network coding based privacy preservation against traffic analysis in multi-hop wireless networks by...

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Network Coding Based Privacy Preservation against Traffic Analysis in Multi-hop Wireless Networks

Yanfei Fan, Yixin Jiang, Haojin Zhu, Jiming Chen, Xuemin (Sherman) Shen

Abstract — Privacy threat is one of the critical issues in multi-hop wireless networks, where attacks such as traffic analysis can be easily launched by a malicious adversary due to the open-air trans-mission. Network coding has the potential to thwart traffic analysis attacks since the coding/mixing operation is encouraged at intermediate nodes. However, the simple deployment of network coding can not achieve the goal once enough packets are collected by the adversaries. On the other hand, the coding/mixing nature precludes the feasibility of employing the existing privacy-preserving techniques, such as Onion Routing, in network coding enabled networks. Different from any ex-isting solutions, in this paper, we propose a novel network coding based privacy-preserving scheme against traffic analysis in multi-hop wireless networks. With homomorphic encryption on Global Encoding Vectors (GEVs), the proposed scheme offers two significant privacy-preserving features, packet flow untraceability and message content confidentiality, for efficiently thwarting the traffic analysis attacks. Moreover, the proposed scheme keeps the random coding feature, and each sink can recover the source packets by inverting the GEVs with a very high probability. Theoretical analysis and simulative evaluation demonstrate the validity and efficiency of the proposed scheme.

Keywords — Network coding; homomorphic encryption; privacy preservation; traffic analysis

I. INTRODUCTION

Wireless networks, such as Wi-Fi, have been widely deployed in the access network area due to their

benefits such as convenience, mobility, and low cost. However, they still suffer from their inherent

shortcomings such as limited radio coverage, poor system reliability, as well as lack of security and pri-

vacy. Multi-hop Wireless Networks (MWNs) are regarded as such a promising solution for extending the

radio coverage range of the existing wireless networks. System reliability can be improved through

multi-path packet forwarding, which is feasible in MWNs.*

However, there exist many security and privacy issues in MWNs. Due to the open-air wireless trans-

mission, MWNs suffer from various kinds of attacks, such as eavesdropping, data modification/injection,

and node compromising; these attacks may breach the security properties of MWNs, including confiden-

tiality, integrity, and authenticity. In addition, some advanced attacks, such as traffic analysis and flow

tracing, can also be launched to compromise the privacy of users, including source anonymity and traffic

* Part of the paper was presented in IEEE Infocom’09.

Downloaded from engine.lib.uwaterloo.ca on 12 May 2011 Page 2 of 28

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 3, MARCH 2011

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secrecy. In this paper, we focus on the privacy preservation issue, i.e., how to prevent traffic analysis/flow

tracing and achieve source anonymity in MWNs.

hunter

sensor

endangered animal

Decapitation Strike

trace back

decoy

(B)(A)

spy

Fig. 1. Privacy threats in MWNs

Among all privacy properties, source anonymity is of special interest in MWNs. Source anonymity

refers to communicating through a network without revealing the identity or location of the source node.

Preventing traffic analysis/flow tracing and provisioning source anonymity is critical for securing MWNs,

especially military related, such as sensor or tactical, networks. Consider a simple example of multicast

communication in military ad hoc networks (which can be regarded as a kind of MWNs), where the nodes

can communicate with each other through multi-hop packet forwarding. If an attacker can intercept the

packets and trace back to the source through traffic analysis, it may disclose some sensitive information

such as the location of some critical nodes (e.g., the commanders) and further impair the location privacy.

Subsequently, the attacker can take some actions to launch the so-called Decapitation Strike to destroy

these critical nodes [23], as shown in Fig. 1 (A). Another example is the event reporting in wireless sensor

networks, where flow tracing can help attackers to identify the location of concerned events, e.g., the

appearance of an endangered animal in a monitored area, and then take subsequent actions to capture or

kill the animals [22], as shown in Fig. 1 (B).

How to efficiently thwart traffic analysis and flow tracing attacks and provide privacy protection in

MWNs is a very challenging issue. Existing privacy preservation solutions, such as proxy-based schemes

[11], [12], Chaum’s mix-based schemes [13], [14], and onion-based schemes [15], [16], may either require

a series of trusted forwarding proxies or result in severe performance degradation in practice, e.g., with an



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end-to-end delay of several minutes. Different from traditional privacy-preserving techniques, in this

work, we study the privacy issue from a brand-new perspective: achieving privacy preservation with

network coding.

Network coding was first introduced by Ahlswede et al [1]. Subsequently, two key techniques, random

coding [2] and linear coding [18], further promote the development of network coding technologies. The

random coding makes network coding more practical, while the linear coding is proven to be sufficient

and computationally efficient for network coding. Currently, network coding has been widely recognized

as a promising information dissemination approach to improving network performance. Primary appli-

cations of network coding include the file distribution and multimedia streaming on P2P overlay networks

[6], data transmission in sensor networks [7], tactical communications in military networks [8], etc.

Compared with conventional packet forwarding technologies, network coding offers, by allowing and

encouraging coding/mixing operations at intermediate forwarders [1], several significant advantages such

as the potential throughput improvement [3], transmission energy minimization [4], and delay minimiza-

tion [5]. Network coding can also work as an erasure coding scheme to enhance the dependability of a

distributed data storage system [27].

The deployment of network coding in MWNs can not only bring the above performance benefits, but

also provide a possible way to efficiently thwart the traffic analysis/flow tracing attacks since the cod-

ing/mixing operation is encouraged at intermediate nodes. Similar to Chaum’s mix-based schemes [13],

[14], network coding provides an intrinsic message mixing mechanism, which implies that privacy pres-

ervation may be efficiently achieved in a distributed manner. Moreover, the unlinkability between in-

coming packets and outgoing packets, which is an important property for preventing traffic analysis/flow

tracing, can be achieved by mixing the incoming packets at intermediate nodes. However, the privacy

offered by such a mixing feature is still vulnerable, since the earliest decoding is still possible and linear

dependence can be easily analyzed. A simple deployment of network coding can not prevent earliest

decoding and traffic analysis since the explicit Global Encoding Vectors (GEVs, also known as tags)

prefixed in the encoded packets provide a back door for adversaries to compromise the privacy of users.

Once enough packets are collected, the adversaries can easily recover the original packets and then con-



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duct the traffic analysis/flow tracing attacks based on these original packets. A naïve solution to address

this vulnerability is to employ link-to-link encryption. This method can prevent traffic analysis in a certain

level, but it also introduces heavy computational overhead, and thus results in significant performance

degradation of the whole system. Additionally, it can not preserve the privacy of users, if some interme-

diate nodes are compromised by malicious adversaries. Such deficiencies motivate us to explore an effi-

cient privacy-preserving scheme for MWNs.

In this paper, based on network coding and Homomorphic Encryption Functions (HEFs) [19], [20], we

propose an efficient privacy-preserving scheme for MWNs. Our objective is to achieve source anonymity

by preventing traffic analysis and flow tracing attacks. To the best of our knowledge, this is the first re-

search effort to use network coding for thwarting traffic analysis attacks and realizing privacy preservation.

The proposed scheme offers the following attractive features: 1) Enhanced Privacy against flow tracing

and traffic analysis: With the employment of HEFs, the confidentiality of GEVs is effectively guaranteed,

which makes it difficult for attackers to recover the GEVs. Even when some intermediate nodes are

compromised, the adversaries still can not decrypt the GEVs, since only the sinks know the decryption key.

The confidentiality of GEVs further brings the implicative benefit of the confidentiality of message con-

tent, because message decoding only relies on the GEVs. Moreover, with message recoding on encrypted

GEVs, the coding/mixing feature of network coding can be exploited in a natural manner to satisfy the

mixing requirements of privacy preservation against traffic analysis. 2) Efficiency: Due to the Homo-

morphism of HEFs, message recoding at intermediate nodes can be directly performed on encrypted

GEVs and encoded messages, without knowing the decryption keys or performing expensive decryption

operations on each incoming packet. The performance evaluation on computational complexity demon-

strates the efficiency of the proposed scheme. 3) High Invertible GEVs: Random network coding is

feasible only if the prefixed GEVs are invertible with a high probability. Theoretical analysis has proven

that the influence the HEFs pose upon the invertible probability of GEVs is negligible. Thus, the random

coding feature is kept in our network coding based privacy-preserving scheme.

The remainder of the paper is organized as follows. In Section II, preliminaries related to the proposed

scheme are given, including network coding, homomorphic cryptosystem, privacy measurement, and



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threat model. In Section III, the proposed privacy-preserving scheme is presented in detail. In Sections IV

and V, the security analysis and performance evaluation/optimization are conducted, respectively. In

Section VI, related works are surveyed, followed by the conclusions in Section VII.

II. PRELIMINARIES

A. Network Coding Model

Unlike traditional packet-forwarding systems, network coding allows intermediate nodes to perform

computation on input messages, making output messages be the mixture of the input ones. This elegant

principle implies a plethora of surprising opportunities, such as random coding [2]. As shown in Fig. 2,

whenever there is a transmission opportunity on an outgoing link, an outgoing packet is formed by taking

a random combination of packets in the current buffer.

Fig. 2. Random coding (mixing) at intermediate nodes

An overview of network coding and possible applications has been given in [5], and packet tagging and

buffering are key for practical network coding. Packet tagging will be introduced later. In practical net-

work coding, source information should be divided into blocks with h packets in each block. All coded

packets related to the kth block belong to generation k and random coding is only performed among the

packets in the same generation. Packets within a generation need to be synchronized by buffering for the

purpose of network coding at intermediate nodes.

Consider an acyclic network (V, E, c) with unit capacity, i.e., c(e)=1 for all e E∈ , meaning that each

edge can carry one symbol per unit time. Assume that each symbol is an element of a finite field qF .

Consider a network scenario with multicast sessions, where a session is comprised of one source s V∈



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and a set of sinks T V⊆ (or one single sink t V∈ ). Let ( , )h MinCut s T= be the multicast capacity, and

1, , hx x be the h symbols to be delivered from s to T.

For each outgoing edge e of a node v, let ( ) qy e ∈F denote the symbol carried on e, which can be

computed as a linear combination of the symbols ( ')y e on the incoming edges 'e of node v, i.e., '

''( ) ( ) ( )ee

y e e y eβ=∑ . The coefficient vector '( ) [ ( )]ee eβ=β is called as Local Encoding Vector (LEV).

By induction, the symbol ( )y e on any edge e E∈ can be computed as a linear combination of the

source symbols 1, , hx x , i.e., 1

( ) ( )h

i iiy e g e x

== ∑ . The coefficients form a Global Encoding Vector

(GEV) 1( ) [ ( ),..., ( )]he g e g e=g , which can be computed recursively as ''( ) ( ) ( ')eee e eβ=∑g g , using the

LEVs ( )eβ . The GEV ( )eg represents the code symbol ( )y e in terms of the source symbols 1, , hx x .

Suppose that a sink t T∈ receives symbols 1( ), , ( )hy e y e , which can be expressed in terms of the

source symbols as

1 1 1 1 1 1

1

( ) ( ) ( )

( ) ( ) ( )

h

t

h h h h h h

y e g e g e x x

G

y e g e g e x x

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (1)

where Gt is called Global Encoding Matrix (GEM) and the thi row of Gt is the GEV associated with ( )iy e .

Sink t can further recover the h source symbols by inverting the matrix Gt and applying the inverse to

1( ), , ( )hy e y e .

In general, each packet can be considered as a vector of symbols 1( ) [ ( ), , ( )]Ne y e y e=y . By likewise

grouping the source symbols into packets ,1 ,[ , , ]i i i Nx x=x , the above algebraic relationships carry over

to packets. To facilitate the decoding at the sinks, each message should be tagged with its GEV ( )eg ,

which can be easily achieved by prefixing the thi source packet ix with the thi unit vector iu . Then, each

packet is automatically tagged with the corresponding GEV, since

''

1

[ ( ), ( )] ( )[ ( '), ( ')]

( )[ , ]

ee

h

i i ii

e e e e e

g e

β

=

=

=

∑∑

g y g y

u x. (2)

The benefit of tags is that the GEVs can be found within the packets themselves, so that the sinks can

compute Gt without knowing the network topology or packet-forwarding paths. Nor is a side channel

required for the communication of Gt. Actually, the network can be dynamic, with nodes and edges being

added or removed in an ad hoc way. The coding arguments can be time varying and random.


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B. Homomorphic Encryption Function

Homomorphic Encryption Functions (HEFs) have the property of homomorphism, which means op-

erations on plaintext can be performed by operating on corresponding ciphertext. For example, suppose

( )E ⋅ is a HEF. It is easy to compute ( )E x y+ from ( )E x and ( )E y without knowing the corresponding

plaintext x and y. To be applicable in the proposed scheme, a HEF ( )E ⋅ needs to satisfy the following

properties:

1) Additivity: Given the ciphertext, ( )E x and ( )E y , there exists a computationally efficient algo-

rithm ( , )Add ⋅ ⋅ such that ( ) ( ( ), ( ))E x y Add E x E y+ = .

2) Scalar Multiplicativity: Given ( )E x and a scalar t, there exists a computationally efficient algo-

rithm ( , )Mul ⋅ ⋅ such that ( ) ( ( ), )E t x Mul E x t⋅ = .

Actually, the scalar multiplicativity can be deduced from the additivity, since 1

( ) ( )t

iE t x E x

=⋅ = ∑ .

Benaloh [19] and Paillier [20] cryptosystems are such two additive HEFs, where the addition on plaintext

can be achieved by performing a multiplicative operation on the corresponding ciphertext, i.e.,

1 2 1 2( ) ( ) ( )E x x E x E x+ = ⋅ . Further, the following two equations can be easily derived.

( ) ( )( ) ( )

i

t

ti i ii i

E t x E x

E t x E x

⋅ =

⋅ =∑ ∏ (3)

C. Privacy Measurement

Currently, privacy measurement consists of two main methods: anonymity set and privacy entropy. In

the anonymity set method, members in an anonymity system is assumed to be uniformly distributed, i.e.,

all members are equally possible to be the target user in probability from the perspective of adversaries.

The privacy entropy method can measure the privacy degree more accurately for those systems where the

members are not uniformly distributed. Following Shannon’s classic definition of information entropy,

the entropy of privacy ( )pE S can be defined as:


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1

1 2 3

1

( ) ( ) log ( )

{ , , , , , , }( ) .

( ) 1

n

p i ii

i n

i i

n

ii

E S p x p x

where

S x x x x x

p x is the prob of being x

p x

=

=

= −

=

=

∑

∑

(4)

The privacy entropy shares some common characteristics with information entropy, such as continuity,

symmetry, and additivity. For example, continuity means that changing the value of one of the prob-

abilities by a very small amount should only change the entropy by a small amount; symmetry means that

if the probabilities of any two objects are swapped, the privacy entropy remains the same. It is easy to

conclude that any reorder of these probabilities will not change the privacy entropy of the system. We will

use the above two approaches to measure the enhanced privacy of the proposed scheme.

D. Threats Model

We consider the following two attack models.

nk

Outside Attacker

e1

e2

e3

e’1

e’2

e’3

e’4

ni

e’1

e’2

e’3

e’4

e1

e2

e3

nj

Inside Attacker

nk

(a) (b)

Fig. 3. Attack Model: (a) Outside Attacker, (b) Inside Attacker

Outside Attacker: An outside attacker can be considered as a global passive eavesdropper who has the

ability to observe all network links, as shown in Fig. 3 (a). An outside attacker can examine the tags and

message content, and thus link outgoing packets with incoming packets. Further, even if messages are

encrypted in an end-to-end manner, it is still possible for a global outside attacker to trace packets by

analyzing and comparing the message ciphertext.

Inside attacker: An inside attacker may compromise several intermediate nodes, as shown in Fig. 3 (b).

Link-to-link encryption is vulnerable to inside attackers since they may already obtain the decryption keys

and thus the message plaintext can be easily recovered.


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Both inside and outside attackers may perform more advanced traffic analysis/flow tracing techniques,

including size correlation, time-order correlation, and message content correlation [21]. Thus, adversaries

can further explore these techniques to deduce the forwarding paths [22].

Without loss of generality, we assume that an anonymous secure routing protocol [23] is deployed to

assist network nodes to determine the forwarding paths. The generation number of a packet can be hidden

by the secure routing scheme so that an outside attacker can not determine the generation of a packet for its

further analysis.

III. THE PROPOSED NETWORK CODING BASED PRIVACY-PRESERVING SCHEME FOR MWNS

In this section, we propose a novel network coding based privacy-preserving scheme for MWNs, which

can efficiently thwart traffic analysis and flow tracing attacks, followed by theoretical analysis on the

invertibility of GEMs.

A. The Proposed Privacy-Preserving Scheme

Though providing an intrinsic mixing mechanism, the original network coding cannot provide privacy

guarantee due to explicit GEVs, since an adversary can recover the original messages as long as enough

packets are collected. Link-to-link encryption is vulnerable to inside attackers since they may already

compromise several intermediate nodes and obtain the secret keys.

An intuitive way to resolve this issue is to keep GEVs confidential to intermediate nodes by encrypting

the GEVs in an end-to-end manner, which can prevent compromised intermediate nodes from analyzing

GEVs or recovering the original messages. Such an intuitive approach, however, cannot prevent the ad-

versaries from tracking the message ciphertext since the “mixing” feature of network coding will be

disabled by the end-to-end encryption.

To address this issue, in this paper, we employ the Paillier cryptosystem [20] as the HEF to apply en-

cryption to GEVs. HEF can not only keep the confidentiality of GEVs, but also enable intermediate nodes

to efficiently mix the coded messages. In the Paillier cryptosystem, given a message m and the public key

(n, g), the encryption function is described as follows,

2( ) (mod )m nE m g r n= ⋅ , (5)


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where r is a random factor in the Paillier cryptosystem. ( )E m satisfies the following homomorphic

property:

1 2 21 2 1 2 1 2( ) ( ) ( ) (mod ) ( )m m nE m E m g r r n E m m+⋅ = ⋅ = + . (6)

Fig. 4. Homomorphic encryption on packet tags

With HEFs, intermediate nodes are allowed to directly perform linear coding/mixing operations on the

coded messages and encrypted tags, as shown in Fig. 4. In other words, due to the homomorphism of the

HEF, linear network coding can be achieved by operating on the encoded messages and the ciphertext of

GEVs, without knowing the decryption keys or performing the decryption operations.

The proposed scheme primarily consists of three phases: source encoding, intermediate random linear

recoding, and sink decoding. Without loss of generality, we assume that each sink acquires two keys, the

encryption key ek and the decryption key dk, from an offline Trust Authority (TA), and the encryption key

ek is published to all other nodes. For supporting multicast, a group of sinks are required to obtain from the

TA or negotiate the key pair in advance [26]; then, they can publish the encryption key and keep the de-

cryption key private in the group.

Fig. 5. Packet Tagging before Source Coding

Source Encoding: Consider that a source has h messages, say 1, , hx x , to be sent out. The source

first prefixes h unit vectors to the h messages, respectively, as illustrated in Fig. 5. After tagging, the

source can choose a random LEV and then perform a linear encoding operation on these messages. Thus,


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one LEV will generate an encoded message with the GEV (which is equal to the LEV temporarily) tagged.

To offer confidentiality for the tags, homomorphic encryption operations are employed on these tags as

follows:

1 2

( ) ( ( )), (1 )( ) [ ( ), ( ), , ( )]i ek i

h

c e E g e i h

e c e c e c e

= ≤ ≤

=c, (7)

where the notation ek denotes the encryption key.

Intermediate Random Linear Recoding: After receiving a number of packets of the same generation,

an intermediate node can perform random linear coding on these packets. To generate an outgoing packet,

firstly, a random LEV [ ]1, , hβ β is chosen independently; then, a linear combination of message content

of the incoming packets is computed as the message content of the outgoing packet, as shown in Fig. 2.

Since the tags of the h incoming packets are in ciphertext format, and an intermediate node has no

knowledge of the corresponding decryption keys, it is difficult for the intermediate node to perform

functions such as earliest decoding to get the original message content. However, due to the homo-

morphism of the encryption function, a linear transformation can be directly performed on the encrypted

tags of the incoming packets to generate a new tag for the outgoing packet, namely,

1

( ) ( ) ( ' )h

i iig e e g eβ

==∑ . (8)

The GEV of a new outgoing packet can be calculated according to equation (7). By utilizing the

homomorphic characteristic of the encryption on GEVs, the ciphertext of the new GEVs for outgoing

packets can be calculated as follows:

1

1

( )1

( ( )) ( ( ) ( ' ))

( ( ) ( ' ))

( ( ' ))i

h

ek ek i ii

h

ek i ii

h eek ii

E g e E e g e

E e g e

E g eβ

β

β=

=

=

=

=

=

∑∏∏

. (9)

The ciphertext of new GEVs can be computed from the ciphertext of GEVs of incoming packets

without knowing the decryption key. Finally, the ciphertext of a new GEV is prefixed to the corresponding

message content to form a new outgoing packet, which is sent out to downstream nodes.

Sink Decoding: After receiving a packet, the sink first decrypts the packet tag using the corresponding

decryption key dk.


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1 2

( ) ( ( )) (1 )( ) [ ( ), ( ), , ( )]i dk i

h

g e D c e i h

e g e g e g e

= ≤ ≤

=g (10)

Once enough packets are received, a sink can decode the packets to get the original messages. Then, the

sink derives the decoding vector, which is the inverse of the GEM, as shown in the following equations.

1

1 2

(mod )[ ( ), ( ), , ( )]T

h

n

e e e

− ⋅ =

=

G G U

G g g g (11)

Finally, the sink can use the inverse to recover the original messages, shown as follows.

1 1

1

( )

( )h h

e

e

−

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

x y

G

x y

. (12)

For random network coding, a key issue is the invertibility of a GEM. We discuss in detail the in-

vertibility of a GEM as follows.

B. Invertibility of a GEM

Let GEM A be comprised of h GEVs with h elements in each GEV. A and *A are the determinant

and the adjoint of the matrix A, respectively. According to the theory of linear algebra, finding the inverse

of a square matrix A is equivalent to solving the corresponding system of linear equation with A being the

coefficient matrix.

Gaussian elimination can be applied to solve a system of linear congruence equations. Due to the

homomorphism of the modulo congruence in terms of the addition, subtraction, and multiplication op-

erations, a system of linear congruence equations can be separated into several single equations with one

unknown in each equation, shown as follows:

1( 1) (mod )h i j

i j ijjA x y M n+

== −∑ . (13)

A system of linear congruence equations is solvable if and only if every independent equation is

solvable. The only difference of solving a system of linear equations and solving a system of linear con-

gruence equations lies in finding the inverse of |A| modulo n. In order to further discuss the solutions, we

formulate a system of linear congruence equations as:

(mod ) ( 1, , )i iA x A n i h= = , (14)


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where 1( 1) (mod )h i j

i j ijjA y M n+

== −∑ .

Theorem 1: A system of linear congruence equations has a unique solution only if 0A ≠ .

Proof: See Appendix A. ■

However, this condition is not a sufficient condition for a system of linear congruence equations to

have a unique solution, because a solution for i iA x A= does not imply a corresponding solution

for (mod )i iA x A n= .

Theorem 2: A system of linear congruence equations has dh solutions if:

0

0(mod ) ( 1, 2, , )i

A

A d i h

⎧ ≠⎪⎨

≡ =⎪⎩ (15)

where gcd( , )d A n= .

Proof: See Appendix B. ■

Corollary 1: A system of linear congruence equations has a unique solution if and only if:

0

gcd( , ) 1A

A n

⎧ ≠⎪⎨ =⎪⎩

(16)

and 1 (mod ) ( 1,2, , )i ix A A n i h−= = .

Proof: See Appendix C. ■

Theorem 2 and Corollary 1 indicate that a solvable system of linear congruence equations does not

imply the invertibility of the corresponding coefficient matrix. A stronger condition, i.e., gcd( , ) 1A n = , is

required for the invertibility of a coefficient matrix A modulo n. The above theorems and corollary gen-

erally hold no matter if n takes the value of a prime number q or the product of two prime numbers p and q.

In section V, we will further give a quantitative analysis on the invertible probability of a coefficient

matrix.


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IV. SECURITY ANALYSIS

In this section, we analyze the privacy preservation from the proposed scheme, as well as an extra

privacy-enhancing policy.

A. Privacy Enhancement

The allowance of earliest decoding conflicts with the basic principle of privacy preservation. Therefore,

the proposed scheme inherently precludes the feasibility of earliest decoding. In the following, we further

discuss what protection the proposed scheme can provide and how well it can provide.

The first protection is on generation numbers. If an adversary attempts to launch a traffic analysis at-

tack, a better method is to identify the packet generations first. However, in the proposed scheme, the

generation numbers are hidden in the secure routing scheme.

The second protection is to resist size correlation and time-order correlation attacks, which are two

widely-used techniques in traffic analysis. In the proposed scheme, every message is trimmed to be of the

same size, making size correlation useless to adversaries. The inherent buffering technique [5] of network

coding can significantly improve the resistance to time-order correlation.

The third protection is to eliminate message content correlation, which is also commonly used in traffic

analysis and flow tracing. Assume that the first protection on generation numbers is compromised, due to

the vulnerability of a secure routing system. To launch content correlation based traffic analysis, adver-

saries must compromise several forwarders to intercept packets of the same generation. Let the number of

packet generations going through these compromised nodes be w during a period of time v. To determine

if an intercepted packet, say md, in some downstream link, is a linear combination of some known packets,

the following steps should be performed:

1) The first step is to choose h columns of linearly independent coded symbols from messages of the

same generation, as shown in Fig. 6. The independence can be tested using Gaussian elimination on the h

vectors. The computational complexity of this step is O(h3) in terms of multiplication operations.


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Encoded Messages

Tags

Original Messages

hh

l l

Fig. 6. Message Content Encoding

2) The second step is to compute the coefficient vector of the linear combination associated to the

message md from the chosen h vectors. With Gaussian elimination, the computational complexity of this

step is O(h2) in terms of multiplication.

3) The third step is to verify if the resultant coefficient vector satisfies all the other l h− columns of

code symbols. The computational complexity is ( ( ))h l h−O in terms of multiplication.

From the above three steps, the whole computational complexity to check if an objective message is the

linear combination of a generation of messages is 3( )h h l+ ⋅O in terms of multiplication. To check w

generations, the computational complexity is 3( ( ))w h h l⋅ + ⋅O in terms of multiplication.

The probability of false positive verification (to incorrectly determine a generation of messages to be

the coding source of a specified message) is considerably low. Because the correlation between messages

of the same generation is uncertain or unknown, it is reasonable to assume that the other (l-h) columns of

code symbols are randomly distributed in the field F. Thus the probability of false positive verification is

( , ) h l

fpvP h l F−= , which is pretty small since l h .

In summary, under the assumption that an adversary can compromise a group of nodes which are as-

sociated with a bunch of Min-Cut edges, computational complexity of 3( ( ))w h h l⋅ + ⋅O in terms of mul-

tiplication is required for the adversary to determine if a downstream packet is coming from one of the w

generations of intercepted packets. If the first protection on generation numbers is not compromised

(adversaries fail to identify the generation of a packet), the computational complexity will increase to 3( ( ))h

whO C h h l+ ⋅ in terms of multiplication. Compared to the conventional network coding with explicit

GEVs, the proposed scheme enhances the privacy preservation so greatly that the traffic analysis attacks

are almost computationally impossible.


- 17 -

B. Privacy-Enhancing Policy

An intuitive and effective measure to enhance privacy is employing end-to-end encryption on original

message plaintext. This measure transforms the original plaintext into ciphertext, which can be regarded

as a random mapping from the plaintext space to ciphertext space. The “random mapping” makes it more

difficult for adversaries to launch content analysis attacks successfully, since the ciphertext gives no spe-

cific knowledge to those without decryption keys. Light-weight symmetric-key cryptosystems can be

chosen as the end-to-end encryption method. In addition, these encryption and decryption operations are

performed only at the two ends of a session, incurring zero overhead to other nodes in the network.

In general, if the original message is in plaintext format, the number k of original message sets may not

be large enough to resist the above backward deduction attacks. The privacy-enhancing policy is to em-

ploy end-to-end encryption on message content, which brings more “confusion” to adversaries. Moreover,

with the change of session keys, the original messages may be mapped to the whole ciphertext space. Thus,

we can regard the message set number ke to be the size of the whole ciphertext space, i.e., ke = |F|h·l.

If privacy entropy [24] is used to measure the privacy enhancement from end-to-end encryption, we

can obtain:

1 1

( ) ( )

( log ) ( log )e

p p e p

k ke ei i i ii i

E E M E M

p p p p= =

Δ = −

= − − −∑ ∑. (17)

where Me and M denote the anonymity set with and without end-to-end encryption, respectively, and eip

and ip denote the appearance probability of element i in the corresponding anonymity set, respectively. If

each message set appears with the same probability, i.e., ( )e ei jp p i j= ≠ and ( )i jp p i j= ≠ , the enhanced

privacy entropy can be computed as follows:

log log

log log

h l

pE F k

h l F k

⋅Δ = −

= ⋅ ⋅ −. (18)

Since the number k is usually much smaller than ke, the increased entropy of privacy is considerably

significant. With the increase of privacy entropy, the size of message space will increase exponentially.

Thus, this measure greatly enhances the privacy of the proposed scheme since it is difficult to traverse the

message space.


- 18 -

V. PERFORMANCE EVALUATION AND OPTIMIZATION

In this section, we evaluate the performance of the proposed scheme in term of invertible probability

and computational overhead. A performance optimization framework is also developed to minimize the

statistical computational overhead.

A. Invertible Probability

Let each element of a LEV be randomly chosen from field qF . The following two theorems hold.

Theorem 3: For a Local Encoding Matrix (LEM), which is comprised of h LEVs with h elements in

each LEV and each element is from the finite field qF , the invertible probability of a GEM (also own h

vectors) will be degraded by 1(1 )h i

q is q−

== −∏ .

Proof: See Appendix D. ■

Corollary 2: The invertibility factor qs imposed by an h×h LEM can be approximated to 1 21 q q− −− −

when h≥4, and the error of this approximation is within the magnitude of 5( )q−O .

This corollary can be easily proofed by expanding the multiplication of the polynomials. This corollary

gives two important implications. Firstly, in practical network coding, the min-cut capacity h is much

larger than the condition required in corollary 2. That is, this corollary can be safely used in practical

network coding. Secondly, the field size q is relatively a large number. Therefore, an amount in the

magnitude 5( )q−O is very small and can be omitted.

Based on Theorem 3 and Corollary 2, the invertible probability of a GEM can be easily calculated. For

a network coding system with a min-cut capacity ( 4)h h ≥ , the invertible probability can be approximated

as 1 2(1 )tq q− −− − , where q is the field size and t is the total encoding number from the source to sinks. In

practical network coding, since q is a relatively large prime number, the above invertible probability can

be further approximated to 11 tq−− .

If the Paillier cryptosystem is chosen as the homomorphic encryption method, the invertible probability

will differ from the result in Theorem 3. In the Paillier cryptosystem, the element is chosen from a ring

nR , where n pq= and p, q are two prime numbers.

Theorem 4: For a LEM, which is comprised of h LEVs, where the elements are randomly chosen from


- 19 -

a ring nR ( n pq= ), the invertible probability of a GEM (also with h vectors) will be degraded by

p q ns s s+ − .

Proof: See Appendix E. ■

According to Corollary 2, sp, sq, sn, and the integral invertibility factor can be approximated to 1 21 p p− −− − , 1 21 q q− −− − , 1 21 n n− −− − , and 1 2 1 2 1 21 p p q q n n− − − − − −− − − − + + , respectively, with the er-

ror in the magnitude of 5 5( )p q− −+O .

If p q≈ , the integral invertibility factor can be approximately reduced to 1 11 p q− −− − , with the error

confined in 2 2( )p q− −+O . If a session performs totally t times of random encoding, the invertible prob-

ability of GEVs at sinks is 1 1(1 )tp q− −− − . In practical network coding, since p and q are relatively large

prime numbers, the above invertible probability can be approximated to 1 11 ( )t p q− −− + . From the above

analysis, it can be seen that the invertible probability is independent of the number of sinks, but dependent

on the total numbers of random coding.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1Invertibility vs. Field Size

Field Size n

Inve

rtib

le P

roba

bilit

y

Sanders's schemeHo's schemeThe proposed scheme

Fig. 7. Invertible Probability versus the Size of the Algebraic Structure

Compared with the random coding schemes in [2] and [18], the invertible probability of the proposed

scheme seems a little bit smaller, as shown in Fig. 7. The difference can be regarded as the overhead of

privacy preservation.

According to the above analysis, the invertible probability decreases with the increase of the total times


- 20 -

of random coding. The simulation results in Fig. 8 match the analytical results pretty well, thus demon-

strate the validity of the performance analysis. The analysis and simulation results of the original random

coding scheme are also shown in the same figure for comparison.

B. Computational Overhead

The computational overhead of the proposed scheme can be investigated respectively from three as-

pects, including source encoding, intermediate recoding, and sink decoding. Since the computational

overhead of the proposed scheme is closely involved with the specific homomorphic encryption algorithm,

in the following analysis, we will take the Paillier cryptosystem as the encryption method when necessary.

Source Encoding Overhead: Consider h GEVs with h elements in each GEV, which form an h×h

2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Invertibility vs. Coding Times (p=19, q=23)

Total Times of Random Coding

Inve

rtib

le P

roba

bilit

y

Analysis on Random CodingSimulation on Random CodingAnalysis on the proposed schemeSimulation on the proposed scheme

2 4 6 8 10 12 14 16 18 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



Inve

rtib

le P

roba

bilit

y


2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



Inve

rtib

le P

roba

bilit

y


2 4 6 8 10 12 14 16 18 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



Inve

rtib

le P

roba

bilit

y


Fig. 8. Invertible Probability vs. Total Times of Random Coding (a. p=19, q=23; b. p=47, q=53; c. p=101, q=103; d. p=241, q=251)


- 21 -

GEM. After source encoding, every element in the GEM is encrypted one by one. Thus, the computational

overhead is O(h2) in terms of encryption operations. Every encryption operation requires 2 exponentia-

tions, 1 multiplication, and 1 modulus operation in the Paillier cryptosystem. Therefore, the computational

complexity is 2( log )h n⋅O in terms of multiplication operations.

Intermediate Recoding Overhead: In intermediate nodes, linear transformation on the elements of

GEVs can only be performed by manipulating the ciphertext of these elements because intermediate nodes

have no knowledge of decryption keys. According to Eq. (8), the computational complexity of producing

one element in new GEVs is h exponentiations and 1h − multiplications on the ciphertext, which is

( log )h n⋅O in terms of multiplications together. Thus, the computational complexity is 2( log )h n⋅O for a

GEV and 3( log )h n⋅O for a GEM with h GEVs in terms of multiplication.

Sink Decoding Overhead: After receiving an encoded message, a sink can decrypt the elements in the

GEV. According to the Paillier cryptosystem, decrypting an element requires 1 exponentiation, 1 multi-

plication, and 1 division operation. Therefore, the computational complexity of decrypting a GEV is

( log )h n⋅O in terms of multiplication operations. Thus, for a whole GEM with h GEVs, the computational

overhead is 2( log )h n⋅O in terms of multiplication.

C. Performance Optimization

As described in the previous subsections, the invertible probability and computational overhead of the

proposed scheme are 1 11 ( )t p q− −− + and 3( log )h n⋅O , respectively. Thus, the statistical computational

overhead for a GEM can be expressed as follows:

3

1 1

( log )1 ( )

h nT

t p q− −

⋅=

− +O . (19)

From Eq.(19), we can see that the computational overhead of the proposed scheme is a monotonically

increasing function of h, i.e., the length of a GEV, for any given n and t. As discussed in Section IV, the

security of the proposed scheme is also monotonically increasing with the increase of h. Thus, a tradeoff

between the security and the computational overhead should be made in practical deployment. A typical

way to make this tradeoff is to set the security requirements first and then choose the minimum h to meet

the requirements. In this way, the minimum computational overhead can be achieved in terms of h.


- 22 -

On the other hand, noticing that n pq= and p q≈ , we can approximate Eq.(19) for any given h and

formulate it as the following optimization problem:

log( , )1 2

: 4, ,1, .

nMinimize g n t

t n

subject to n n is an integer

where t t is an integer

=−

≥≥

(20)

After solving the ordinary differential equation 0g

n

∂=

∂, we have: 2 2

1 4 ( 1/ 2 )n t LambertW e t= − ⋅ and 2 2

2 4 (1/ 2 )n t LambertW e t= ⋅ . Since the Lambert-W function has infinite branches in the complex plane,

we only consider the branches which have real-valued solutions with real arguments. In addition, since

1t ≥ (t means the total encoding number) and the Lambert-W function is single-valued in the real plane

for a positive argument, we can determine that 2 2 22 4 (1/ 2 ) 1/n t e t e< ⋅ ⋅ = , which is in conflict with the

condition 4n ≥ . Thus, 2n can be elimilated for further consideration.

1n is double-valued in the real plane since the argument of the function LambertW(x), 1/ 2x e t= − ⋅ , is

in the region of ( 1/ , 0)e− . We denote the double-valued results as

2 21( , ) 4 ( , 1/ 2 ) , 1,0n k t t LambertW k e t k= − ⋅ = − , (21)

where k can be any integer in the complex plane. For a real-valued solution of 1n , k can only be 0 and -1.

Similarly, we can determine that 1 1 1(0, ) (0, ) 0.215tn t n t =< = and this principal value is also not in ac-

cordance with the prescribed condition 4n ≥ . Finally, we can determine by checking that the result

1( 1, )n t− is the point where the objective function ( , )g n t achieves its minimum for any given parameter t.

For example, given 5t = , we can get three real-valued results as follows:

1

1

2

( 1,5) 2390.936334,(0,5) 0.1460863321,(0,5) 0.1260584706,

n

n

n

− =⎧⎪ =⎨⎪ =⎩

(22)

where only 1( 1,5)n − meets the prescribed condition 4n ≥ . Fig. 9 shows the computational overhead

versus the size of algebraic structure.


- 23 -

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08Total Random Coding Number: t = 5

The Size of Algebraic Structure: n

Sta

tistic

al C

ompu

tatio

nal O

verh

ead:

g

n1(0,5)

1000 2000 3000 4000 5000 6000 7000 8000 9000 10009.75

9.8

9.85

9.9

9.95

10

10.05

10.1

10.15

10.2

10.25Total Random Coding Number: t = 5

The Size of Algebraic Structure: n

Sta

tistic

al C

ompu

tatio

nal O

verh

ead:

g

Mininum Point n1(-1,5)

Fig. 9. Computational Overhead vs. the Size of Algebraic Structure (t=5)

After obtaining the minimum point 1( 1, )n t− of the objective function, we can find the closest positive

integer n which is the product of two primes p and q, i.e., n pq= . The integer n can then be substituted

into Eq. (19) to achieve the minimum computational overhead.

VI. RELATED WORKS

Many privacy-preserving schemes have been proposed, and they can be classified into three categories:

proxy-based, mix-based, and onion-based. Proxy-based schemes include Crowds [11] and Hordes [12].

The common characteristic of these schemes is that they all employ one or more network nodes to issue

service requests on behalf of the originator. In Crowds, for example, servers and even crowd members can

not distinguish the originator of a service request, since it equally likely originate from any member of the

crowd.

Chaum’s mix based schemes include MorphMix [13] and Mixminion [14]. The common feature of

these schemes is to employ techniques such as shaping which divides messages into a number of

fixed-sized chunks, and mixing which caches incoming messages and then forwards them in a randomized

order. These two techniques can be used to prevent attacks such as size correlation and time-order cor-

relation.

Onion-based schemes include Onion Routing [15] and Onion Ring [16]. The common feature of this


- 24 -

kind of schemes is the chaining technique, which chains onion routers together to forward messages hop

by hop to the intended recipient. The characteristic of this technique is that every intermediate onion router

only knows about the router directly in front of and behind itself, respectively, which can protect user

privacy if one or even several intermediate onion routers are compromised.

Network coding has privacy-preserving features, such as shaping, buffering, and mixing. However,

network coding suffers from two primary types of attacks, pollution attacks [10] and entropy attacks [9].

Pollution attacks can be launched by untrusted nodes or adversaries through injecting polluted messages

or modifying disseminated messages, which is fatal to the whole network due to the rapid propagation of

pollution. In entropy attacks, adversaries forge non-innovative packets that are linear combinations of

“stale” ones, thus reducing the overall network throughput.

To secure network coding, some solutions have been proposed and they can be divided into two

categories according to different theoretical bases. Information-theory based schemes [8] can only detect

or filter out polluted messages at sinks, not at forwarders. Cryptography-based solutions include homo-

morphic hashing [9], homomorphic signatures [10], and secure random checksum [9]. These solutions

either require an extra secure channel [9], or incur high computation overhead [10].

In summary, existing studies on secure network coding mainly focus on detecting or filtering out

polluted messages [10]. Little attention has been paid to the privacy issues, especially on how to protect

the encoded messages from tracking or traffic analysis.

VII. CONCLUSIONS

In this paper, we have proposed an efficient network coding based privacy-preserving scheme against

traffic analysis and flow tracing in multi-hop wireless networks. With the light-weight homomorphic

encryption on Global Encoding Vectors (GEVs), the proposed scheme offers two significant pri-

vacy-preserving features, packet flow untraceability and message content confidentiality, which can ef-

ficiently thwart the traffic analysis attacks such as flow tracing. Moreover, with homomorphic encryption,

the proposed scheme keeps the essence of random linear network coding, and each sink can recover the

source messages by inverting the GEVs with a very high probability. The quantitative analysis on privacy


- 25 -

enhancement and computational overhead, and the simulative evaluation, demonstrate the effectiveness

and efficiency of the proposed scheme. In our future work, we will further improve the performance of the

proposed privacy-preserving scheme, such as the invertible probability of GEVs.

APPENDIX

A. Proof of Theorem 1.

Proof: From the theory of modulo congruence, for 1n > , the mapping : nf is a homomorphic

mapping in terms of the addition, subtraction, and multiplication operations. Therefore, for each solution

to (mod )i iA x A n= , there must be one or more solutions to ( )i iA x A k n k= + ⋅ ∈ .

According to the theory of linear algebra, the necessary and sufficient condition for

( )i iA x A k n k= + ⋅ ∈ to have a unique solution is 0A ≠ . Thus, the necessary condition for a system

of linear congruence equations to have a unique solution is 0A ≠ . ■

B. Proof of Theorem 2.

Proof: According to Theorem 1, 0A ≠ is a necessary condition for a system of linear congruence

equations to have a unique solution. Therefore, the original system of linear congruence equations can be

transformed to (mod )( 1,2, , )i iA x A n i h= = by only applying addition and multiplication operations,

which are preserved by the homomorphic mapping : nf .

In equations (mod )i iA x A n= , variables ( 1, 2, , )ix i h= can be solved independently from each

other. Thus, for every independent equation (mod )i iA x A n= , solutions can be solved by employing the

theory of linear congruence equations.

A linear congruence equation (mod )ax b n= is solvable if and only if the congruence

0(mod )b d= with gcd( , )d a n= is solvable, where gcd( , )a n denotes the greatest common divisor of the

two operands a and n. Let one solution to the linear congruence equation be 0 /x n d s< = . Then the so-

lutions are 0 0 0, , , ( 1)x x x s x d s= + + − . If 1d = , then there is only one unique solution which is less than

n.

According to the theory of linear congruence equations, if 0(mod ) ( 1,2, , )iA d i h≡ = , then every

equation (mod )i iA x A n= has d solutions. The solutions to the system of linear congruence equations


- 26 -

are the combination of these independent solutions to every single congruence equation. The combination

number is hd . ■

C. Proof of Corollary 1.

Proof: From Theorem 2, if gcd( , ) 1d A n= = , the system of linear congruence equations has 1hd =

solution, which is also the unique solution to the system. In addition, according to the congruence theory,

when gcd( , ) 1A n = , the modular inverse of the integer A , which is denoted as 1A

− , can be calculated

using the extended Euclidean algorithm. The result satisfies the following equation: 1 1(mod )A A n− = .

The unique solution to the system of linear congruence equations is 1 (mod )i ix A A n−= ( 1,2, , )i h= .

The solution can be expressed in a matrix form as 1 1 * (mod )A A X X A A Y n− −= = . ■

D. Proof of Theorem 3.

Proof: The invertibility factor imposed by any h LEVs depends only on the linear dependence of the h

LEVs themselves. Firstly, the elements in the first LEV can be any combination except for all zeros.

Therefore, the invertibility factor imposed by the first LEV is 1 hq−− . For the second LEV, it is required to

consider the linear dependence on the first one, where the all-zero situation is included. Thus, the in-

vertibility factor imposed by the second LEV is 11 / 1h hq q q −− = − . For the third LEV, considering its

linear dependence on the former two, the invertibility factor is 2 21 / 1h hq q q −− = − . Similarly, for the

remaining LEVs, the invertibility factors are 31 hq −− , 41 hq −− , , 11 q−− , respectively.

Therefore, the integrated invertibility factor imposed by the whole LEM can be calculated by multi-

plying these individual factors together as follows:

1(1 )h i

q is q−

== −∏ . ■

E. Proof of Theorem 4.

Proof: The problem can be decomposed into two separate sub-problems in terms of the prime number

p and q, respectively. As for the sub-problem in terms of the prime number p, there is a mapping from the

original problem to the problem modulo p. According to Theorem 3, the imposed invertibility factor is

1(1 )h i

p is p−

== −∏ . Similarly, the invertibility factor imposed by the sub-problem of prime number q is

1(1 )h i

q is q−

== −∏ .


- 27 -

The above two sub-problems have overlap, which happens on these points where the number is con-

gruent to zero modulo n. The invertibility factor related to the overlap area is 1(1 )h i

n is n−

== −∏ . Ac-

cording to the union principle of the set theory, the integral invertibility factor can be calculated as fol-

lows:

1 ((1 ) (1 ) (1 ))p q n p q ns s s s s s− − + − − − = + − . ■

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