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Enforce Truth-Telling in Wireless Relay

Networks for Secure Communication

Shuhang Liu∗, Rongqing Zhang∗, Lingyang Song∗, Zhu Han†, and Bingli Jiao∗

∗School of Electronics Engineering and Computer Science, Peking University, Beijing, China.

†Electrical and Computer Engineering Department, University of Houston, Houston, TX, USA.

Abstract—To ensure security in data transmission is one of the

most important issues for wireless relay networks. In this paper,

we consider a cooperative network, consisting of one source node,

one destination node, one eavesdropper node, and a number

of relay nodes. Specifically, the source selects several relay

nodes which can help forward the signal to the corresponding

destination to achieve the best security performance. However,

the relay nodes may have the incentive not to report their true

private channel information in order to get more chance to be

selected and gain more payoff from the source. We employ a

self-enforcing truth-telling mechanism into the network to solve

this cheating problem. By adding a transfer payoff to the total

payoff of each selected relay node, we prove that each relay node

would get its maximum expected payoff only when it tells the

truth to the source. And then, an optimal secrecy capacity of the

network can be achieved. Simulation results verify the efficiency

of the proposed mechanism.

I. INTRODUCTION

Security and privacy protection is one of the most im-

portant issues in wireless communications due to the broad-

cast nature of wireless channels. In recent years, besides

traditional cryptographic mechanisms, information-theoretic-

based physical layer security have been developing fast. The

concept of wiretap channel was first introduced by Wyner

[1], who showed that perfect secrecy of transmitted data

from the source to the legitimate receiver is achievable in

degraded broadcast channels. In follow-up work, Leung-Yan-

Cheong and Hellman further investigated the secrecy capacity

in the Gaussian wiretap channel [2]. Later, Csisz´ ar and K¨ orner

extended Wyner’s work to non-degraded broadcast channels

and found an expression of secrecy capacity [3].

When considering a wireless relay network, the realization

of secrecy capacity is much more complicated. In [4], [5],

the authors demonstrated that cooperation among relay nodes

can dramatically improve the physical layer security in a

given wireless relay network. The channel state information

(CSI) is assumed to be known at both the transmitter and

the receiver in [6]. However, in practice, the relay node

always measures its own channel gains and distributes the

information to others through a control channel. There is

no guarantee that it reveals its private information honestly.

Hence, the most crucial problem is how to select efficient

relay nodes to optimize the total secrecy rate in the network,

while some selfish relay nodes may report false information

to the source in order to increase their own utilities. In [7]

truth-telling is assured by using the threat of punishment, and

in [8], [9] reputation methods are designed to achieve this

goal. However, all these methods need a delicate and complex

detection scheme to monitor and catch the liar nodes, which

is difficult to be realized because too much information needs

to be exchanged.

With the help of game theory [10], [11] and its therein ap-

plications in wireless communications [12], [13], we propose a

self-enforcing truth-telling mechanism to achieve the Bayesian

Nash Equilibrium [14] and solve the possible cheating problem

in relay networks. We focus on a system in which all the

channels are orthogonal and each relay node’s private channel

information is unknown by others. After properly adding a

transfer payoff function into the total payoff, each relay node

would have no incentive to report false information which can

lead to a loss in its own expected total payoff. We prove that

the unique equilibrium is achieved on the condition that all the

relay nodes report the truth. In other words, the competing

relay nodes are enforced to obey the selection criterion and

cooperate with each other honestly, and thus, no extra cost

would be paid by the source since the total transfer payoff of

all the relay nodes equals zero. Simulation results show that

the relay nodes can get their maximum utilities when they

all report their true channel information to the source and

any cheating leads to a loss in the total secrecy rate of the

system as well as the expected total payoff. We also observe

that the more relays the source selects, the more expected

total payoff each relay node can gain. In addition, we prove

with simulations that the best strategy for each relay node is

to improve its own physical channel condition to enlarge its

secrecy rate and always report the truth to the source.

The remainder of this paper is organized as follows. In

Section II, the system model for a relay network is described.

In Section III, we propose a self-enforcing truth-telling mech-

anism to enforce relay nodes to reveal the true private infor-

mation and prove that it is the unique equilibrium. Simulation

results are shown in Section IV, and the conclusions are drawn

in Section V.

II. SYSTEM MODEL

Consider a cooperative network shown in Fig. 1, consisting

of one source node, one destination node, one eavesdropper

node, and 퐼 relay nodes, which are denoted by 푆, 퐷, 퐸,

and 푅푖, 푖 = 1,2,...,퐼, respectively. When the source node

broadcasts a signal 푥, all the relay nodes in the network

could receive it but only 푁 (푁 ≤ 퐼) nodes can decode this

signal correctly due to their different geographical conditions.

Then these 푁 relay nodes report their own channel gains

of both relay-destination and relay-eavesdropper links to the

The First International Workshop on Security in Computers, Networking and Communications

978-1-4244-9920-5/11/$26.00 ©2011 IEEE1088

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Fig. 1.System model for a relay network with one eavesdropper.

source node. We assume that all the channels in the network

are orthogonal, and they have the same bandwidth, which is

denoted by 푊. The source node wishes to gain the highest

secrecy rate by properly selecting some efficient relay nodes

based on their reported channel information. We denote the

number of selected relay nodes by 퐾 (퐾 ≤ 푁) and the set

of 퐾 relay nodes by 풦.

At the destination node the received signal from the 푖-th

relay node (푅푖∈ 풦) can be expressed as

푦푟푖,푑=

√푃푟푖ℎ푟푖,푑푥 + 푛푟푖,푑,

(1)

and at the eavesdropper node the received signal can be

expressed as

√푃푟푖ℎ푟푖,푒푥 + 푛푟푖,푒,

where 푃푟푖denotes the transmit power of relay node 푅푖under

the power constraint 푃푟푖≤ 푃푚푎푥, ℎ푟푖,푑 is the channel gain

between 푅푖 and 퐷, and ℎ푟푖,푒 is the channel gain between

푅푖 and 퐸. We assume that ℎ푟푖,푑 and ℎ푟푖,푒 contain both

the path loss and the Rayleigh fading factor. Without loss

of generality, we also assume that all the links have the

same noise power which is denoted by 휎2. The decode-and-

forward (DF) protocol is used for relaying.

The signal-to-noise-ratio (SNR) at the destination node is

푦푟푖,푒=

(2)

SNR푟푖,푑=

푃푟푖ℎ2

휎2

푟푖,푑

,

(3)

and the SNR at the eavesdropper node is

SNR푟푖,푒=푃푟푖ℎ2

푟푖,푒

휎2

.

(4)

The channel capacity for relay 푅푖to destination 퐷 is

퐶푖,푑= 푊 log2(1 + SNR푟푖,푑).

(5)

Similarly, the channel capacity for relay 푅푖 to eavesdropper

퐸 is

퐶푖,푒= 푊 log2(1 + SNR푟푖,푒).

(6)

Then, the secrecy rate achieved by 푅푖can be defined as [15]

퐶푖,푠= (퐶푖,푑− 퐶푖,푒)+,

(7)

where (푥)+= max{푥,0}.

Therefore, the total secrecy rate achieved by the 퐾 selected

relay nodes can be written as

∑

III. SELF-ENFORCING TRUTH-TELLING MECHANISM

In this section, we propose a self-enforcing truth-telling

mechanism to guarantee that each relay node reports its true

information to the source during the process of relay selection.

We define 휋 as the price per unit of secrecy rate achieved by

the relay node. The relay nodes in the network are assumed to

be rational and fair-minded, which means that although they

are selfish, none is malicious. The objective of the relay nodes

is to maximize their own payoff under the payoff allocation

scheme set by the source. The source selects 퐾 best relay

nodes according to their reported channel information and the

destination would calculate the payoff allocation of these relay

nodes according to their real secrecy rate.

We assume that the channel gain is private information

of each relay node, and thus, the source is unable to know

whether the reported information is true or not. Since only the

relay nodes selected by the source for secure data transmission

could get the payoff, they may not report their true information

to the source in order to win greater opportunity to be

selected. In this condition, it may cause unfairness in selection

and damage the expected payoff of those unselected. It also

decreases the total secrecy rate of the system as well as the

total payoff paid by the destination. It can be expressed as

퐶푠=

푅푖∈풦

퐶푖,푠.

(8)

ˆ퐶푠≤˜퐶푠,

(9)

and

ˆ퐷 ≤˜퐷,

(10)

whereˆ퐶푠andˆ퐷 represent the total secrecy rate and the total

payoff calculated according to the information reported by the

relay nodes, respectively, while˜퐶푠and˜퐷 represent the total

secrecy rate and the total payoff when all the relay nodes report

the truth, respectively.

In the network, each relay node reports its own channel

information (ℎ푟푖,푑,ℎ푟푖,푒) to the source. Assume

{(˜ℎ푟1,푑,˜ℎ푟1,푒

is a realization of channel gains at one time slot and the relay

nodes report their information

{(ˆℎ푟1,푑,ˆℎ푟1,푒

to the source, which may not be the true information, but the

source will make the selection based on it. Define 푅푖’s private

information as

)

,

(˜ℎ푟2,푑,˜ℎ푟2,푒

)

,...,

(˜ℎ푟푁,푑,˜ℎ푟푁,푒

)}

)

,

(ˆℎ푟2,푑,ˆℎ푟2,푒

)

,...,

(ˆℎ푟푁,푑,ˆℎ푟푁,푒

)}

푔푖= {ℎ푟푖,푑,ℎ푟푖,푒}.

(11)

Due to the channel orthogonality, the payoff of 푅푖can be

expressed as

{

퐷푖=

휋퐶푖,푠,

0,

푅푖∈ 풦,

otherwise.

(12)

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The total payoff from the destination can be expressed as

퐷 =

푁

∑

푖=1

퐷푖.

(13)

Firstly, we prove that no equilibrium can be achieved under

this situation.

Proposition 1: Assuming that 푅푖does not know other re-

lays’ secrecy rate, but knows that it obeys a certain probability

density function defined as 푝(ˆ퐶푗,푠) (0 ≤ˆ퐶푗,푠< ∞,푗 ∕= 푖),

then it has an incentive to exaggerate itsˆ퐶푖,푠to ∞ to get the

maximum expected total payoff.

Proof: 푅푖’s expected payoff (12) can also be expressed

as

퐷푖(ˆ 푔푖) = 휋˜퐶푖.푠푃(푅푖∈ 풦),

(14)

where 푃(푅푖∈ 풦) represents the probability of 푅푖∈ 풦. Since

푃(푅푖∈ 풦) ∝ˆ퐶푖,푠and whenˆ퐶푖,푠→ ∞, 푃(푅푖∈ 풦) → 1, 푅푖

gets its maximum payoff at infinity. This indicates that each

relay node has an incentive to report false channel information,

resulting in larger secrecy rate than reality, to the source,

and thus, there is no equilibrium under this kind of payoff

allocation.

To prevent relay nodes reporting distorted information,

we can use either the reputation method or the threat of

punishment [7]–[9]. However, it needs a delicate and complex

detection scheme to catch the liar node. To this end, we

propose a much easier and effective self-enforcing truth-telling

mechanism to solve this problem. By using this mechanism,

honest relay nodes gain the maximum payoff, as any cheating

in the process would lead to a decrease in expected payoff.

We add another part of payoff, defined as the transfer payoff

푡푖(ˆ 푔1, ˆ 푔2⋅⋅⋅ ˆ 푔푁) = Φ푖(ˆ 푔푖) −

1

푁 − 1

푁

∑

푗=1,푗∕=푖

Φ푗(ˆ 푔푗)

(15)

to make the total payoff of 푅푖as

푈푖(ˆ 푔푖) = 퐷푖(ˆ 푔푖) + 푡푖(ˆ 푔1, ˆ 푔2⋅⋅⋅ ˆ 푔푁),

(16)

where

Φ푖(ˆ 푔푖) =

푁

∑

푗=1,푗∕=푖

퐸[퐷푗(ˆ 푔푖)]

(17)

represents the sum of the other relay nodes’ expected payoff

given the reported information ˆ 푔푖.

We calculate the total transfer payoff and get

푁

∑

푁

∑

푖=1

푡푖(ˆ 푔1, ˆ 푔2⋅⋅⋅ ˆ 푔푁)

=

푖=1

Φ푖(ˆ 푔푖) −

1

푁 − 1

푁

∑

푖=1

푁

∑

푗=1,푗∕=푖

Φ푗(ˆ 푔푗)

=0.

(18)

This implies that the proposed scheme can realize a payoff

reallocation among the relay nodes, and no extra cost requires

to be paid by the system.

If one relay node claims a higherˆℎ푟푖,푑 or a lowerˆℎ푟푖,푒

than the reality to make its secrecy rate larger, it may get

larger chance to be selected by the source, but also will pay

a higher transfer payoff to those unselected. On the contrary,

if one relay node reports a lower secrecy rate than reality it

will receive the compensation from other relay nodes at the

cost of smaller chance to be selected. By adding this transfer

function, we can prove that only when the relay nodes report

the true information of their channels, they can gain the largest

expected payoff. There is only one equilibrium under this kind

of payoff allocation.

Proposition 2: By using the transfer function (15) to

balance the payoff allocation, relay node 푅푖can gain its largest

expected total payoff when it reports its true private channel

information:

ˆ 푔푖= ˜ 푔푖(ˆℎ푟푖,푑=˜ℎ푟푖,푑,ˆℎ푟푖,푒=˜ℎ푟푖,푒).

(19)

Proof: Without loss of generality, we consider the total

expected payoff of 푅1. Since 푅1only knows its own channel

information, its expected total payoff can be expressed as

퐸[푈1(ˆ 푔1)] =퐸[퐷1(ˆ 푔1) + 푡1(ˆ 푔1, ˆ 푔2⋅⋅⋅ ˆ 푔푁)]

=퐸[퐷1(ˆ 푔1)] + 퐸[

푁

∑

푗=1,푗∕=푖

퐷푗(ˆ 푔1)]

−

1

푁 − 1

푁

∑

푁

∑

푗=1,푗∕=푖

Φ푗(ˆ 푔푗)

=퐸[

푖=1

퐷푖(ˆ 푔1)] −

1

푁 − 1

푁

∑

푗=1,푗∕=푖

Φ푗(ˆ 푔푗).

(20)

We can see that there are two terms in the right side of (20).

The first one represents the total expected payoff paid by the

destination when 푅1reports ˆ 푔1as its channel information (The

expectation is calculated by 푅1 itself). Since the other term

being independent of ˆ 푔1, only this one decides the expected

total payoff of 푅1. As we have discussed above, the total

payoff paid by the destination is based on the real secrecy

rate and the maximum payoff can be achieved when the best

퐾 relay nodes are selected by the source. That is to say,

only when all the relay nodes tell the truth, the source can

select the best 퐾 relay nodes and the destination pays the

maximum payoff. Any cheating leads to a decrease in the

relay nodes’ total payoff, which meansˆ퐷 ≤˜퐷. Therefore,

퐸[푈1(ˆ 푔1)] could reach the maximum when 푅1 reports its

true channel information. Then, we can conclude that 푅1will

report ˆ 푔1 = ˜ 푔1 to maximize its own expected total payoff.

Similarly, each relay node in the system has an incentive to

report its true channel gain (ˆ 푔푖= ˜ 푔푖) to the source. Thus, the

unique equilibrium is achieved in this condition.

IV. SIMULATION RESULTS

In this section, we provide simulation results of the pro-

posed self-enforcing truth-telling mechanism. Specifically, to

simplify the calculation and simulation, we assume that each

relay node first calculates its own secrecy rate according to

its channel information and then reports it to the source.

Not considering the process of calculating 휋퐶푖,푠, we give

random values 푥푖 to indicate 휋퐶푖,푠(푖 = 1,2,...,푁), which

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00.511.5 22.533.54

−0.2

0

0.2

0.4

0.6

0.8

reported secrecy capacity

expected total payoff

R1

R2

R3

R4

Fig. 2.Expected total payoff when different secrecy capacities are reported.

would not affect the source’s selection result. Furthermore,

we assume that though 푅푖does not know other relay nodes’

channel information, it knows that each reported value obeys

the probability density function: 푒−푥푖(푥푖 ∈ [0,∞) and

+∞

∫

Firstly, we consider a system with 푁 = 4 relay nodes and

from which the source chooses 퐾 = 2 relay nodes for data

transmission. A random sample of these relay nodes’ secrecy

rate is obtained as 1.0132,0.6091,0.3885,1.3210 and the price

per unit of secrecy rate 휋 = 1 is assumed.

Fig. 2 shows the variation of 푅푖’s expected total payoff

when the reported values change. Given that the other three

nodes are honest, 푅푖(푖 = 1,2,3,4) could get its maximum

total payoff when reporting the truth. From Fig. 2 we can

observe that when they all tell the truth, the larger the true

value of secrecy rate one relay node has, the more the expected

total payoff it gains. For example, 푅4has the largest secrecy

rate (˜퐶4,푠= 1.3210) and its expected total payoff is the largest

and up to 0.7732 when it reports the true value. It is higher

than the other three relay nodes’ payoff even though it is not

as much as 휋퐶4,푠= 1.3210, which is paid by the destination.

Fig. 3 shows the expected transfer payoff of 푅1, 푅2, 푅3,

and 푅4. They are all monotone decreasing because the larger

the reported value is, the more transfer payoff should be paid

to others. From Fig. 3 we also find that 푅1’s and 푅4’s transfer

payoffs are negative while the other two’s are positive when

they tell the truth. This is because 푅1 and 푅4 are actually

selected by the source and get payoff from the destination

while 푅2and 푅3are not. By using the self-enforcing truth-

telling mechanism, the relay nodes which have smaller secrecy

rate will get compensations from those which have larger ones.

It can balance the payoff allocation of the system and benefit

those in worse channel conditions. Furthermore, we calculate

the expected transfer payoff of 푅푖 when they all report the

truth: 푡1= −0.1247, 푡2= 0.1570, 푡3= 0.2831, 푡4= −0.3154

and 푡1+ 푡2+ 푡3+ 푡4 = 0, which is in accord with (18):

0

푒−푥푖푑푥푖= 1).

0 0.511.522.53 3.54

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

reported secrecy capacity

expected transfer payoff

R1

R2

R3

R4

Fig. 3.

reported.

Expected transfer payoff when different secrecy capacities are

00.5 11.5 2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

secrecy capacity of R1

expected total payoff of R1

Fig. 4.Expected total payoff of 푅1.

푁 ∑

0.

푖=1푡푖(ˆ 푔1, ˆ 푔2⋅⋅⋅ ˆ 푔푁) =

푁 ∑

푖=1Φ푖(ˆ 푔푖) −

1

푁−1

푁 ∑

푖=1

푗=푁

∑

푗=1,푗∕=푖

Φ푗(ˆ 푔푗) =

Secondly, we consider that when 푅1enlarges its real secrecy

rate by some methods, while 푅2, 푅3, and 푅4hold the same

value as we have assumed above. From Fig. 4 we can see

that when 푅1’s real secrecy rate increases, the total payoff it

could get will increase at the same time. As the value of 퐶1,푠

changes from 0 to 2, the total payoff changes from 0.2391 to

1.4316. Therefore, 푅1has a strong inclination to increase its

own real secrecy rate even though it has to pay more transfer

payoff to other relay nodes then.

Lastly, we focus on the effect of the value 퐾 on the

expected total payoff. From Fig. 5 we can observe that when

the value 퐾 increases, the expected total payoff of 푅1 also

increases. The larger 퐾 is, the lower the decline rate of

the expected total payoff becomes, which makes the curves

smoother. Considering the transfer payoff of 푅1 shown in

Fig. 6, we find that when 퐾 is smaller (퐾 = 1), the transfer

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00.511.52 2.533.54

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

reported secrecy capacity

expected total payoff

K=1

K=2

K=3

Fig. 5. Expected total payoff of 푅1at different 퐾.

00.5 11.5 2 2.533.54

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

reported secrecy capacity

expected transfer payoff

K=1

K=2

K=3

Fig. 6.Expected transfer payoff of 푅1at different 퐾.

payoff decreases faster and gets a larger negative value quickly.

This explains why 푅1’s expected total payoff decreases faster

when 퐾 decreases. So if the source increases the number of

relays which are selected to forward signals, all the relay nodes

in the system could gain larger utilities.

V. CONCLUSIONS

In this paper, we have proposed a self-enforcing truth-

telling mechanism to guarantee that each relay node tells

the truth during the process of relay selection taking secure

data transmission into consideration. By adding a transfer

payoff we found that each relay node gets its maximum

utility only when it reports its true channel information, and

any deviation from the truth will lead to a loss in its own

expected total payoff as well as the total secrecy rate of the

network. Simulation results verify that the relay nodes have no

incentive to report false information after adding the transfer

payoff. We also observed that increasing the number of relay

nodes selected by the source and increasing one relay node’s

real secrecy rate are two ways to increase the expected total

payoff, which also indicates that trying to improve one’s own

condition but not cheating the source is able to get more

payoff.

ACKNOWLEDGMENT

This work was partially supported by the US NSF CNS-

0953377, CNS-0905556, and CNS-0910461, and also by

the National Nature Science Foundation of China under

grant number 60972009 and 61061130561, as well as the

National Science and Technology Major Project of China

under grant number 2009ZX03003-011, 2010ZX03003-003,

2010ZX03005-003, and 2011ZX03005-002.

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