O N THE O PTIMAL P LACEMENT OF M IX Z ONES : A G AME -T HEORETIC A PPROACH Mathias Humbert LCA1/EPFL January 19, 2009 Supervisors: Mohammad Hossein Manshaei Julien Freudiger Jean-Pierre Hubaux Mini-project of the Security and Cooperation in Wireless Networks course

M OTIVATIONS Pratical case study on location privacy Use of the relevant information from Lausanne’s traffic data Game-theoretic model evaluating agents’ behaviors a priori Incomplete information game analysis 2

O UTLINE Lausanne traffic: a case study System model and mixing effectiveness Game-theoretic approach Game results: A complete information game Numerical evaluations An incomplete information game Conclusion and future work 3

L AUSANNE DOWNTOWN 4 Intersections’ statistics stored in 23 matrices (size = 5x5) Place Chauderon Place Chauderon: 23 intersections Traffic matrix:

S YSTEM MODEL Road network with N intersections Mobile nodes vs. Local passive adversary Nodes’ privacy-preserving mechanisms (at intersection i): Active mix zone (cost = c i m ) c i m = c i p + c i q = pseudonyms cost + silence cost Passive mix zone (cost = c i p ) Adversary’s tracking devices:: Sniffing station (cost = c s ) Mobility parameters: Relative traffic intensity λ i Mixing effectiveness m i 5 mix Traffic matrix:

M IXING EFFECTIVENESS Mixing: uncertainty for an adversary trying to match nodes leaving the active mix zone to the entering ones => normalized entropy => relative traffic intensity 6 Smallest mixing between Chaudron & Bel-Air: m i = 0 (no uncertainty for the adversary) Greatest mixing at place Chaudron: m i = 0.74

G AME - THEORETIC APPROACH G = {P, S, U} 2 players: {mobile nodes, adversary} Nodes’ strategies s n,i (intersection i): Active mix zone (AMZ) Passive mix zone (PMZ) Nothing (NO) Adversary’s strategies s a,i : Sniffing station (SS) Nothing (NO) Payoffs: 7 Adversary Nodes 0 < λ i, m i, c i m, c s < 1

C OMPLETE INFORMATION GAME FOR ONE INTERSECTION Pure-strategy NE [theorem 1]: Mixed-strategy NE: Probabilities: p i = (λ i -c s ) /λ i m i 1- p i q i = min(c i q /λ i m i, 1) 0 1- q i mixed-strategy Nash equilibrium 8

N INTERSECTIONS - GAME Global NE = Union of local NE Global payoffs at equilibrium defined as Number of sniffing stations = W s (upper bound) Game = two maximisation problems: 9 Nodes Adversary

N INTERSECTIONS - GAME Algorithm converging to an equilibrium [theorem 2] 10 Remove sniffing stations at mixed NE first Remove sniffing stations at pure NE (Start with smallest adversary’s payoff) The nodes normally take advantage of the absence of sniffing station to deploy a passive mix zone As u i a = 0 at mixed-strategy NE and assuming (wlos) that m 1 < m 2 < … < m n

11 N UMERICAL RESULTS : LOW PLAYERS ’ COSTS Fixed (normalized) costs and unlimited nb of sniffing stations: limited Fixed (normalized) costs and limited nb of sniffing stations (W s = 5):

12 N UMERICAL RESULTS : MEDIUM SNIFFING COST Fixed (normalized) costs and unlimited nb of sniffing stations: limited Fixed (normalized) costs and limited nb of sniffing stations (W s = 5):

I NCOMPLETE INFORMATION GAME FOR ONE INTERSECTION Assumptions: Nodes do not know the sniffing cost Instead, they have a probability distribution on cost’s type Theorem 3: one pure-strategy Bayesian Nash equilibrium (BNE) with strategy profile defined by: 13 with (probability that the adversary installs a sniffing station) defined using the probability distribution on cost’s type Suboptimal BNE, such as (AMZ, NO) or (PMZ, SS) for nodes’ payoff can occur if nodes’ belief on sniffing station cost’s type is inacurrate

N INTERSECTIONS INCOMPLETE INFORMATION GAME Potential algorithm to converge to a Bayesian Nash equilibrium (ongoing work): 14 Complete knowledge for the adversary => remove sniffing stations leading to smallest payoffs at BNE Nodes know W s => put passive mix zones where adversary’s expected payoffs are the smallest

C ONCLUSION AND FUTURE WORK Prediction of nodes’ and adversary’s strategic behaviors using game theory Algorithms reaching an optimal (Bayesian) NE in complete and incomplete information games In incomplete information game, significant decrease of nodes’ location privacy due to lack of knowledge about adversary’s payoff Concrete application on a real city network Nodes and adversary often adopting complementary strategies Future work Evaluation of the incomplete information game with the real traffic data and various probability distributions on sniffing station cost 15

N UMERICAL EVALUATION OF OPTIMAL STRATEGIES WITH VARIABLE COSTS 16 1) Unlimited number of SS: 2) Limited number of SS:

B ACKUP : M IXING E FFECTIVENESS COMPUTATION Mixing: uncertainty for an adversary trying to match nodes leaving the active mix zone to the entering ones => entropy => relative traffic intensity Dfdf dfd 17

B ACKUP : B AYESIAN NE FOR THE I NCOMPLETE I NFORMATION G ONE INTERSECTION Nodes do not know the sniffing cost Instead, they have a probability distribution on cost’s type Theorem 3: one pure-strategy Bayesian Nash equilibrium (BNE) with strategy profile defined by: 18 With (probability that the adversary installs a sniffing station) defined using the cdf of the cost’s type: Suboptimal BNE, such as (AMZ, NO) or (PMZ, SS) for nodes’ payoff can occur if nodes’ belief on sniffing station cost’s type is inacurrate

B ACKUP : M OTIVATION Master project [1]: study of mobile nodes’ location privacy threatened by a local adversary Application of this work on a practical and real example Collaboration with people of TRANSP-OR research group at EPFL Lausanne’s traffic data based on actual road measurements and Swiss Federal census (more on this in next slide) Selection of the relevant information from the traffic data New game-theoretic model in order to exploit the provided data and evaluate nodes’ location privacy Incomplete information game to better model the players’ knowledge on payoffs and behaviors of other participants 19 [1] M. Humbert, Location Privacy amidst Local Eavesdroppers, Master thesis, 2009