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An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks



Xiaoliang Wang, Yuling Liu and Hengfu Yang
 
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ABSTRACT

In P2P network, the existing researches focus on protecting the security of information transmission, or ensuring users’ privacy. Security certification is a basic demand of P2P network, including the general authenticity, credibility, integrity and so on. Privacy protection for transactions is the high-level requirement of security, including confidentiality, copyright management, access control. One-sided pursuit of certification will affect the privacy rights of users while one-sided pursuit of anonymous will bring a series of anonymous abuse problems. However, little research has paid attention to urgency of this dilemma, leading to the emergence of a large number of problems in the applications. This study presents an anonymous authentication scheme based on homomorphic encryption, called FHET (Fully homomorphic encryption trust). FHET makes use of trust certificates and can also be combined with existing P2P reputation system which effectively prevents the selfish behavior of peers and ensures scalability, portability and practicality.

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  How to cite this article:

Xiaoliang Wang, Yuling Liu and Hengfu Yang, 2012. An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks. Information Technology Journal, 11: 613-619.

DOI: 10.3923/itj.2012.613.619

URL: https://scialert.net/abstract/?doi=itj.2012.613.619
 
Received: November 23, 2011; Accepted: January 14, 2012; Published: February 22, 2012



INTRODUCTION

Privacy is a fundamental right of citizens so that the anonymity has an extensive applications in present society (Liu, 2012; Shao et al., 2008; Zaidan et al., 2011). P2P software has become one of the most popular applications (Jiang et al., 2009; Peng and Zheng, 2010; Xie et al., 2009; Ye et al., 2011) but the open environment of P2P communication and resource sharing also brings users more privacy concerns (Chen et al., 2011; Modarresi et al., 2008, 2009). For this concern, many researchers have studied anonymous mechanisms of P2P networks and achieved great successes (Freedman and Morris, 2002; Goldschlag et al., 1999; Rennhard and Plattner, 2002).

How to design a secure authentication mechanism for P2P networks is also a research hotspot (Mekki and Fezza, 2009). In order to ensure an available response from resource owners, a lot of trust model (Damiani et al., 2002; Kamvar et al., 2003; Xiong and Liu, 2004) come into being. The trust model can effectively verify the identity of unknown peers. However, these identity-based trust models are based on a particular assumption: a peer must know the real identities of partner peers. This restriction leads to a dilemma that between trust mechanism and anonymous mechanism seems to exist contradictory.

So far, FBST (Wang and Sun, 2009) and CST (Wang et al., 2010b) schemes are based on credential-based trust systems and can not fully integrate the existing P2P reputation system. In other words, although these trusts between peers can be built due to other peers’ introduction but this certificate does not contain information about the reputation of partner peer which will lead to the system prone to free riders or other selfish behavior. Whether there exists a new anonymous authentication scheme which not only meets the anonymous, security and certification requirements and also compatible with the existing mature P2P reputation mechanisms to limit peers’ selfish behavior (Wang et al., 2010a). So in this study, FHET scheme is proposed.

HOMOMORPHIC ENCRYPTION BASED ON ALL ANONYMOUS AUTHENTICATION ALGORITHM

This section describes our anonymous authentication scheme FHET (Fully homomorphic encryption trust). In FHET, the system considers the distributed P2P network environment and uses the homomorphism of fully homomorphic encryption to ensure anonymity, authentication and traceability features in unstructured P2P networks.

Network structure: FHET scheme applies to super-peer model of unstructured P2P networks like PALMS-SP (Hoong and Matsuo, 2008). Firstly, all peers are divided into logical groups called Autonomous Domains (AD). Each autonomous domain chooses the best performance peer as the super-peer (SP). Other peers in the autonomous domain are called Normal Peers (NP). Super-peer keeps resource information of normal peers. Normal peers rely on super-peer when searching and accessing resource information, then directly contact the resources peers. This super-peer network architecture has many advantages, such as decreasing search time and bandwidth, self-management, load balancing, etc. (Oh et al., 2008).

Safe assumption: As the discredited super-peer can not be entrusted under the fully homomorphic encryption, it is necessary to limit the credibility of the super-peer. In this scheme, there is a basic premise that super-peers must meet “Honest but curious” assumption.

Honest: Super-peer must be able to faithfully perform all operations of system encryption process and will not deliberately discard packets
Curious: Super-peer may want to peep into the data content, so the whole encryption process should not disclose any plaintext information to super-peer

“Honest but curious” assumption applies to most situations but not to the weak security case.

Basic algorithm: The specific target of FHET is to protect the peer’s identity, data content, privacy rights and trace of anonymous abuse peer, while also addressing the selfish behavior of P2P peers.

Phase 1 Initialization: In this phase, the system uses CDC partition algorithm (Ramaswamy et al., 2005) to divide every peer into a certain logical area. This logic area is called Autonomous Domain (AD). In every AD, system uses SOBIE (Liu et al., 2008) algorithm to select the best peer as the super-peer (SP). We assume in initialization phase the system is in the relative safety state.

In this phase, each super-peer uses integer-based homomorphic encryption (Van-Dijk et al., 2010) to generate a private key. This generation is as follows. Suppose super-peer of autonomous domain B is SPB.

SPB selects an odd p as the his private key of homomorphic encryption, called skB. p is a binary sequence of length η, its value is odd. That is to say:

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks

Then system uses SOB to generate the public key of homomorphic encryption in autonomous domain B, called pkB. Detailed steps are as follows. According to the η bit odd p, SPB designs the following function:

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks

where, q is a large random integer and r is a small random integer. Notice that r should be very smaller so that it can make 2r+m«p.

Public key pkB is a sequence of bits, pkB = <x0, x1,..., xτ, where each number:

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks

After sorted, x0 of the sequence is the largest number and also an odd, while rp (x0) is an even. If selected number does not meet the above restriction, system continues to re-select it until x0 is eligible to this requirement.

After this, every super-peer mutually exchange and store each other’s public key of homomorphic encryption.

In addition, the normal peers within the same autonomous domain use anonymous multicast way to send their own identity information to local super-peer.

Phase 2 Resource index stored by the fully homomorphic encryption: Like general super-peer mode, the super-peer in FHET will collect the list of resources of each local peer and form the resource index for the retrieval operation. This mode will transfer calculation and resource research overload from normal peers to super-peers which reduces the burden of normal peers. However, this super-peer mode also brings some security risks: calculation process and resource index will disclose the peer privacy. Therefore, taking data privacy into account, the local normal peers upload their own resource indexes using fully homomorphic encryption. Suppose a normal peer, called u in autonomous domain A. Like initialization phase, u generates its public/private key pair pku and sku and then uses fully homomorphic encryption algorithm to encrypt resource index. Finally u anonymously sends encrypted index and public key pku to the local super-peer by multicast. The detailed homomorphic encryption algorithm will be introduced in the next section. As a result of multicast and homomorphic encryption process, although in the previous stage SPA has stored all the peers identities but in the process of resource index phase it does not know the real identity of u and resource index content so that privacy of data and u have been protected. After SPA has collected those related information, it will save the them and put pku into the local public key set.

Phase 3 Anonymously research and download of resources: At the beginning of system, the search and exchange of peer resources are limited between the neighbor peers. All neighbor peers record others’ reputation value according to every transaction and exchange each others’ public key homomorphic encryption. Only after good reputation is built, a peer can search and exchange resource within non-neighbor peers. It must notice that P2P anonymous schemes are considered in the non-neighbor peers situation. Regardless of which mode adopted by anonymous P2P networks, previous peer’s IP address is always known to successor peer, so the most of anonymous P2P schemes only consider multi-hop anonymity rather than adjacent neighbor peers’ anonymity. We also follow the rule.

When a normal peer u needs to publish its query message within non-neighbor peers, it sends the query to SPA by multicast, attaching its own public key pku and resource query qu. Because of anonymity of multicast, SPA does not know the real identity of u. Once receiving local multicast of u, SPA verifies it with the public key encryption and judges whether u is in the local domain. This process relies on public key set which has already been uploaded by the resource peer in the phase 2. In other words, SPA verifies whether pku is in the public key set. If not, it returns the non-accepting response to u. The aim is to prompt local peers to choose sharing resources instead of selfish behavior. If successful, it proved that this is a query from the local peer, so SPA accepts the query of u and starts to look for related resource. There are two cases: local search and cross-domain search.

Local search: Firstly, SPA searches resource in local domain. In this phase, it uses traditional Gerard vector space model (Salton et al., 1975) of information retrieval to express the query and computes similarity of query and local resource index. The process is as follows: SPA changes this query into segmentations and stem of the word and obtains plaintext sequence of keywords and then uses different users’ public key to encrypt those sequences respectively. Weight vector of keywords is used to represent resource. This weight is obtained by traditional information retrieval methods which is the normalized form of word frequency multiplied by logarithmic of inverted document frequency. By the use of encrypted word frequency and inverted document frequency, SPA can get resource weight and then utilizes it to determine whether the required resource is in the local domain. For SPA has stored resource indexes of local normal peers, it can judge whether the resource is kept in the local area. If the resource is locally stored, the query will be sent to the resource peer by multicast. Then the resource peer also uses the multicast to send resources within the domain to let u receive resources anonymously.

Cross-domain search: If the resource is external, SPA signs the query and forward it to the neighboring autonomous domain.

If the query with signature passed by partner super-peer SPB, SPB broadcasts this query in his local domain. If not, SPB also forwards the query to the next super-peer until query reaches the autonomous domain where the resource exists.

Suppose a peer in domain B, called v, has required resources. SPB informs SPA and asks SPA a recommended credibility value of u as the trust certificate. Every domain has its own reputation threshold. We assume the reputation threshold of domain B is ThreshholdB, so if only the recommended credibility value of u exceeds ThreshholdB, the query of u can be met.

The generation process of trust certificate is shown in Fig. 1. To protect peer privacy, the generation of recommended credibility values uses fully homomorphic encryption. Owing to public key exchange of super-peers in the initialization phase, SPA can publicly broadcast pkB of SPB and pku of u within the local autonomous domain and requires the neighbor peers of u to provide the recommended values of u.

Each peer checks received pku in its the neighbor public key set and if it find u is its neighbor peer, it will serve as a referee peer for recommended credibility values of u, called Referee Peer (RP).

Let’s assume that a referee peer is called RPi, 1≤i≤k. According to every historical transaction record of u, RPi will compute a score for u within a certain range of integers. Finally, RPi summarizes all the scores to calculate a mean value as recommended credibility value of u. The detailed process is as follows. Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks donates recommended credibility value of u. RPi makes binary serialization of Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks, in which each bit uses the following full homomorphic encryption algorithm. RPi chooses a random subset S⊆{2,..., τ} and a random integer r within the range of -2ρ, 2ρ. Then a homomorphic encryption bit of recommended values is obtained:

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks
Fig. 1: Trust certificate generation process

Similarly, RPi can complete every bit of the recommended values and attain encrypted recommended values. Then, RPi uses public key pkB of SPB to encrypt its own identity and obtains Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks. Finally, RPi sends Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks to SPA as well as sending Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks to u. Here FHET simplifies the calculation of recommended value. But after the existing mature P2P reputation algorithms are improved, they can be also applied to FHET.

To get k neighbors’ recommended credibility values, SPA makes use of additive homomorphism of encryption to get the total reputation value of u:

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks

where, k is the number of neighbor peers. Then SPA uses multiplication to get all the mean reputation value:

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks

Owing to homomorphic encryption, although, SPA does not know the private key of SPB and can not decrypt recommended credibility values of u, it also summarizes recommended credibility values of u.

After u has obtained referee ID s from enough referee peers, it adopts fully homomorphic encryption to deal with ID s and obtains referee peers’ identities set:

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks

where, k is the number of neighbor peers. Then u sends IDRP-u and pku to SPA by multicast.

After these processes, SPA owns a mean recommended credibility value of u, donated as Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks and referee identities set IDRP-u. Since it does not know the private key of SPB, it do not know the credibility value of u as well as the referee identities, so that privacy of peers is protected. But SPA meets the “Honest but curious” assumption, so SPA will create a trust certificate of u so as to recommend u to SPB. SPA builds a triple including Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks, IDRP-u and query qu signed by its own private key signature.

The process of cross-domain anonymous access is shown in Fig. 2. If the required resource is in the foreign domain, SPA will use Onion Routing approach (Goldschlag et al., 1999) to contact super-peer SPB attaching the above signed triple. After SPB receives the signed triple, it verifies the signature and decrypts Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks and then check whether it is greater than ThreshholdB. If successful, from the local public set SPB finds the public key of u, then encrypts query qu and multicasts it in the local domain.

Phase 4 Resources access: Upon v receives the encrypted query of u from the local super-peer SPB. v checks whether the query resource is own resource. If not, it shows that resource has been updated and v informs SPB. SPB adjusts the related index stored in its catalogs and re-forward the query.

If v has the required resources, v sends resource to SPA via onion routing. Finally, SPA multicasts it in local domain to allow u to anonymously access this resource.

Phase 5 The discovery of malicious peers: In FHET scheme, if anonymous abuse exists, we assume the malicious peer is u, u attacks v by anonymous mechanism.

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks
Fig. 2: Anonymous Access Cross-domain

In this case, v will apply to SPB. SPB contacts SPA and sends the IDRP-u of u to SPA. Since the number of peers in a domain is not large and in the initialization phase all the peers identities already have stored in super-peer, so SPA can use exhaustive method to find out the identities of referee peers recommending u. After this, SPA can cooperate with those RPs and link this malicious transaction with u so that malicious peer is tracked. It should be noticed that the exhaustive method owns high computational complexity to ensure the trace mechanism will not be abused in the general case.

ALGORITHM ANALYSIS

Authentication security analysis: Since recommended values and referee peers identities are included in packet encrypted by full homomorphic encryption, the authentication mechanism is ensured. The ability of authentication depends on the security of the full homomorphic encryption algorithm used in the FHET. Because of space constraint, this section simplifies the formal homomorphism proof of the above algorithm. Some parameters restrictions and strict proof are in the reference (Van-Dijk et al., 2010). We assume there are two plaintext bits, m1 and m2. After the homomorphic encryption algorithm is done, c1 = q1p+2r1+m1, c2 = q2p+2r2+m2. For addition operation, there is c1+c2 = (q1+q2)p+2(r1+r2)+(m1+m2). By the selection of parameters, we can make 2(r1+r2)+(m1+m2) much smaller than p, so we can get c1+c2 mod p = 2(r1+r2)+(m1+m2), which proves that this algorithm is additive homomorphism. For multiplication operation, c1xc2 = (c1q1+q1c2+q1q2)p +2(2 r1r2+r1m2+m1r2)+m1m2. Also by the selection of parameters, we can make 2(2 r1r2+r1m2+m1r2)+m1m2 much smaller than p, so we can get c1xc2 mod p = 2(r1r2+r1m2+m1r2)+m1m2, which proves that this algorithm is multiplication homomorphism. Moreover, after SPB obtains triple, the reputation of u can be verified. Finally, the signature of SPA provides the proof for the credibility value of u and other peers can verify this signature. In fact, recommended credibility value of u provides u an access to external resources
Anonymity and privacy: In FHET, u uses pku as public key of full homomorphic algorithm to encrypt all the resources of its, while anonymously multicasting them to the local super-peer SPA which makes the external peers not directly know the detailed index content so as to protect the privacy of remote storage

When communicating with non-neighbor peers, normal peer u sends query to SPA by multicast. Because of anonymity of the multicast, SPA does not know the real identity of the partner peer and the privacy of the anonymous peer is protected. On the other hand, because that summation of the credibility value is used by full homomorphic algorithm, although, SPA can obtain average recommendation credibility Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks and referee identities set:

Image for - An Anonymous Authentication Scheme Based on Fully Homomorphic Encryption in P2P Networks

it still can not know the detailed content and confidentiality of credibility value is protected. The reason is as follows. If a certain peer wants to peek into encrypted credibility value of u, it must deduce private key of u from its public key. The complexity of cracking encryption is equal to attacking approximate integer gcd which is difficulty (Van-Dijk et al., 2010).

For the different super-peer SPB, it can not deduce the referee identities from encrypted credibility value of u without collaboration of SPA, which play a very good privacy protection for referee peers.

For the Man-in-the-Middle attacker, the transmission of information in the transaction process uses multicast or onion routing, so they can not break communication anonymity of transaction.

Finally, v communicates with SPB by multicast so that SPB who owns public key of v can not link resources with the identity of v.

Traceability analysis: In FHET, if anonymous abuse happens, v will apply to SPB. Then SPB contacts SPA and with the help of the public key they can cooperate to track the malicious peer. This is because pku exists in the generation of credibility value and anonymity is limited in the transaction. If malicious attack appears, the credibility value can be used to track the real identity of u
Prevent selfish behavior of peer: In FHET, for local search phase, without sharing resource, the public key of u should not exist in public key set of SPA, so u can not query resources among non-neighboring peers which limits its selfish behavior and encourages it to actively sharing its own resources. For the cross-domain search related to reputation threshold , SPB needs to verify the credibility value of u which makes u will change selfish behaviors, such as “reap without sowing” and “free riders”, to actively sharing resources in order to obtain a higher reputation value among neighbor peers.

CONCLUSION AND FUTURE WORK

Since P2P networks need duouble requirements of security and privacy, this study presents an anonymous authentication trust (Fully homomorphic encryption trust, FHET) based on full homomorphic encryption. This proposal improves FBST (Wang and Sun, 2009) and CST (Wang et al., 2010b) schemes which are based on credential-based trust systems and can not fully integrate the existing P2P reputation system. It owns much stronger privacy protection for P2P networks users. FHET not only meets the anonymous, security and certification requirements but also is compatible with the existing mature P2P reputation algorithm which can limit the selfish behavior of peers. Therefore, FHET is more scalable and practical.

ACKNOWLEDGMENTS

This study is supported by Natural Science Foundation of Xiangtan University (No. 10XZX16, 11QDZ41), Scientific Research Fund of Hunan Provincial Education Department (No. 11C1215), Scientific Research Fund of Hunan Science and Technology Department (No. 2011GK3205), Hunan Provincial Natural Science Foundation of China (No. 10JJ4041), National Natural Science Foundation of China (No. 61073191, 61103215).

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