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Information 2013, 4(3), 262-282; doi:10.3390/info4030262

Article
Lightweight Proofs of Retrievability for Electronic Evidence in Cloud
Xiuli Song 1,* and Hongyao Deng 2
1
Department of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2
School of Mathematics & Computer Science, Yangtze Normal University, Chongqing 408000, China; E-Mail: hydeng_2004@163.com
*
Author to whom correspondence should be addressed; E-Mail: songxl@cqupt.edu.cn; Tel./Fax:+86-2362461404.
Received: 27 March 2013; in revised form: 4 June 2013 / Accepted: 19 June 2013 /
Published: 5 July 2013

Abstract

: Proofs of Retrievability (PoR) is one of the basic functions of electronic evidence preservation center in cloud. This paper proposes two PoR schemes to execute the workflow of evidence preservation center, which are named Finer Grained Proofs of Retrievability (FG-PoR) and More Lightweight Proofs of Retrievability (ML-PoR). The two PoR schemes do not use multi-replication technology or erasure code technology, but employ the verification tags and signatures to implement provable data possession and data recovery dual functions. When some data blocks have been lost in Archive Storage Area (ASA), FG-PoR can recover each data block of evidence matrix, but ML-PoR can only recover a column of evidence matrix. The analysis results show our two PoR schemes do not only provide the integrity verification guarantee but also have robust recovery guarantee to electronic evidence in cloud. The two schemes can allow for lower computation and storage costs than other similar schemes; moreover, ML-PoR can provide lower costs than FG-PoR.
Keywords:
cloud storage; proofs of retrievability; Nyberg-Rueppel signature; electronic evidence preservation; computation and storage costs

1. Introduction

In a general operating environment, electronic evidence is derived from systematic data and network data. Systematic data include system logs, audit records, temporary or hidden files, hard disk drive exchange partition, etc. Network data include logs of firewall, Intrusion Detection Systme (IDS), and router, E-mail information, real-time chats, network monitoring records, etc. Electronic evidence may provide massive information and resource bases in various forms. In cloud computing environments, it provides users with flexible services in a transparent manner. One fundamental aspect of service is that data is being centralized or outsourced into a “cloud”, which is a collection of devices and resources connected through the Internet. Electronic evidence is a special kind of data, and it must be integrated before it is sent to a court of law. When it is being collected from the cloud, it should be fixed in a specific form and stored safely on the local or cloud server so as to prevent them from being destructed by nature and man. This operating process is called electronic evidence preservation [1].

Cloud forensics is a cross discipline of cloud computing and electronic forensics. Cloud computing is a shared collection of configurable networked resources (e.g., networks, servers, storage, applications and services) that can be reconfigured quickly with minimal effort [2]. Electronic forensics is the application of computer science principles to recover electronic evidence for presentation in a court of law [3]. Cloud forensic investigations are likely to involve evidence acquisition, preservation and analysis in a cloud. From the investigators’ perspective, storing electronic evidence remotely in a cloud in a flexible, on demand manner brings appealing benefits: relief of the burden of storage management, access to data with independent geographical locations, and avoiding capital expenditure on hardware, software and personnel maintenance [4]. Also, it can reduce the litigants to the objectivity of the evidence, simplify the review procedure of evidence in court, and effectively assess the fairness and justice in the administration.

While cloud computing makes these advantages more appealing than ever, it also brings new and challenging security threats to the outsourced electronic evidence. On one hand, electronic evidence has its own vulnerability, such that it can more easily be deleted, forged, altered and removed than traditional printed evidence. On the other hand, cloud computing is open and has a virtual operation environment for all users. When the outsourced electronic evidence is stored to a cloud, it might be unclear as to where evidence is processed within the cloud, and such processing might occur in different jurisdictions. Once electronic evidence is extracted from the cloud and is sent to the court, it must be true, reliable and integrated, in accord with legal requirements [5]. So how to ensure the integrity of electronic evidence in cloud is a considerable challenge to both cloud service providers and research institutes.

2. Related Work

When the users send their data to the Cloud Storage Server (CSS), they should pay a fee to obtain an appropriate storage space to preserve their data. The CSS is not fully trusted because the storage devices are under the control of the Cloud Service Provides (CSP), not the users. The importance of ensuring the integrity of data in untrusted storage servers has been highlighted by some researchers, and they have proposed two basic schemes to check the availability and integrity of cloud storage data. The two basic schemes are called provable data possession (PDP) [6] and proofs of retrievability (PoR) [7]. A PDP scheme can only verify the integrity of a file on the server, but cannot ensure the retrievability of a file [8]. However, a PoR scheme is a challenge response protocol. In the protocol, the server can demonstrate to the client that a file is intact and retrievable without any loss and corruption.

The provable data possession (PDP) scheme was built upon by Ateniese et al. [6]. In the scheme, the user could utilize RSA-based homomorphic tags to challenge the server. The server proved possession of the file by sending back a few randomly sampled blocks of the file. In a subsequent study, Curtmola et al. [9] described a multiple replica PDP (MR-PDP) scheme, which ensured that multiple replicas of the user’s files were stored at the untrusted storage server. Barsoum [10] constructed two efficient multi-copy PDP (EMC-PDP) protocols which were called Deterministic EMC-PDP (DEMC-PDP) and Probabilistic EMC-PDP (PEMC-PDP). PEMC-PDP depended on spot checking by validating a random subset of the file blocks instead of validating all the blocks in DEMC-PDP to reduce computation and storage overhead.

The proofs of retrievability (PoR) scheme was first proposed by Juels et al. [7], and this scheme used spot-checking and error-correcting codes to ensure both “possession” and “retrievability” of the files on remote servers. Shacham and Waters [11] also proposed a PoR scheme. In this scheme, the file was encoded using erasure code and was split into some blocks. The server used homomorphic algorithm to compact all data tags into a short tag, and it was taken as the response to the client. Bowers et al. [8] introduced a retrievable scheme for cloud storage data, which permitted a set of servers to prove to a client that a stored file was intact and retrievable. In [2], Cong Wang et al. utilized the homomorphic token to ensure the integrity of erasure-coded data with additional feature of data error localization. In a subsequent work, Qian Wang et al. [12] allowed a third party auditor to verify the integrity of the data stored in cloud based on Merkle hash tree.

To cloud forensics, several works have already been published in this field. Wolthusen [13] noted that when attempting to locate evidence in a distributed and complex environment such as the cloud, the distributed and virtual nature of the cloud would likely increase the difficulty of evidence collection, making tracing activity and re-construction of evidence more challenging. Grispos et al. [14] analyzed how established digital forensic procedures would be invalidated and discussed several new research challenges in the cloud environment. Birk and Wegener [15] assessed whether it was possible for the customer of cloud computing services to perform a traditional digital investigation from a technical point of view. Furthermore, they discussed possible solutions and possible new methodologies helping customers to perform such investigations.

In above works, literature [6,12] were the PDP schemes. They could only check the integrity of the file through the server sending back random samples of the file. Literature [2,7,8,11] were all the PoR schemes based on erasure codes. They used erasure codes technology to compact many data blocks into fewer redundancy. This approach added storage costs of redundant data and computation costs of encoding and decoding. Literature [9,10] were the PoR schemes based on multiple replica. In the two schemes, each data object needed to be created some copies, and the size of data objects and their copies were the same. Unfortunately, the storage costs of the copies in the two schemes was too high. Literature [13,14] discussed the potential benefits and challenges of cloud computing for electronic forensic investigations. Literature [15] mentioned possible solutions and methodologies of technology to face the challenges. But these three papers [13,14,15] did not all provide technical detail on how to solve the problem. This paper proposes two verification schemes of retrievability for preservation of electronic evidence in cloud computing environment. The two schemes are kept light-weight by employing verification tags and signatures with neither the use of erasure codes nor the use of mutiple-replica.

3. An Electronic Evidence Preservation Center in Cloud

Due to outdated equipment and the lack of technical support, it is impossible that electronic evidence could be preserved perfectly by the courts or the public security organs. This means electronic evidence will usually be stored in a special preservation center. In general, a traditional electronic evidence preservation center uses some technologies, such as encryption algorithm, digital signature, time stamp and digital digest to ensure the security of electronic evidence [1]. In a cloud computing environment, except for these conventional technologies, privacy protection, PDP and PoR technologies should be considered in a preservation center. This paper designs two PoR schemes for application in a cloud preservation center for electronic evidence. The architecture of the electronic evidence preservation center is shown in Figure 1.

Information 04 00262 g001 1024
Figure 1. The architecture of electronic evidence preservation center in cloud.

Click here to enlarge figure

Figure 1. The architecture of electronic evidence preservation center in cloud.
Information 04 00262 g001 1024

From Figure 1, the preservation center contains multiple functional areas: Center Management Area (CMA), Classification Preservation Area (CPA), Archive Storage Area (ASA), Evidence Recovery Area (ERA). When Law Authorities or Users (LAU) send electronic evidence to the preservation center, they only interact with the CMA, so it will greatly improve the work efficiency.

3.1. Center Management Area (CMA)

In the electronic evidence preservation center, electronic evidence would be stored using a classification storage method by the CPA. Each evidence file is divided into fixed-size chunks, and ASA will store these chunks. Each chunk will be assigned a label at the time of creation. The CMA stores all metadata associated with classification evidence. Also, the CMA provides the services of electronic evidence reception, notary and forensic services for the law authorities and users (LAU), which is an external service window of the electronic evidence preservation center. Moreover, it executes business process supervision and coordination and management to other function areas. It will be taken as a Trusted Audit Center (TAC) in our PoR schemes.

3.2. Classification Preservation Area (CPA)

If the CMA has received electronic evidence from the LAU, it will forward the evidence to CPA to store temporarily. When electronic evidence has been collected from selected mobile devices, including mobile phones, MP3/4, PDA, etc., it will be recorded as mobile terminal equipment evidence to be preserved. When electronic evidence is extracted from computer systematic data, including a system log, audit records, temporary files, exchange files, or hidden files, it will be recorded as computer system evidence to be preserved. When electronic evidence has been collected from all kinds of private computers, notebook computers, storage servers, or the hard drives of this equipment, it will be collected as hardware equipment evidence to be preserved. Electronic evidence may be attained from network servers and hosts, such as webpage browsing historical records, cookies, favorites, cache information, etc. It may also be attained from network communication data including logs of firewalls, IDS, routers, Email information, real-time chat and network monitoring records, etc. This evidence will be recorded as online network evidence to be preserved. After electronic evidence has been classified over a long time span in the CPA, it will be divided into chunks to be sent to the ASA. The CPA will store the metadata of all chunks, such as the tables, mapping the labels to chunk locations.

3.3. Archive Storage Area (ASA)

In this area, there are some Chunk Storage Servers (CSS) Information 04 00262 i002 in the cloud. These servers will store all chunks of the evidence file, and each chunk is split into some blocks. In FG-PoR schemes, each block has a corresponding tag, and all blocks and tags will be stored in the ASA. The ASA will provide safe, reliable, efficient storage services for massive electronic evidence in the cloud. When the CMA asks the CPA about the integrity of a stored evidence file, the CPA will query the metadata for the locations of the desired chunks and construct proofs of retrievability to prove that the file is intact and retrievable.

3.4. Evidence Recovery Area (ERA)

When the LAU needs their electronic evidence, it sends a request message to the CMA. The CMA accesses the chunks of evidence file by querying the CPA for the locations of the desired chunks; if the chunks are not being operated on, the CPA replies with the locations, and the CMA then contacts the ERA. The ERA retrieves the evidence from CSS Information 04 00262 i003, and uses hash function to check the integrity of the evidence file. Then it sends the evidence to the CMA, and further forwards it to the LAU. The CMA could also check the integrity of the evidence file by running verifying algorithm VeriResp( ). If the ERA or the CMA has found some error data blocks of the evidence matrix, the ERA will use the retrieved algorithm RetrData( ) to recover these data blocks.

4. Proofs of Retrievability for Electronic Evidence

From Figure 1, we know that CPA, ERA and CMA will coordinate to ensure the integrity and retrievability of electronic evidence. Moreover, the CMA will play the role of Trusted Audit Center (TAC) in collaboration. The workflow of LAU, CPA, ASA, ERA and CMA as the five functional areas is shown in Figure 2.

Information 04 00262 g002 1024
Figure 2. The workflow of five functional areas. “Send( )” means that the sender sends the evidence file to the receiver; “Requ( )” means that the sender sends the request message to the receiver; “Chal( )” means that one side sends the challenge message to the other side; “Resp( )” means that one side sends the response values to the other side; “Query” means that the CPA or the ERA queries chunks of the ASA; “Feedback” means that the ASA gives feedback messages of chunks and blocks.

Click here to enlarge figure

Figure 2. The workflow of five functional areas. “Send( )” means that the sender sends the evidence file to the receiver; “Requ( )” means that the sender sends the request message to the receiver; “Chal( )” means that one side sends the challenge message to the other side; “Resp( )” means that one side sends the response values to the other side; “Query” means that the CPA or the ERA queries chunks of the ASA; “Feedback” means that the ASA gives feedback messages of chunks and blocks.
Information 04 00262 g002 1024

In our work, we provide two PoR schemes: Finer Grained Proofs of Retrievability (FG-PoR) and More Lightweight Proofs of Retrievability (ML-PoR) to execute the workflow of the electronic evidence preservation center. There are six algorithms in the three functional areas of CPA, ERA and CMA. The CMA has three algorithms TagSigGen( ), KeyGen( ), and VeriResp( ); the CPA has an algorithm RespGen( ); the ERA has two algorithms RetrData( ) and VeriHash( ). These six algorithms will be described in Section 4.1. It is assumed the LAU has an evidence file Information 04 00262 i005 and will send it to the ASA to store. The CMA will send challenge message Chal( ) to the CPA so as to make regular checks on the integrity and availability of electronic evidence at appropriate intervals. The CPA will query the metadata for the locations of the challenged chunks and compute response values to send back to the CMA. When the CMA has found out some data blocks are incorrect, it will tell the information to the ERA, then the ERA uses retrieved algorithm RetrData( ) to recover electronic evidence. When all incorrect data blocks have been recovered, the ERA will send them back to the ASA again.

4.1. Notation and Preliminaries

Let Information 04 00262 i005 be an electronic evidence file: it is divided into Information 04 00262 i006 chunks, and each chunk is split into Information 04 00262 i007blocks: Information 04 00262 i008, where each block Information 04 00262 i009. These evidence blocks may be expressed in the following matrix form

Information 04 00262 i257

In execution processes of the workflow of five functional areas, our two schemes FG-PoR and ML-PoR both consist of six algorithms. We start with the precise definition of FG-PoR scheme and ML-PoR scheme.

Definition 1. FG-PoR scheme and ML-PoR scheme are both collection of six polynomial-time algorithms (KeyGen, TagSigGen, RespGen, VeriResp, VeriHash, RetrData) such that:

Information 04 00262 i010 KeyGen( ). This algorithm is run by the CMA to generate public key Information 04 00262 i011 and private key Information 04 00262 i012.

Information 04 00262 i013 TagSigGen Information 04 00262 i014. This algorithm is run by the CMA. It takes as input the private key Information 04 00262 i015 and the evidence blocks set Information 04 00262 i005, and outputs the tags set Information 04 00262 i016 and signatures set Information 04 00262 i017.

Information 04 00262 i018 RespGen Information 04 00262 i019. This algorithm is run by the CPA. It takes as input the evidence blocks subset Information 04 00262 i020 and the tags subset Information 04 00262 i021, and returns a response value Information 04 00262 i022 as output. It will query the ASA whether the ASA is actually storing all evidence blocks intact or not.

Information 04 00262 i023 VeriResp Information 04 00262 i024. This algorithm is run by the CMA. It takes as input the public key Information 04 00262 i025, the response value Information 04 00262 i022 and signatures subset Information 04 00262 i026, It will output 1 if the integrity of all evidence blocks is verified, otherwise will output 0.

Information 04 00262 i023 VeriHash(H,H′). This algorithm is run by the ERA. It takes as input two sets of hash values of evidence blocks Information 04 00262 i027 and H′. Information 04 00262 i027 is saved previously, and H′ is computed when it is needed. If Information 04 00262 i027 and H′ are equal, outputs 1, otherwise outputs 0.

Information 04 00262 i028 RetrData(Ω′, S′). This algorithm is run by the ERA. It takes as input the set of tags Ω′ and the set of signatures S′ of the error evidence blocks returned from the ASA, and outputs the correct evidence blocks Information 04 00262 i029.

4.2. Nyberg-Rueppel Signature Scheme

The Nyberg-Rueppel signature scheme first appeared in the literature in [16], named after its authors K. Nyberg and R.A. Rueppel. Its security is based on the intractability of discrete logarithm problem. Camenisch et al. [17] described this scheme as follows:

The parameters consist of a prime Information 04 00262 i030, a prime factor Information 04 00262 i031 of Information 04 00262 i032, and an element Information 04 00262 i033 of order Information 04 00262 i031. The signer’s private key is a random element Information 04 00262 i034, while the corresponding public key is Information 04 00262 i035.

To sign a message Information 04 00262 i036, the signer selects Information 04 00262 i037 at random and computes Information 04 00262 i038 and Information 04 00262 i039 as follows:

Information 04 00262 i040
Information 04 00262 i041

The pair Information 04 00262 i042 is the signature of the message Information 04 00262 i043. The signer sends the message Information 04 00262 i044 and the signatures Information 04 00262 i045to the receiver.

When the receiver has received Information 04 00262 i044 and the signatures Information 04 00262 i042, to verify the validity of a signature, it checks that the following equality holds:

Information 04 00262 i046

If the equation is true, then the signature is valid. Otherwise, the signature is invalid.

4.3. Finer Grained Proofs of Retrievability (FG-PoR)

We propose two schemes, Finer Grained Proofs of Retrievability (FG-PoR) and More Lightweight Proofs of Retrievability (ML-PoR), which are both based on the Nyberg-Rueppel signature scheme. The execution processes of FG-PoR consist of the following six steps. Before describing the execution processes, we first give a definition for FG-PoR scheme.

Definition 2. A FG-PoR scheme built on the six algorithms (KeyGen, TagSigGen, RespGen, VeriResp, VeriHash, RetrData) can guarantee data possession. Also, it can recover each data block of all chunks of the evidence file.

4.3.1. Key Generation

The CMA runs the KeyGen( ) algorithm to generate key pair Information 04 00262 i047. Chooses two primes Information 04 00262 i048 and Information 04 00262 i031, Information 04 00262 i031is a factor of Information 04 00262 i032, and an element Information 04 00262 i033 of order Information 04 00262 i031. Chooses a secret random element Information 04 00262 i049, and sets Information 04 00262 i050. Chooses a secret key Information 04 00262 i051, Information 04 00262 i052 is the length of the key Information 04 00262 i053. Thus the secret key is Information 04 00262 i054 and the public key is Information 04 00262 i055.

4.3.2. Tags and Signatures Generation

Given evidence blocks Information 04 00262 i056, the CMA runs the TagSigGen( ) algorithm to create a tag Information 04 00262 i057 and a signature Information 04 00262 i058 for each block Information 04 00262 i059 as

Information 04 00262 i258
where Information 04 00262 i060 Information 04 00262 i061 Information 04 00262 i062, Information 04 00262 i063 is a pseudo-random function. The tags set may be expressed in the following matrix form
Information 04 00262 i259

The CMA sends the set of evidence blocks Information 04 00262 i064 and the set of corresponded tags Information 04 00262 i065 to the CPA; the CPA categorizes the chunks and sends all blocks to the ASA, and only saves metadata of the chunks on it. Then the CMA computes Information 04 00262 i066 Information 04 00262 i067 and sends hash values set Information 04 00262 i068 to the ERA. Finally, the CMA deletes all copies of Information 04 00262 i029, Information 04 00262 i016 and Information 04 00262 i027. It preserves only signatures set Information 04 00262 i069 and metadata of the evidence file on its own storage. The storage distribution of the electronic evidence blocks and their tags on the CSS of the ASA is given in Figure 3.

Information 04 00262 g003 1024
Figure 3. The storage distribution of evidence blocks and tags on the Chunk Storage Servers (CSS) in Finer Grained Proofs of Retrievability (FG-PoR).

Click here to enlarge figure

Figure 3. The storage distribution of evidence blocks and tags on the Chunk Storage Servers (CSS) in Finer Grained Proofs of Retrievability (FG-PoR).
Information 04 00262 g003 1024

4.3.3. Challenge Choice

After the CMA has sent challenge values to the CPA, and the CPA has given back response values to the CMA, the CMA will check the integrity of all evidence blocks by the response values. The challenge values are Information 04 00262 i070, where Information 04 00262 i071 is the identity number of the evidence file Information 04 00262 i005, and it may be expressed as Information 04 00262 i072. Information 04 00262 i073 is the number of challenged columns of the evidence matrix, Information 04 00262 i074. Information 04 00262 i075 is the number of challenged rows of the evidence matrix, Information 04 00262 i076. Information 04 00262 i077 are three fresh keys and are chosen randomly for each challenge.

Let Information 04 00262 i063 be a pseudo-random function, Information 04 00262 i078 be a pseudo-random permutation. At each challenge, both the CMA and the CPA use key Information 04 00262 i079 to generate indices of challenged columns Information 04 00262 i080 Information 04 00262 i081, also use key Information 04 00262 i082 to generate indices of challenged rows Information 04 00262 i083 Information 04 00262 i084. They further use key Information 04 00262 i085 to derive Information 04 00262 i086 coefficients Information 04 00262 i087 Information 04 00262 i061, Information 04 00262 i088, Information 04 00262 i089.

4.3.4. Response Generation

The CPA runs the RespGen( ) to generate response values to prove that the ASA is still preserve all evidence blocks intact. The ASA has held evidence blocks Information 04 00262 i090 and corresponded tags Information 04 00262 i065 in Section 4.3.2. Grounded on indices of challenged columns and rows Information 04 00262 i091, Information 04 00262 i092, the CPA chooses subset of evidence blocks Information 04 00262 i093 and subset of tags Information 04 00262 i094 from the ASA by querying the metadata of evidence chunks, then it generates response values Information 04 00262 i095 based on Information 04 00262 i096 and Information 04 00262 i097.

The subset of evidence blocks Information 04 00262 i098 and the subset of tags Information 04 00262 i097 may be expressed in the following matrix form

Information 04 00262 i260

The computational processes of response values Information 04 00262 i095 are as follows

Information 04 00262 i261

The CPA takes response values Information 04 00262 i099 as a proof that the ASA possesses electronic evidence Information 04 00262 i005, finally the CPA sends Information 04 00262 i100 to the CMA.

4.3.5. Response Verification

After the CMA has received response values Information 04 00262 i101 from the CPA, it takes out indices of challenged columns Information 04 00262 i091, challenged rows Information 04 00262 i102, and coefficients Information 04 00262 i103. Then the CMA chooses the subset of signatures Information 04 00262 i104 from the set of signatures Information 04 00262 i105which has been saved previously. Further, the CMA computes

Information 04 00262 i262

Now it runs verify algorithm VeriResp( ) to check the following equation

Information 04 00262 i263

If the above equation is true, the verify algorithm returns 1, the CMA believes that the ASA preserves well evidence blocks set Information 04 00262 i005. Otherwise, the verify algorithm returns 0.

The above equation holds because:

Information 04 00262 i264

Further

Information 04 00262 i110

4.3.6. Evidence Retrieve

At a later time, the LAU needs its evidence file Information 04 00262 i005, it sends a request message Requ Information 04 00262 i111 to the CMA. The CMA will forward message Requ Information 04 00262 i111 to the CPA and the ERA. The ERA queries the chunks and the blocks of the evidence file from the ASA. After the ERA has got feedback message Information 04 00262 i112, it will use hash function to compute the hash value of each element of evidence matrix Information 04 00262 i113 to get the set Information 04 00262 i114. Each element of the set Information 04 00262 i115 is calculated as following

Information 04 00262 i265

The ERA runs VeriHash( ) algorithm to compare the set of hash values Information 04 00262 i116 with Information 04 00262 i117, which has been saved in Section 4.3.2. If Information 04 00262 i118, then Information 04 00262 i119, it means that all evidence blocks are intact. If one or several hash values of the set are not equal, then this means that these evidence blocks may have been altered in network transmitting or on the ASA storage. In Section 4.3.5, when the verify algorithm returns 0, it shows also that some evidence blocks may be incorrect in the ASA.

Assume checked out evidence block Information 04 00262 i120, in order to get original evidence block Information 04 00262 i121, the ERA queries its tag Information 04 00262 i122 from the ASA, and asks the CMA to send back corresponding signature Information 04 00262 i123. As long as tag Information 04 00262 i122 and Information 04 00262 i123 are not damaged, the ERA will use RetrData( ) algorithm to recover evidence block Information 04 00262 i124

Information 04 00262 i266

In fact

Information 04 00262 i125

4.4. More Lightweight Proofs of Retrievability (ML-PoR)

We modify FG-PoR scheme to attain a More Lightweight Proofs of Retrievability (ML-PoR) scheme. It consists of the following six steps, but it has a weaker recovery guarantee than FG-PoR. We give a definition for ML-PoR scheme that is described as follows.

Definition 3. A ML-PoR scheme built on the six algorithms (KeyGen, TagSigGen, RespGen, VeriResp, VeriHash, RetrData) can guarantee data possession, and it can recover each chunk of the evidence file.

4.4.1. Key Generation

Key generation is the same as FG-PoR, and the secret key is Information 04 00262 i126 and the public key is Information 04 00262 i127.

4.4.2. Tags and Signatures Generation

Let Information 04 00262 i128 be a pseudo-random function, the CMA uses secret key Information 04 00262 i129 to derive random sequence

Information 04 00262 i267

Given the evidence blocks Information 04 00262 i130, the CMA computes

Information 04 00262 i268

The CMA runs the TagSigGen( ) algorithm to create a tag and a signature for each Information 04 00262 i131 as

Information 04 00262 i269

Further, the CMA computes hash value for each column of evidence matrix Information 04 00262 i005 as

Information 04 00262 i270

Here, the storage distribution of the electronic evidence blocks and their tags on the CSS of the ASA is shown in Figure 4.

Information 04 00262 g004 1024
Figure 4. The storage distribution of evidence blocks and tags on the Chunk Storage Servers(CSS) in More Lightweight Proofs of Retrievability (ML-PoR).

Click here to enlarge figure

Figure 4. The storage distribution of evidence blocks and tags on the Chunk Storage Servers(CSS) in More Lightweight Proofs of Retrievability (ML-PoR).
Information 04 00262 g004 1024

4.4.3. Challenge Choice

Here, challenge values are Information 04 00262 i132. Both the CMA and the CPA use Information 04 00262 i078keyed with Information 04 00262 i133 to generate indices of challenged rows Information 04 00262 i083 Information 04 00262 i089, and use Information 04 00262 i134 keyed with Information 04 00262 i135 to derive coefficients Information 04 00262 i136 Information 04 00262 i137 Information 04 00262 i089.

4.4.4. Response Generation

The CPA chooses the subset of evidence blocks Information 04 00262 i138 and the subset of tags Information 04 00262 i139 to computes

Information 04 00262 i271

The CPA takes response values Information 04 00262 i095 as a proof that the ASA possesses electronic evidence Information 04 00262 i005, and the CPA sends response values Information 04 00262 i095 to the CMA.

4.4.5. Response Verification

After the CMA has received the response values Information 04 00262 i095 from the CPA, it chooses the subset of signatures Information 04 00262 i140 from the set of signatures Information 04 00262 i141 and computes

Information 04 00262 i272

Then it runs VeriResp( ) algorithm to check the following equation

Information 04 00262 i273

4.4.6. Evidence Retrieve

After the ERA has queried evidence blocks Information 04 00262 i142 from the ASA, it uses hash function to compute hash value of each column of evidence matrix Information 04 00262 i113 to get Information 04 00262 i143 Information 04 00262 i144

Information 04 00262 i274

The ERA runs VeriHash( ) algorithm to compare the set of hash values Information 04 00262 i145 with Information 04 00262 i146, if Information 04 00262 i118, then Information 04 00262 i119, so this means that all evidence blocks are intact. If one or several hash values are not equal, then it means that some column vectors of evidence matrix have been altered in network transmitting or on the ASA storage.

Assume the ERA has checked Information 04 00262 i147 Information 04 00262 i148, it means tth column elements Information 04 00262 i149 have been altered. To recover Information 04 00262 i149, the ERA queries the ASA to get the set of tags Information 04 00262 i150. As long as the set of tags Information 04 00262 i016 is not damaged, the ERA will use Information 04 00262 i016 and the set of signatures Information 04 00262 i151 to recover Information 04 00262 i131.

Information 04 00262 i275

In each row of evidence matrix, except for the element Information 04 00262 i152, other elements are intact. So the ERA further computes following equation to recover Information 04 00262 i152 Information 04 00262 i153.

Information 04 00262 i276

When some data blocks have been lost in the ASA, FG-PoR can recover each data block of evidence matrix, but ML-PoR can only recover a column of evidence matrix.

5. Security and Performance Analysis

By generating a tag to each row of evidence matrix instead of generating a tag to each element of evidence matrix, the ML-PoR scheme can reduce computation costs and storage costs of the set of tags compared to the FG-PoR scheme. On the other hand, the ML-PoR scheme uses XOR operation to converge all elements of a row of evidence matrix into Information 04 00262 i131, so it adds extra computation costs. To reduce explaining duplication, for security analysis, we will only focus on the FG-PoR scheme. For performance analysis, we will consider both the FG-PoR scheme and ML-PoR scheme.

5.1. Security Analysis

In this section, we present a security analysis for our FG-PoR scheme. Depending on the hardness of the Discrete Logarithm Problem (DLP), we reduce the security of our FG-PoR to the security of DLP, and model hash function H( ) as random oracles. In order to facilitate the discussion, we merge the CPA with the ASA into an area, which is called Cloud Storage Area (CSA).

Definition 4. Discrete Logarithm Problem (DLP): Given Information 04 00262 i154 and Information 04 00262 i155 of order Information 04 00262 i031, and set Information 04 00262 i156, compute Information 04 00262 i157. It is pointed out that no probabilistic algorithm could solve DLP with non-negligible advantage within polynomial time.

Theorem 1. As Information 04 00262 i158 is a random oracle, by the definition of a random oracle, the CSA can guess hash values Information 04 00262 i159 on the premise Information 04 00262 i160 with only negligible probability.

Proof 1. Let us assume that the CSA has lost some of evidence blocks, but preserves well all tags, it can be proved that the CSA can’t pass the CMA’s possession verification.

Assume challenged subset of evidence blocks is Information 04 00262 i093, but the CSA has lost evidence blocks Information 04 00262 i161, where Information 04 00262 i162, so the CSA forges evidence blocks Information 04 00262 i163 to replace Information 04 00262 i164, and computes

Information 04 00262 i277
where Information 04 00262 i165.

As the set of tags Information 04 00262 i166 is stored perfectly in the CSA, the subset of challenge tags Information 04 00262 i167 is also stored perfectly. Though some evidence blocks are forged, the values of Information 04 00262 i168 based on the subset of challenge tags Information 04 00262 i097 are no change.

Therefore, the CSA generates response values Information 04 00262 i169.

After the CMA has received response values Information 04 00262 i169, he computes the value of Information 04 00262 i170, and verifies the relation Information 04 00262 i171 whether it is true or not. In Section 4.3.5, we have proved the relation

Information 04 00262 i172

To make the equation Information 04 00262 i173 true, unless the CSA can solve the random oracle. This means it can find hash values Information 04 00262 i174 and Information 04 00262 i175 to let Information 04 00262 i159 on the premise Information 04 00262 i160, but this is not feasible [18]. In view of this, the CMA thinks that evidence blocks have been altered on the CSA.

Theorem 2. If the DLP is hard in our ML-PoR scheme, then there is no CSA that can forge an evidence block, corresponded tag and signature to pass the verification equation except by true evidence block to compute response values.

Proof 2. Let Information 04 00262 i176, Information 04 00262 i177 and Information 04 00262 i178 be the malicious CSA’s forged evidence block, corresponded tag and signature, and Information 04 00262 i179, Information 04 00262 i180 and Information 04 00262 i181 be the expected values from an honest CSS. If the forged values Information 04 00262 i182, Information 04 00262 i183 and Information 04 00262 i184 make the equation true, then we can find a solution to the DLP.

In our FG-PoR, the expected values Information 04 00262 i179, Information 04 00262 i180 and Information 04 00262 i185 satisfy the following equation

Information 04 00262 i186

Also,

Information 04 00262 i278

Assume Information 04 00262 i187, then have Information 04 00262 i188, Information 04 00262 i189 and Information 04 00262 i184 satisfy the following equation

Information 04 00262 i190

Also,

Information 04 00262 i279

Obviously Information 04 00262 i191, otherwise Information 04 00262 i192, which contradicts our assumption.

Since the Equations (21) and (22) are both valid, dividing the two equations, we obtain

Information 04 00262 i193

As Information 04 00262 i194 and Information 04 00262 i195, have Information 04 00262 i196, Information 04 00262 i197. Therefore we have found a solution to DLP Information 04 00262 i198.

From Proof 2, no CSA can forge whichever evidence block, corresponded tag and signature to satisfy Equation (21), so the CSA can only use a true set of blocks, tags and signatures to compute response values Information 04 00262 i095.

Remark 1. Our FG-PoR scheme ensures all tags and signatures are different. Firstly, the CMA uses random number Information 04 00262 i199 to generate tag Information 04 00262 i200 for each evidence block Information 04 00262 i201. Then it uses random number Information 04 00262 i202 to blind tag Information 04 00262 i200 to get signature Information 04 00262 i203. Even if the contents of two evidence blocks are the same, they have different indices, so their tags and signatures are different. It avoids evidence blocks of different indices having the same tags and signatures.

Remark 2. Our FG-PoR scheme ensures challenged blocks and response values of each challenge are different. When the CMA gives a challenge information Information 04 00262 i204 to the CSA, including key Information 04 00262 i205. The CSA uses pseudo-random permutation Information 04 00262 i078 keyed with Information 04 00262 i079 to generate indices of challenged columns Information 04 00262 i206 and keyed with Information 04 00262 i133 to generate indices of challenged rows Information 04 00262 i207. In each challenge, the key Information 04 00262 i208 are different, so Information 04 00262 i091 and Information 04 00262 i209 are different, finally, challenged subset of evidence blocks Information 04 00262 i093 are not the same.

Further, the CSA uses pseudo-random function Information 04 00262 i063 keyed with Information 04 00262 i135 to derive coefficients Information 04 00262 i210, and uses coefficients Information 04 00262 i210 to generate response values. In each challenge, Information 04 00262 i085 is chosen randomly, so coefficients Information 04 00262 i211 are derived randomly. Moreover, challenged subset of evidence blocks Information 04 00262 i098 are not the same, and then response values Information 04 00262 i095 of each challenge are not the same. It avoids the CSA to use its own expected challenge blocks to calculate the response values, or using previous response values instead of response values is needed in this challenge.

Remark 3. Our FG-PoR scheme ensures robust evidence recovery function. When the ERA thinks that the set of evidence blocks Information 04 00262 i005 has been altered in the CSA, he will ask the CSA to send back the set of tags Information 04 00262 i065. Assume evidence block Information 04 00262 i121 is incorrect; the ERA takes Information 04 00262 i212 from the set of signatures Information 04 00262 i213, and uses following equation to recover Information 04 00262 i214

Information 04 00262 i280

Thus, our FG-PoR scheme has good robustness; electronic evidence can be stored intact in an evidence preservation center in the cloud.

5.2. Performance Analysis

Comparing our FG-PoR and ML-PoR with the DEMC-PDP [10], PEMC-PDP [10] and [11](Section 6), to the five schemes, communication costs are mainly composed of the costs of challenge and response values. In [11](Section 6), the verifier takes the set of indices and random values Information 04 00262 i215 as challenge values and sends them to the storage server. Moreover, the storage server returns Information 04 00262 i216 as response values to the verifier, so communication costs of [11](Section 6) are the highest in the five schemes. The communication costs of DEMC-PDP, FG-PoR and ML-PoR are roughly equivalent.

To computation costs, we ignore the costs that the storage server and the verifier derive challenge blocks indices Information 04 00262 i217, Information 04 00262 i218 and random coefficient Information 04 00262 i210 in the five schemes. To tags and signatures generation, response generation and response verification three steps, the computation costs of five schemes are listed in Table 1. In Table 1, the operation symbols denote meaning: H: hash function operation; A: addition operation; M: multiplication operation; E: exponentiation operation; P: pairing operation; X: xor operation

Table Table 1. Comparison of communication, computation and storage costs for the five schemes.

Click here to display table

Table 1. Comparison of communication, computation and storage costs for the five schemes.
Communication, Computation and Storage CostsDEMC-PDP [10]PEMC-PDP [10][11](Section 6)FG-PoRML-PoR
Communication costs of challenge values Information 04 00262 i219 Information 04 00262 i220 Information 04 00262 i221 Information 04 00262 i222 Information 04 00262 i223
Communication costs of response values Information 04 00262 i224 Information 04 00262 i225 Information 04 00262 i226 Information 04 00262 i227 Information 04 00262 i227
Computation costs of tags and signatures generation Information 04 00262 i228 Information 04 00262 i229 Information 04 00262 i230 Information 04 00262 i231 Information 04 00262 i232
Computation costs of response generation Information 04 00262 i233 Information 04 00262 i234 Information 04 00262 i235 Information 04 00262 i236 Information 04 00262 i237
Computation costs of response verification Information 04 00262 i238 Information 04 00262 i239 Information 04 00262 i240 Information 04 00262 i241 Information 04 00262 i242
Storage costs of file blocks and tags Information 04 00262 i243 Information 04 00262 i244 Information 04 00262 i245 Information 04 00262 i246 Information 04 00262 i247
Computation costs of encoding and decodingNoNoYesNoNo

As some different parameters are used in the above five schemes, it is difficult to compare clearly which scheme is optimal in computation costs. So we consider a concrete example to compare the differences of these schemes in computation costs and storage costs. In the five schemes, assume the size of modulus is the same Information 04 00262 i248, each has 1024 bits, Information 04 00262 i031 is a 160-bit prime. Given an 80 MB evidence file Information 04 00262 i005 that has 640,000 data blocks, each block is 1 Kbits (1024 bits). The parameters of five schemes are described as follows:

  • DEMC-PDP [10], PEMC-PDP [10]:

    • the number of file blocks is Information 04 00262 i249;

    • the number of copies is Information 04 00262 i250;

    • the number of challenged blocks is Information 04 00262 i251.

  • [11]( Section 6):

    • the number of file blocks is 640,000;

    • the number of encoded blocks is 32,400;

    • the number of columns is the same as the number of rows in matrix Information 04 00262 i252;

    • the number of challenged columns Information 04 00262 i251.

  • Our FG-PoR, ML-PoR:

    • the number of file blocks is 640,000;

    • the number of columns is the same as the number of rows in matrix Information 04 00262 i253;

    • the number of challenged columns is the same as the number of challenged rows Information 04 00262 i254.

Our concrete example is conducted on the system Windows 7 with two Intel Core 2 processors running at 2.4 GHz each, and 4 GB of RAM. In our implementation, we use the GNU Multiple Precision Arithmetic Library Edition 4.2.1 and OpenSSL version 1.0.0 cryptographic library and choose SHA1 for Hash function. We choose a 160-bit group order for the elliptic curve group to get 80-bit security level. It has been described in [6] that if the server deletes 1% of data file, the verifier only needs to check for 460 random blocks of the file so as to detect sever misbehavior with probability larger than 99%. So we choose Information 04 00262 i255 to achieve a high probability of assurance.

From Table 1, we know the communication costs in [11](Section 6) are the highest in the five schemes. Moreover, the communication costs of all schemes are much lower than computation costs and storage costs. The computation costs of tags and signatures generation have slight impact on the overall system performance, because the generation task of tags and signatures is completed only once during the files life time, which may be many years. Therefore, we only consider computation costs of response generation, computation costs of response verification, and storage costs of file blocks and tags.

To achieve more intuitive and clear directions, we count computation costs of response generation and computation costs of response verification. Also, the sum is called computation costs of response generation and verification. Here, computation costs are running times of the operation in Table 1. In our implementation, the computation times of response generation and verification of DEMC-PDP [10], PEMC-PDP [10], [11](Section 6), FG-PoR, and ML-PoR are 8388,623.63 ms, 724.79 ms, 254,200.72 ms, 279,620.18 ms and 698.56 ms. The comparison result of computation costs of the five schemes are shown in Figure 5.

Figure 5 indicates computation costs of DEMC-PDP [10] are apparently higher than the other four schemes. This was due to the fact that DEMC-PDP [10] and PEMC-PDP [10] store multi-copies of the file to the server; moreover, DEMC-PDP [10] depends on checking by validating all file blocks. To [11](Section 6), computation costs of response generation and verification are slightly lower than our FG-PoR scheme. The communication costs of PEMC-PDP [10] and ML-PoR are roughly equivalent and are much lower than the other three schemes.

The storage costs are storage space of file blocks, tags, signatures and coding. In our implementation, the storage space of file blocks and tags DEMC-PDP [10], PEMC-PDP [10], [11](Section 6), FG-PoR, and ML-PoR are 25,600,000 Kbits, 13,440,000 Kbits, 673,220 Kbits, 1280,000 Kbits and 640,800 Kbits. The comparison result of storage costs of file blocks and tags of the five schemes are shown in Figure 6.

Information 04 00262 g005 1024
Figure 5. Computation costs of response generation and verification of the five schemes.

Click here to enlarge figure

Figure 5. Computation costs of response generation and verification of the five schemes.
Information 04 00262 g005 1024
Information 04 00262 g006 1024
Figure 6. Storage costs of file blocks and tags of the five schemes.

Click here to enlarge figure

Figure 6. Storage costs of file blocks and tags of the five schemes.
Information 04 00262 g006 1024

From Figure 6, we know that the storage costs of file blocks and tags of DEMC-PDP [10] and PEMC-PDP [10] are apparently higher than the other three schemes, and the storage costs of DEMC-PDP [10] are highest in all schemes. The storage costs of [11](Section 6) and ML-PoR are roughly equivalent; moreover, the storage costs of ML-PoR are lowest in all schemes.

In five schemes, only [11](Section 6) uses the technologies of encoding and decoding. The [11](Section 6) first applies the erasure codes to encode the file, and then splits encoded file into Information 04 00262 i256sectors. It provides provable data possession and data recovery dual functions, but its erasure of codes adds extra computation costs and storage costs, so the total costs of [11](Section 6) are higher than our FG-PoR and ML-PoR. DEMC-PDP [10] and PEMC-PDP [10] use multi-replication technology to achieve provable data possession and data recovery functions, but storage costs are too high. Also, the computation costs of DEMC-PDP [10] are the highest of the five schemes. Our ML-PoR generates only a tag to each row of evidence matrix, rather than generating a tag to each element of evidence matrix. Therefore, it reduces computation costs and storage costs compared to that of the FG-PoR. In overall performance, ML-PoR is superior to the other four schemes.

6. Conclusions

Proofs of Retrievability (PoR) to cloud storage data are mainly based on multi-replication technology and erasure code technology [19]. PoR based on multi-replication technology is required to create some copies of the same size for each data block, so the server needs to provide extra storage space for these copies, such as in the schemes in [9,10]. PoR based on erasure code technology needs to blend some data blocks into less redundant blocks, so it saves storage space, but the encoding and the decoding operation add computation costs, such as in the schemes in [8,11]. This paper proposes two PoR schemes—FG-PoR and ML-PoR—for the storage of electronic evidence in the cloud. The two PoR schemes do not use multi-replication technology or erasure code technology, and the two technologies are replaced by employing verification tags and signatures. Therefore, FG-PoR andML-PoR have lower computation costs and storage costs than other similar schemes. Moreover, they not only can ensure the integrity of electronic evidence, but also provide a robust evidence recovery guarantee.

Acknowledgments

This work is partially supported by Natural Science Foundation of Chongqing Science & Technology Commission of China under Grant No. 2011jjA1350 and No. 2011jjA40031. Science & Technology Research Foundation of Education Committee of Chongqing of China under Grant No. KJ110505, and Found of Innovation Scheme of Postgraduate Education of Chongqing University of Posts and Telecommunications of China under Grant No. Y201107.

Conflict of Interest

The authors declare no conflict of interest.

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