1. Jinbei Zhang, Luoyifu, Xinbing Wang
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
Good afternoon, everyone. Thank you for coming. My name is Jinbei,Zhang. I am from Shanghaijiaotong University. Today my topic is about the impact of secrecy on capacity in large-scale wireless networks. This is a joint work with Luoyi Fu and our supervisor Prof. Xinbing Wang.

2. Outline Introduction Network Model and Definition
Motivations
Related works
Objectives
Network Model and Definition
Secrecy Capacity for Independent Eavesdroppers
Secrecy Capacity for Colluding Eavesdroppers
Discussion
Conclusion and Future Work
The outline is listed as follows.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

3. Motivations Secrecy is a Major Concern in Wireless Networks.
Mobile Phone Wallet
Military networks …
Since there are more and more wireless netwroks deployed, secrecy is a major concern due to wireless channels' broadcast nature especially in military appliactions.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

4. Related works – I/II Properties of Secrecy Graph
And as is shown in previous work, even a small number of eavesdroppers will cause dramatic decreasing in nodes' connectvity. So it is challenging to ensure secrect transmissions in wireless networks especially in large-scale wireless networks.
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
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

5. 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]
when the network is mobile, legitimate receiver can get closer to the transmitter than eavesdroppers and hence secrecy can be guarenteed. when the network is static, the secrecy can be guarrenteed if the transmitter can sense the eavesdroppers in a region which is called guard zone, and reference 13 study how long the sense range should be. And reference 12 uses artificial noise and fading gain to ensure secrect transmission. Artificial noise is used to degrade channels of eavesdroppers near transmitter and fading gain can be used to differ the channels of legitimate receiver and eavesdroopers near the receiver. However, there stil exists some problems.
[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 [13] O. Koyluoglu, E. Koksal, E. Gammel, “On Secrecy Capacity Scaling in Wireless Networks”, submitted to IEEE Trans. Inform. Theory, Apr
[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
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

6. How to effectively improve the secrecy capacity?
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?
First, channel state information is hard to obtain in reality and artificial noise not only degrade eavesdroppers' channel but also legitimate reveivers'. And the cost of fading gain is quite large since there should be many nodes to choose to ensure a good fading gain. Furthermore, until now the upper bound of secrecy capacity is still unknown due to the complex secrecy graph. And the impact of other models will also be inverstigated in this work.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

7. Outline Introduction Network Model and Definition
Secrecy Capacity for Independent Eavesdroppers
Secrecy Capacity for Colluding Eavesdroppers
Discussion
Conclusion and Future Work
Now i will present the models and defition of this paper.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

8. Network Model and Definition – I/II
Legitimate Nodes:
Self-interference cancelation[16] adopted
3 antennas per-node
CSI information unknown
Eavesdroppers:
Location positions unknown
As the shortcomings of artificial noise and fading gain i mentioned, how to effectily improve the secrecy capacity. Fading gain maybe not a good choice due to the large cost. So is it possible that using artificial noise to degrade eavesdroppers' channels while has little impact on the legitimate receivers? Yes, it is if you use self-interference cancellation which is introduced in reference 17. We assume that each legitimate nodes is equipped with 3 antennas. When it acts as a receiver, two antennas are used to generate artificial noise. The distances between these two antennas and the receive antenna differ half the wavelength and hence the noise will be cancelled at the receive antenna while the noise will have a negative impact on the eavesdroppers.
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
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

9. Network Model and Definition – II/II
Extended networks:
n nodes randomly distributed in a network with size n.
Static
Physical channel model
where
Definition of secrecy capacity
The network is extened, static and we use physical channel model here. The SINR of legitimate receiver and eavesdroppers are expressed as follows where T denotes the set of concurrent transmiiter and R denote the legitimate receiver. Note that the noise of legitimate receiver is ignored here while it can not be eliminated by the eavesdropper. Definition of secrecy capacity is given as follows. To compute its lower bound, we should first study the upper bound of the rate of eaavesdroppers and then the lower bound of legitimate nodes.
where
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

10. 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

11. 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
Received Power
where is the Euclidean distance between legitimate node
t and node r and is the distance between legitimate node
t and eavesdropper e.
And the rate of eavesroppers can be upper bounded as follows where dte Note tha t the first equation is bounded by the receive power of eavesdroppers and the latter part can be proved as follows. when dte and dtr are both greater than 1, the result holds and other cases can be proved similarly.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

12. Independent Eavesdroppers
Case 1: when and both greater 1,
Case 2-4:
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

13. 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 , where is a constant.
. Now we will compute the rate of legitimate receivers. the rate of legitimate receiver is on the order of d power minus alpha when the concurrent transmission range satisfy this constraint.
Combined with the rate of eavesdroppers, we can compute the secrecy rate.
when choosing and is a constant.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

14. 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 , so that the receiver can receive at a rate of securely from the transmitter.
Choose
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

15. Independent Eavesdroppers
Highway System
Draining Phase
Highway Phase
Delivery Phase
We use highway system to transmit the message. And the bottleneck is still on the highway phase where d equals a constant. And the node should relay the message of root n. So the secrecy capacity is omega one over root n. Since the capacity without secrecy concern is also omega one over root n, this secrecy capacity is obiviously optimal.
Theorem 2: With n legitimate nodes poisson distributed, the achievable per-node secrecy throughput under the existence of independent eavesdroppers is
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

16. 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
[18] P. Gupta and P. Kumar, “The Capacity of Wireless Networks”, in IEEE Trans. Inform. Theory, Vol. 46, No. 2, pp , Mar
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

17. 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

18. 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 per-node secrecy capacity is
Now we will consider the colluding case. First we consider each eavesdrooper is equipped with multiple antennas, and the analysis is similar to previous one, we omit it here.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

19. Colluding Eavesdroppers
Eavesdroppers Collude
Assume that each eavesdropper equipped with one antenna and different eavesdroppers can collude to decode the message.
Now we will consider the colluding case. First we consider each eavesdrooper is equipped with multiple antennas, and the analysis is similar to previous one, we omit it here. Now we assume differnent eavesdroppers can collude to decode a message and we upper bound the eavesdroppers' SINR first. Divide the network into equal regions with size f(n). Denote the density of eavesdroppers as phi e n. Combined with lemma 1, we can get the following equation. After carefully choose the value of r_1, P_r and concurrent transmission range k, we can obtain the lower bound of secrecy capacity as follows.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

20. Colluding Eavesdroppers
Lower Bound
Theorem 5: 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
And note that if the density of eavesdroppers is very small e.g, smaller than 1/n, according to our results in independent case, the secrecy capacity should be theta 1/root n. So this result is not tight and this is due to a constaint in previous analysis. We re-examine this problem from another perspertive.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

21. Colluding Eavesdroppers
Lower Bound
Lemma 5: When the intensity of the eavesdroppers is for any constant β>0, partitioning the network into disjoint regions with constant size c and denoting by the number of nodes inside region i, we have
where
Theorem 6: If eavesdroppers are poisson-distributed in the network with intensity for any constant β>0, the per-node secrecy capacity is
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

22. Colluding Eavesdroppers
Upper Bound
Next we will study the upper bound of secrecy capacity and we should lower bound the SINR eavesdroppers can obtain first. There are many concurrent transmission region in the network and the figure shouws one. N_ei denotes the number of eavesdroppers inside the region which at least 1/4 k and at most 3/4 k distance from the transmiter. And the mean number of N_ei is 1/ So there exists at least one concurrent ttransmission region whose N_ei is greater than this. To snsure a positive rate, k should be grateer than .. and hence the upper bound of secrecy rate is derived.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

23. Colluding Eavesdroppers
Upper Bound
If the noise is outside the dotted line, the SINR of eavesdroppers will be little affected. So the noise should be inside the dotted area between 1/4 k and ¾ k. Further partition the dotted area into 2 parts, then noise inside red line have little impact of the eavesdroppers outside the red line and the versus also hold. Hence, the artificial noise should be generated almost inside the whole dotted line, and its impact on the legitimate receiver is expressed as this. So our upper bound holds.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

24. 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
The result of upper bound and lower bound are both given in the figure and it can be seen that they fit well.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

25. Outline Introduction Network Model and Definition
Secrecy Capacity for Independent Eavesdroppers
Secrecy Capacity for Colluding Eavesdroppers
Discussion
Conclusion and Future Work
Next i will present some extensions of this work.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

26. 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
The netwrok we study is possion network and it of coz can be applied to random networks since when nodes'number converges to infinity, these two netwrks are almost the same. And we give an altenative proof which is differnent from that in Ref 27.
We also study the secrecy capacity using our scheme in other secnarios such as ...
[27] M. Penrose, “Random Geometric Graphs”, Oxford Univ. Press, Oxford, U.K., 2003.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

27. Discussions Multicast Secrecy Capacity
Corollary 1. Assume that legitimate nodes and eavesdroppers
are independent poisson distributed with parameter 1 and respectively. For each legitimate node, k − 1 nodes are randomly chosen as its destinations. For independent eavesdroppers case, the aggregated multicast secrecy is when
and is when
[24] X. Li, “Multicast Capacity of Wireless Ad Hoc Networks”, in IEEE/ACM Trans. Networking, Vol. 17, No. 3, pp , 2009.
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

28. 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 and the corresponding delay is For colluding case, the per-node secrecy capacity is
and the corresponding delay is
[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 ,
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

29. 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 For colluding
case, the per-node secrecy capacity is and the
corresponding delay is
[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
Asymptotic Analysis on Secrecy Capacity in Large-Scale Wireless Networks

30. 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

31. Conclusions
We derive the upper bound for secrecy capacity in large-scale 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

32. 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

33. Thank you !

34. Impact of Secrecy on Capacity in Large-Scale Wireless Networks
Backup
Revolve on its own
Using 4 antennas
Impact of Secrecy on Capacity in Large-Scale Wireless Networks