Transcript Slides PPT

Impact of Secrecy on Capacity in Large-Scale
Wireless Networks
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
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Motivations
 Secrecy is a Major Concern in Wireless Networks.
 Mobile Phone Wallet
 Military networks
 …
Impact of Secrecy on 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.
Impact of Secrecy on 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.
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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
 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
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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
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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.
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Network Model and Definition – II/II
 Network Model:
 Extended networks
 Static
 Physical channel model
SIN R ij 
SIN R ie 
Pt l ( x i , x j )
N0 

k T \{ i }
Pt l ( x k , x j ) 

k  R \{ i }
Pr l ( x k , x j )
Pt l ( x i , x e )
N 0   k T \{ i } Pt l ( x k , x e )   k  R Pr l ( x k , x e ) where

l ( x i , x j )  m in (1, d ij )
 Definition of secrecy capacity
Rij  Rij  Rie  log 2 (1  SIN R ij )  log 2 (1  SIN R ie )
s
where S IN R ie  m ax e S IN R ie
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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
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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

R e  m in(
Pt d te
N0
,
Pt
Pr

(1  d tr ) )
Received Power
where d tr is the Euclidean distance between legitimate node
t and node r and d te is the distance between legitimate node
t and eavesdropper e.
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Independent Eavesdroppers
Case 1: when d te and d re both greater 1,
SIN R e 


Pt l ( x t , x e )
N0 

k T \{ i }
Pt l ( x t , x e )
＝
Pｒl ( xｒ , x e )

Pｄ
t te
Pｒ( ｄrt  d te )
Pt l ( x k , x e ) 

t te

ｒ re

kR
Pr l ( x k , x e )
Pｄ
Pｄ


Pt
Pr
(1  d tr )

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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 c 2 Pt d , where c 2
is a constant.
R ( d )  log(1 
 log(1 
S (d )
)
N 0  I (d )

Pt ( c ( d  1))
N 0  c1 ( Pt  Pr )( kc )
'

 c 2 Pt ( c ( d  1))

 c 2 Pt d
1
when choosing k   ( P
r

)

)
and c 2 is a constant.
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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 R s ( d )   ( d    4 ) ,
so that the receiver can receive at a rate of R s ( d ) securely from
the transmitter.
Rs (d ) 

Choose
Pr  2
c3
1
(k  d )
1
2
(k  d )
2
d
( R ( d )  Re )
( c 2 Pt d
2
c2
Rs (d )   (d

 c3
Pt

d )
Pr
1
k   ( Pr )   ( d )
2
  4
)
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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
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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.
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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
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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
1
per-node secrecy capacity
.
 s( n ) is
(
A( n )
)

2

n
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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.

R e  m in(
SI NRe  SI NRei


Pt d te
Pt
,
N0
Pr

(1  d tr ) )

SI NR1j 
j e1
 
SI NRi j
i  2 j   ei

 2f ( n ) e( n )SI NRe 1 
 2f ( n )
e
( n )SI NRei
i 2
Pt
 2f ( n ) e( n )
Pr
 2 e( n ) ( r 1
2
Pt
Pr



(1  drt ) 
 2f ( n )
( n)
Pt r i  1
e
N0
i 2

(1  drt ) 
Pt
N0
2
r1

i


2
)
i 1
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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
2
 2


1
 e( n )   2 ) ,  e( n )  ( l og  n )
 (
n

 s( n )  
2
 2

1

(
l og  n ) ,  e( n )  O( l og  n )

n

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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
N ei
constant size c and denoting by
the number of nodes inside
region i, we have
P( N ei  v ,  i )  1
where
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
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Colluding Eavesdroppers
 Upper Bound
Nei 
SI NRe 
1
2
 e( n ) k
Sj

N0  I
J
(
 N ei
3k
4
)
j

N0  I
 c 11 e( n ) k
2
j
2
1
k   e( n )   2
Impact of Secrecy on Capacity in Large-Scale Wireless Networks
 s( n )  (
1
k
2
)
n
22
Colluding Eavesdroppers
 Upper Bound
Theorem 7: Consider the wireless network B where
legitimate nodes and eavesdroppers are independent
 e( n )
poisson distributed with parameter 1 and
respectively,
the per-node secrecy capacity is
2


1
 e( n )   2 ) ,  e( n )  ( 1)
O(

n
 s( n )  
1

O(
) ,  e( n )  O( 1)

n

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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
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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.
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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.
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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
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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
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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
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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.
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Future Work
 Secrecy capacity under active attacks
 The impact of dense networks
 The impact of heterogeneity networks
 The impact of social networks
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Thank you !
Backup
 Details on the Models of Legitimate nodes
 Revolve on its own
 Using 4 antennas
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