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
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