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  <front>
    <journal-meta />
    <article-meta>
      <title-group>
        <article-title>Location Privacy in Location-Based Services: Beyond TTP-based Schemes</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Agusti Solanas</string-name>
          <email>agusti.solanas@urv.cat</email>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Josep Domingo-Ferrer</string-name>
          <email>josep.domingo@urv.cat</email>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Antoni Mart´ınez-Ballest´e</string-name>
          <email>antoni.martinez@urv.cat</email>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>Rovira i Virgili University Department of Computer Engineering and Maths UNESCO Chair in Data Privacy Av. Pa ̈ısos Catalans 26 E-43007 Tarragona</institution>
          ,
          <addr-line>Catalonia</addr-line>
          ,
          <country country="ES">Spain</country>
        </aff>
      </contrib-group>
      <abstract>
        <p>Location-Based Services (LBS) are gaining importance due to the advances in mobile networks and positioning technologies. Nevertheless, the wide deployment of LBS can jeopardise the privacy of their users, so ensuring user privacy is paramount to the success of those services. This article surveys the most relevant techniques for guaranteeing location privacy to LBS users. The rigid dichotomy between schemes which rely on Trusted Third Parties (TTP-based) and those which do not (TTP-free) is emphasised. Also, the convenience of both approaches is discussed and some ideas on the future of location privacy in these services are sketched.</p>
      </abstract>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>-</title>
      <p>
        The Information Society rests on the Information and Communications
Technologies (ICT). Location-Based Services (LBS) are becoming an important ICT
and will be eventually available anywhere anytime. LBS provide users with highly
personalised information accessible by means of a variety of mobile devices that
are able to locate themselves, e.g. by using a GPS or a fixed network
infrastructure with GSM [
        <xref ref-type="bibr" rid="ref1">1</xref>
        ]. Mobile devices are ubiquitous and services related to the
user’s current location proliferate. Examples of LBS are location-based tourist
information [
        <xref ref-type="bibr" rid="ref2">2</xref>
        ], route guidance [
        <xref ref-type="bibr" rid="ref3">3</xref>
        ], emergency assistance [
        <xref ref-type="bibr" rid="ref4">4</xref>
        ], location-based
advertising [
        <xref ref-type="bibr" rid="ref5">5</xref>
        ], etc.
      </p>
      <p>
        The extensive deployment of ubiquitous technology is not without privacy
drawbacks. By sending their locations, LBS users could endanger their security
and privacy because, for example, an attacker could determine their location and
track them. This tracking capability of attackers opens up many computer-aided
crime possibilities (harassment, car theft, kidnapping, etc.). Also, if an attacker
impersonates an LBS provider, the traffic patterns of LBS users could be
influenced by false information, and the users’ location could be compromised [
        <xref ref-type="bibr" rid="ref6">6</xref>
        ].
There are also other attacks which aim to identify users by means of the
locations contained in their queries. By identifying users, attackers can link queries
to real identities. In those ways, attackers can obtain detailed profiles of the users
and send them undesired advertisements or even harass them. Some examples
of techniques/attacks for identifying users are the restricted space identification
(RSI) attack and the observation identification (OI) attack. The RSI attack
consists in linking locations to identities by using queries which are submitted from
a restricted space (e.g. if a user submits queries from his garage in a suburban
house, it is easy to link those queries to his real identity by looking up who lives
in that house, for example by means of a phonebook). Similarly the OI attack
links queries to identities by observing where users are (i.e. the attacker knows
the user’s location because she can see him) and correlating this information
with the location contained in their queries [
        <xref ref-type="bibr" rid="ref7">7</xref>
        ].
      </p>
      <p>
        Several countries have taken legal initiative to cope with privacy problems
related to electronic communications. In Europe, the European directive on Data
Protection and Privacy [
        <xref ref-type="bibr" rid="ref8">8</xref>
        ] agrees on a set of measures to assure the privacy of
the users of telecommunications technologies such as LBS. Similarly, the
Wireless Privacy Protection Act [
        <xref ref-type="bibr" rid="ref9">9</xref>
        ] does the same in the US. Unfortunately, all these
measures regulate well-established business models but they can hardly be
applied to the new LBS that arise in ad-hoc networks created and dismantled on
the fly.
      </p>
      <p>
        Although there are many other relevant topics related to LBS (e.g profile
anonymisation [
        <xref ref-type="bibr" rid="ref10 ref11">10, 11</xref>
        ], trajectories analysis [
        <xref ref-type="bibr" rid="ref12 ref13">12, 13</xref>
        ], privacy in location-based
community services [
        <xref ref-type="bibr" rid="ref14">14</xref>
        ], etc.), in this article we concentrate on the methods to
protect the location privacy of LBS users who send their location to an LBS
provider.
1.1
      </p>
    </sec>
    <sec id="sec-2">
      <title>Contribution and plan of this article</title>
      <p>In this article, we provide a survey of the most relevant and recent schemes
designed to offer location privacy to LBS users. We analyse, organise and classify
them in two main groups: (i) TTP-based schemes and (ii) TTP-free schemes.
Moreover, we sketch some ideas on the future of location privacy in LBS and
some lines for future research.</p>
      <p>The rest of the article is organised as follows. In Section 2 we suggest a
classification of the methods for location privacy in LBS proposed in the literature.
Section 3 is devoted to the analysis of TTP-based schemes and Section 4 studies
TTP-free approaches. Finally, Section 5 contains a brief discussion and some
suggestions for future research.
2</p>
      <sec id="sec-2-1">
        <title>Classification of methods for location privacy in LBS</title>
        <p>In the simplest form of communication between an LBS user (U ) and an LBS
provider (P ), the former sends a simple query (Q) containing an ID, his location
(L) and a request for information (I) that he wants to retrieve from P . Thus, a
simple query sent from U to P can be Q = {IDU , L, I} = {IDU , xU , yU , “Where
is the closest bus station?”} (cf. Figure 1.A). By sending their current locations
to P , LBS users assume that P manages their data honestly and refrains from
any misuse. However, LBS providers cannot always be trusted and more complex
communication schemes are needed.</p>
        <p>Most of the solutions proposed in the literature to address the location
privacy problem are based on Trusted Third Parties (TTP), i.e. entities which fully
guarantee the privacy of their users. Although this approach is widely accepted,
it simply moves users’ trust from LBS providers to intermediate entities. By
doing so, LBS providers are no longer aware of the real locations and identities
of the users; trust and, by extension, power are handed over to intermediate
entities such as brokers, pseudonymisers or anonymisers. The problem is that
users are not necessarily satisfied by completely trusting intermediate entities or
providers, especially after the recent scandals related to the disclosure of personal
data by this kind of trusted entities1 (cf. Figure 1.B).</p>
        <p>The main difference between the simple communication scheme and the
TTPbased one is that in the latter the set of intermediate entities can be expected
to be smaller than the number of service providers. Therefore, intermediate
entities can be well-known and the risk of trusting a dishonest entity is lessened.
However, due to the above mentioned scandals, many users would prefer to trust
1 In Autumn 2007, several data privacy disasters happened in the UK connected to
Her Majesty’s Revenue and Customs. Two computer disks full of personal data on
25 million British individuals disappeared; HMRC also lost another disk containing
pension records of 15,000 people and a laptop containing personal data on 400 people.
In 2006 in the U.S, data on 26.5 million people were stolen from the home of an
employee of the Department of Veterans Affairs, and queries by 658,000 users were
disclosed by the AOL search engine.
TTP-based</p>
        <p>TTP-free
Simple</p>
        <p>Policy-based</p>
        <p>PIR-based
Pseudonym-based</p>
        <p>Anonymity-based</p>
        <p>Collaboration-based Obfuscation-based
nobody, which leads to TTP-free schemes. These represent a substantial change
of paradigm (cf. Figure 1.C). Instead of trusting a third party, users collaborate
to protect their privacy. As it is explained in Section 4, there is not even need to
trust the users one collaborates with. Figure 2 depicts our proposed classification
of location privacy methods. The main aim of the classification is to emphasise
the rigid dichotomy between these two paradigms: (i) TTP-based methods and
(ii) TTP-free methods. In the following sections we review some of the most
relevant representatives of TTP-based and TTP-free methods.
3</p>
      </sec>
      <sec id="sec-2-2">
        <title>Privacy in TTP-based schemes</title>
        <p>TTP-based schemes are very common because they are easy to understand/develop,
and because, in general, they offer a reasonable trade-off between efficiency,
accuracy and privacy. Moreover, some of the ideas used in these schemes arose in
more mature fields like e-commerce.</p>
        <p>In the simple scheme described in Section 2, users send their location
information and queries directly to the LBS provider. In this scheme, whatever
location privacy LBS users can get depends on the honest behaviour of the LBS
provider.</p>
        <p>In the following sections we concentrate on some TTP-based schemes that
aim to protect the location privacy of the users.
3.1</p>
      </sec>
    </sec>
    <sec id="sec-3">
      <title>Policy-based schemes</title>
      <p>Policy-based schemes are one step forward in LBS privacy with respect to the
simple scheme. Although the conceptual framework is the same (i.e. a user
submits queries to a provider), in this case, providers adhere to a set of privacy
policies known by users. Hence, if providers do not properly follow their privacy
policies, users have the right to ask for a compensation and/or take legal action
against providers.</p>
      <p>
        Privacy policies are legal notices that contain statements defining what
service providers can do with their users’ personal data. Privacy policies are
published by service providers, and users decide whether such policies are acceptable
to them. These policies refer to many concepts and specific languages are used
to define them [
        <xref ref-type="bibr" rid="ref15 ref16">15, 16</xref>
        ]. Users reach an agreement with providers about which
data are collected, what are these data used for and how they can be distributed
to third parties. In this kind of schemes, privacy is understood as the ability of
individuals to decide when, what, and how information about them is disclosed
to others. Ideally, users can choose amongst a variety of policies. So, depending
on the selected policy, users can save some money but, in return, providers can
distribute/sell some of their data.
      </p>
      <p>
        These schemes are widely used on the Internet by e.g. e-commerce sites which
define their privacy policies in e.g. P3P (Platform for Privacy Preferences) [
        <xref ref-type="bibr" rid="ref17">17</xref>
        ].
They have been used for automotive telematics [
        <xref ref-type="bibr" rid="ref18">18</xref>
        ], and the Geopriv
(Geographic Location/Privacy) Charter of the IETF proposes their use for LBS
also [
        <xref ref-type="bibr" rid="ref19">19</xref>
        ]. A recent study on the use of policies and access control techniques
can be found in [
        <xref ref-type="bibr" rid="ref20">20</xref>
        ].
3.2
      </p>
    </sec>
    <sec id="sec-4">
      <title>Pseudonymisers</title>
      <p>Pseudonymisers are the simplest intermediate entity between LBS users and
providers. Pseudonymisers receive queries from users and, prior to forwarding
them to LBS providers, they replace the real IDs of the users by fake ones (i.e.
pseudonyms). In this way, the real user IDs remain hidden to the provider, but
pseudonymisers must store the real IDs and their corresponding pseudonyms
in order to forward the answers from the providers to the users. Clearly, users
must completely trust pseudonymisers, because the latter see all the location
information on the former.</p>
      <p>
        The main problem of this technique is that an attacker (e.g. the LBS provider
herself) can infer the real identity of the user by linking the user location with
e.g. a public telephone directory (e.g. by using the aforementioned RSI or OI
attacks [
        <xref ref-type="bibr" rid="ref7">7</xref>
        ]).
3.3
      </p>
    </sec>
    <sec id="sec-5">
      <title>Anonymisers</title>
      <p>Anonymisers are the most sophisticated option in TTP-based location privacy.
Instead of taking care of policies or users’ identifiers, anonymisers assume that
communications are anonymous, i.e. LBS providers do not require an ID to
answer queries2. Anonymisers aim to hide users true identity with respect to
emitted location information. In this section we concentrate on techniques that
hide the location information of users and we assume that identifier abstraction
is already guaranteed.
2 If this assumption was not made, it would be easy to track a given LBS user by
simply checking the ID or the pseudonym (like in the case of pseudonymisers).</p>
      <p>
        A very common way to hide the real location of the users from the LBS
provider is by using the k-anonymity property. k-Anonymity is an interesting
approach to face the conflict between information loss and disclosure risk, suggested
by Samarati and Sweeney [
        <xref ref-type="bibr" rid="ref21 ref22 ref23 ref24">21–24</xref>
        ]. Although it was designed for application in
databases by the Statistical Disclosure Control (SDC) community, k-anonymity
has been adapted to LBS privacy. In this context, we say that the location of
a user is k-anonymous if it is indistinguishable from the location of another
k − 1 users. So, the fundamental idea behind k-anonymisers is to replace the
real location of the user by cloaking areas in which at least k users are located.
Anonymisers transform locations (x, y) at time t to ([x1, x2], [y1, y2], [t1, t2])
where ([x1, x2], [y1, y2]) is the rectangular area containing (x, y) between times
t1 and t2 such that t ∈ [t1, t2]. By doing so, LBS providers cannot easily
determine which of the k users in the cloaking area is really submitting the query.
      </p>
      <p>
        Many examples of this kind of approach and other similar ones based on
cloaking can be found in the literature [
        <xref ref-type="bibr" rid="ref25 ref26 ref7">7, 25, 26</xref>
        ]. One of the most recent advances
in anonymisers is proposed in [
        <xref ref-type="bibr" rid="ref27">27</xref>
        ], where an extension of a previous anonymiser
version [
        <xref ref-type="bibr" rid="ref25">25</xref>
        ] is proposed. The proposed anonymiser allows a user to define his
personal privacy requirements, i.e. the number k of users amongst which he
wants to be anonymised, and the maximum delay and location perturbation he
is willing to accept. The proposal is resilient against identification attacks such
as RSI and OI. However, it has some important drawbacks which, as we explain
in the next section, can be avoided by TTP-free approaches: (i) the architecture
relies on a TTP, so that the user must completely trust the platform mediating
between him and the LBS provider; (ii) it is assumed that LBS providers are
not malicious but semi-honest, which might turn out to be too much of an
idealisation; and (iii) the architecture is centralised, which makes it vulnerable
to Denial of Service (DoS) attacks.
      </p>
      <p>
        In [
        <xref ref-type="bibr" rid="ref28">28</xref>
        ] a similar method called PrivacyGrid is described. Although the anonymiser
described in [
        <xref ref-type="bibr" rid="ref27">27</xref>
        ] and the PrivacyGrid approach are very similar, the latter seems
to be more efficient due to the cloaking techniques based on grids (i.e. bottom-up,
top-down and hybrid) that it uses. Moreover PrivacyGrid adds the l-diversity
property to the already considered k-anonymity one. By doing so, the privacy
of LBS users is improved. Although PrivacyGrid seems to improve the proposal
in [
        <xref ref-type="bibr" rid="ref27">27</xref>
        ], it mainly suffers from the same shortcomings.
      </p>
      <p>Current research on anonymisers focuses on improving the efficiency of the
intermediaries and designing highly personalised services able to guarantee the
privacy of the users.
4</p>
      <sec id="sec-5-1">
        <title>Privacy in TTP-free schemes</title>
        <p>Due to the shortcomings of the TTP-based schemes, other methods that do not
rely on TTPs have been proposed. First, we consider the collaboration methods
that aim to obtain the same results (e.g. k-anonymity, l-diversity, efficiency)
than the ones based on TTP. Then, we pay attention to the methods based on
the obfuscation of the real location without collaboration. Finally we point out
a new location privacy trend based on Private Information Retrieval (PIR).
4.1</p>
      </sec>
    </sec>
    <sec id="sec-6">
      <title>Collaboration-based methods</title>
      <p>
        In [
        <xref ref-type="bibr" rid="ref29">29</xref>
        ], the first collaborative TTP-free algorithm for location privacy in LBS is
proposed. The user perturbs his location by adding zero-mean Gaussian noise to
it. Then the user broadcasts his perturbed location and requests neighbours to
return perturbed versions of their locations. Amongst the replies received, the
user selects k −1 neighbours such that the group formed by the locations of these
neighbours and his own perturbed location spans an area A satisfying Amin &lt;
A &lt; Amax, where Amin is a privacy parameter (the minimum required area for
cloaking) and Amax is an accuracy parameter (the maximum area acceptable
for cloaking). Finally, the user sends to the LBS the centroid of the group of
k perturbed locations including his own. Since users only exchange perturbed
locations, they do not need to trust each other for privacy. On the other hand,
perturbations tend to cancel out each other in the centroid, so accuracy does
not degrade3. This method does not achieve k-anonymity because the centroid
is only used by a single user to identify himself. In addition, due to the noise
cancellation, users cannot use this method several times without changing their
location. In [
        <xref ref-type="bibr" rid="ref30">30</xref>
        ], a similar peer-to-peer scheme for location privacy is presented.
Its main idea is to generate cloaking areas as in [
        <xref ref-type="bibr" rid="ref29">29</xref>
        ]: users must find other users
in their cover range and share their location information. Once this information
is known, users can send their queries to LBS providers using the cloaking area
instead of their real locations. The main shortcoming of this proposal is that
users must trust other users because they exchange their real locations. Thus, a
malicious user can easily obtain and publish the location of other users. Although
we classify this technique as a TTP-free technique, it can also be understood as
a distributed TTP-based scheme, where each user is a TTP.
      </p>
      <p>
        In [
        <xref ref-type="bibr" rid="ref31">31</xref>
        ], the authors propose a method based on Gaussian noise addition to
compute a fake location that is shared by k users (unlike in [
        <xref ref-type="bibr" rid="ref29">29</xref>
        ]). Thus, all k
users use the same fake location and the LBS provider is unable to distinguish
one user from the rest, so that their location becomes k-anonymous. This method
was extended to support non-centralised communications in [
        <xref ref-type="bibr" rid="ref32">32</xref>
        ]. The proposal
is based on a stack of modules that progressively increase the privacy achieved
by users. The basic module is equivalent to the method described in [
        <xref ref-type="bibr" rid="ref30">30</xref>
        ] where
users have to trust each other because they share their location. Once they know
the locations of other users, they can compute a centroid that they use as their
fake location. In order to allow users to exchange their location without trusting
other peers, a second module that perturbs the location is added. This module
adds Gaussian noise with zero mean to the real location of users. As explained
above, the centroid of locations perturbed with zero-mean Gaussian noise is quite
similar to the centroid of unperturbed locations. However, if this procedure is
3 The average of k zero-mean perturbations with variance σ2 has zero mean and
variance σ2/k.
repeated several times with static users (i.e. users that do not change their
location substantially), their real location could be deduced because of the noise
cancellation (this is the main problem of [
        <xref ref-type="bibr" rid="ref29">29</xref>
        ]). To prevent this, the protocol uses
privacy homomorphisms [
        <xref ref-type="bibr" rid="ref33">33</xref>
        ] to guarantee that users cannot see the real locations
of other users whilst still being able to compute the centroid. Finally, a module
that distributes users in a chain is added to avoid denial of service attacks to the
central user. At the end of the protocol users become k-anonymous and their
location privacy is secured. However, the main problem of this proposal is that
it cannot provide a lower bound of the location error.
4.2
      </p>
    </sec>
    <sec id="sec-7">
      <title>Obfuscation-based methods</title>
      <p>
        Obfuscation is a TTP-free alternative to collaboration-based methods.
Obfuscation can be understood as the process of degrading the quality of information
about a user’s location, with the aim to protect that user’s privacy [
        <xref ref-type="bibr" rid="ref34">34</xref>
        ]. Some
methods like the ones described in previous sections (e.g. cloaking methods) can
be understood as special kinds of obfuscation because they basically modify the
location information in several ways to improve user’s privacy. However, we
classify them in different categories because they need TTPs and/or achieve other
properties such as k-anonymity or l-diversity.
      </p>
      <p>
        In [
        <xref ref-type="bibr" rid="ref35">35</xref>
        ] an obfuscation method based on imprecision is presented. The space
is modelled as a graph where vertices are locations and edges indicate adjacency.
Hence, in order to obtain an imprecise location, the user sends a set of vertices
instead of the single vertex in which he is located. The LBS provider cannot
distinguish which of the vertices is the real one. The article proposes negotiation
algorithms that allow users to increase the QoS whilst maintaining their privacy.
The main problem of this technique is that users and providers must share the
graph modelling the space (cf. [
        <xref ref-type="bibr" rid="ref36">36</xref>
        ] for a comprehensive approach to imprecision
in location systems). Some other recently proposed obfuscation methods can be
found in [
        <xref ref-type="bibr" rid="ref37">37</xref>
        ], where the real location of LBS users is replaced by circular areas
of variable centre and radius.
      </p>
      <p>
        SpaceTwist [
        <xref ref-type="bibr" rid="ref38">38</xref>
        ] is the most recent proposal for non-collaborative TTP-free
location privacy. SpaceTwist generates an anchor (i.e. a fake point) that is used
to retrieve information on the k nearest points of interest from the LBS provider.
After successive queries to the LBS provider, SpaceTwist is able to determine
the closest interest point to the real location whilst the LBS provider cannot
derive the real location of the user. The main advantages of this method are: (i)
no TTP and no collaboration are needed; (ii) the closest interest point is always
found; (iii) the location of the user is hidden in a controlled area. However, due
to the lack of collaboration, this method is not able to achieve the k-anonymity
and/or the l-diversity properties.
4.3
      </p>
    </sec>
    <sec id="sec-8">
      <title>PIR-based methods</title>
      <p>
        A totally different approach to TTP-free LBS privacy is proposed in [
        <xref ref-type="bibr" rid="ref39">39</xref>
        ]. In
that article, Private Information Retrieval (PIR) is used to provide LBS users
with location privacy. Although the idea of using PIR techniques is promising,
the proposed approach requires the LBS provider to co-operate with users by
following the PIR protocol; this prevents the use of this method in real
environments, where LBS providers simply answer queries containing a location or
an area without any regard for location privacy. However, if this shortcoming
was solved and without significant computation and efficiency penalties, using
collaborative PIR amongst peers (i.e. users) could be a really promising future
research line.
5
      </p>
      <sec id="sec-8-1">
        <title>Discussion and future work</title>
        <p>In the above sections we have reviewed some of the most recent and relevant
contributions to location privacy protection in LBS. There is a clear
distinction between TTP-based schemes and the TTP-free ones. Although TTP-based
schemes are the most common ones, TTP-free schemes seem superior in terms
of privacy due to the following shortcomings of intermediate TTPs: (i) TTPs
are critical points which can be attacked; (ii) TTPs are bottlenecks; (iii) There
must be many users subscribed to a TTP for the latter to be able to compute
suitable cloaking regions (offering sufficient privacy and accuracy).</p>
        <p>In general TTP-based schemes are weak because users rely on a single trusted
entity. This entity can be impersonated by a bogus TTP created by the attacker,
in which case all the information shared by users with the bogus TTP falls in
the hands of the attacker. A way to mount such an attack is to tamper with
transmitters or use a more powerful signal.</p>
        <p>Despite being inferior regarding privacy, TTP-based schemes are easier to
implement than collaborative-based methods because all the infrastructure
required by users to circumvent the use of a TTP is not necessary. However,
obfuscation-based methods are also easy to implement. We believe that there is
room in the market for both approaches.</p>
        <p>
          The use of k-anonymity and l-diversity properties must be carefully
considered because in some scenarios they are insufficient to preserve user’s privacy [
          <xref ref-type="bibr" rid="ref40">40</xref>
          ].
In our opinion, there are a lot of opportunities for synergy between future work
in PIR and TTP-free LBS privacy. Indeed, current PIR techniques face the (very
serious) limitation of needing co-operation from the database server in following
the PIR protocol. If practical PIR protocols are developed which do not need
such a co-operation, it will be possible to use them for TTP-free location privacy:
if a query can be submitted to a non-co-operative commercial LBS server in such
a way that the latter does not learn what the query is about (i.e. the location
supplied by the user), then one obtains a TTP-free LBS privacy protocol.
        </p>
      </sec>
      <sec id="sec-8-2">
        <title>Acknowledgements</title>
        <p>The authors are solely responsible for the views expressed in this article, which
do not necessarily reflect the position of UNESCO nor commit that organisation.
This work was partly supported by the Spanish Ministry of Education through
projects TSI2007-65406-C03-01“E-AEGIS” and CONSOLIDER INGENIO 2010
CSD2007-0004 “ARES”, and by the Government of Catalonia under grant 2005
SGR 00446.</p>
        <p>The first author thanks Dr. Paul A. Karger (IBM T. J. Watson Research
Center) for his help in providing relevant references and material.</p>
      </sec>
    </sec>
  </body>
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