,⋯,Ys}. |
4. With X1, Y1, the adversary can compute A1,B1 as following:
X(Cϵμ)2,11=A(Cϵμ)1,1(Cϵμ)2,11B(Cϵ)1,1(Cϵμ)2,11,Y(Cϵμ)1,11=A(Cϵμ)2,1(Cϵμ)1,11B(Cϵ)2,1(Cϵμ)1,11, |
then
X(Cϵμ)2,11Y(Cϵμ)1,11=A(Cϵμ)1,1(Cϵμ)2,11B(Cϵ)1,1(Cϵμ)2,11A(Cϵμ)2,1(Cϵμ)1,11B(Cϵ)2,1(Cϵμ)1,11=B(Cϵ)1,1(Cϵμ)2,11B(Cϵ)2,1(Cϵμ)1,11=B(Cϵ)1,1(Cϵμ)2,1−(Cϵ)2,1(Cϵμ)1,11. |
5.Due to the group order p is publicly known and thus the following holds. Let Z1=X(Cϵμ)2,11Y(Cϵμ)1,11 then B1=Z((Cϵ)1,1(Cϵμ)2,1−(Cϵ)2,1(Cϵμ)1,1)−1modp1.
6. Similarly
X(Cϵ)2,11=A(Cϵμ)1,1(Cϵ)2,11B(Cϵ)1,1(Cϵ)2,11,Y(Cϵ)1,11=A(Cϵμ)2,1((Cϵ)1,11B(Cϵ)2,1(Cϵ)1,11, |
then
X(Cϵ)2,11Y(Cϵ)1,11=A(Cϵμ)1,1(Cϵ)2,11B(Cϵ)1,1(Cϵ)2,11A(Cϵμ)2,1((Cϵ)1,11B(Cϵ)2,1(Cϵ)1,11=A(Cϵμ)1,1(Cϵ)2,1−(Cϵμ)2,1(Cϵ)1,11. |
7. Due to the group order p is publicly known and thus the following holds. Let
W1=X(Cϵ)2,11Y(Cϵ)1,11, |
then
A1=W((Cϵ)2,1−(Cϵ)1,1)−1modp1. |
8. By using the above same method, the adversary can compute A2, B2, ⋯, ⋯, As, Bs.
With A1, B1, A2, B2, ⋯, ⋯, As, Bs, the adversary can forge any data block's authenticator as the following.
1. First the adversary (the IS or the cloud) can obtain
Ti=(H3(name||i)⋅s∏j=1ucijj)ktag(1≤i≤n). |
Note here cij is public known to all.
2. With
Ti=(H3(name||i)⋅s∏j=1ucijj)ktag, |
and A1, B1, A2, B2, ⋯, ⋯, As, Bs, the adversary can compute
TiBci11Bci22Bci23⋯Bci2n=(H3(name||i)⋅s∏j=1ucijj)ktagBci11Bci22Bci23⋯Bci2n=(H3(name||i)⋅s∏j=1ucijj)ktag(uktag1)ci1(uktag2)ci2(uktag3)ci2⋯(uktagn)ci2=(H3(name||i))ktag. |
Then it forges any data block's authenticator as following
3. Let ^cij be the forged encrypted j-th sector of the i-th data block, then the adversary compute the following:
(H3(name||i))ktag⋅(B^ci11B^ci22B^ci23⋯B^ci2n)=(H3(name||i)⋅s∏j=1u^cijj)ktag, |
which is a valid authenticator for the any forged encrypted sector.
4. This means that the cloud can modify the outsouced encrypted data block and its corresponding authenticator to be any other one, which obviously breaks the security of cloud auditing protocols.
First, we will review the core idea for updating the authenticator in [10]. Next, we will analyze why this core idea is not secure. Finally, we will present an improved method.
● Now we review the core idea in [10]. In the original proposal, (H(i)⋅uCi)x is the authenticator. Assume σi=(H(i)⋅uCi)x is the original authenticator and σi=(H(i)⋅uC′i)x is the new corresponding authenticator. Denote △i=σ′i/σi=(H(i)⋅uC′i)x(H(i)⋅uCi)x=(u(C′i−Ci))x=(ux)(C′i−Ci). Thus, σi⋅(ux)(C′i−Ci)=σ′i. The cloud can compute σ′i given σi and (ux)(C′i−Ci). However, the inverse of (C′i−Ci) can be calculated by the adversary and thus it is not secure. For example, for C∗i, the corresponding authenticator σ∗i=σi⋅(ux)(C′i−Ci)(C′i−Ci)−1⋅(C∗i−Ci)=σi(ux)(C∗i−Ci) can be forged. For safety, a blind factor r is introduced. (ux)(r⋅C′i−Ci) is first computed and then uploaded by the user to the storage cloud. New authenticator σ′i=σi⋅(ux)(r⋅C′i−Ci) whenever there are changes regarding the data popularity.
● However, the attack above shows that their idea of using (ux)(r⋅C′i−Ci) instead of (ux)(C′i−Ci) is still not secure. The reason is the following: if the cloud knows (ux)(r⋅C′i−Ci) for many such 1≤i≤n, it can compute (ux)r and (ux) easily. And thus it can forge any authenticator updates (ux)(r⋅C′any−Cany) easily. Furthermore, it also can forge authenticator σi=(H(i)⋅uCany)x easily for any block Cany, thus their core idea is not secure.
● We improve their core idea by modifying (ux)(r⋅C′i−Ci) to be ((ux)(ri⋅C′i−Ci),uri) for many such 1≤i≤n, in this way, the cloud can not compute (ux)ri(1≤i≤n) and (ux) easily. And thus it can not forge authenticators any more.
Building upon the improved core idea, we have developed an improved cloud auditing scheme which is outlined below:
1. Setup: For the sake of comparison, it is worth noting that this algorithm is identical to the corresponding algorithm presented in [10].
2. Join: This algorithm is the same as the corresponding algorithm in [10].
3. Upload: This algorithm is the same as the corresponding algorithm in [10].
4. AuthGen: With a ciphertext of file C={c1,c2,⋯,cn} (specially C is Cϵμ or Cϵ) and a secret key ktag←Z∗p, the key v←gktag is computed and published by the user. For each ciphertext block ci(1≤i≤n), the authenticator Ti is generated by Ui and uploaded to the cloud.
(a) u1,u2,⋯,us are s generators of G1, which are chosen by Ui, r1,r2,⋯,rn←Z∗p are also randomly chosen by Ui.
(b) Let τ0=name||n||vr1||vr2||⋯||vrn||u1||u2||⋯||us. A signing key ssk←Z∗p and the corresponding verification key Pssk←gssk are randomly generated by the user. The file tag is τ←τ0||SSigssk(τ0).
(c) For each data block the authenticator is computed by Ui as
Ti=(H3(name||i)⋅s∏j=1ucijj)ktag. |
(d)
{uktag(r1(Cϵμ)1,1−(Cϵ)1,1)1,uktag(r1(Cϵμ)1,2−(Cϵ)1,2)2,⋯,uktag(r1(Cϵμ)1,s−(Cϵ)1,s)s}, |
{uktag(r2(Cϵμ)2,1−(Cϵ)2,1)1,uktag(r2(Cϵμ)2,2−(Cϵ)2,2)2,⋯,uktag(r2(Cϵμ)2,s−(Cϵ)2,s)s}, |
⋯⋯⋯⋯⋯⋯, |
{uktag(rn(Cϵμ)n,1−(Cϵ)n,1)1,uktag(rn(Cϵμ)n,2−(Cϵ)n,2)2,⋯,uktag(rn(Cϵμ)n,s−(Cϵ)n,s)s} |
are computed by Ui and sent to IS, we denote them as Upd.
(e) The file tag and {Ti}1≤i≤n are sent by Ui to the cloud.
5. PopulartityChange: This algorithm is the same as the corresponding algorithm in [10] except
T′i=Ti⋅s∏j=1uktag(ri(Cϵμ)i,j−(Cϵ)i,j)j. |
Note here we use ri instead of r in the exponentiation.
6. ProofGen: With the {ci}1≤i≤n, {Ti}1≤i≤n as the input,
● the auditing challenge is generated by TPA as the following:
(a) The file tag gained by the TPA from the cloud and using the key Pssk it checks whether the correctness of signature on τ0. TPA rejects and halts if the signature is not correct.
(b) Otherwise, filename name, n, vr1,vr2,⋯,⋯,vrn and {u1,u2,⋯,us} are recovered by the TPA. Then c(1≤c≤n) is chosen by him, which is the number of the challenged blocks.
(c) k1←Z∗p, k1←Z∗p are randomly picked by the TPA.
(d) The challenge chal=(c,k1,k2) is sent by the TPA to the cloud.
● The cloud computes lt=πk1(t),at=ϕk2(t)(1≤t≤c) after receiving chal from the TPA. And then the proof T=∏ct=1Tatlt, ηj=∑ct=1at⋅clt,j,1≤j≤s is computed.
● ProofVerify: With the proof P=(T,η) and the challenge massage chal=(c,k1,k2), TPA computes lt=πk1(t),at=ϕk2(t)(1≤t≤c). Then the below verification equations are checked
e(T,g)=e(c∏t=1(H3(name||lt)ats∏j=1uηjj),v), |
e(T,g)=c∏t=1e(H3(name||lt)ats∏j=1uηjj,vrt). |
If one of them passed, the proof is valid.
The reasons why this improved proposal can resist the attack above is explained as follows: From the Upd,
{uktag(r1(Cϵμ)1,1−(Cϵ)1,1)1,uktag(r1(Cϵμ)1,2−(Cϵ)1,2)2,⋯,uktag(r1(Cϵμ)1,s−(Cϵ)1,s)s}, |
{uktag(r2(Cϵμ)2,1−(Cϵ)2,1)1,uktag(r2(Cϵμ)2,2−(Cϵ)2,2)2,⋯,uktag(r2(Cϵμ)2,s−(Cϵ)2,s)s}, |
⋯⋯⋯⋯⋯⋯, |
{uktag(rn(Cϵμ)n,1−(Cϵ)n,1)1,uktag(rn(Cϵμ)n,2−(Cϵ)n,2)2,⋯,uktag(rn(Cϵμ)n,s−(Cϵ)n,s)s}, |
the adversary can not obtain the below values anymore
{uktag(r(Cϵμ)1,1−(Cϵ)1,1)1,uktag(r(Cϵμ)1,2−(Cϵ)1,2)2,⋯,uktag(r(Cϵμ)1,s−(Cϵ)1,s)s},{uktag(r(Cϵμ)2,1−(Cϵ)2,1)1,uktag(r(Cϵμ)2,2−(Cϵ)2,2)2,⋯,uktag(r(Cϵμ)2,s−(Cϵ)2,s)s},{uktag(r(Cϵμ)3,1−(Cϵ)3,1)1,uktag(r(Cϵμ)3,2−(Cϵ)3,2)2,⋯,uktag(r(Cϵμ)3,s−(Cϵ)3,s)s},⋯⋯⋯,{uktag(r(Cϵμ)n,1−(Cϵ)n,1)1,uktag(r(Cϵμ)n,2−(Cϵ)n,2)2,⋯,uktag(r(Cϵμ)n,s−(Cϵ)n,s)s}, |
it can only obtain
{uktag(r1(Cϵμ)1,1−(Cϵ)1,1)1,uktag(r1(Cϵμ)1,2−(Cϵ)1,2)2,⋯,uktag(r1(Cϵμ)1,s−(Cϵ)1,s)s},{uktag(r2(Cϵμ)2,1−(Cϵ)2,1)1,uktag(r2(Cϵμ)2,2−(Cϵ)2,2)2,⋯,uktag(r2(Cϵμ)2,s−(Cϵ)2,s)s},{uktag(r3(Cϵμ)3,1−(Cϵ)3,1)1,uktag(r3(Cϵμ)3,2−(Cϵ)3,2)2,⋯,uktag(r3(Cϵμ)3,s−(Cϵ)3,s)s},⋯⋯⋯{uktag(rn(Cϵμ)n,1−(Cϵ)n,1)1,uktag(rn(Cϵμ)n,2−(Cϵ)n,2)2,⋯,uktag(rn(Cϵμ)n,s−(Cϵ)n,s)s}, |
from these values, the adversary can not compute ur1ktag1, uktag1, ur2ktag2, uktag2, ⋯,⋯,⋯, urnktagn, Bs=uktag2 anymore. Thus the above attack can not work anymore.
In 2019, Hou et al. proposed an auditing scheme. However, in this paper, we demonstrate that their proposal is not secure. The main reason for this is that the core idea of their updated authenticator algorithm is vulnerable. Specifically, if the cloud storage server obtains many values of (ux)(r⋅C′i−Ci) for 1≤i≤n, it can easily compute (ux)r and (ux), which allows it to forge an authenticator σi=(H(i)⋅uCany)x for any block Cany. This attack is a generalization of attacks on many cloud storage auditing protocols based on the discrete logarithm hard problem, as shown in [31,32]. It highlights the need for caution when designing cloud storage auditing protocols using cryptographic techniques, as these schemes have rich algebraic structure that may result in vulnerabilities.
To address these shortcomings, we have developed an improved cloud storage auditing scheme based on Hou et al.'s proposal. Our updated authenticator algorithm now uses (ux)(ri⋅C′i−Ci) for 1≤i≤n, which makes it impossible for the adversary to compute (ux)r and (ux). We have also analyzed why our improved scheme is secure. We hope that our work will help future researchers avoid similar shortcomings in their own cloud storage auditing schemes.
This work is supported by the Key Research and Development Program of Xianyang City(No. L2022ZDYFSF061), Scientific Research Funding of Xianyang Vocational & Technical College on "Research on Key Technologies for Secure Outsouced Cloud Storage"(Grant No.2021KJB03).
The authors declare there is no conflict of interest.
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