a game-theoretic approach - Security and Cooperation in Wireless
Download
Report
Transcript a game-theoretic approach - Security and Cooperation in Wireless
Mini-project of the Security and Cooperation in Wireless Networks course
ON THE OPTIMAL PLACEMENT OF MIX ZONES:
A GAME-THEORETIC APPROACH
Mathias Humbert
LCA1/EPFL
January 19, 2009
Supervisors:
Mohammad Hossein Manshaei
Julien Freudiger
Jean-Pierre Hubaux
MOTIVATIONS
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
OUTLINE
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
LAUSANNE DOWNTOWN
Intersections’ statistics stored in 23 matrices (size = 5x5)
Place Chauderon
Place Chauderon:
Traffic matrix:
23 intersections
4
SYSTEM MODEL
Road network with N intersections
Mobile nodes vs. Local passive adversary
Nodes’ privacy-preserving mechanisms (at intersection i):
Active mix zone (cost = cim)
Passive mix zone (cost = cip)
Adversary’s tracking devices::
Traffic matrix:
cim = cip + ciq = pseudonyms cost + silence cost
Sniffing station (cost = cs)
Mobility parameters:
Relative traffic intensity λi
Mixing effectiveness mi
mix
5
MIXING 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:
mi = 0 (no uncertainty for the adversary)
Greatest mixing at place Chaudron: mi = 0.74
GAME-THEORETIC APPROACH
G = {P, S, U}
2 players: {mobile nodes, adversary}
Nodes’ strategies sn,i (intersection i):
Active mix zone (AMZ)
Passive mix zone (PMZ)
Nothing (NO)
Adversary’s strategies sa,i :
Sniffing station (SS)
Nothing (NO)
0 < λi, mi, cim, cs < 1
Payoffs:
Active mix zone
Nodes
Passive mix zone
Nothing
Adversary
Sniffing station
Nothing
(λimi-cip-ciq ; λi(1-mi)-cs)
(λi-cip-ciq ; 0)
(-cip
; λi-cs)
(0 ; λi-cs)
(λi- cip
; 0)
(0 ; 0)
7
COMPLETE INFORMATION GAME FOR ONE INTERSECTION
Probabilities:
Probabilities:
pi = (λi-cs) /λimi
1- pi
Sniffing station/SS
Nothing/NO
Active mix zone
AMZ
(λimi-cip-ciq ; λi(1-mi)-cs)
(λi-cip-ciq ; 0)
1- qi
Passive mix zone
PMZ
(-cip ; λi-cs)
(λi- cip ; 0)
0
Nothing
NO
(0 ; λi-cs)
(0 ; 0)
qi = min(ciq/λimi, 1)
Pure-strategy NE [theorem 1]:
Mixed-strategy NE:
8
N INTERSECTIONS-GAME
Global NE = Union of local NE
Global payoffs at equilibrium defined as
Number of sniffing stations = Ws (upper bound)
Game = two maximisation problems:
Nodes
Adversary
9
N INTERSECTIONS-GAME
Algorithm converging to an equilibrium [theorem 2]
As uia = 0 at mixed-strategy NE and assuming (wlos) that m1 < m2 < … < mn
Remove sniffing stations at
mixed NE first
Remove sniffing stations at
pure NE (Start with smallest
adversary’s payoff)
10
The nodes normally take advantage of the absence of
sniffing station to deploy a passive mix zone
NUMERICAL RESULTS: LOW PLAYERS’ COSTS
sniffing
stations:
Fixed (normalized) costs and unlimited
limited nbnb
of of
sniffing
stations
(Ws = 5):
11
NUMERICAL RESULTS: MEDIUM SNIFFING COST
Fixed (normalized) costs and unlimited
limited nbnb
of of
sniffing
sniffing
stations
stations:
(Ws = 5):
12
INCOMPLETE 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:
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
13
N INTERSECTIONS INCOMPLETE INFORMATION GAME
Potential algorithm to converge to a Bayesian Nash
equilibrium (ongoing work):
Complete knowledge for the
adversary => remove sniffing stations
leading to smallest payoffs at BNE
Nodes know Ws => put passive
mix zones where adversary’s
expected payoffs are the smallest
14
CONCLUSION 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
Concrete application on a real city network
In incomplete information game, significant decrease of nodes’
location privacy due to lack of knowledge about adversary’s payoff
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
NUMERICAL EVALUATION OF OPTIMAL STRATEGIES
WITH VARIABLE COSTS
2) Limited
1)
Unlimited
number of SS:
16
BACKUP: MIXING EFFECTIVENESS COMPUTATION
Mixing: uncertainty for an adversary trying to match
nodes leaving the active mix zone to the entering ones
=> entropy
=> relative traffic intensity
Dfdf
Dfdf
dfd
17
BACKUP: BAYESIAN NE FOR THE INCOMPLETE
INFORMATION GAME @ 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:
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
18
BACKUP: MOTIVATION
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’
19
knowledge on payoffs and behaviors of other participants
[1] M. Humbert , Location Privacy amidst Local Eavesdroppers, Master thesis, 2009