Asymptotic Analysis on Secrecy Capacity in
Large-Scale Wireless Networks
Jinbei Zhang, Luoyifu, Xinbing Wang
Department of Electronic Engineering
Shanghai Jiao Tong University
Aug. 13, 2013
Outline
 Introduction
 Motivations
 Related works
 Objectives
 Network Model and Definition
 Secrecy Capacity for Independent Eavesdroppers
 Secrecy Capacity for Colluding Eavesdroppers
 Discussion
 Conclusion and Future Work
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Motivations
 Secrecy is a Major Concern in Wireless Networks.
 Mobile Phone Wallet
 Military networks
 …
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Related works – I/II
 Properties of Secrecy Graph
Cited From [5]
Cited From [5]
[4] M. Haenggi, “The Secrecy Graph and Some of Its Properties”, in Proc. IEEE ISIT,
Toronto, Canada, July 2008.
[5] P. C. Pinto, J. Barros, M. Z. Win, “Wireless Secrecy in Large-Scale Networks.” in Proc.
IEEE ITA’11, California, USA, Feb. 2011.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Related works – II/II
 Secrecy Capacity in large-scale networks,
 Mobile Networks [16]
 Guard Zone [13]
 Artificial Noise+Fading Gain(CSI needed) [12]
Cited from [12]
[16] Y. Liang, H. V. Poor and L. Ying, “Secrecy Throughput of MANETs under Passive and
Active Attacks”, in IEEE Trans. Inform. Theory, Vol. 57, No. 10, Oct. 2011.
[13] O. Koyluoglu, E. Koksal, E. Gammel, “On Secrecy Capacity Scaling in Wireless
Networks”, submitted to IEEE Trans. Inform. Theory, Apr. 2010.
[12] S. Vasudevan, D. Goeckel and D. Towsley, “Security-capacity Trade-off in Large
Wireless Networks using Keyless Secrecy”, in Proc. ACM MobiHoc, Chicago, Illinois, USA,
Sept. 2010.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Objectives
 Several questions arise:
 CSI information is difficult to obtain
 Artificial noises also degrade legitimate receivers’ channels
 Cost on capacity is quite large to utilize fading gain
How to effectively improve the
secrecy capacity?
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Outline
 Introduction
 Network Model and Definition
 Secrecy Capacity for Independent Eavesdroppers
 Secrecy Capacity for Colluding Eavesdroppers
 Discussion
 Conclusion and Future Work
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Network Model and Definition – I/II
 Legitimate Nodes:
 Self-interference cancelation[16] adopted
 3 antennas per-node
 CSI information unknown
 Eavesdroppers:
 Location positions unknown
 CSI information unknown
Cited from [17]
[17] J. I. Choiy, M. Jainy, K. Srinivasany, P. Levis and S. Katti, “Achieving Single Channel,
Full Duplex Wireless Communication”, in ACM Mobicom’10, Chicago, USA, Sept. 2010.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Network Model and Definition – II/II
 Network Model:
 Extended networks:
n nodes randomly distributed in a network with size n.
 Static
 Physical channel model
SINRij 
SINRie 
Plt ( xi , x j )
N 0   kT \{i} Plt ( xk , x j )   kR \{i} Pr l ( xk , x j )
Plt ( xi , xe )
l ( xi , x j )  min(1, dij )
where
N0  kT \{i} Plt ( xk , xe )  kR Pl
(
x
,
x
)
r
k
e
 Definition of secrecy capacity
Rijs  Rij  Rie  log2 (1  SINRij )  log 2 (1  SINRie )
where SINRie  maxe SINRie
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Outline
 Introduction
 Network Model and Definition
 Secrecy Capacity for Independent Eavesdroppers
 Lower Bound
 Upper Bound
 Secrecy Capacity for Colluding Eavesdroppers
 Discussion
 Conclusion and Future Work
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Independent Eavesdroppers
 Capacity for Eavesdroppers
Lemma 1: When a legitimate node t is transmitting to a legitimate
receiver r, the maximum rate that an independent eavesdropper e
can obtain is upper-bounded by

Pd
P
Re  min( t te , t (1  dtr ) )
N0 Pr
Received Power
where dtr is the Euclidean distance between legitimate node
t and node r and dte is the distance between legitimate node
t and eavesdropper e.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Independent Eavesdroppers
Case 1: when dte and d re both greater 1,
Plt ( xt , xe )
SINRe 
N 0   kT \{i} Plt ( xk , xe )   kR Pr l ( xk , xe )

Plt ( xt , xe ) Pｄ

＝ t te
Pl
Pｒｄre
ｒ ( xｒ, xe )

Pｄ
Pt

t te


(1

d
)
tr

Pｒ( ｄrt  dte )
Pr
Case 2-4:
dte  1, dre  1; dte  1, dre  1; dte  1, dre  1.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Independent Eavesdroppers
 Capacity for Legitimate Nodes
Lemma 2: When a legitimate node t is transmitting to a legitimate
receiver which is located d cells apart, the minimum rate that the

legitimate node can receive is lower-bounded by c2 Pd
, where c 2
t
is a constant.
S (d )
)
N 0  I (d )
Pt (c(d  1)) 
 log(1 
)

N 0  c1 ( Pt  Pr )(kc)
 c2' Pt (c(d  1)) 

 c2 Pd
t
R(d )  log(1 
1
when choosing k  ( Pr ) and c 2 is a constant.

Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Independent Eavesdroppers
 Secrecy Capacity for Each Cell
Theorem 1: For any legitimate transmitter-receiver pair which is
spaced at a distance of d cells apart, there exists an Rs (d )  (d  4 ),
so that the receiver can receive at a rate of Rs (d ) securely from
the transmitter.
1
( R (d )  Re )
2
(k  d )
Pt 
1


(c2 Pd
 c3 d )
t
2
(k  d )
Pr
Rs (d ) 
Choose P  2 c3 d 2
r
c2
1
k  ( Pr )  (d 2 )
Rs (d )  (d  4 )
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Independent Eavesdroppers
 Highway System
 Draining Phase
 Highway Phase
 Delivery Phase
Rs (d )  (d  4 )
Theorem 2: With n legitimate nodes poisson distributed, the
achievable per-node secrecy throughput under the existence of
1
independent eavesdroppers is ( ) .
n
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Independent Eavesdroppers
 Optimality of Our Scheme
Theorem 2: When n nodes is identically and randomly located in a
wireless network and source-destination pairs are randomly
chosen, the per-node throughput λ(n) is upper bounded by ( 1 ) .
n
[18] P. Gupta and P. Kumar, “The Capacity of Wireless Networks”, in IEEE Trans. Inform.
Theory, Vol. 46, No. 2, pp. 388-404, Mar. 2000.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Outline
 Introduction
 Network Model and Definition
 Secrecy Capacity for Independent Eavesdroppers
 Secrecy Capacity for Colluding Eavesdroppers
 Lower Bound
 Upper Bound
 Discussion
 Conclusion and Future Work
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Colluding Eavesdroppers
 Eavesdroppers Collude
 Assume that the eavesdropper can employ maximum ratio
combining to maximize the SINR which means that the correlation
across the antennas is ignored.
Theorem 4: If eavesdroppers are equipped with
A(n) antennas, the
2

s( n ) is( 1 A( n)  ) .
per-node secrecy capacity
n
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Colluding Eavesdroppers
 Eavesdroppers Collude
 Assume that each eavesdropper equipped with one antenna and
different eavesdroppers can collude to decode the message.

Pd
P
t te
Re  min(
, t (1  dtr ) )
N0 Pr
SI NRe  SI NRei


j e 1
SI NR1j 


i  2 j ei
 2f ( n ) e( n )SI NRe1 
SI NRi j

 2f ( n) ( n)SI NR
e
i 2
P
 2f ( n ) e( n ) t (1  d r t ) 
Pr
ei

Pr
2f ( n ) e( n ) t i 1

N0
i 2

Pt
Pt 2 

 2 e( n ) ( r
(1  d r t ) 
r1  i
Pr
N0
i 1
2
1


2
)
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Colluding Eavesdroppers
 Lower Bound
Theorem 5: Consider the wireless network B where legitimate nodes and
eavesdroppers are independent poisson distributed with parameter 1
 eand
( n)
respectively, the per-node secrecy capacity is

(

s( n)  
(


1
n
1
n
 e( n)

l og
2


2
 2
 2

) ,  e( n)  ( l og
 2

n) ,  e( n)  O( l og
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
n)
n)
20
Colluding Eavesdroppers
 Lower Bound
Lemma 5: When the intensity of the eavesdroppers is e( n )=O( n   ) for
any constant β>0, partitioning the network into disjoint regions with
Nei the number of nodes inside
constant size c and denoting by
region i, we have
where
P( Nei  v , i )  1
1
v    +1
 
Theorem 6: If eavesdroppers are poisson-distributed in the network with
intensity e( n)  O( n   ) for any constant β>0, the per-node secrecy
capacity is ( 1 ) .
n
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Colluding Eavesdroppers
 Upper Bound
Nei
1
  e( n )k 2
2
SI NRe 
N
J
0
Sj
Ij
3k 
)
4
 Nei
N0  I j
(
 c11 e( n )k
2 
k   e( n )
1
 2
s( n)  (
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
1
k
2
)
n
22
Colluding Eavesdroppers
 Upper Bound
k 2 e( n)k   (1)
k   e( n )
s( n)  (
1
 2
1
k
2
)
n
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Colluding Eavesdroppers
 Upper Bound
Theorem 7: Consider the wireless network B where
legitimate nodes and eavesdroppers are independent
 e( n ) respectively,
poisson distributed with parameter 1 and
the per-node secrecy capacity is

O(

s( n )  
O(

1
n
1
n
 e( n )

2
 2
) ,  e( n )  (1)
) ,  e( n )  O(1)
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Outline
 Introduction
 Network Model and Definition
 Secrecy Capacity for Independent Eavesdroppers
 Secrecy Capacity for Colluding Eavesdroppers
 Discussion
 Conclusion and Future Work
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Discussions
 Secrecy Capacity in Random Networks
 Random networks: total node number is given
 Poisson networks: node numbers in different regions are
independent
 When n goes to infinity, they are the same in the sense of
probability
 Our results still hold in random networks
[27] M. Penrose, “Random Geometric Graphs”, Oxford Univ. Press, Oxford, U.K., 2003.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Discussions
 Multicast Secrecy Capacity
Corollary 1. Assume that legitimate nodes and eavesdroppers
are independent poisson distributed with parameter 1 and  e( n )
respectively. For each legitimate node, k − 1 nodes are randomly
chosen as its destinations. For independent eavesdroppers case,
n
l og  4 n)
k l og n
n
when
k

(
) .
n)
l og n
the aggregated multicast secrecy is (
k  (
n
)
l og n
and is ( l og  4
when
[24] X. Li, “Multicast Capacity of Wireless Ad Hoc Networks”, in IEEE/ACM Trans.
Networking, Vol. 17, No. 3, pp. 950-961, 2009.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Discussions
 Secrecy Capacity in i.i.d Mobility Networks
Corollary 2. Consider a cell-partitioned network under the
two-hop relay algorithm proposed in [19], and assume that
nodes change cells i.i.d. and uniformly over each cell every
timeslot. For independent eavesdroppers case, the per-node
secrecy capacity is
(1) and the corresponding delay is ( n)
For colluding case, the per-node secrecy capacity is ( f ( e( n ) ) )
and the corresponding delay is (
n
).
f ( e( n) )
[19] M. J. Neely and E. Modiano, “Capacity and Delay Tradeoffs for Ad Hoc Mobile
Networks”, in IEEE Trans. Inform. Theory, Vol. 51, No. 6, pp. 1917-1937, 2005..
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Discussions
 Secrecy Capacity under Random Walk Networks
Corollary 3. Under random walk mobility model, nodes can
only move to adjacent cells every timeslot. For independent
eavesdroppers case, the per-node secrecy capacity is
and the corresponding delay is
(1)
( n l og n) . For colluding
case, the per-node secrecy capacity is ( f ( e( n ) ) ) and the
corresponding delay is ( n l og n ) .
f ( e( n) )
[30] A. Gamal, J. Mammen, B. Prabhakar, and D. Shah, “Throughput-delay trade-off in
wireless networks”, In Proceeding of IEEE INFOCOM, Hong Kong, China, Mar. 2004.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Outline
 Introduction
 Network Model and Definition
 Secrecy Capacity for Independent Eavesdroppers
 Secrecy Capacity for Colluding Eavesdroppers
 Discussion
 Conclusion and Future Work
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Conclusions
 We derive the upper bound for secrecy capacity in largescale wireless networks by capturing the underling SINR
relationship of eavesdroppers and legitimate nodes.
 The proposed scheme is order optimal for both the
independent eavesdroppers and the colluding case.
 Our model relies weakly on the specific settings such as
traffic pattern and mobility models of legitimate nodes
and can be flexibly applied to more general cases and
shed insights into the design and analysis of future
wireless networks.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Future Work
 Secrecy capacity under active attacks
 The impact of heterogeneity networks
 The impact of dense networks and CR networks
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks
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Thank you !
Backup
 Revolve on its own
 Using 4 antennas
Impact of Secrecy on Capacity in Large-Scale Wireless Networks
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