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

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 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 SP_B.

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

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

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 pk_B is a sequence of bits, pk_B = <x₀, x₁,..., x_τ, where each number:

After sorted, x₀ of the sequence is the largest number and also an odd, while r_p (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 pk_u and sk_u and then uses fully homomorphic encryption algorithm to encrypt resource index. Finally u anonymously sends encrypted index and public key pk_u 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 SP_A 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 SP_A has collected those related information, it will save the them and put pk_u 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 SP_A by multicast, attaching its own public key pk_u and resource query q_u. Because of anonymity of multicast, SP_A does not know the real identity of u. Once receiving local multicast of u, SP_A 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, SP_A verifies whether pk_u 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 SP_A 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, SP_A 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: SP_A 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, SP_A can get resource weight and then utilizes it to determine whether the required resource is in the local domain. For SP_A 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, SP_A signs the query and forward it to the neighboring autonomous domain.

If the query with signature passed by partner super-peer SP_B, SP_B broadcasts this query in his local domain. If not, SP_B 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. SP_B informs SP_A and asks SP_A 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 Threshhold_B, so if only the recommended credibility value of u exceeds Threshhold_B, 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, SP_A can publicly broadcast pk_B of SP_B and pk_u 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 pk_u 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 RP_i, 1≤i≤k. According to every historical transaction record of u, RP_i will compute a score for u within a certain range of integers. Finally, RP_i summarizes all the scores to calculate a mean value as recommended credibility value of u. The detailed process is as follows. donates recommended credibility value of u. RP_i makes binary serialization of , in which each bit uses the following full homomorphic encryption algorithm. RP_i chooses a random subset S⊆{2,..., τ} and a random integer r within the range of -2^ρ, 2^ρ. Then a homomorphic encryption bit of recommended values is obtained:

Fig. 1:	Trust certificate generation process

Similarly, RP_i can complete every bit of the recommended values and attain encrypted recommended values. Then, RP_i uses public key pk_B of SP_B to encrypt its own identity and obtains . Finally, RP_i sends to SP_A as well as sending 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, SP_A makes use of additive homomorphism of encryption to get the total reputation value of u:

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

Owing to homomorphic encryption, although, SP_A does not know the private key of SP_B 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:

where, k is the number of neighbor peers. Then u sends ID_RP-u and pk_u to SP_A by multicast.

After these processes, SP_A owns a mean recommended credibility value of u, donated as and referee identities set ID_RP-u. Since it does not know the private key of SP_B, it do not know the credibility value of u as well as the referee identities, so that privacy of peers is protected. But SP_A meets the “Honest but curious” assumption, so SP_A will create a trust certificate of u so as to recommend u to SP_B. SP_A builds a triple including , ID_RP-u and query q_u 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, SP_A will use Onion Routing approach (Goldschlag et al., 1999) to contact super-peer SP_B attaching the above signed triple. After SP_B receives the signed triple, it verifies the signature and decrypts and then check whether it is greater than Threshhold_B. If successful, from the local public set SP_B finds the public key of u, then encrypts query q_u and multicasts it in the local domain.

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

If v has the required resources, v sends resource to SP_A via onion routing. Finally, SP_A 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.

Fig. 2:	Anonymous Access Cross-domain

In this case, v will apply to SP_B. SP_B contacts SP_A and sends the ID_RP-u of u to SP_A. 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 SP_A can use exhaustive method to find out the identities of referee peers recommending u. After this, SP_A 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

When communicating with non-neighbor peers, normal peer u sends query to SP_A by multicast. Because of anonymity of the multicast, SP_A 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, SP_A can obtain average recommendation credibility and referee identities set:

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 SP_B, it can not deduce the referee identities from encrypted credibility value of u without collaboration of SP_A, 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 SP_B by multicast so that SP_B who owns public key of v can not link resources with the identity of v.

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.