Anirban Basu
email: basu@kddilabs.jp

Juan Camilo Corena
email: corena@kddilabs.jp

Jaideep Vaidya
email: jsvaidya@rutgers.edu

Jon Crowcroft
email: jon.crowcroft@cl.cam.ac.uk

Shinsaku Kiyomoto
email: kiyomoto@kddilabs.jp

Stephen Marsh
email: stephen.marsh@uoit.ca

Yung Van Der Sype

Toru Nakamura
email: tr-nakamura@kddilabs.jp

Yung Shin Van Der Sype
email: yungshin.vandersype@law.kuleuven.be

Lightweight Practical Private One-Way Anonymous Messaging

Keywords: privacy, anonymity, message, routing, http, cloud

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The art of providing online anonymity has been popular with the use of proxies to avoid revealing the actual message sender to the recipient. Through the principles of Chaum mixes [START_REF] Chaum | Untraceable electronic mail, return addresses, and digital pseudonyms[END_REF] (and other mix networks based on that), the identities of senders are delinked from the actual messages through a mixing process. Secret sharing through Dining Cryptographers networks [START_REF] Chaum | The dining cryptographers problem: Unconditional sender and recipient untraceability[END_REF] also provides sender and recipient anonymity. Onion routing [START_REF] Goldschlag | Hiding routing information[END_REF] ensures that encrypted messages are relayed through a pre-determined path of intermediate hops such that the actual senders cannot be identified. Probabilistic forwarding (e.g., Crowds [START_REF] Reiter | Crowds: Anonymity for web transactions[END_REF]) also delinks messages from their actual senders without a preset forwarding path.

The Pfitzmann and Hansen terminology [START_REF] Pfitzmann | A terminology for talking about privacy by data minimization: Anonymity, unlinkability, undetectability, unobservability, pseudonymity, and identity management[END_REF] defines three types of anonymity in terms of unlinkability as: (a) sender anonymity, (b) recipient anonymity and (c) relationship anonymity. Sender anonymity means that a particular message is not linkable to a sender, and a particular sender cannot be linked to any message. Likewise, recipient anonymity means that a recipient and one or more messages are unlinkable. Relationship anonymity, a weaker construct, underscores that it is impossible to trace who communicates with whom. This is implied by either sender anonymity or recipient anonymity or both. The authors also define unobservability in terms of undetectability as: (a) sender unobservability, (b) recipient unobservability and (c) relationship unobservability. Unobservability means that it is impossible to detect who sends (sender unobservability) or who recieves (recipient unobservability) or that a certain set of senders and recipients are at all communicating (relationship unobservability). Unobservability is expressed in terms of undetectability, which is established through indistinguishability of messages. This means that the messages sent out or received are indiscernible from random noise. The Pfitzmann and Hansen terminology states that unobservability implies anonymity. Furthermore, similar to relationship anonymity, sender and recipient unobservability both imply relationship unobservability.

In this paper, we propose a novel one-way anonymous message routing protocol, which guarantees the privacy of the messages and the anonymity of the sender through fundamentals of public key cryptography with homomorphic properties. Our proposed scheme is usable in scenarios such as anonymous microblogging and anonymous surveys. We present a practical HTML5 clouddeployed prototype, which piggybacks messages over HTTP(S), thus requiring no configurations at the network level and making it easy to run on mobile devices. We provide a theoretical analysis of the anonymity guarantees in our protocol as well as performance results of our prototype. The proposed protocol is pertinent to systems where the sender's identity and the message itself require protection, not only from external attackers but also from the recipient.

From a legal perspective, the one-way anonymous routing system will fall under the substantive protection of the national applicable law. As the system may typically route personal data globally, the question arises whether the strict EU legal framework might hinder the practical usability of the system for initially non-EU related processing operations. Article 25 of Directive 95/46/EC provides that the transfer of personal data outside the EU is limited to third countries with an adequate level of protection. Thus, if the routing is considered as a data transfer, EU data protection law could potentially bring an additional set of complex data protection requirements to ensure this adequate level of protection. Yet, it is pointed out that the transit or simple routing of data is not the same as the transfer of the data. Consequently the mere routing of data through the one-way anonymous routing system does not establish a data transfer under EU law 6 . Nevertheless, it is still possible that EU law is applicable in cases when, e.g., the personal data of EU citizens is processed, the controller is established in a EU Member States.

Related work

Research in the last three decades has made considerable advancements in facilitating private and anonymous communication over the Internet. Anonymous communication systems can be broadly classified into two categories. (1) Highlatency systems provide strong anonymity but are only suitable for applications that are tolerant of the rather long delays in end-to-end message delivery.

(2) Low-latency systems offer better performance and can be used with interactive bi-directional communication, such as Web browsing. Analogous to the nature of packet transport over UDP and TCP, high-latency and low-latency systems are sometimes referred to as message-based and connection-based systems respectively. Apart from these two systems, anonymous communication have been modelled as multiparty computation such as the Dining Cryptographer Networks [START_REF] Chaum | The dining cryptographers problem: Unconditional sender and recipient untraceability[END_REF] and secret sharing schemes [START_REF] Kikuchi | Sender and recipient anonymous communication without public key cryptography[END_REF].

High-latency message-based systems: David Chaum introduced the idea of a mix [1] -a building block in several high-latency anonymous communication systems. The mix is characterised by the grouping of several encrypted input messages into a batch and then forwarding, after decryption, some or all messages in the batch. This necessitates that the public key of the mixing node be known by the senders. The fact that the mix node decrypts messages does not jeopardise privacy or secrecy of messages because each message could be encrypted further with the public key of its intended recipient. The original Chaum mix outputted messages in lexicographical order while more recent schemes output messages in random order [START_REF] Danezis | Mixminion: Design of a type iii anonymous remailer protocol[END_REF][START_REF] Moeller | Mixmaster protocol version[END_REF]. Mix networks address the problem of a single point-offailure of the original Chaum mix by making messages pass through an ordered sequence of mixes that constitute a path. The paths are typically either free routes or mix cascades.

Low-latency connection-based systems: Work on low-latency anonymisation dates back to traffic redirection through a centralised and trusted proxy, which hides the IP address of the actual source host from the destination. This concept is used by the commercial Anonymizer service. Unlike mixes, a proxy forwards all incoming traffic immediately without any packet reordering. The proxy is trusted not to reveal information about original senders. Onion routing [START_REF] Goldschlag | Hiding routing information[END_REF][START_REF] Syverson | Anonymous connections and onion routing[END_REF][START_REF] Reed | Anonymous connections and onion routing[END_REF] utilises traffic redirection through a static set of pre-determined onion routers, each of which maintains cryptographic keys of one upstream and one downstream router. A message sent through onion routing nodes has its layers of encryption are peeled off until it is sent off to the final recipient. To cater for performance, public key cryptography is only used to establish an encrypted circuit, while faster symmetric encryption is used to transfer the actual data. Churn in onion routing nodes limits the scalability of the protocol. Onion routing has been shown [START_REF] Syverson | Towards an analysis of onion routing security[END_REF] to be detrimental to sender or recipient anonymity if either the first or the last router is compromised. Tor [START_REF] Dingledine | Tor: The second-generation onion router[END_REF] represents the current state-of-the-art in the evolution of onion routing. Crowds [START_REF] Reiter | Crowds: Anonymity for web transactions[END_REF] and AP3 [START_REF] Mislove | Ap3: Cooperative, decentralized anonymous communication[END_REF] make use of probabilistic forwarding through a randomly established path while the response from the recipient is relayed back to the sender using the same already established path. Denoting the total number of relay nodes by N , the number of collaborating malicious nodes by C and the forwarding probability at each node as p f , the Crowds protocol defined a property called probable innocence, which means that the first collaborating malicious node's predecessor is no more likely to be the true sender of that message than not if N ≥ p f (C+1) p f -1/2 . Tarzan [START_REF] Freedman | Tarzan: A peer-to-peer anonymizing network layer[END_REF] sends messages through a dynamically created encrypted tunnel through multiple hops in a peer-to-peer network, adding overhead costs of creating the tunnel. Similar to Tarzan, MorphMix [START_REF] Rennhard | Introducing morphmix: peer-to-peer based anonymous internet usage with collusion detection[END_REF] is a peer-to-peer low-latency anonymity system where any node in the network relay traffic for any other node.

Anonymous message routing

The crux of protecting the identity of any node from being traced as the sender is to ensure that an egress encrypted message is always different from an ingress encrypted message and yet both are of the same size, see Figure 1. This implies that genuinely different messages are indistinguishable from those that look different only in the encrypted domain. Neither the sender nor any intermediate node should be able to decrypt the messages in any identifiable form. Thus, even the sender Alice, herself, is unable to identify her own message if it is forwarded back to her because there is no characteristic of the message (e.g., message signature) that remains unchanged in the multi-hop forwarding process. This unlinkability provides deniability to the sender: if Alice is to deny sending a message then there is no way to prove otherwise.

Throughout the remainder of this paper, it is assumed that all messages sent out in the network are of the same size. Messages of varying sizes should be split, padded and sequenced if necessary for re-construction. It is also assumed that nodes participating in the message relay network intermittently send out random messages containing just noise of the same size as the actual messages. The recipient will be able to filter the noise from the actual messages upon

E(f (m 1 , . . . , m n )) ≡ g(E(m 1 ), . . . , E(m n ))
. Fully homomorphic schemes [START_REF] Gentry | Computing arbitrary functions of encrypted data[END_REF] are mathematically sound in terms of security but their computational requirements often make them unfeasible for practical applications. The Paillier public-key cryptosystem [START_REF] Paillier | Public-key cryptosystems based on composite degree residuosity classes[END_REF], and its variant, the Damgård-Jurik cryptosystem [START_REF] Damgård | A generalisation, a simpli. cation and some applications of paillier's probabilistic public-key system[END_REF], have practical implementations and both exhibit only additively homomorphic properties: the encryption of the sum of two plaintext messages m 1 and m 2 is the modular product of their individual ciphertexts, i.e., E(m

1 + m 2 ) = E(m 1 ) • E(m 2 ));
and the encryption of the product of one plaintext message m 1 and another plaintext multiplicand π is the modular exponentiation of the ciphertext of m 1 with π as the exponent, i.e., E(m 1 • π) = E(m 1 ) π .

Unlinkable message forwarding

Any relay node n i , connected to a centralised or distributed node inter-connectivity registry, can forward messages on behalf of another node. We describe how a relay node can perform unlinkable message forwarding of short messages as well as arbitrary length messages. Note that short messages are those that fit within the plaintext key space of the homomorphic cryptosystem that is used while messages that do not fit are termed as long messages. It is possible to break long messages into multiple short messages, each fitting into the plaintext key space of the homomorphic cryptosystem. This will necessitate that multiple homomorphic operations be run over every part of the long message. To reduce the number of computationally expensive homomorphic operations, symmetric encryption can be used to encrypt longer messages.

The length of the messages plays a vital role in their indistinguishability. The implementation of our proposed protocol should use the message length for all messages, breaking down the large messages if necessary and allowing for re-construction at the recipient's end. If the implementation chooses a message length that fits within the plaintext space of the homomorphic cryptosystem then The final recipient can decrypt the message m from the ciphertext sent by a relay node because it possesses the equivalent private key for the public key used in the homomorphic encryption. The main limitation of this approach is that m must be contained within the plaintext domain of the additively homomorphic cryptosystem. Thus, with a 2048-bits Paillier cryptosystem, the maximum size of the message is 256 bytes. Even if it is possible to encrypt an arbitrary length message by breaking it up into small blocks (where each block fits into the plaintext domain of the cryptosystem), the encryption function of an additively homomorphic cipher is relatively slow in comparison with an equivalent strength symmetric cipher.

Unlinkability in long messages with symmetric and homomorphic encryption: Arbitrary length messages can be encrypted by any symmetric block cipher or stream cipher. The general idea is to (re-)encrypt the ciphertext at each relay node, so that the ingress message and the egress message look different but are of the same size. Since the ingress message at any forwarding node is already encrypted, the node simply needs to encrypt it again with a random symmetric cipher key. This key and any other relevant parameters are then added to a homomorphically encrypted header, where the size of the header also remains the same. The generalised representation of the message and its headers is shown in Figure 2(a). The size of the header, but not the actual message, is limited by the plaintext size of the homomorphic cryptosystem.

Let the size of the symmetric key be denoted as |k| and that of the other relevant parameters (e.g., initialisation vector in a block cipher) as |p| while Algorithm 1 Forwarding algorithm with homomorphic obfuscation and symmetric message encryption at relay node n i .

Require: Additively homomorphic encryption function, E, for the final message recipient R. Require: Symmetric encryption function E. Require: Ingress encrypted message (m)n i from node ni-1.

1: if | (m)n i | ≤ the the ciphertext space of E then 2: E(m)n i ← (m). 3: Compute ciphertext E(m)n i+1 ← E(m)n i • E(0). 4:
Egress message (m)n i+1 ← E(m)n i+1 . 5: else 6: Split (m)n i into header E(hn i ) and symmetric ciphertext cn i . 7:

Generate random symmetric encryption key kn i and parameters pn i .

8:

Compute header E(hn i+1 ) ← E(hn i ) k||p represents their bitwise concatenation, which is of size |k| + |p|. All relay nodes use same length keys and parameters. Assume that the encrypted header at the i-th relay is E(h ni ) where initially, h n1 = k n1 ||p n1 (here, n 1 is the actual sender). Each relay node n i adds the symmetric cipher information as A limitation of this approach compared to the one with purely homomorphic encryption is that the forwarding operation can be continued so long as the size of h ni+1 is less than the plaintext space of the additively homomorphic cipher. However, it is not possible for any node n i to know, in the ciphertext domain, if h ni or h ni+1 are within the maximum size of the plaintext domain of the homomorphic encryption. Once the size is exceeded, the left shift operation will lose information about symmetric keys making it impossible for the recipient to decrypt the message. The number of hops that a message goes through can be controlled by the forwarding probability. Compute ciphertext E(m)n 1 .

E(h ni+1 ) = E(h ni ) 2 |k|+|p| • E(k ni ||p ni ).

3:

Egress message (m)n 1 ← E(m)n 1 . 4: else 5:

Generate random symmetric encryption key kn i and parameters pn i . 6:

Compute header E(hn 1 ) ← E(kn 1 ||pn 1 ). 7:

Compute ciphertext cn 1 from the symmetric encryption with the key and parameters kn 1 and pn 1 , i.e., cn 1 ← E(m).

8:

Egress message (m)n 1 ← E(hn 1 )||cn 1 . 9: end if 10: Obtain, from the intermediary relay service S, a list Ln of nodes that the message can be forwarded to. 11: Send (m)n 1 to random node ni ∈ Ln through S.

instance of a 128-bit AES cipher at each hop. The tags and the initialisation vector (IV) of the AES cipher are both set to 64-bits, which are valid parameters for the CCM or the Galois/counter mode implemented by the Stanford Javascript Crypto Library.

Algorithm 1 describes the forwarding protocol at a relay node n i .

Step 3 for short messages and Steps 8 and 9 for long messages achieve the desired unlinkability of the ingress and egress messages.

Unlinkable message sending

The actual sender always sends the message in encrypted form to a random relay node. Since the messages sent out are of the same size, and some of the messages sent out are just random noise, it is impossible to tell by looking at an egress message that the node sending out that message is the actual sender, or simply a relay node forwarding a message from someone else. The protocol for sending out a message is similar to that for forwarding messages in terms of the cryptographic operations. Algorithm 2 describes the protocol that takes place on the sender node.

Adversarial analysis

Traffic analysis [START_REF] Danezis | Introducing traffic analysis[END_REF] is regarded as the de-facto yardstick to determine resilience of anonymous communications systems against attackers. Ignoring the contents of the message, traffic analysis aims to derive information regarding senders and recipients from the traffic metadata, such as packet arrival times and message lengths. Below we discuss several of the key attacks along with the resilience of our system to these attacks.

One of the key attacks in traffic analysis is the timing attack. In this attack, a passive global adversary who can observe the ingress and egress connections from the anonymous relay network may be able to link the inputs and outputs based on their patterns of packet inter-arrival times. Although there are delays in message delivery in our protocol, such delays are not as large or consistent as those in high-latency systems. In our system, the order in which messages are sent can get disrupted due to the forwarding probability, thus contributing to a different order at the arrival point. Nodes can also inject noisy traffic with messages of the same length as the original messages to obfuscate the message arrival pattern at the recipient's end.

Another important attack is the replay or tagging attack [START_REF] Pries | A new replay attack against anonymous communication networks[END_REF]) in which the adversary controls the entry and the exit points of the anonymiser network and is able to tag a message at the entry point only to be able to detect it at the exit point. The attack is able to break the Tor network7 because of the presence of pre-established paths. In our model, the node at which the message enters or exits the anonymiser network is not fixed. Therefore our system is more robust to this attack, since (when a large number of nodes exist in the network), the attacker will have to control a large proportion of the nodes in order to ensure that most messages will enter and exit the network through a node under the control of the adversary.

In the predecessor attack, if the paths for message routing are observed over time, the attacker can guess with reasonable accuracy the true senders of the messages. In our protocol, the paths are not constant due to the probablistic routing. Furthermore, depending of the frequency and origins of the noisy messages, the attacker will find it hard to differentiate the senders of actual messages from senders of noise, given that the senders of actual messages may also send out noise, and vice-versa. However, our scheme is not secure against predecessor attacks by design. Given a large number of collaborating Sybil [START_REF] Douceur | The sybil attack[END_REF] nodes under control of a single attacker, the anonymity of the sender will fall back to k-anonymity whereby the attacker can be confident that the sender is one out-of-the k remaining non-malicious nodes. This can be dangerous if k is small.

In the fingerprinting attack, the attacker utilises the variation of the size and the rate in which the messages are sent. In the disclosure attack [START_REF] Kedogan | Limits of anonymity in open environments[END_REF] or the refined statistical disclosure attack [START_REF] Danezis | Statistical disclosure attacks[END_REF], the attacker uses set intersections over time to determine the recipient in an anonymous communication. Through our protocol, adding noisy messages in the network at certain unpredictable rates can reduce the effectiveness of such attacks. In addition, our system is not concerned with recipient anonymity.

Analysis of anonymity

We now present a theoretical analysis of anonymity. Diaz et al. [START_REF] Diaz | Towards measuring anonymity[END_REF] formulated a measure of anonymity in terms of Shannon entropy. Our anonymous message routing protocol resembles a Crowds-style probabilistic forwarding. With the probability of any node forwarding a message set to p f (thus, the probability of it sending to the final recipient as 1 -p f ) and the number of forwarding nodes under the control of the attacker set to C (e.g., Sybil nodes), the entropy of the system, H(X), after the attack has happened is formulated as shown as

H(X) = N -p f (N -C -1) N log 2 N N -p f (N -C -1) +p f N -C -1 N log 2 N p f
The maximum entropy is given as

H M = log 2 (N -C).
The degree of anonymity d of the relay network is given by d = H(X) H M . The condition d = 0 applies to the special cases: (a) where there is only one user, i.e., N = 1; and (b) when C = N -1. In all other cases, it is evident that 0 ≤ d ≤ 1 Diaz et al. [START_REF] Diaz | Towards measuring anonymity[END_REF]. Setting N = 10 and N = 100, Figure 3 illustrates the degree of anonymity, d, with respect to the number of corrupt nodes, C, under the control of the attacker. The higher the forwarding probability p f , the better the degree of anonymity.

However, we have already seen that increasing the forwarding probability is not ideal for messages sent using symmetric encryption because: (a) a high forwarding probability implies a lower chance of the message reaching the final recipient; and (b) with symmetric encryption, a message passing through more than a certain number of hops is rendered undecryptable when it reaches the final recipient.

Denoting the message forwarding probability by p f , the probability of the message reaching the k-th hop (instead of reaching the final recipient) is p k-1 f . This is because the sender will never send the message directly to the recipient, so the probability of the message passing through the first hop is 1, followed by the forwarding probability at each hop. Assume that the maximum allowed value of k is some threshold k th for some Paillier cryptosystem. Thus, any k > k th will corrupt a message. Therefore, the probability that a message is corrupted is synonymous with the probability of the message passing through the (k th +1)-th hop; and is given as

p mc = p (k th +1)-1 f = p k th
f . Similarly, the average number of hops that a message passes through is given as n = 1 + 1 1-p f and is illustrated in exists because the message will pass through at least one hop, i.e., the sender will never send it directly to the recipient. We have seen that with a 128-bits AES password, 64-bit tags and 64-bit initialisation vector, every message requires, at the time of forwarding, 256-bits of additional space in the 2048-bits Paillier plaintext space. Thus, with a 2048-bits Paillier cryptosystem, the message can go through a total of 8 hops (including the original sender) while the total number of hops is 4 and 16 for a 1024-bits Paillier cryptosystem and a 4096-bits Paillier cryptosystem respectively. Thus, k th = 4 (1024-bits), k th = 8 (2048-bits) and k th = 16 (4096-bits). For the message to be corrupt, it should pass through at least a k th + 1 relay. Figure 4(a) shows the probability of message corruption versus the forwarding probability considering 128-bits AES password, 64-bits tags and 64-bits initialisation vectors.

Diaz et al. [START_REF] Diaz | Towards measuring anonymity[END_REF] also proposed the degree of anonymity from the perspective of the sender as the probability p H of a message passing only through honest nodes, given as p H = 1 -C N -p f (N -C) . For N = 10 and N = 100, Figure 5 illustrates the probability that a message passes through honest nodes, p H , with respect to the number of corrupt nodes, C, under the control of the attacker. The higher the forwarding probability p f , the worse the value of p H .

A HTMLand Google App Engine implementation

To illustrate the practical viability of our proposal, we have implemented it as two mobile web applications and deployed on the Google App Engine for Java. One application (WSA-CRouter) is responsible for forwarding the messages while the other (WSA-MB) is a generic message recipient. The front-end of the former, which is where the anonymisation algorithms run, is implemented in HTML5 technologies: Javascript and JQuery mobile. The backends of both applications are based on simple Java servlets and the no-SQL key-value datastore of the App Engine. The role of the cloud is that of an untrusted registry providing inter-connectivity between the relay nodes. We have tested our application on different HTML5 compliant browsers on various devices: multiple desktop computers, tablet computers and mobile phones. We have used two Javascript libraries from Stanford University: the JSBN (and JSBN2) and the SJCL for supporting cryptographic operations. Due to a bug in SJCL's conversion of bit arrays to strings (base64 or hex encoded), we had to pad, on the left, the concatenation of AES parameters with 4-bits set to 1 each. Thus, with a 128-bit AES as shown in the example in Figure 2(b), we actually had a 260-bits shift size instead of the 256-bits as illustrated. This constrains our maximum hop limit to 7 instead of 8 as one would expect from a 2048-bit plaintext space of Paillier.

Anonymous messaging over HTTP(S)

The two applications WSA-CRouter (the message forwarding registry and frontend) 8 and WSA-MB (the public message recipient) 9 are both hosted on the Google App Engine for Java. The front-end of WSA-CRouter is run on each node in the diagram. For simplicity, the private keys corresponding to various public keys used in the messaging are stored by the WSA-MB application on the App Engine. In reality, this should not happen because a cloud insider can get hold of these keys and decrypt messages while these are being forwarded.

The application WSA-CRouter is a centralised registry service that routes messages between the different relay nodes. In essence, this type of registry can be decentralised over a structured P2P networks, such as Chord [START_REF] Stoica | Chord: A scalable peer-to-peer lookup service for internet applications[END_REF] or Pastry [START_REF] Rowstron | Pastry: Scalable, decentralized object location, and routing for large-scale peer-to-peer systems[END_REF]. This is especially useful if the connections to a centralised registry are blocked by a network admininstrator. In our prototype, WSA-CRouter implements the Google Channel API service to notify connected relay nodes of messages bound for them. In terms of functionality, the Channel API is close to the proposed Web Sockets standard (RFC 6455). The backend of WSA-CRouter running the cloud receives messages sent to it by a node; looks up the next hop address on that message and sends it on to the next hop if it is connected. The Algorithms 1 and 2 run entirely on the relay node client-side. The front-end of WSA-CRouter receives a message from another node sent through the cloud. Either it randomly forwards the message on to another connected and active relay node; or it sends the message to the final recipient. Figure 6 shows how a message, originally sent by Eve, passes through Dave, Bob and Alice to the recipient WSA-MB service. To the recipient, the message appears to have originated from Alice, when in reality it is from Eve.

If the message has been encrypted with a purely homomorphic obfuscation then it is impossible for the recipient to tell who the actual sender of the message is. However, if the message is encrypted with homomorphic and symmetric obfuscation and the message has not been corrupted because of passing through many hops then the recipient, upon decrypting, can tell that the number of hops the message has passed through. This will reveal that in the most likely case, the message did not originate at the node from which the final recipient received the message.

The backend of the WSA-MB service, upon receiving the encrypted message, looks up the public key sent with it and finds the corresponding private key. Depending on the type of obfuscation used, it decrypts either the message or its homomorphically encrypted headers and stores it in the datastore. A client Javascript application -the front-end of WSA-MB -can then retrieve the decrypted messages or headers, decrypting in rounds, the headers if required to retrieve the underlying plaintext messages.

Performance evaluation

We describe the performance of our prototype in terms of the time taken for message forwarding on the client-side front-end and that of various functions on the cloud-side back-end. Throughout the experiment, we have used 2048-bit Paillier keys and 128-bit AES keys with 64-bit initialisation vectors and 64-bit tags. The connections to the WSA-CRouter web application have been also encrypted over HTTPS. The iPhone 5S has the worst forwarding performance and this can be attributed to its less powerful hardware as well as the Javascript library on Chrome for iOS 7 compared to the computers. It is to be noted that Internet Explorer performs fairly badly too on reasonably high-end hardware. Use-cases such as anonymous surveys are tolerant to the relatively long delay (due to the over-aminute delay per hop) if all the hops en-route are similar low-powered devices. However, microblogging such as a commentary of live events may not be able to tolerate large delays.

Client-side performance

Cloud-side performance Both WSA-CRouter and WSA-MB applications have been deployed on the F2 instance class (1200MHz CPU, 256MB RAM) of the Google App Engine for Java. The message forwarding delay in the Table 2(a) is negligible compared to the delays on the client-side (Table 1).

Table 2(b) shows the performances of the different functions in the WSA-MB application. The functions Message decryption essentially has the same performance irrespective of the size of the message because it only decrypts the homomorphic header, leaving the header to be shifted and decrypted on the client-side; or it decrypts the homomorphic message. Thus, both cases involve just one 2048-bits Paillier decryption along with parsing the messaging and looking up the datastore for the private key corresponding to the public key. 6 Conclusions and future work

In this paper, we have proposed a private one-way anonymous messaging protocol and a real world prototype to demonstrate the applicability of our protocol in lightweight HTML5-enabled browsers on mobile devices. Our protocol is suitable for particular application scenarios where the sender's identity needs to be protected not only from external attackers but also from the recipient. Our protocol uses probabilistic forwarding through a random set of relay nodes and protects the anonymity of the sender and the privacy of the messages through principles in homomorphic cryptosystems. We are in the process of implementing a real-world anonymous survey system based on our prototype. In the future, we plan to look into the construction of the reverse path. We also plan to deal with churn that can cause message loss and intentional message loss.

Fig. 1 .

 1 Fig. 1. The incoming and the outgoing encrypted messages to and from a node look completely different, offering no linkability.decrypting. We now briefly introduce the notion of homomorphic encryption before describing how we use it to devise unlinkability.Homomorphic encryption -a brief primer: Homomorphic encryption allows computing over encrypted data without requiring the knowledge of either the actual data or any results produced through the computation. Depending on the type of computational operations supported, homomorphic cryptosystems are classified as: (1) additive, (2) multiplicative, (3) somewhat homomorphic (e.g., allowing a number of additions and one multiplication), and (4) fully homomorphic. Denoting the encryption of a plaintext message m as E(m), in the generalised case of fully-homomorphic encryption, for any function f in the plaintext domain there exists a function g in the ciphertext domain, such that E(f (m 1 , . . . , m n )) ≡ g(E(m 1 ), . . . , E(m n )). Fully homomorphic schemes[START_REF] Gentry | Computing arbitrary functions of encrypted data[END_REF] are mathematically sound in terms of security but their computational requirements often make them unfeasible for practical applications. The Paillier public-key cryptosystem[START_REF] Paillier | Public-key cryptosystems based on composite degree residuosity classes[END_REF], and its variant, the Damgård-Jurik cryptosystem[START_REF] Damgård | A generalisation, a simpli. cation and some applications of paillier's probabilistic public-key system[END_REF], have practical implementations and both exhibit only additively homomorphic properties: the encryption of the sum of two plaintext messages m 1 and m 2 is the modular product of their individual ciphertexts, i.e., E(m 1 + m 2 ) = E(m 1 ) • E(m 2 )); and the encryption of the product of one plaintext message m 1 and another plaintext multiplicand π is the modular exponentiation of the ciphertext of m 1 with π as the exponent, i.e., E(m 1 • π) = E(m 1 ) π .

  The encrypted header and actual data part of a message that is forwarded on by nodes. 128-bits AES key 64-bits AES IV 64-bits AES tags 256-bits shift size 2048-bits Paillier plaintext size (b) Inside the 2048-bits Paillier encrypted header associated with a 128bit AES encrypted message.

Fig. 2 .

 2 Fig. 2. Homomorphic headers and parameters of the symmetric cipher the message forwarding protocol does not need to use symmetric encryption, thus avoiding the limitation on the number of forwarding hops that we describe below.

  The operation in the encrypted domain is equivalent, in the plaintext domain, to a left shift of h ni by |k| + |p| bits and a placement of k ni ||p ni in the rightmost |k| + |p| bits of h ni to produce h ni+1 . Since the additively homomorphic encryption guarantees semantic security, E(h ni+1 ) and E(h ni ) are indistinguishable. Similarly, the symmetrically encrypted egress and ingress ciphertexts at any relay c ni+1 and c ni are also indistinguishable. The egress encrypted header of E(h ni+1 ) and the egress ciphertext c ni+1 at any relay node n i together achieve the desired unlinkability.The recipient, possessing the equivalent private key for the public key used in the homomorphic encryption, can decrypt the encrypted header, right shift by the size of |k| + |p| and recover the actual message m in rounds by recovering the symmetric cipher key and other parameters for each round.

Figure 2 (

 2 b) shows how the encrypted header looks like for a 2048-bit Paillier homomorphic cryptosystem and with an Algorithm 2 Algorithm for dispatching a message at the sender node n 1 . Require: Additively homomorphic encryption function, E, for the final message recipient R. Require: Symmetric encryption function E. Require: Plaintext message m. 1: if |m| ≤ the plaintext space of E then 2:

Fig. 3 .

 3 Fig.3. The degree of anonymity d[START_REF] Diaz | Towards measuring anonymity[END_REF] with respect to the number of corrupt nodes C.

Figure 4 (

 4 b). The extra unity in the equation The probability of message corruption versus the forwarding probability. Three Paillier plaintext sizes are shown: 1024, 2048 and 4096 bits. It is assumed that 128-bits AES ciphers are used with 64-bits initialisation vectors and 64-bits tags. The average number of hops a message passes through versus the forwarding probability.

Fig. 4 .

 4 Fig. 4. Hops and message corruption

Fig. 5 .

 5 Fig.5. The probability of a message passing through honest nodes pH ([START_REF] Diaz | Towards measuring anonymity[END_REF]) with respect to the number of corrupt nodes C.

Fig. 6 .

 6 Fig. 6. The flow of a message from Eve to the final recipient.

  2 |kn i |+|pn i | • E(kn i ||pn i ). 9: Compute ciphertext cn i+1 ← E(cn i ) with the key and parameters kn i and pn i respectively such that |cn i+1 | = |cn i |. 10: Egress message (m)n i+1 ← E(hn i+1 )||cn i+1 . 11: end if 12: With probability p f th , send (m)n i+1 to node ni+1 ∈ Ln through S. 13: With probability 1 -p f th , send (m)n i+1 to final message recipient R.

Table 1

 1

	demonstrates the performance of message
	forwarding on different browsers and platforms. In our set of test devices, we
	had:
	(a) (mobile) Chrome 35.0.1916.38/iOS 7.1.1 running on a iPhone 5S;
	(b) (desktop) IE 10.0.9200.16899S/Windows 8.0 running on a hardware consist-
	ing of a 3GHz Intel Core i7 processor and 16GB RAM;
	(c) (desktop) Firefox 29.0/Ubuntu Linux 14.10 running on a hardware consisting
	of a 3GHz Intel Core i7 Extreme processor and 16GB RAM;
	(d) (laptop) Chrome 35.0.1916.114/Mac OS X 10.9 running on a Macbook Air
	hardware consisting of a 1.8GHz Intel Core i7 processor and 8GB RAM.

Table 1 .

 1 The mean forwarding time for different platforms using the same Javascript implementations of 2048-bits Paillier and 128-bits AES encryptions.

	Platform	Mean forwarding time
	(mobile) Chrome/iOS	70.9s
	(desktop) IE/Windows	17.2s
	(laptop) Chrome/OS X	4.06s
	(desktop) Firefox/Linux	1.89s

Table 2 .

 2 Performances of message forwarding and cryptographic functions (a) The performance of the WSA-CRouter implementation on the Google App Engine. The performance of the WSA-MB implementation on the Google App Engine using 2048-bits Paillier encryption.

	Function	Performance	(b) Function	Performance
	Message forwarding	25ms	Key generation	3.88s
			Message receipt	0.13s
			Message decryption	1.16s

UK Information Commissioner Office, "The eight data protection principle and international data transfers", 2010, http://ico.org.uk/for_organisations/guidance_ index/ ~/media/documents/library/Data_Protection/Detailed_specialist_ guides/international_transfers_legal_guidance_v4_290410.ashx, 1.3.2.

See: https://blog.torproject.org/blog/one-cell-enough

See: https://wsa-crouter.appspot.com.

See: https://wsa-mb.appspot.com.


Natasha Dwyer
email: natasha.dwyer@vu.edu.au

Stephen Marsh
email: stephen.marsh@uoit.ca

Social Network Culture Needs the Lens of Critical Trust Research

Trust is essential to the success of the social networks that are aggregating and applying masses of information about us. In this position paper, we argue that a critical approach to exploring trust and social networks is required; this entails genuinely working in the interests of users and acknowledging the power relations and wider social context of this form of technology that is impacting more and more of our everyday life. Without a critical approach, digital environments may become monopolised by corporate interests.

Introduction

The digital traces left by individuals can now easily be collected, crosschecked and stored as part of a phenomenon known as social networking and what is loosely described as 'big data'. Because this complex set of information can be used as a form of social control, social science researchers argue that we need to handle social network data and culture critically; to question the ways the data is gathered and used and to produce alternative means to understand and participate in the phenomenon [START_REF] Dalton | What does a critical data studies look like, and why do we care? Seven points for a critical approach to "Big Data[END_REF]. But what does this mean for trust researchers? In this position paper, we contend that critical approaches to trust and social networks require researchers to work genuinely in the interests of users and to respond to the power relations connected with social networks. We suggest that trust researchers should follow the lead provided by privacy researchers, in acknowledging that the interests of an individual can clash with those of governments and corporations. Before we outlay our argument, we describe what social network data is and explain the central role trust plays in the success of the social networks that produce the data. As a final point, we discuss how social network data can be of use to trust researchers.

Social Network Data

Social network data is generally considered to be information generated at networks designed for exchange between users, for example, messages exchanged at websites such as Twitter and Facebook. Social network content has existed in some form for some time. More recently, content collected from disparate locations can be united, compared and stored at a magnitude not previously possible. The digital data provided by individuals can be analysed and applied in a variety of ways. For instance, social network users, such as subscribers to Facebook, now receive advertisements for products their friends have bought: the underlying idea behind the strategy is that individuals trust the recommendations of their friends so the technique is an effective way to convince users to buy a product or accept a message. Social network analysis, used by industry and the academy, explores the links between individuals, behavior and artifacts (for instance, messages) to find patterns, such as the flows of information and influence.

A social network occurs when a computer connects individuals or organizations, according to Garton [START_REF] Garton | Studying online social networks[END_REF], so the data could include material as diverse as logs indicating which individuals share a printer to transcripts of credit card transactions. Self-tracking systems, applications that gather data about individuals' health performance also now need to be considered as a type of social network content. Systems are now available that enable individuals to track and compare their personal health indicators such as eating habits, sleep, blood pressure etc. There is the option to share this information with others via broadcasts on social networks. Daly [START_REF] Daly | The Law and Ethics of'Self Quantified'Health Information: An Australian Perspective[END_REF] outlines how although these systems were once the domain of a few 'enthusiasts', interested in the response of their bodies of the course of a day, the information is now in the hands of multinationals who are beyond the reach of local laws describing how personal information should be protected. Another ramification of self-tracking technology is that now, according to Lupton [START_REF] Lupton | Beyond Techno-Utopia: Critical Approaches to Digital Health Technologies[END_REF], individuals are not trusting their own insights about their bodies and health because it's easier to trust the 'numbers over physical sensations'. Self-tracking systems teach us to adhere to expected societal norms regarding sleeping, eating, and drinking, which are reinforced through the act of sharing data. One aim of these systems is to use personal data as a form of motivation to improve one's health through self-reflection, guilt and peer pressure [START_REF] Lupton | Beyond Techno-Utopia: Critical Approaches to Digital Health Technologies[END_REF]. Soon employers may demand access to their employees' health information and decide on who is trustworthy on the basis of these results [START_REF] Walston | Understanding the factors affecting employees' perceived benefits of healthcare information technology[END_REF].

Social Networks and Trust

Trust plays a central role in the continuing flow of social network content to adapt Fukuyama's famous line [START_REF] Fukuyama | Trust: The social virtues and the creation of prosperity[END_REF], trust greases the wheels of the networks. If a user does not trust the information received from a network, it will not be passed on [7]. The business model of the social network sites depends on users creating, sharing and consuming content. IBM's CEO, Ginni Rometty in 2014 described the opportunity for the sale of individual private data as the goldmine for the 21 st century [START_REF] Sathi | Engaging Customers Using Big Data: How Marketing Analytics Are Transforming Business[END_REF]. So social network sites are designed to provoke high levels of disclosure from users. For instance, participation can be set up in a popularity contest framework where users strive to receive more attention in the form of 'likes' and 'shares' etc. [START_REF] Yu | Toward predicting popularity of social marketing messages[END_REF]. The website 'Dark Patterns' (www.darkpatterns.org) is a collection of instances where the designer has applied a solid understanding of human nature in order to coerce the user into doing something that is not in the user's individual interest, for instance, disclosing personal information. An example provided by Dark Patterns is Yahoo's Hotjob site. In order to interact, the user is required to answer a series of personal information, even though the information required by a potential hirer is already provided in the job application. As the user is required to complete the information fields in order to progress an application, there is pressure to comply.

The uneven power relations propagated by social networks means that individuals can be manipulated and controlled, whether by marketing companies, governments, or any other entity that has access to the data and the means to analyse it. This is known as 'information injustice'. Individuals are often unaware of the data traces they are leaving and the value of the information they leave behind [START_REF] Johnson | From open data to information justice[END_REF], as demonstrated by the disturbing effect of the viral website 'Take This Lollipop' (www.takethislollipop.com). Upon entry to this site, the user is asked to provide permission for an app to access the user's personal Facebook account. This is a common request from many websites, so users are accustomed to agreeing without much consideration. Once permission is granted, there is a film sequence of a grimy house with a menacing looking man typing on computer. The camera moves closer and we see what he is looking at. Integrated into the film image is the personal Facebook profile view of the user that contains information that the user has not yet set to public display. This includes information that the user may not have given directly to Facebook, such as birth date, that Facebook has collected from one of its partner organisations. The intention of 'Take This Lollipop' is to shock. Even if an individual is alert to the circulation of personal data via a social network, it is difficult for an individual to retract a data trace once it is distributed [START_REF] Johnson | From open data to information justice[END_REF]. The exploitation of users is built into the design of technology, as price for use. When someone uses technology to conduct some sort of activity, such as a searching, buying products, or catching up with friends, the user enters into an arrangement where personal data is collected as a condition of use of the technology [START_REF] Langlois | Meaning in the Age of Social Media[END_REF].

As trust plays a central role in the creation of successful social networks, it is easy to see how the work of trust researchers is attractive to owners of digital environments such as corporations and governments. Trust research can be used to exploit users and to create environments that are commercially successful by giving the appearance of trustworthiness. If trust researchers take their work seriously and believe it has an impact on the direction of both the research field and industry, then an implication is that the work could be used to improve the profits and control of a private company. For instance, [12] study trust interactions on social network sites from a user's perspective, in particular, how a user can improve social capital in these environments. They recommend that as well as having many 'loose ties' users should stay visible in these spaces and remain attentive to their contacts. Social networks such as Twitter and Facebook could use this advice to encourage users to keep engaging with their systems. Similarly, Lui et al [13], working with the notion that different users in a social network seek a range of trust evidence, seek a means to calculate tailored trust ratings, which they refer to as Quality of Trust. However, a social network developer could use this research to persuade users to buy the products they endorse. There may be researchers who may be comfortable with their work servicing the support of the status quo. However, for those researchers are not, we argue that a critical approach is required. Of course, as Gupta [START_REF] Gupta | Your Papers please": personal and professional encounters with surveillance[END_REF] points out, it is possible that any idea and critique can be subsumed by the mainstream to maintain the current state of affairs, but this does not mean that resistance is useless.

There are other roles that trust research can play and one importance stance is to work in the interests of users first, not commerce. To argue for a critical approach to technology is not to declare the technology as dystopic. Rather it is to understand technology as a 'double-edged sword' reliant on 'the context and comportment' of a particular scenario including the actions of and interplay between digital environment owners, developers, designers, and users [15].

A Critical Approach to Trust and Social Networks

So what does a critical approach to studying and designing trust consist of? A critical perspective entails two key actions by researchers that are reviewed in this section. Firstly, an acknowledgement is required of the power relations inherent in the domain of social network data, trust and research. Acknowledgement entails questioning whose biases and expectations are served by the production of digital systems and the research that surrounds digital technology [16]. Acknowledgement can lead to research and design that deciphers technology as a social construct that enables a range of social relations not just those that suit governments and corporations [START_REF] Verbeek | The nature of design practice and implications for interaction design research[END_REF]. (Alternatively, acknowledgement of the power relations can lead to the researcher identifying that there is limited possibility for research with a critical perspective, a discussion beyond the scope of this paper).

However, according to Stolterman [18], scientists and engineers are not well-suited to undertake a consideration of the socio-political framework their work exists in. They are unaccustomed to such a practice as their training involves focusing on one problem and isolating it away from context much as possible. Zelenko and Felton [16] add that designers are also not well suited to consideration of the wider milieu that their work exists in as they are trained to acquiesce to the instructions of the client. But education and practice is changing. The trust research field, including practitioners from a range of disciplines, now conceptualise trust as a social phenomenon. The next step is to move beyond conceptualising trust as a scenario between individuals or groups and consider the wider social and political structures impacting on individual interactions. An example of research that includes a consideration of social context is the work of Pearson and Tsiavos [START_REF] Pearson | Taking the Creative Commons beyond copyright: developing Smart Notices as user centric consent management systems for the cloud[END_REF] who explore 'smart notices'; a means for individuals to control their information and to set the expectations for the products and services they seek. Although they are working in a corporate environment (HP Bristol), these authors place their research in a sociotechnical context, the Creative Commons movement, and design around the inherent power relations that occur between an individual and a network.

Secondly, researchers need to commit to working genuinely in users' interests; addressing information imbalances, making users aware of their trust interactions, enabling users to negotiate trust on their own terms and learning from users. Trust researchers can learn from privacy researchers who have a natural inclination to study a scenario from the perspective of users' welfare. As Krontiris [START_REF] Krontiris | Trust and privacy in mobile experience sharing: future challenges and avenues for research[END_REF] review, privacy research has a long history of conceptualisation as tussle between an individual and others who might gain advantage from private knowledge about that individual. Some protection for individuals is legislated and there is also a long history of privacy advocacy that provides a framework that researchers' work can fit into. Some privacy researchers explore the risks that users are exposing themselves to by interacting with social networks. For instance, Nurse et al [START_REF] Nurse | Inferring social relationships from technology-level device connections[END_REF] investigate the hazards from the occurrence of incidental individual interactions online. Some actions are as innocuous as printing from a device. A mass of data is left behind that can be used to infer about that individual's 'real world social relationships', without the individual having any knowledge of the disclosure. Those inferences could be used in a myriad of ways depending on who had access to the data. Netter et al [START_REF] Netter | Pernul G Privacy Settings in Online Social Networks--Preferences, Perception, and Reality[END_REF] add that there is often a mismatch between a user's perception of privacy controls in a social network and what is really occurring. To solve this problem, they have developed identity management software to assist a user to handle disclosures made online. Basu et al [START_REF] Basu | Privacy-preserving collaborative filtering for the cloud[END_REF] suggest a system that could work without the disclosures and privacy trade-offs users are accustomed to offering in order to participate. Games are becoming a location for mass amounts of social network data, as players reveal physiological and psychological data (for instance, response time, prioritisation of strategies and attention rate) and Martinovic et al [START_REF] Martinovic | You are what you play": Breaching privacy and identifying users in online gaming[END_REF] argue that this is an area that needs attention from privacy researchers.

Readdressing information imbalance is to strive so that individuals have as much access and control to the data generated by networks as do powerful bodies such as governments and private corporations. Mann [START_REF] Mann | Lo R Cyborglogging with camera phones: Steps toward equiveillance[END_REF] is a proponent of this technique. His response to surveillance is 'sousveillance', that he defines as 'the recording of an activity by a participant in the activity' to 'reverse the otherwise one-sided panoptic gaze'. In this vision, individual users adopt the recording technologies in the spaces they inhabit and combine their resources to form lobby groups. We see these types of community groups already happening in the form of police observation groups such as Copwatch in Canada and FITwatch7 in the United Kingdom [START_REF] Goldsmith | Policing's new visibility[END_REF]. The result is a challenge to traditional sources of power, where trust is qualified rather than just given. Individuals are enabled to form trust decisions using their own data rather than information mediated by news sources. Of course, there are shortcomings to Mann's vision. For instance, there would need to be a change in the governance framework of a society so that individuals can access information that is currently closed, and also resourcing so that individuals can obtain the required technology. However, these limitations should not mean that the idea is dismissed. As Goldsmith [START_REF] Goldsmith | Policing's new visibility[END_REF] points out, the potential for individuals to perform some level of sousveillance increases with every improvement in camera technology.

Another example of practice that works genuinely in users' interests is the creation of digital environments that make users aware of the implications of their social network interactions. This alternative is a shift away from the majority of trust research that as far as possible claims that trust should be managed in the background and automatically handled by the digital system. The underlying objective is that trust in a digital environment should be a fluid state that it is not explicitly addressed by the user. This type of design can allow participants to attend to activities, whether that is shopping, finding a date or exploring a medical issue [START_REF] Marsh | Device Comfort for mobile health information accessibility[END_REF]. In contrast, we, the authors (as outlined in [START_REF] Marsh | Device Comfort for mobile health information accessibility[END_REF]), wish to create designs that make users mindful of their interactions so users become contributors to the wisdom of the digital ecosystems they inhabit (in other words, the users are 'strong links'). In particular, we wish to design a device that combines several factors (such as time, nearby devices and previous user preferences) in order to communicate an overall 'comfort level' to the user for the user to interpret. What the user does with this guidance is ultimately up to the user. The idea is to provide an opportunity for the user to have a 'second thought'. Awareness can be created by 'obstructive interfaces', which could be as simple as a message to the user, "Are you sure that you really want to do this?' Storey et al [START_REF] Storer | Encouraging second thoughts: Obstructive user interfaces for raising security awareness[END_REF] have prepared a range of interface elements such as persistent, oversized, and insistent stop buttons and messages that require a deliberate hand action to dismiss. Different user's require tailor made obstructions, add [START_REF] Murayama | The structure of the sense of security, Anshin[END_REF]. Nuanced communication styles for obstruction are a subject of further exploration. In previous research by the authors, [START_REF] Dwyer | What can the hashtag# trust tell us about how users conceptualise trust?[END_REF] investigate Twitter messages coded as 'trust' by users as a means to understand how trust is conceptualised by users in social media networks.

Practitioners and researchers wishing to learn from users or to create systems that draw attention to the trust implications of interactions can learn from the research field of critical interactive design (see [START_REF] Boehner | Interfaces with the ineffable[END_REF] for an overview), which is a subset of the larger practice known as Human computer interaction (HCI). Critical interactive design is working on similar notions of bringing in the user as an active participant and aims to create a space whereby a "user" can make a reflective choice about interaction. Within critical interaction design, there is the practice of 'seamful design' that aims to present to the user the 'seams' of a design; its constituent parts, and how it integrates with other systems [START_REF] Boehner | Interfaces with the ineffable[END_REF]. The practice is a response to the automation culture we discussed earlier in this paper, the mission of mainstream technology designed to create interconnected, distributed systems that deliver ease and convenience seamlessly that are not noticed by the use. Within a seamful design, the biases, contradictions and problems of technology are exposed, rather than smoothed over by a design and a spirit of critique is engendered [START_REF] Boehner | Interfaces with the ineffable[END_REF]. The role of researchers and designers is to identify which seams, out of all the possible data, will be important to the user and how best to present the seams.

As a final note, we add that as well as the trust community having something valuable to contribute to social network culture, social network data can be of use to trust researchers. Although social network data can be difficult to interrogate and reproduce scientifically (due to the control retained by the owners of social network sites), the data contains users discussing issues of importance to them within a 'real world' context. Trust researchers recognise how difficult it is to gather data from participants about trust either in laboratories, isolated from the impact of demands of everyday life or in the context of 'real' situations. In contrast, within social networks there are users around the world providing their view on trust in the form of publically accessible conversations in text format. Rather than a top-down view of trust, an abstract understanding developed by a researcher, social network data offers a 'bottom-up' view of trust from the perspectives of individuals. Sardana and Cohen [32] use social network data for this purpose and insert material from users to substantiate their trust models.

Conclusion

In conclusion, a critical approach to social media data use is necessary if social networks are not to become monopolised by commercial interests telling us who to trust and what trust means. A critical approach needs to: acknowledge and respond to the wider power relations that social networks generate and work genuinely in users' interests. Trust researchers can learn from the orientation of privacy researchers, to put the individual user first. Some trust researchers are already exploring from a user's perspective, investigating how to readdress information imbalances, to make users more attentive to the implications from interacting with social networks, and to enable users to form trust choices about their trust negotiations within social networks on their own terms.


Natasha Dwyer
email: natasha.dwyer@vu.edu.au

Stephen Marsh
email: stephen.marsh@uoit.ca

The Detail of Trusted Messages: Retweets in a Context of Health and Fitness

Our aim is to know more about the content of a message that is trusted in order to create template messages that users can configure within a system we are designing. To this end, we examine messages that have been forwarded within the social network Twitter in the context of health and fitness. Using content analysis we divide the messages into 3 categories of trust evidence: continuity, competence and motivation. Motivational messages were the most common.

Introduction

What evidence do users seek in order to trust another in social network? In this paper, we investigate trust from a user's perspective and using data provided by users theorise about evidence they seek. To explore our question, we start with and build on Metaxas et al's claim [START_REF] Metaxas | Do Retweets indicate Interest, Trust, Agreement?[END_REF] that a message that is forwarded on is likely to be a trusted message. From the social network site Twitter, we collected forwarded messages (otherwise known as 'retweets'). Twitter conversations revolving around the Nike Training Club, a fitness programme originated by the company Nike is used as a case study. The messages have been categorised into three sets using Cofta's [START_REF] Cofta | Proceedings of the 8th international conference on Electronic commerce: The new e-commerce: innovations for conquering current barriers, obstacles and limitations to conducting successful business on the internet[END_REF] dimensions of trust; continuity, competence and motivation.

Our focus is the context of health, in particular, group fitness apps that integrate with social networks. The purpose of this research is to inform the development of health-orientated digital systems (apps and accompanying social media environments) that negotiate trust in the user's interest. Our impetus is to use social network analysis as a form of user study. By studying the messages users create and circulate using Twitter, coined by [START_REF] Marres | Scraping the Social? Issues in live social research[END_REF] as the 'language of the wire', we wish to explore how users conceptualise trust and consider trust on their own terms. We intend to build our own trust-enabling system for health messages. It is likely that our users will wish to configure the messages we offer. To create a base template, which users can work with, knowledge is required of the type of content and delivery style users find relevant to trust.

This research is important, as more and more people will turn to digital tools as a way to support their exercise. In the times of a recession, gymnasium memberships are one of the first items to be removed from a weekly budget [START_REF] Kelley | The Impact of Fitness Technology on Health Outcomes[END_REF]. Digital environments could deliver health and fitness communications much better than what is currently available, deconstructing how users function in existing environments can point to fruitful directions [START_REF] Conroy | Behavior Change Techniques in Top-Ranked Mobile Apps for Physical Activity[END_REF].

Background Research

Like other industries, those in the health and fitness domain have adopted social media networks, such as Twitter, Facebook and Instagram as a means to communicate, develop knowledge, and research. It can be a way to research how best to develop a new digital product, which is what we are using the system for, to understand how to create a mobile application interface (an app). There is now a strong connection between apps and social networks. Users can discuss their apps on social networks via their laptops and work computers and also digital environments are technically interconnected, users can participate in social networks from within an app.

In the context of health informatics and digital environments, understanding trust is crucial. Trust can mean that the listener to a health message not only hears a health message but acts upon it. [START_REF] Clayman | Providing health messages to Hispanics/Latinos: understanding the importance of language, trust in health information sources, and media use[END_REF]. Currently, users and stakeholders have limited ways to gauge trust and health advice in the digital environment [START_REF] Albrecht | Transparency of health-apps for trust and decision making[END_REF]. The result has an impact on health care resources; Kim and Kim [START_REF] Kim | Physicians' perception of the effects of Internet health information on the doctor-patient relationship[END_REF] itemise inappropriate health service adoption, increased time inefficiency and patients incorrectly concerned about symptoms. There is a complex pattern of influence that occurs in social networks that has changed the nature of health communications [START_REF] Lalli | Trust, Inequalities and Health Literacy: The Tangle Meeting with Dr Google[END_REF]. Rather than communicating with their audiences via broadcasts, such as television advertisements, health communicators need to listen and develop understanding, the traditional actions associated with building trust.

The majority of the work studying social networks in the trust research area is from a quantitative perspective, analysing patterns of behaviour and developing algorithms. For instance, Golbeck [START_REF] Golbeck | Generating predictive movie recommendations from trust in social networks[END_REF] models how users evaluate the trustworthiness of another user whilst Tavakolifard et al [11] study who are the most influential message creators. Suh et al [START_REF] Suh | Want to be retweeted? large scale analytics on factors impacting retweet in twitter network[END_REF] study the contextual elements associated with a trusted social network message, such as the amount of follows, the length of time a social network account as existed and how active a message sender is (in contrast, to our work that studies the content of messages). This early work, developing quantifiable measures of trust in social networks, has led to research by several researchers ([11], [START_REF] Metaxas | Do Retweets indicate Interest, Trust, Agreement?[END_REF]) that ascertain a link between a forwarded message and trust.

A retweeted message is likely to be a trusted message, according to recent research by Metaxas et al [START_REF] Metaxas | Do Retweets indicate Interest, Trust, Agreement?[END_REF] who studied retweeted messages and then surveyed those who sent the messages. Retweeting is the practice of broadcasting a message from another user, on the social network Twitter (on which we base our own study). The convention is for a user to copy the original message that the user wishes to broadcast and precede the words with the text RT. The retweet can also contain the name of the originating author preceded with the '@' symbol. An example is 'RT @NTC: I've been training for 2 years'. Retweeting is the main way information is distributed on Twitter [START_REF] Suh | Want to be retweeted? large scale analytics on factors impacting retweet in twitter network[END_REF]. Certainly, as Metaxas et al [START_REF] Metaxas | Do Retweets indicate Interest, Trust, Agreement?[END_REF] acknowledge, there are a complex set of reasons why users may send on a message, such as the desire to be the first in a network to distribute news; but trust is a common denominator underlying a range of intentions. Metaxas et al study the act of retweeting in general. Their claim linking retweeting and trust has been used by [START_REF] Csefalvay | Gamergate series 2: Retweets[END_REF], who uses the notion to analyse the Gamegate phenomenon (a particular episode where female video game professionals where subjected to misogynistic abuse). In the domain of journalism, [START_REF] Finn | A system for monitoring the propagation of rumors on twitter[END_REF] use Metataxas et al's claim to build a system designed for journalists that verifies Twitter messages which may be relevant to a news report. The research we describe in this paper applies Metaxas et al's claim to the domain of health and fitness. We seek detail regarding the type of content in messages that are deemed as trustworthy in order to understand the trust evidence that users value.

According to Cofta [START_REF] Cofta | Proceedings of the 8th international conference on Electronic commerce: The new e-commerce: innovations for conquering current barriers, obstacles and limitations to conducting successful business on the internet[END_REF], there is general agreement between researchers that trust evidence falls into three dimensions: continuity, competence and motivation. Those considering whether to trust or not, trustors, look for evidence across these three categories. The evidence may lie in elements such as the past actions or reputation of the party they are considering whether to trust, the trustee. Continuity evidence is connected with social embededness, how connected is a trustee to relevant communities? Considerations such as reputation and the length of time a trustee has been connected with a community are pertinent. The evidence dimension of competence refers to whether the trustee has the ability and skill to fulfill the requirements of the interaction. Finally, the dimension of motivation has to do with shared interest: Does the trustee have an interest in working towards the welfare of the trustor? Castelfranchi and Falcone [START_REF] Castelfranchi | Trust theory: A socio-cognitive and computational model[END_REF] use the label 'belief of unharmfulness' to describe when the trustor feels positive about the trustee's intention. According to Kranz [16], motivation can be expressed by providing personalised feedback to users of a digital health service.

In order to use these dimensions of evidence as a lens to understand social network messages, a structure is required. Health communication and trust researchers, McNeill and Briggs [START_REF] Mcneill | Understanding Twitter Influence in the Health Domain: A social-psychological contribution[END_REF] provide a structure that allows a conceptualization of trust (such as Cofta's) to understand the micro-detail of trust evidence, in social network messages. McNeill and Briggs [START_REF] Mcneill | Understanding Twitter Influence in the Health Domain: A social-psychological contribution[END_REF] provide a precedent for understanding how the evidence in social network messages can build a picture of trust or distrust. All messages are framed in terms of a metamessage, a predominant message that influence the receiver's worldview. For instance, a 'motivational' frame evokes support and encouragement to the recipient of a message. An example of a 'continuity' frame is when a message conveys a sense that the message sender is connected to a community. A 'competence' frame could suggest a level of skill or experience.

Methodology

The methodology involved a process of collecting messages from Twitter, in particular coding and categorizing the messages using content analysis. To develop a system to code the material, we used Cofta's dimensions of trust evidence. The social network, Twitter, was used for this research because there is a large amount of public data available, making it popular with a range of authors (see Bruns for an overview). It needs to be mentioned that the use of Twitter is controversial, the business model behind Twitter means that it is not possible to query the nature of the sample Twitter offers to researchers, meaning that it is difficult to develop scientific claims. For this reason, Twitter data collection is likened to stepping in the river; it's a different experience every time. However, this does not mean that the data is useless or should not be considered. Rather, we argue, the data has a limitation that must be acknowledged; the ability to make generalisations is constrained. As the aim of this paper is to develop insights into how users conceptualise trust evidence for a design context, the limitations of Twitter do not problematise our research. In order to download our sample from Twitter, we used the Twitter Archiving Google Spreadsheet (TAGS) system developed by Martin Hawksey. TAGS uses Twitter's REST API to access the data. Our sample was 539 messages collected over a 12-day period in late 2014. We collected data until we had in excess of 500 messages.

The collected messages were all associated with the Nike Training Club (NTC). Run by the clothing company Nike, the NTC programme is an app and a social media presence where users can undertake custom exercise workouts on their own or with others and share their reflections. In particular, we collected retweeted messages marked by users with the hashtag #NTC. As mentioned in the literature review, a retweet is a message written by one user and forwarded by another. It is marked by the text 'RT', usually at the start of the message. (The text 'via' is sometimes used but is not common). A hashtag, denoted by the symbol '#' preceding a term, is a convention on Twitter that users implement when they want their message to be part of a public conversation (in our example the hashtag text looked like this: #NTC). When a user includes a hashtag in a Twitter message, it is likely that the user is aware that this means that others can search for that certain hashtag and view their message alongside other messages labeled with that hashtag.

Messages around #NTC were selected as the focus of our research to find users discussing fitness in a community environment. As the inclusion of the word 'club' in the name suggests, #NTC aims to develop a community around fitness, so there is a lot of data with users discussing exercise together and motivating each other; displaying trustworthiness. Other fitness programmes tend to have users lodging their accomplishments with minimal interaction or conversation with others. The wide exposure of Nike brand means there is a large volume of data to be tapped into. There was a stark difference between the quantities of messages collected across different days. The difference is due to whether the Nike department controlling #NTC issued a message that resonated in the NTC community and forwarded by members. Within our message collection, there are also messages originally generated by community members not part of the NTC official team, but these messages are small in comparison with the coverage that the Nike itself can generate.

There are precedents for academic researchers studying #NTC. #NTC is one of only three digital systems that currently meet medical researcher, Padmasekara [START_REF] Padmasekara | Fitness apps, a Valid Alternative to the Gym: a pilot study[END_REF] recommendations for what a fitness digital systems should be; free, and not requiring any special equipment (which are qualities we plan with our app). Researchers Yoganathan and Duwaraka [START_REF] Yoganathan | Persuasive Technology for Smartphone Fitness Apps[END_REF] have also used digital systems produced by Nike as a focus of their study. Less corporate orientated programmes were also reviewed for possible inclusion in our research, such as #ZombieRun, but interestingly, we found that within these conversations, users tended to discuss a range of issues beside fitness that were difficult to code in a meaningful fashion.

To analyse our data, content analysis (CA) was used, to filter and categorise the messages. Trust researchers across several domains use CA as a systematic way to form valid elucidations from a body of text data, whether the data is a television news transcript or a body of social network communiqué [START_REF] Riff | Value co-creation and purchase intention in social network sites: The role of electronic Word-of-Mouth and trust-A theoretical analysis[END_REF]. For example, business researchers, [21] use the technique to understand the link between informal social network recommendations and purchases. In the domain of Public Relations, [START_REF] Waters | Tweet, tweet, tweet: A content analysis of nonprofit organizations' Twitter updates[END_REF] investigates how not-for-profit organisations use Twitter. Health communication researchers Sillence et al [START_REF] Sillence | Trust and mistrust of online health sites[END_REF] use CA to explore the content of websites to understand user's trust perceptions. The method can give a sense of how users 'really' feel about an issue in the context of their daily lives [START_REF] Riff | Value co-creation and purchase intention in social network sites: The role of electronic Word-of-Mouth and trust-A theoretical analysis[END_REF].

To adopt meaningful categories to allocate the messages and guide the basis by which a message was allocated to a particular category, we were informed by the work of Cofta [START_REF] Cofta | Proceedings of the 8th international conference on Electronic commerce: The new e-commerce: innovations for conquering current barriers, obstacles and limitations to conducting successful business on the internet[END_REF] to divide trust evidence into three categories, as discussed in the literature review of this paper. Depending on the emphasis of the message, the twitter messages in our corpus were allocated into one of the three categories: continuity, competence and motivation. The two researchers crosschecked coding decisions.

The problems that affect all content analysis and qualitative coding techniques were also encountered during this project. There is no means within a small-scale research design with limited resourcing to validate the category choices made by researchers. Sometimes the nature of a message means that it is difficult to easily categorise the content. Alternatively, sometimes a message could be placed into more than one category, and it is a judgment call on the part of the researcher to allocate the message. However, as qualitative coding can gather detail informing a research question that other methods cannot access, we adopted the method to inform our research.

Findings and Discussion

During the process of categorising retweeted twitter messages, we observed users displaying and considering trust. In this section we discuss how the three categories of trust evidence, continuity, competence and motivation were expressed in the messages. The most significant category in our data set was motivational messages (total of 329 messages), followed by continuity messages (total of 119 messages) and finally the category of competence was represented by 89 messages. However, an exploration of the data from a quantitative perspective will be the focus of another paper. Some of these messages may have been distributed by Nike's marketing department rather than by individual users. However, the origin of the message is not important to us, it is the validation of messages expressed through the act of retweeting a message. Nike's marketing department may know how to design messages that are trusted by their target audience and our research taps into their findings.

To examine the data from a qualitative perspective, we build on the claim of McNeill and Briggs [START_REF] Mcneill | Understanding Twitter Influence in the Health Domain: A social-psychological contribution[END_REF], mentioned in the literature review, that there is an essence to a message that can be categorised. We combine this work with that of Verbeek [START_REF] Verbeek | Moralizing technology: Understanding and designing the morality of things[END_REF], a technology philosopher who presents a design methodology that focuses on the values within the messages produced by technology design. In Verbeek's view, technology and design is a mediator. Users shape their own communications using technology and technology design changes in response to user needs. Verbeek's work is useful because it is a means to link analysis of technology and design with a wider socio-political context. In particular, Verbeek identifies 'user logic', which underlies how users interpret and value messages they exchange, and this notion underlies our following analysis of our data.

In the context of fitness digital communities, the continuity dimension of trust was expressed in messages demonstrating that the writer has social connections and is part of a community. For example: RT Got my best friends in the #NikeZoom campaign with me!!! RT Together soon! RT @liztrinnear: @EvaRedpath you yelling "don't give up, push harder girls" in my head, gets me through my #NTC app workouts RT @SuperNoodleRach: Thanks kailoha for my Waanngg yoga mat _Ù÷! This morn's @nikewomen #ntc session -post Christmas http://t.co/saHcG1dTC5 Some writers hint at their contribution to a community as well as their connection: RT @geniebouchard: So amazing to inspire young girls to be active! Had so much fun working out & dancing! @nikewomen @mariepurvis #NTC Several messages use a sense of community as a way to promote an event. For instance:

RT @NikeSF: The city is our playground. Step up to 2015 with Nike Training Club. Tonight at 6:30pm.

#NTC

The dimension of competence is demonstrated in messages that communicate the message writer's level of skill to those in the network. The design of the Twitter technology allows users to broadcast and share their fitness exploits. An example message is: RT @surayafaye: I just smashed 45 minutes with Nike+ Training Club http://t.co/0zEnHSVGns #NTC #TrainLikeAMachine #ChristmasWorkOut Some display their competence by providing expert advice to others. For instance: RT Trainer tip from @evaredpath: Strengthen your core and hamstrings with The Roll Up Nike, who design the #NTC campaign, has capitalised on the popularity of new media formats to display competence and provide tools for users to capture and distribute their achievements, as these messages indicate: RT @SkyDigg4: Check out my Zoom in 5 #NTC workout with @NikeWomen!! RT @nikewomen: Earn your selfie. Share minutes, milestones and your personal bests in the new #NTC app.

Finally, the motivation category of trust evidence in the context of our research was expressed in the messages as encouragement to and support for others to improve their skill and commitment levels. For example, the message might be to inspire others to tackle a particular fitness goal. A message illustrating this point is: RT @nikewomen: Challenge the weather one rep at a time with @evaredpath on #Vine

The intention conveyed in these messages resonates with Hardin's [25] claim that those who are trusted demonstrate an intention of concern for others and to work in their interests; in other words, 'encapsulated interest'. Some examples from our data, of messages written to inspire others, include: RT Get ready to relax and recharge with my @NikeWomen ab workout!! Coming soon to the #NT RT @nikewomen: @imagin_IT_ive Congrats on your #NTC milestone. Keep getting after it. RT @vshuguet: A big shout out for the #Nutanix french team. Thanks for being awesome guys! Sometimes the message is written in the format of a question, which is a way to draw in the reader. For example: RT @nikewomen: @thefaradaykage No better time to break them in. Which #NTC session did you smash?

Trust researchers such as [25] and [26] outline how difficult it is for individuals to establish encapsulated interested in the 'real', offline world as individuals may need to demonstrate binding commitment to each other and expose themselves to an element of risk. However, we see on Twitter, that it is easy to express motivation to assist another with their goals. What we may be seeing here is the development of familiarisation. Building on authors such as Luhmann and Möllering, Frederiksen (2014) [26] argues that familiarization is part of the process of demonstrating encapsulated interest, the development of 'from one of being strangers to then acquaintances and finally friends involves the transition from risk to trust. Perhaps the writers of the Twitter messages are establishing the basis of trust by broadcasting gestures of support. If this is correct, it explains why Twitter has the potential to be a powerful marketing tool.

So how can we use the insights from our analysis of the Twitter messages? Commercial researchers would suggest that these insights inform the type of messages a product-owner can send out in order to increase the appearance of trustworthiness. However, the aim of our project is to design systems that enable users to negotiate trust on their own terms. The emphasis is on how users communicate with each other in a system. Users may need to decide who to socialise and train with in a system, or whose training advice to accept. Using the insights we have gathered, we will design base messages, guided by what users consider important in the formation of trust (which will form the basis of another paper) and what they expect or prefer from others. In terms of Verbeek's [START_REF] Verbeek | Moralizing technology: Understanding and designing the morality of things[END_REF] design methodology, this is 'script logic' (as distinct from 'user logic') the design of norms into technology as a base for users to appropriate. Verbeek describes the design process as modest, not an autocratic steering of user behaviour but rather an activity that creates tools for others to adapt during use.

Conclusion

In conclusion, using content analysis, we explored the trust evidence users prefer as represented by 'retweeted' social network messages. Some messages demonstrated trust by expressing continuity and connections with community. Others indicated competence and a level of skill. The most popular type of trusted communications contained motivational messages, for instance, encouraging others to commit to their fitness training. These insights will be used to design template messages for a digital system we are creating that aims to enable users to negotiate trust on their own terms.


L Xie

S Baguet

B Prabel

R Dufour

PARAMETRIC ANALYSIS OF THE NONLINEAR BEHAVIOR OF ROTATING STRUCTURES

Keywords: non-linear dynamics, rotor dynamics, parametric analysis, stability boundary, balance harmonic method

An efficient frequency-domain method is presented for the rapid parametric analysis of stability changes in nonlinear rotating systems which are modeled by three-dimensional finite elements. This method provides directly the stability boundary with respect to parameters such as the system nonlinearity or excitation level. Firstly, the response curve is calculated by combining Harmonic Balance Method (HBM) and continuation. Then stability of equilibrium solutions is determined by Floquet theory. The singular points where a stability change often arises are detected with the sign change of the Jacobian determinant and then located through a penalty method that increases the solving equation system by a completing constraint. Tracking these points, which provides an efficient way to analyze parametrically the nonlinear behavior of a system, can be fulfilled, once again, by the continuation technique.

Numerical methods

Basic rotordynamics modeling is a classical and efficient way to for design and predictive maintenance in most industrial applications. However, when we have to take into consideration of non-linearity, such as a crack, rotorstator contact, hydrodynamic bearing, etc, a three-dimensional finite element model will define more precisely the geometry and the interaction of system. Due to the large number of DOFs and the broad range of study frequency, the computation time can be quite prohibitive. The Harmonic Balance Method (HBM) is thus employed due to its efficiency in predicting steady state behavior. With the help of continuation method, all dynamic equilibrium solutions of nonlinear systems are determined.

Then, determining the local stability of a periodic solution is particularly interesting in an engineering context since only stable solutions are experimentally encountered. Moreover, a change in the stability can lead to significant, qualitative, and possibly dramatic changes in the system response. Therefore, Floquet theory is used for stability analysis of periodic solutions.

In order to assess the influence of parameters on the dynamic behavior in a more economic way, the direct parametric analysis is necessary for numerical investigation of nonlinear systems. Singular points which include the limit(turning) points and bifurcation points, are often accompanied by a change of stability. The determinant of the Jacobian matrix is monitored because its sign change indicates whether a limit point or bifurcation point has emerged. Then the singular points are located by adding a new constraint equation to the solving system. Next, applying once again the continuation method to the augmented system brings about direct tracking of singular points as a function of nonlinear parameters or excitation level. Thus, parametric analysis of the nonlinear behavior of a dynamical system is achieved.

Numerical applications

Nonlinear Jeffcott rotor

The first test case is a modified Jeffcott rotor which can come into contact with a stator that is modeled as an added stiffnessGroll and Ewins (2001). The equations of motion are shown below

mẍ + c ẋ + kx + k c (1 - h r )(x -µysign(v rel )) = p b ω 2 cosωt mÿ + c ẏ + ky + k c (1 - h r )(µxsign(v rel ) + y) = p b ω 2 cosωt (1)
where k c is the stiffness of contact surface, h is the clearance between the rotor and the stator, r = x 2 + y 2 is the radial displacement, p b is the imbalance amplitude and v rel is the relative velocity between the rotor and the stator at the contact point. When r < 0, there is no rub between the rotor and the stator, k c = 0.

Assessments are carried out with the friction coefficient µ as the varied parameter. The sign change of Jacobian determinant helped to locate the singular points (magenta and green points as shown in Fig. 1) while the stability assessment gave information for stable and unstable solutions (blue and red points). However, the stability change next to the excitation frequency 0.3 (Fig. 1(b)) doesn't cause a sign change in the Jacobian determinant. In fact, it's related to the regime change of the system from synchronous full annular rub motion to partial rub motion, which means the stable periodic motion changes to quasi-periodic motion. This fact indicates that monitoring the sign change of Jacobian is efficient to provide information on simple singular points only (saddle-node, asymmetric, and pitchfork bifurcations), but it is insufficient to predict all the stability change of system solutions. As observed in fig. 1(c), the continuation technique succeeds to locate directly all the bifurcation points (half-green half-magenta line). (1998) was also tested (see Fig. 2). This rotor is composed of 13 Timoshenko beam elements. Two linear isotropic mountings are located at both ends of the rotor. The aim is to calculate its response with an imbalance force located on disk 2. The rotor makes contact with a circular, static, and rigid stator located in the vicinity of disk 2. The contact model is identical to that described in last example with the initial clearance h. The equation of motion is written in a finite element form:

Finite element nonlinear rotor

Mẍ(t) + C(ω) ẋ(t) + Kx(t) + f c (x) = p(ω, t) (2)
where M, C(ω) and K represent the mass, damping and stiffness matrices respectively. C(ω) includes the gyroscopic matrix, which varies with ω. The calculations for finite element rotor are carried out in Cast3mCAST3M (2014). Figure .3(a) represents the response curve with stable and unstable branches for the friction coefficient µ = 0. The black line stands for linear response. Two turning points are also detected. The response curve for µ = 0.03 is shown in Fig. 3(b). Same as in Fig. ??, the system has passed stable and unstable phases due to complex motion changes due to the strong non-linearity arising from contact. Similarly, not all bifurcations have been detected.

In Fig. 3(c) the magenta line is directly calculated with the continuation technique and covers all the bifurcation points with respect to µ (from 0 to 0.2). This example illustrates that the continuation technique is able to locate directly the bifurcation points in applications of finite element analysis.

  Forced response curve of Jeffcott rotor for µ = 0.12 (magenta: turning point; green: saddle-node bifurcation) Forced response curve of Jeffcott rotor for µ = 0.2 (magenta: turning point; green: saddle-node bifurcation) (c) Singular point tracking of Jeffcott rotor as a function of friction coefficient µ (green: stable limit point; magenta: unstable limit point)

Figure 1 :

 1 Figure 1: Assessment of Jeffcott rotor for m = 1, c = 5, k = 100, k c = 2500, h = 0.105, p b = 0.1 (blue: stable solution; red: unstable solution)

Figure 2 :

 2 Figure 2: Multi-DOFs finite element rotorA finite element nonlinear rotor model[START_REF] Lalanne | Rotordynamics prediction in engineering[END_REF] was also tested (see Fig.2). This rotor is composed of 13 Timoshenko beam elements. Two linear isotropic mountings are located at both ends of the rotor. The aim is to calculate its response with an imbalance force located on disk 2. The rotor makes contact with a circular, static, and rigid stator located in the vicinity of disk 2. The contact model is identical to that described in last example with the initial clearance h. The equation of motion is written in a finite element form:

Figure 3 :

 3 Figure 3: Assessments of finite element rotor (blue: stable solution; red: unstable solution)