Statistical and Combinatorial Analysis of the TOR Routing Protocol

Structural Weaknesses Identiﬁed in the TOR Network

Eric Filiol

, Nicolas J.

1,2

and Maxence Delong

Laboratoire de Virologie et de Cryptologie Op

erationnelles, ESIEA, Laval, France

Department of Defense, Paris, France

Keywords:

TOR Network, Routing Protocol, Power Law, Anonymity Network.

Abstract:

In this paper, we present the results of a deep analysis of TOR routing protocol from a statistical and combi-

natorial point of view. We have modeled all possible routes of this famous anonymity network exhaustively

while taking different parameters into account with the data provided by the TOR foundation only. We have

then confronted our theoretical model with the reality on the ground. To do this, we have generated thousands

of roads on the TOR network and compared the results obtained with those predicted by the theory. A last step

of combinatorial analysis has enabled us to identify critical subsets of Onion routers (ORs) which 33%, 50%,

66% and 75% of the TOR traﬁc respectively depends on. We have also managed to extract most of the TOR

relay bridges which are non public nodes managed by the TOR foundation. The same results as for the ORs

have been observed.

1 INTRODUCTION

The various computer networks such as the Internet

today allow everyone to have many means of com-

munication. However, this access is only possible if

the user has an IP address. This condition therefore

allows entities that have access to the network ﬂows,

to analyze it and thus to retrieve information about

particular users.

When communicating in a secure and anonymous

manner, it is necessary to protect not only the con-

ﬁdentiality of information (NATO’s COMSEC) but

also to protect the communication channel (NATO’s

TRANSEC) in order to reduce the risk of a targeted

attack (Defense Science Board, 2015).

Communities have been formed to defend the ano-

nymity of each other and have developed networks

whose purported purpose is to enable people to na-

vigate without anyone being able to identify the ori-

gin and destination of the communication. These net-

works therefore seem to be an ideal tool for criminals

and terrorists of all kinds, as they can carry out their

actions with little trace.

The purpose of this article is therefore to study a

particular network of anonymization, both for clients

and services: The Onion Router (TOR for short). Its

purpose is to protect the conﬁdentiality, integrity and

availability of exchanges within its organization. This

network relies on a community that uses a particular

network protocol. The open source code is documen-

ted, but many aspects such as route construction are

only brieﬂy documented. This is contrary to the prin-

ciples of security, since TOR’s security of this net-

work is based on a considerable amount of obscurity

and secrecy. Therefore, would it be possible for an at-

tacker who controls a limited number of routers to be

able to control and impact a signiﬁcant part of the traf-

ﬁc and hence more or less would be able to damage

TOR’s security?

Our study of the architecture and functioning of

this network has focused on TOR’s routing protocol.

We have established a theoretical statistical model of

this network, based solely on information provided

by the Tor Foundation (TOR Foundation, 2014b) and

the deep analysis of the TOR source code. We have

shown that the statistical law governing the route ge-

neration is a power law and not a normal or uniform

law, the latter laws being more compatible with the le-

vel of security claimed for such a network. The analy-

sis of a large number of routes via the Tor network has

conﬁrmed the validity of our model. In a second part,

the analysis of this problem from the point of view of

graph theory and sub-problems of optimal paths and

vertex cover has made it possible to identify particular

reduced subsets of nodes on which a large proportion

of the TOR trafﬁc depends. Taking control of these

Filiol E., J. N. and Delong M.

Statistical and Combinatorial Analysis of the TOR Routing Protocol - Structural Weaknesses Identiﬁed in the TOR Network.

DOI: 10.5220/0006634705070516

In Proceedings of the 4th International Conference on Information Systems Security and Privacy (ICISSP 2018), pages 507-516

ISBN: 978-989-758-282-0

nodes by an attacker could signiﬁcantly damage the

security of the TOR network.

The paper is organized of follows. Section 2

shortly presents what the TOR network is and how it

works. Then Section 3 presents our statistical analy-

sis of the TOR routing protocol. Section 4 reﬁnes our

results through a combinatorial analysis. Before con-

cluding in Section 6, we discuss in Section 5 how our

results may impact the TOR general security. All data

related to this study are publicly available on (Filiol

et al., 2017).

2 DESCRIPTION OF THE TOR

PROTOCOL

The TOR project is a complex architecture and due

to lack of space we will only give a brief des-

cription of its structure and how it works, for the

paper to be self-contained. The interested reader

will ﬁnd detailed information in (TOR Foundation,

2014b; TOR Foundation, 2014d; TOR Foundation,

2014a).

The Tor Project Inc. is a non-proﬁt foundation loca-

ted in Massachusetts. It was founded in December

2006 by seven computer scientists, including Roger

Dingledine and Nick Mathewson. It is primarily re-

sponsible for software maintenance — including the

Tor Browser and Tail — for the Tor anonymous net-

work.

The Tor protocol is directly derived from a pro-

ject developed within the United States Naval Rese-

arch Laboratory in which Paul Syverson participated

(Goldschlag et al., 1996; Syverson et al., 1997). Its

purpose was to provide the armed forces with a net-

work that provided TRANSEC and COMSEC. The

Tor project was born in 2004 thanks to this labora-

tory which opened the source code and provided the

necessary funding. Today the organization is funded

by donors and government agencies, including the US

Department of State Bureau of Democracy and the

German Federal Foreign Ofﬁce (see (Delong et al.,

2018) for a deep OSINT analysis of the Tor project

and foundation).

2.1 TOR Architecture

The Tor network consists of about 12,000 routers di-

vided into two families, relays and bridges. The Tor

Foundation publishes almost all the information in the

ﬁrst category, and quite nothing for the second cate-

gory in order to prevent a state from denying access to

the network because it does not have the IP addresses

of these hidden nodes.

The Tor network is organized into several levels

where routers have multiple roles. These depend on

the trust placed in them by the network, which is par-

ticularly dependent on the time of activity. The diffe-

rent roles by level are:

• Four Authority levels are identiﬁed:

– Nine Directory authorities and Bridge authori-

ties. They list all the routers — relays or brid-

ges — in the network, analyze the bandwidth,

assign them a particular weight — depending

on age, bandwidth and stability. They also as-

sign speciﬁc roles to routers and establish, sign

and publish a document called consensus ﬁle.

The IP addresses of these servers are listed di-

rectly in the source code of TOR.

– Directory caches that download authorities’

data to cache them for distribution to customers

who request them.

– Fallback directory mirrors which are cache ser-

vers whose IP addresses are also written in the

code. They allow clients to download the con-

sensus ﬁle during initialization connection.

– HSDir which are servers equivalent to the Inter-

net DNS servers. They hold the information to

contact a hidden service such as the addresses

of the introduction points, deﬁned below.

• For the TOR routers:

– the Exit node is the third router of a circuit in-

tended to transmit the network stream to the In-

ternet.

– the Non exit node is a router that only processes

incoming or internal ﬂows in Tor.

– the Entry node is the ﬁrst router to which a

client addresses itself to go through Tor.

– the Guard node is an input router to the Tor net-

work. It also has the role of non exit router and

can therefore process internal network ﬂows.

– the Bridge node (TOR Foundation, 2014d) is a

router whose information is only partially pu-

blished and whose purpose is to allow custo-

mers to bypass censorship. Currently, they are

acting as entry routers. As for this particu-

lar class, we have developed an automatic ex-

traction procedure which is non public. At the

present time, a ﬁrst list of more than 2,500 brid-

ges (on a total number of 3,000-3,500) have

been extracted (Filiol et al., 2017) as well as

many information about the way bridges are

managed behind the scene. Recently the Tor

foundation has increased the number of bridges

but this does not change anything with respect

to our extraction procedure.

• To provide services within the network:

– The hidden service is a server that will only

accept incoming connections through a parti-

cular protocol. The initiators of the connection

will not be able to learn the IP address of the

service in order to preserve its anonymity.

– The introduction point is a router (relay type)

connected to a service hidden by a particu-

lar circuit and which allows a client to submit

an appointment request for establishing a con-

nection.

– The rendezvous Point is a router (relay type)

that connects two independently built circuits,

one by a client and the other by a hidden ser-

vice to allow end-to-end exchanges.

2.2 TOR Protocol Description

The Tor protocol manages two types of communica-

tions, either to the Internet network (external circuits,

three routers) or to its internal network (internal cir-

cuits, six routers) through which an end-to-end ﬂow

will be exchanged between a client and a server.

To build TOR routes (or circuits), clients and servers

of the Tor network build routes consisting of three

second level routers. A route consists of an entry rou-

ter, a relay non-exit router and a relay speciﬁc router

depending on the type of communication.

From the client, the connection to the third server is

either to an exit relay (connection to the Internet) or

any relay (connection with a hidden service). In this

latter case, a special protocol is used to establish the

junction between two independently constructed cir-

cuits that meet at a point called rendezvous point (or

RP).

3 STATISTICAL ANALYSIS OF

THE TOR ROUTING

PROTOCOL

We essentially focuses on the public routers in the rest

of the papers (external circuits). As far as bridges are

concerned, it is sufﬁcient to say that we have obtained

similar results (6-node circuits) as for public routers.

Other information are not public.

3.1 Consensus and Details Files

The Tor routing protocol relies on the consensus ﬁle

which contains all the available TOR routers

. It is

The ﬁle can be downloaded on https://metrics.

torproject.org/collector.html

renewed every hour. It is downloaded by the Tor pro-

tocol during each ﬁrst communication and then every

two hours. This ﬁle is created, signed and published

by the authority routers. It is also possible to retrieve

enriched information, including probabilities by rou-

ter type, by downloading the details ﬁle

. Essential

information we need is:

• the router name and its ﬁngerprint (in fact the

SHA-1 value of its public key).

• IP address with useful ports.

• Router role and type.

• A weight called consensus weight which is voted

by the authority servers and depends in particular

on the bandwidth.

• Probabilities of choosing this type of router

(guard, middle, exit) when establishing a route.

• An exit policy when the router belongs to the exit

type as well as the ownership family, i. e. all

the routers that belong to the same owner. This

is provided by the latter if he wishes.

• A published bandwidth.

A very ﬁrst analysis (see Figure 1) clearly shows that

the distribution of routers is far from being uniform

and exhibits a wide dispersion. In the minds of the

Figure 1: middle router probability distribution.

general public, the choice of each router is made rand-

omly. This popular belief is false insofar as it implies

that each node has the same probability of being cho-

sen.

It remains to determine how the Tor protocol uses the

weight assigned to a router when selecting a node of

a circuit (TOR Foundation, 2014c). First, it will use

a random value (use of the OpenSSL library) and is

considered to be of quality.

Building a circuit requires the choice of three sepa-

rate routers. In the case of an external route, the pro-

tocol will have to choose a guard, a middle and a exit

node. To make this choice, the protocol groups rou-

ters by type. If necessary, it eliminates routers that

https://metrics.torproject.org/onionoo.html

are not speciﬁcation-compliant. Then, it computes the

different sums of the weights and chooses a random

value r between 0 and this sum. Finally, it scrolls

through the list of the type considered by summing

weights up to i-th rank such as

∑

j=i−1

j=0

weight

< r

and

j=i

∑

j=0

weight

> r.

This choice is not a priori questionable since it

weighs certain routers according to their stability, du-

ration of network activity and available bandwidth.

However, this weighting is already partly undermi-

ning the supposed equiprobability. After this ﬁrst ob-

servation, it is important to study the extent to which

this particular weight inﬂuences possible choices. In-

deed, if it is possible to favor certain routers, it is the-

refore possible for the Tor foundation to restrict the

choice of routers to particular subsets.

3.2 Theoretical Statistical Model

Since the open source code of Tor is very large and

complex, we ﬁrst implement the automatic extraction

of all relevant data the writing of routers given by

the details ﬁle and to analyze them. It has then been

possible to ﬁnd the links between probabilities and

weights. Indeed, if we sort the routers by type guard,

middle, exit, it appears that we have a constant ratio:

consensus weight

router

probability

router

= C

router type

The constant value C

router type

depends on the router

type as follows:

• guard nodes:

guard

∑

i∈guards

consensus weight

;

• middle nodes with guard ﬂag set:

middle guard

guard

×C

middle guard

guard

−C

middle guard

• middle nodes without any ﬂag set:

middle seul

∑

i∈middles

consensus weight

;

• exit nodes:

exit

∑

i∈exits

consensus weight

;

Moreover, we established other relationships between

those values:

guard

≈

middle guard

middle seul

middle guard

≈C

middle seul

×(1 +

√

All these relationships have been used to calculate the

probability of choosing the next router knowing the

choice of the precedent ones. Indeed, the speciﬁca-

tions of the Tor network require that two routers on

the same circuit do not belong to the same x.y.z.w/16

subnetwork (class B) as well as to the same (Johnson

et al., 2013) family. Finally, the Tor network seeks to

optimize its own workload. This is why the consen-

sus ﬁles are established every hour in order to change

these weights by router. This has the effect of inﬂuen-

cing client choices. Let us precise that a highly multi-

threaded implementation enables to process any new

version of the consensus ﬁle in less than 5 minutes

(around 10 billions of possible routes).

In order to synthetize all these information and build

our model, we used the following procedure:

1. Get the details ﬁle.

2. Count the router numbers (relay and bridges).

3. For the relay family, sort routers according to their

type (guard, middle, exit) with respect to their

weight.

4. For bridge routers as well as for nodes in the third

position of an internal circuit, calculate an empiri-

cal probability and then sort all the routers in each

category with respect to it.

5. Calculate the sum of weights or probabilities on

parts of each type.

The results obtained show that the impact of weight is

not negligible. Indeed, in the case of middle and exit

routers, the probability of choice follows a power law

that favors a minority since about 80% of the total

weight is given to 20% of the nodes. The detailed

mathematical results are given in Appendix.

It is therefore possible, with a good probability, to

restrict the study of Tor routes to a part of them by

considering only sub-sets of each type (Table 1).

After having determined the predominant para-

meter in the choice of routers and knowing the con-

straints imposed in the construction of a circuit, it is

possible to generate all the routes and to accumulate

for each router the conditional probabilities of being

chosen given the circuit. By noting K the routers of

a type which is compatible with the previous nodes,

the weight of router k and C

the constant of the

type considered, the probability P

of the router i ∈ K

is computed as follows:

The cumulation of these probabilities gives an occur-

rence for each of the roles for each router. The results

of these occurrences for approximately 1 billion and

10 billion generated external circuits are presented in

Table 1: Fraction of ORs involved in different percentage of the Tor routes/trafﬁc.

Proba Guard fraction Middle fraction Exit fraction Number of routes

50% 1/2 81,6% 1/4 78,1% 1/4 78% 308 millions

66% 1/2 81,6% 1/3 85,5% 1/2 95,5% 823 millions

75% 2/3 91,0% 1/3 85,5% 1/2 95,5% 1,1 billion

100% All 100% All 100% All 100% 9,9 billions

Appendix. A second important parameter is the num-

ber of routes to which a router may belong. Indeed,

some routers may belong to fewer circuits than ot-

hers but this difference is compensated by the weight.

Conversely, some nodes may belong to all roads but

are downgraded because of their low weight.

3.3 Testing of the Statistical Model

Our theoretical model must now be compared with

what is observed within the Tor network in order to

validate it or not. In order to have a representative

sample with a margin of error of 1% and a conﬁ-

dence level of 99%, we must consider at least 16,500

TOR routes. We wrote a python script, using the Stem

library developed by the Tor foundation. We have

then extracted more than 60,000 routes considering

a change of the router of entry or not

The comparison between the results obtained from the

data extracted and processed by the algorithm and all

generated routes validates our model. Figure 2 repre-

sents the occurrences obtained for the different types

of routers when generating routes and shows that, ac-

cording to the established model, they follow a power

law. Therefore, the weight assigned to each router is

the parameter that inﬂuences the choice made by the

protocol (see Table 2). It is also important to look at

the case of internal circuits (used to access to hidden

services). In this case, the probability of the third rou-

ter on the circuit cannot be established directly from

the data provided by the Tor foundation. It was the-

refore necessary to identify the proportions of each

type of router (guard, middle, exit) positioned in third

position. Thus, it became possible to determine the

constants necessary for calculating probabilities simi-

larly to what was found for an external circuit since

the probabilities of the third routers verify the equa-

tion:

∑

i∈guard

poids

guard

∑

i∈middle

poids

middle

∑

i∈exit

poids

exit

= 1

The statistical analysis of the 60,000 routes revealed

the following fractions for router types: about

are

By default, the conﬁguration freezes the entry router

during connection initialization for a period of two to nine

months

of Guard type, about

are Middle routers and about

are Exit nodes. These results are empirical but they

identify the signiﬁcant proportions of relay routers to

be considered in the calculation of routes in the case

of a study of internal circuits. Similar to external ci-

rcuits, the study of routes can be restricted to subsets

of nodes (see Table 3). It is important to note the ratio

— in the order of 30 — that exists between the total

number of routes and the number that we can study

while considering 50% of the trafﬁc. Finally, we have

compared the observed and theoretical rankings of

each router type according to the number of occur-

rences. The theoretical and observed results are not

exactly identical but are very close. In other words,

each ranking shows the same list for the most used

routers in a slightly different order. This may be due

to the fact that theoretical computation are made from

the details ﬁle while the route generation took about

two days. As a result, the weighting of each router

may have slightly evolved during the votes in order

to homogenize ﬂows within the network. In addition,

the study was conducted on a small but representative

sample size but it is acceptable that on a large num-

ber of simulations, the results would converge to the

theory.

4 COMBINATORIAL ANALYSIS

OF THE TOR ROUTING

PROTOCOL

The statistical approach has enabled the development

of a model that has highlighted subsets of important

routers within the Tor network through which a sig-

niﬁcant portion of trafﬁc transits. However, it is pos-

sible to reﬁne these subset if the network is modeled

differently. Indeed, the previous study of routes has

focused only on routers and not on the links between

nodes. The purpose of this section is therefore to ana-

lyze how these links can favor some routes over ot-

hers. This will also reveal that signiﬁcant fractions of

the ﬂow are managed by a particular subset of nodes.

Without loss of generality, this new model deals with

at external circuits only and when the input router is

Figure 2: Observed frequencies of router types (guard on the left, middle in the center and exit on the right) with a log-log

scale.

Table 2: Comparison of the theoretical model and the TOR reality.

Guard Middle Exit

Ratio Theorical Virtually Theorical Virtually Theorical Virtually

1/4 58,16% 56,44% 78,1% 78,36% 78,01% 77,82%

1/3 67,40% 68,82% 85,49% 85,24% 86,22% 87,09%

1/2 81,59% 81,68% 94,05% 94,47% 95,53% 96,14%

2/3 91,02% 90,51% 98,44% 98,90% 98,98% 99,13%

3/4 94,29% 93,70% 99,35% 99,64% 99,66% 99,64%

Table 3: Fraction of ORs involved in different percentage of the Tor routes/trafﬁc for internal circuits.

Proba Guard fraction Middle fraction Relay fraction Number of routes

50% 1/2 81,6% 1/4 78,1% 1/4 81,1% 2,7 billion

66% 1/2 81,6% 1/3 85,5% 1/2 95,5% 7,2 billion

75% 2/3 91,0% 1/3 85,5% 1/2 95,5% 9,6 billion

100% All 100% All 100% All 100% 86,6 billion

Figure 3: Graph model for the Tor routing protocol.

a guard. Additional results can be obtained by con-

tacted the ﬁrst author.

We modeled the network as a graph (Figure 3) to ana-

lyze the problem from a combinatorial perspective.

Initially, the vision in vertex cover seemed to be the

right solution, but given the weightings that cannot be

taken into account effectively in this type of problems,

we opted for an optimal path search. The purpose

of graph modeling is to try to determine a relations-

hip between the links guard-middle and middle-exit.

Thus, routers are represented by vertices and the links

by edges, the latter being characterized by a weight.

The calculation of these weights was a real difﬁculty,

since it had to reﬂect the previous results obtained

while being sufﬁciently discriminating to extract a

particular relationship between the guard-middle and

middle-exit links. The weight considered here is the-

refore a function of the number of routes that use this

link as well as the weight of the destination routers.

Indeed, previous calculations have shown that these

two factors are predominant.

Let us note N the number of routes using edge i → j

(Figure 3), K the set of all destination routers which

are compatible with the source router i and p

the

weight of the router k. The the weight P

i j

of the edge

i → j is computed as follows:

i j

n=N

∑

n=1

∑

k∈K

Since each i → j edge has its own weight of P

i j

, we

calculate the ratio between the total sum of the weig-

hts of the links guard-middle and the total sum of the

links middle-exit. This ratio is taken as a reference to

calculate the proportions of each type of routers. In

this case, the ratio is about three.

Therefore, in order to determine a subset of exit rou-

ters, it will be necessary to determine a subset of

middle routers and then calculate the total weight of

the links guard-middle considered (the determination

Figure 4: Algorithm description for router links analysis.

of the middle nodes leads to the determination of the

guard nodes). In our case, we consider the stron-

gest links. Then, to determine the totality of the exit

routers, the total weight of middle-exit links is de-

ducted from the one obtained above and, considering

the most favorable links, the weights of the latter are

summed until they obtain a value approaching that

calculated (Figure 4).

This modeling has enabled to reﬁne the number

and identity of routers of each type that manage a sig-

niﬁcant proportion of the Tor trafﬁc (Table 4) These

Table 4: Number of top-signiﬁcant routers per percentage

Tor trafﬁc.

Fraction Guard Middle Exit Total

33% 755 899 166 1065

50% 1030 1217 246 1463

66% 1326 1617 342 1959

75% 1507 1882 425 2307

results thus reﬁne the ratios between the guard-middle

and exit types. Tables 1, 2 and 3 did not take into ac-

count the ratio calculated here. Finally, the route ana-

lysis conﬁrms the ratio found between the number of

routers in the middle node and the number found in

the exit node subsets. Indeed, for three samples of

about 5,000 routes in which the input router is ﬁxed

— according to the default conﬁguration — as well

as for a sample of the same size in which there is an

input node rotation, the number of routers per type is

given in Table 5.

Table 5: Number of top-signiﬁcant routers for basic conﬁ-

guration.

Guard Middle Exit

Sample 1 (4920) 1 2123 475

Sample 2 (4915) 1 2085 488

Sample 3 (4865) 1490 1976 478

5 DISCUSSION ABOUT THE TOR

SECURITY

Results presented in Section 4 (especially Tables 4

and 5) prove that it is enough to target a reduced

subset of nodes (only 1,463 nodes (1,217 middle, of

which 1,030 guards and 246 exits) to control 50% of

the trafﬁc in Table 4) . The interest for an attacker to

identify these routers lies in the fact that it can take

control of them and then analyze the trafﬁc to corre-

late the ﬂows (Johnson et al., 2013). Thus, it would

be able to remove the anonymity of users — clients

and hidden services — by tracking packets end-to-end

through special patterns identiﬁed or set up in the pac-

kets. It seems that the FBI operated in this way when

it dismantled the silk road in 2014 (TOR Foundation,

2015). The list of subsets we have identiﬁed is avai-

lable in (Filiol et al., 2017). Similarly, in the case of

internal circuits as well as in the case where the in-

put router is a bridge, it is possible to identify similar

subsets.

These node subsets of higher interest are then privi-

leged targets for DDoS (mandatorily from outside the

Tor network) or coordinated targeted malware attacks.

Therefore, it is interesting to carry out a security ana-

lysis of the ports and services which are opened for

each of them, since they can constitute potential en-

try points (a vulnerability scan to look for 0-day vul-

nerabilities is also a required step). We have used a

massive scan tool designed in our lab to analyze top

signiﬁcant routers involved in 50% of the Tor traf-

ﬁc, as well as the ﬁrst thousand ports and a few ot-

her frequently used ones. Table 6 summarizes the re-

sults. The results show that two thirds of routers have

Table 6: Open ports for top signiﬁcant routers controlling

50% of Tor trafﬁc.

Port 21 22 23 25 53

Number 70 631 17 573 121

Fraction (%) 4.7 42.1 1.1 38.2 8.1

Port 80 109 110 135 139

Number 843 3 64 561 561

Fraction (%) 56.2 0.2 4.3 37.4 37.4

Port 443 445 1133 1134

Number 811 577 10 7

Fraction (%) 54.1 38.5 0.7 0.5

open ports because they also serve as SSH, SMTP,

DNS, HTTP, POP, HTTPS servers. In addition, many

routers have open ports that expose them particularly

well. Ports 135,139 and 445 are of high interest for

attackers. In addition, these ports are possible targets

that can be overwhelmed by syn ﬂooding attacks. This

will cause a signiﬁcant evolution of the consensus ﬁle,

which may then favor some potentially trapped rou-

ters. Indeed, during the vote, the authorities will me-

asure a low bandwidth.

As far as bridge routers are concerned, a corrupted

machine in middle position could count the number

of network frames it is relaying when constructing a

circuit. It could then determine its own position and

hence compare the IP address of the machine right be-

fore its own position with those listed by the founda-

tion. It would enable to determine whether the input

router is a bridge or not. In order to prevent such an

attack, it is envisaged to always insert a guard router

between the bridge and the middle router. Thus, hid-

den nodes would be drowned within network users.

6 CONCLUSION

Our study tends to prove that the security of the Tor

network is not optimal. Indeed, since overall security

is based on the individual good practices of the router

owners, the maximum level that will be reached will

not exceed that of the weakest link. The latter can also

be set up voluntarily to identify particular users.

It is also important to note that local security (of no-

des and servers, protocol strength...) is not sufﬁcient

as soon as we deal with a worldwide infrastructure.

Having a more global and higher point of view is also

important. In this respect, our study has showed the

Tor infrastructure too much relies actually on a redu-

ced number of nodes.

Our future work will mostly focus on two aspects.

First, we intend to optimize our multi threaded con-

sensus ﬁle processing algorithm. The aim is to be

able to process any new version of such ﬁle and ex-

tract top signiﬁcant node subsets within less than one

minute. Second to go on working on the bridge mana-

gement and its evolution and to reﬁne our extraction

algorithm.

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2017-09-12.

APPENDIX

Statistical Model for the TOR Routing

Protocol

The results obtained seem to indicate that the distri-

bution of TOR routes follows a power law distribu-

tion (general case including Pareto, Zipf, Mandelbrot

laws). We will limit ourselves to the discrete case

(however, when the number of data is large enough

—which is our case — it is possible to work with the

continuous version of this law (Clauset et al., 2009).

For more detailed information of these laws the reader

can refer to (Clauset et al., 2009; Alstott et al., 2014)

we also used for statistical analysis.

A discrete variable X follows a Power law if its

probability density is given by

p(x) = P[X = x] = C.x

−α

where α is a constant parameter called power para-

meter of the law and C is the proportionality factor.

In practice, most of the phenomena obey a power law

for some x ≥ x

min

. The constant C is given by the

value x

min

using the fact that

∞

min

p(x)dx = 1

This results in C =

α−1

−α+1

min

On a graph with logarithmic scales (log-log repre-

sentation), the graph of a power law is a line since

when noting y = P[X = x] we can write

log(y) = α. log(x) + log(C)

Another useful representation is the inverse of the

so-called cumulative cumulative distribution function

(P[X[X > x] = 1 − F(x)). This is the one we will

use here to compare the theoretical law obtained

with the empirical law of data. In the following

we will limit ourselves to the estimation of the pa-

rameters α (maximum likelihood method, validation

by the Kolmogorov-Smirnov adequacy test) and x

min

(exhaustive estimation on all the values minimizing

the D value of the Kolmogorov-Smirnov test).

It should be noted that in few cases, the theory

suggests that empirical data can be described by two

laws, without any clear distinction between the two.

In our case. In three cases, the log-normal law has

been identiﬁed as a possible alternative, but relatively

close to the power law. Let us recall that a random

variable X follows a log-normal law if Y = ln(X ) fol-

lows a normal law of average µ and standard deviation

σ (denoted LOG −N (µ, σ)).

For all possible routes (around 10 billions) we

have results given in Table 7 and in Figure 5. For

Table 7: Results for all possible TOR routes (D represents

the Kolmogorov-Smirnov distance between data and mo-

del).

Data α x

min

Total nodes occurrences 1.89 1850 0.03

Guard nodes occurences 3.16 5239 0.04

Middle nodes occurences 3.09 1223 0.03

Exit nodes occurrences 2.45 21361 0.05

Guard nodes occurences, let us mention the fact that

the LOG −N (7.38, 0.956) law is a possible alterna-

tive law.

Table 8 and Figure 6 provides the results for the

top one billion routes. For the Guard nodes occu-

rences, we have LOG − N (5.99, 0.661) as possible

alternative law while for the Exit nodes occurrences,

LOG −N (8.47, 0.827) can be also an alternative law.

Table 8: Results for 1-billion top TOR routes (D represents

the Kolmogorov-Smirnov distance between data and mo-

del).

Data α x

min

Total nodes occurrences 1.87 404 0.03

Guard nodes occurences 3.09 519 0.04

Middle nodes occurences 2.89 285 0.03

Exit nodes occurrences 2.99 11083 0.04

All OR OR guard

OR middle OR exit

Figure 5: Inverse cumulative density functions of data compared to the ﬁtted law. The red dotted line (resp. green and blue)

describes the power law (resp. log-normal and exponential law).

All OR OR guard

OR middle OR exit

Figure 6: Inverse cumulative density functions of data compared to the ﬁtted law (1-billion top TOR routes). The red dotted

line (resp. green and blue) describes the power law (resp. log-normal and exponential law).