Jinbei Zhang, Luoyifu, Xinbing Wang
Department of Electronic Engineering
Shanghai Jiao Tong University
Mar 15, 2012

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
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 2

Motivations
 Secrecy is a Major Concern in Wireless Networks.
 Mobile Phone Wallet
 Military networks
 …
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 3

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.
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 4

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.
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 5

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
 What’s the upper bound of secrecy capacity?
 What’s the impact of other network models?
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 6

Outline
 Introduction
 Network Model and Definition
 Secrecy Capacity for Independent Eavesdroppers
 Secrecy Capacity for Colluding Eavesdroppers
 Discussion
 Conclusion and Future Work
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 7

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.
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 8

Network Model and Definition – II/II
 Network Model:
 Extended networks
 Static
 Physical channel model
SINR ij

N
0
 

\{ }
Pl x x t
( , i j
)
Pl x x t
( , ) k j
 
SINR ie

N
0
 

\{ }
( , e
)
Pl x x t
( , k e
)
 
P l x x r
( , k e
)
P l x x r
( , k j
) where l x x i j
)
 min(1, d
  ij
)
 Definition of secrecy capacity
R s ij

R ij

R ie
 log (1
2

SINR ij where SINR ie
 max e
 
SINR ie

2

SINR ie
)
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 9

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
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 10

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
R e
 min(
Pd
  t te ,
P
N P
0 r t (1
 d tr

) )
Received Power d is the Euclidean distance between legitimate node t and node r d t and eavesdropper e .
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 11

Independent Eavesdroppers
SINR e



N
0
 

\{ } ｒ
( ,
P ｄ t te

) P
P t
( , t e
)
＝
P ｒ
( ｄ rt
 d te
)
 
Pl x x t
( , t e
)
Pl x x t
( , k e
) ｄ   t te
  ｒ ｄ re

P
P t r
(1

 d
 tr
)

P l x x r
( , k e
)
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 12

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 c Pd
  c is a constant.
( )



 log(1

N
0

I d
)
( (

1)) t
  log(1

' c P c d
2 t
( (

1))
  c Pd
2 t
N
0
 c P
 t

P r
)( kc )
 
)
 
( P r
1
) c
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 13

Independent Eavesdroppers
 Secrecy Capacity for Each Cell
Theorem 1: For any legitimate transmitter-receiver pair which is
( )
 
( d

4 so that the receiver can receive at a rate of securely from s the transmitter.
s


1
( k
 d )
2
1
( k
 d ) 2
( ( )

R e
)
( c Pd
2 t
   c
3
P
P r t d

)
Choose
P r

2 c
3 c
2 d
2
 s
( )
 
( d k P r

1
 
( )
 
( d
2
)
4 )
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 14

Independent Eavesdroppers
 Highway System
 Draining Phase
 Highway Phase
 Delivery Phase s
( )
 
( 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
)
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 15

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 pernode throughput λ(n) is upper bounded by .
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.
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 16

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
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 17

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
 s n
( )

2 per-node secrecy capacity is .
n
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 18

Colluding Eavesdroppers
 Eavesdroppers Collude
 Assume that each eavesdropper equipped with one antenna and different eavesdroppers can collude to decode the message.




SI NR e

SI NR ei
R e
 min(
Pd
  t te ,
P
N P
0 r t (1
 d tr

) ) j

 e 1
SI NR
1 j


  i

2 j
 ei
SI NR i j
 e e 1
 i



2
 e ei f n
 e n
P
P r t (1

2
 e n r
1
2
P t
P r
(1
 d r t
)
 d r t
)


 i



2
P t
N
0 f n
 e n r
1
2
  i



1 i
Pr t i
 

1


2 )
N
0
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 19

Colluding Eavesdroppers
 Lower Bound
Theorem 5: Consider the wireless network B where legitimate nodes and eavesdroppers are independent poisson distributed with parameter 1 and
 e n respectively, the per-node secrecy capacity is
 s n


(
1
 e n


2

2 ) ,

 e n
 
( l og
 

 n
(
1 n l og

2
 n ) , e n

O ( l og
 


2

 n
2
) n )
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 20

Colluding Eavesdroppers
 Lower Bound
 e
( )= (
 
) any constant β>0, partitioning the network into disjoint regions with
N constant size c and denoting by the number of nodes inside region i, e i we have
( ei
 v ,
 i )

1 where v

+1
Theorem 6: If eavesdroppers are poisson-distributed in the network with intensity for any constant β>0, the per-node secrecy e capacity is . n
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 21

Colluding Eavesdroppers
 Upper Bound
N ei

1
2
 e n k 2


SI NR e
 
J
S j
N
0

I j
N ei
3 k
( )
4
 
N
0

I j c
11
 e n k
2
  k
  e n

1

2
Impact of Secrecy on Capacity in Large-Scale Wireless Networks
 s n
 
(
1
2 k n
)
22

Colluding Eavesdroppers
 Upper Bound
Theorem 7: Consider the wireless network B where legitimate nodes and eavesdroppers are independent poisson distributed with parameter 1 and respectively, the per-node secrecy capacity is
 s n

 O (
1
 e
 n


2

2 ) ,
 e
 

 O n
(
1
) , ( )

(1) e n O n n
 
(1)
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 23

Outline
 Introduction
 Network Model and Definition
 Secrecy Capacity for Independent Eavesdroppers
 Secrecy Capacity for Colluding Eavesdroppers
 Discussion
 Conclusion and Future Work
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 24

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.
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 25

Discussions
 Multicast Secrecy Capacity
[24] X. Li, “Multicast Capacity of Wireless Ad Hoc Networks”, in IEEE/ACM Trans.
Networking, Vol. 17, No. 3, pp. 950-961, 2009.
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 26

Discussions
 Secrecy Capacity in i.i.d Mobility Networks
[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..
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 27

Discussions
 Secrecy Capacity under Random Walk Networks
[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.
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 28

Outline
 Introduction
 Network Model and Definition
 Secrecy Capacity for Independent Eavesdroppers
 Secrecy Capacity for Colluding Eavesdroppers
 Discussion
 Conclusion and Future Work
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 29

Conclusions
 In the non-colluding case, the optimal per-node secrecy capacity is achievable in the presence of eavesdroppers.
 In the colluding case, we establish the relationship between the secrecy capacity and the tolerable number of eavesdroppers. More importantly, we first derive the upper bound for secrecy capacity which is achievable.
 We identify the underlying interference model to capture the fundamental impact of secrecy constraints. This model relies weakly on the specific settings such as traffic pattern and mobility models of legitimate nodes. Hence, our study can be flexibly applied to more general cases and shed insights into the design and analysis of future wireless networks.
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 30

Future Work
 Secrecy capacity under active attacks
 The impact of dense networks
 The impact of heterogeneity networks
 The impact of social networks
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 31

Backup
 Details on the Models of Legitimate nodes
 Revolve on its own
 Using 4 antennas
Impact of Secrecy on Capacity in Large-Scale Wireless Networks 33