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Optimizing Mixing in Pervasive Networks: A Graph-Theoretic Perspective

This research explores the optimization of privacy in pervasive networks through mixing techniques. Using a graph-theoretic framework, we address the Mix Cover problem, focusing on the deployment of mix zones in vehicular ad hoc networks (VANETs). We analyze the challenges posed by location privacy and propose algorithms to optimize the placement of mix zones to minimize costs while ensuring comprehensive coverage. The work draws on existing literature and provides a comprehensive evaluation of different optimization strategies to enhance user privacy in location-aware services.

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Optimizing Mixing in Pervasive Networks: A Graph-Theoretic Perspective

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  1. Optimizing Mixing in Pervasive Networks: AGraph-Theoretic Perspective MurtuzaJadliwala, Igor Bilogrevic and Jean-Pierre Hubaux ESORICS, 2011

  2. Wireless Trends Smart Phones Vehicles • Always on • Background apps Cameras Watches Passports

  3. Peer-to-Peer Wireless Networks 1 2 Identifier Message

  4. Examples VANETs • Social networks Nokia Instant Community • Urban Sensing networks • Delay tolerant networks • Peer-to-peer file exchange

  5. Location Privacy Problem Monitor identifiers used in peer-to-peer communications a b c

  6. Location Privacy Attacks Pseudonym • Pseudonymous location traces • Home/work location pairs are unique [1] • Re-identification of traces through data analysis [2,4,3,5] • Attack:Spatio-Temporalcorrelation of traces Identifier Message [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

  7. Location Privacy with Mix ZonesPrevent long term tracking ? b 1 a 2 Mix zone • 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

  8. 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

  9. 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

  10. Outline • Mix Cover (MC) Problem • Algorithms • Evaluation and Results

  11. 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 3 7 9 2 4 8 6 3 6 4 10 2 2 8 7 8 6 2 3 2 6 7 3 2 12 9 2 2 9 5 2 1 8 1 2 9 1 4 2 2 5 4 4 1

  12. 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 3 10 7 9 6 2 4 8 6 3 6 4 10 2 2 8 7 8 6 2 7 3 2 6 7 3 2 12 9 2 2 9 9 5 2 1 8 1 2 9 1 4 2 2 2 5 4 4 1 4 6x6 + 2x5 + 7x12+ 10x8 + 4x1 + 9x9 = 295

  13. Why Mix Cover? Mix zone deployment that provides two guarantees: • Privacy guarantee • All roads are covered at least at one end • Nodes go without mixing over at most one intersection • Cost guarantee • Minimum network-wide mixing cost A mix cover provides both these!

  14. 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

  15. Outline • Mix Cover (MC) Problem • Algorithms • Evaluation and Results

  16. Three Algorithms • Optimization using Linear Programming • “Divide and Conquer” approach • Largest Demand First • Smallest Demand First 1 2 3

  17. Integer Program Formulation Cost guarantee Privacy guarantee Capacity requirement 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

  18. Largest Demand First (LDF) • For each edge, replace smaller demand with larger demand • Round off the demands to the closest power of 2 • Divide into subgraphs based on the rounded edge demands • Obtain for each • For all , , where • Output

  19. 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!

  20. 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

  21. Outline • Mix Cover (MC) Problem • Algorithms • Evaluation and Results

  22. 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

  23. Solution Quality Ratio of LDF/SDF solution cost to naïve strategy cost • 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 v/e v/e LDF Florida SDF LDF Virginia SDF

  24. Execution Efficiency Duration (in seconds) of algorithm execution • 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 LDF Florida SDF LDF Virginia SDF

  25. 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

  26. 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

  27. Backup Slides

  28. 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

  29. bluetoothtracking.org

  30. Pleaserobme.com

  31. Mix networks vs Mix zones Alice home Mix network Mix Zones Mix node Mix node Alice work Bob Alice Mix node

  32. 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

  33. Solution Size Number of vertices in the final solution • 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 v/e v/e LDF Florida SDF LDF Virginia SDF

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