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![Page 1: Optimizing Mixing in Pervasive Networks: A Graph-Theoretic Perspective Murtuza Jadliwala, Igor Bilogrevic and Jean-Pierre Hubaux ESORICS, 2011.](https://reader030.fdocuments.net/reader030/viewer/2022033105/56649e2f5503460f94b1f953/html5/thumbnails/1.jpg)
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
a
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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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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
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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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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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4610
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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 [email protected]
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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