Optimizing Mixing in Pervasive Networks: AGraph-Theoretic Perspective

Murtuza Jadliwala, Igor Bilogrevic and Jean-Pierre Hubaux

ESORICS, 2011

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Wireless TrendsSmart Phones Vehicles

Watches Cameras Passports

• Always on• Background apps

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Peer-to-Peer Wireless Networks

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MessageIdentifier

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Examples

• Urban Sensing networks• Delay tolerant networks• Peer-to-peer file exchange

VANETs • Social networks

Nokia Instant Community

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Location Privacy Problem

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b

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Monitor identifiers used in peer-to-peer communications

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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

MessageIdentifier

[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

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Location Privacy with Mix ZonesPrevent long term tracking

Mix zone

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ab?

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

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

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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

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Outline

1. Mix Cover (MC) Problem

2. Algorithms

3. Evaluation and Results

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Graph-Theoretic Model

• Intersections Vertices (V)

• Roads Edges (E)

• Mixing cost at intersection Vertex weight (w)

• Node intensity on road or demand Edge weight (d)– One for each direction, for

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72

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66

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𝐺≡ (𝑉 ,𝐸 ,𝑤 ,𝑑 )

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Mix Cover (MC) Problem

• Determine a subset and a capacity s.t.– at least one of or

– , for all covered by

(capacity indicates the largest demand the intersection can handle)

– Total weighted cost is minimized

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66

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46 10

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6x6 + 2x5 + 7x12+ 10x8 + 4x1 + 9x9 = 295

𝐺≡ (𝑉 ,𝐸 ,𝑤 ,𝑑 )

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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!

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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

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Outline

1. Mix Cover (MC) Problem

2. Algorithms

3. Evaluation and Results

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Three Algorithms

• Optimization using Linear Programming

• “Divide and Conquer” approach– Largest Demand First

– Smallest Demand First

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Integer Program Formulation

Privacy guarantee

Capacity requirement

Cost guarantee

where mixing cost at vertex decision variable indicating selected capacity of vertex

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

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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 based on the rounded edge demands

4. Obtain for each

5. For all , , where

6. Output

𝐺 ′ ≡ (𝑉 ,𝐸 .𝑤 .𝑑 ′ )𝐺≡ (𝑉 ,𝐸 .𝑤 .𝑑 )

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LDF – Combinatorial Results

• A solution to MC problem on is also a solution for

• Result 3: , where is the optimal solution and

• Result 4: LDF is a linear time -approximation algorithm for mix cover where is approximation ratio of

• Proofs in the paper!

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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 incident on it

– In , choose capacity

• Result 5: SDF is a time -approximation algorithm for mix cover where is approximation ratio of

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Outline

1. Mix Cover (MC) Problem

2. Algorithms

3. Evaluation and Results

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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

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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

LDFSDFLDFSDF

Ratio of LDF/SDF solution cost to naïve strategy cost

v/ev/e

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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

LDFSDFLDFSDF

Duration (in seconds) of algorithm execution

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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

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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

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BACKUP SLIDES

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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

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bluetoothtracking.org

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Pleaserobme.com

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Mix ZonesMix network

Mix networks vs Mix zones

Mixnode

Mixnode

Mixnode

Alice Bob

Alice home

Alice work

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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

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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

LDFSDFLDFSDF

Number of vertices in the final solution

v/ev/e