Optimizing Mixing in Pervasive Networks: A Graph-Theoretic Perspective

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Nov 24, 2013 (3 years and 8 months ago)

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Optimizing Mixing in Pervasive Networks: A

Graph
-
Theoretic Perspective

Murtuza

Jadliwala
, Igor
Bilogrevic

and Jean
-
Pierre
Hubaux

ESORICS, 2011

Wireless Trends

Smart Phones

Vehicles

Watches

Cameras

Passports


Always
on


Background
apps

2

Peer
-
to
-
Peer Wireless Networks

1

Message

Identifier

2

3

Examples



Urban Sensing networks



Delay tolerant networks



Peer
-
to
-
peer file exchange

VANETs


Social networks

Nokia Instant Community

4

Location Privacy Problem

a

b

c

Monitor identifiers used in peer
-
to
-
peer communications

5

Location Privacy Attacks


Pseudonymous location traces


Home/work location pairs are unique [1]


Re
-
identification of traces through data analysis [2,4,3,5]



Attack:

Spatio
-
Temporal

correlation of traces


Message

Identifier

[1] P.
Golle

and K. Partridge.
On the Anonymity of Home/Work Location Pairs
. Pervasive Computing, 2009

[2] A. Beresford and F.
Stajano
.
Location Privacy in Pervasive Computing
. IEEE Pervasive Computing, 2003

[3] B. Hoh et al.
Enhancing Security & Privacy in Traffic Monitoring Systems
. Pervasive Computing, 2006

[4] B. Hoh and M.
Gruteser
.
Protecting location privacy through path confusion
. SECURECOMM, 2005

[5] J.
Krumm
.
Inference Attacks on Location Tracks
. Pervasive Computing, 2007

Pseudonym

6

Location Privacy with Mix Zones

Prevent long term tracking

Mix zone

2

1

a

b

?

Change identifier in
mix zones

[6,7]



Key used to sign messages is changed



MAC address is changed

[6] A. Beresford and F.
Stajano
.
Mix Zones: User Privacy in Location
-
aware Services
. Pervasive Computing and Communications Workshop, 2004

[7] M.
Gruteser

and D.
Grunwald
.
Enhancing location privacy in wireless LAN through disposable interface identifiers: a quantitative analysis
.


Mobile Networks and Applications, 2005

7

Mix
-
zone Placement in Road Networks


Mix zone placement most effective at
intersections [8]



Enables mixing (covers) at roads
leading
in
and out of the intersection



Mix
-
zones incur cost


Communication loss


Routing delays


Cost vary from intersection to intersection



How to place mix
-
zones?


All roads are covered


Overall cost is minimized


Mix Cover

problem

8

[
8
]
L.
Buttyan
, T.
Holczer
, and I.
Vajda
.
On the
effectiveness
of changing
pseudonyms to
provide location privacy in
VANETs
. ESAS
2007

Previous Work on Mix zone Placement


Optimization Approach [9]


Mixing effectiveness using a flow
-
based metric


Given upper bound on mix zones, max. distance between them and
cost, where to place mix zones that maximizes mixing effectiveness


Do not address the coverage problem



Game
-
theoretic Approach [10,11]


Game
-
theoretic model of optimal attack and defense strategies


Only consider local, and not network
-
wide, intersection characteristics

[9]
J.
Freudiger
, R.
Shokri
, and J
-
P.
Hubaux
.
On the optimal placement of mix
zones
. PETS
2009

[10]
M.
Humbert
, M. H.
Manshaei
, J.
Freudiger
, and J
-
P.
Hubaux
.
Tracking games
in mobile
networks
.
GameSec

2010

[11] T.
Alpcan

and S.
Buchegger
.
Security games for vehicular networks
. IEEE
Transactions
on Mobile Computing,
2011





9

Outline

1.
Mix Cover (MC) Problem

2.
Algorithms

3.
Evaluation and Results

10

Graph
-
Theoretic Model


Intersections



Vertices (
V
)


Roads



Edges (
E
)


Mixing
cost

at intersection


Vertex weight (
w
)


Node intensity on road or
demand



E
dge weight (
d
)


One for each direction, for



,

,

𝑒
=
(


𝑒
,



𝑒
)


8

9

8

7

12

8

1

5

4

3

6

3

3

2

2

7

2

2

2

2

7

2

6

2

3

9

1

2

4

1

4

10

2

9

5

4

1

6

6

2

2

9

8

4

𝐺

𝑉
,
𝐸
,

,


11

Mix Cover (MC) Problem


Determine a subset
𝑉
𝑀𝐶

𝑉

and a
capacity


,



𝑉
𝑀𝐶

s.t.






,

,


at least one of


or


𝑉
𝑀𝐶





𝑉
𝑀𝐶
,



max
(


𝑒
)

for all


covered by


(
capacity indicates the largest demand
the intersection can handle)


Total weighted cost



.

𝑥
𝑥

𝑉
𝑀𝐶

is
minimized


8

9

8

7

12

8

1

5

4

3

6

3

3

2

2

7

2

2

2

2

7

2

6

2

3

9

1

2

4

1

4

10

2

9

5

4

1

6

6

2

2

9

8

4

6

10

7

2

4

9

6
x
6

+
2
x
5

+
7
x
12
+
10
x
8

+
4
x
1

+
9
x
9
=
295

12

𝐺

𝑉
,
𝐸
,

,


Why Mix Cover?

Mix zone deployment that provides two guarantees:

1.
Privacy guarantee


All roads are
covered

at least at one end



Nodes go without mixing over at most one intersection

2.
Cost guarantee


Minimum network
-
wide mixing cost


A mix cover provides
both these!

13

Combinatorial Properties


Generalization of Weighted Vertex Cover (WVC) problem



Different from the Facility Terminal Cover (FTC) [13]
generalization of WVC


In FTC, each edge has only a single demand



Result 1
: Mix Cover problem is NP
-
hard


No efficient algorithm for finding optimal solution, even finding a good
approximation seems hard


Proof by polynomial
-
time reduction from WVC

[13]
G.
Xu
, Y. Yang, and J.
Xu
.
Linear Time Algorithms for Approximating the
Facility Terminal
Cover Problem.

Networks 2007

14

Outline

1.
Mix Cover (MC) Problem

2.
Algorithms

3.
Evaluation and Results

15

Three Algorithms


Optimization using Linear Programming



“Divide and Conquer” approach


Largest Demand First


Smallest Demand First

16

Integer Program Formulation

Privacy guarantee

Capacity requirement

Cost guarantee

where






mixing cost at vertex







decision variable indicating selected capacity of vertex


𝑧

𝑒

decision variable for vertex


covering edge


Result 2
: LP relaxation of the above IP can guarantee a polynomial
-
time 2
-
approximation for the Mix Cover problem

17

Largest Demand First (LDF)

1.
For each edge, replace smaller
demand with larger demand

2.
Round off the demands to the
closest power of 2

3.
Divide into
subgraphs

𝐺
𝑘

based on
the rounded edge demands
2
𝑘

4.
Obtain

𝐺
𝑘
=
WVC−2Approx
(
𝐺
𝑘
)

for each
𝐺
𝑘

5.
For all



𝐺
𝑘

,

𝑀𝐶
=
(

,


)

,
where


=

max
{
2
𝑘
|

𝑘

s.t.



𝐺
𝑘
}


6.
Output

𝑀𝐶

𝐺


𝑉
,
𝐸
.

.



𝐺

𝑉
,
𝐸
.

.


18

LDF


Combinatorial Results


A solution to MC problem on
𝐺′

is also a solution for
𝐺



Result 3
:

𝐺


2 
(
𝐺
)
, where


is the
optimal solution and

=
max
{


𝑒



𝑒
,



𝐺
}




Result 4
: LDF is a linear time
4

-
approximation
algorithm for mix cover where


is approximation ratio of
WVC−2Approx



Proofs in the paper!

19

Smallest
Demand First
(SDF
)


LDF highly sub
-
optimal


chosen capacity depends on larger edge
demand value



SDF similar to LDF, except


In step 1, replace larger edge demand value by smaller value


Additional step
: For each vertex, remember the largest edge demand


𝑚𝑎𝑥

incident on it


In

𝑀𝐶
, choose capacity


=
max
{
max
2
𝑘

𝑘

s.t.



𝐺
𝑘
,


𝑚𝑎𝑥
}





Result 5
: SDF is a

(

)

time
4

-
approximation algorithm for mix
cover where


is approximation ratio of
WVC−2Approx


20

Outline

1.
Mix Cover (MC) Problem

2.
Algorithms

3.
Evaluation and Results

21

Experimental Setup


Input graph constructed using real vehicular traffic data


2 US states, Florida and Virginia


3

sizes of road network, 25%, 65% and 100% of total state
municipalities


3 different distributions of vertex weight, constant (1), uniform
(between 1 and 100) and Gaussian (mean=50,
sd
=10)


Edge demands chosen from real traffic intensities



Algorithms implemented in
MATLAB,
executed on multi
-
core computer



Results average over 100 runs


22

Solution Quality


Naïve solution: Select all vertices in final solution







SDF outperforms LDF in both cases for all graph sizes


SDF achieves as low as 34% of the cost of
the naïve
solution


Performance
best for uniform
vertex weight distribution and
worst for
constant distribution

Florida

Virginia

LDF

SDF

LDF

SDF

Ratio of LDF/SDF solution cost to naïve strategy cost

23

v/e

v/e

Execution Efficiency






SDF runs slower compared to LDF in both cases for all graph
sizes


Algorithms fastest when vertex weight constant and worst
when selected from a Gaussian distribution

Florida

Virginia

LDF

SDF

LDF

SDF

Duration (in seconds) of algorithm execution

24

Results for LP
-
based Algorithm


Too slow for large graphs



Executed on reduced Florida graph of 515 and 1024 vertices



For 515 vertices
,
ratio
of
solution
cost
compared to
naïve strategy
improves to
0.24

(better than LDF and SDF)



Execution time is
twice

compared to LDF and
four times
that of SDF



For 1024 vertices, execution time increased by a factor of
20






25

Conclusion


Mix Cover: cost
-
efficient mix zone placement that guarantees mixing
coverage



Modeled as a generalization of weighted vertex cover problem


Never been studied


Model general enough and applicable to other scenarios



Approximation algorithms using


Linear programming


LDF and SDF based on “Divide and Conquer” approach



Results


Proposed algorithms provide solution quality and execution time guarantees


Experimentation using real data and standard computation resources show feasibility





murtuza.jadliwala@epfl.ch

26

BACKUP SLIDES

27

How to obtain mix zones?


Silent mix zones


Turn off transceiver


Passive mix zones


Where adversary is absent


Before connecting to Wireless Access Points


Encrypt communications


With help of infrastructure


Distributed

28

bluetoothtracking.org

29

Pleaserobme.com

30

Mix Zones

Mix network

Mix networks
vs

Mix zones

Mix

node

Mix

node

Mix

node

Alice

Bob

Alice

home

Alice

work

31

Assumption


Central authority periodically computes optimal mix
cover
offline


Knows the (dynamic) node or traffic intensity on roads


Knows mixing cost at each intersection



Nodes
or
vehicles access the latest mix cover
computation from the central authority

32

Solution Size









SDF performs better than LDF in Florida


LDF performs better than SDF in Virginia


Algorithms do not optimize solution size; depends on road network
topology


Solution size between 46% and 58% of the total number of vertices

Florida

Virginia

LDF

SDF

LDF

SDF

Number of vertices in the final
solution

v/e

v/e

33