Bayesian Inference and Traffic Analysis
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1 ayesian Inference and Traffic nalysis armela Troncoso George Danezis September-November 008 Microsoft Research ambridge/ KU Leuven(OSI
2 nonymous ommunications Tell me who your friends are... => nonymous communications to hide communication partners High latency systems (e.g. anonymous r ers use mies [haum 8]: hide input/output relationship MIX MIX MIX
3 nonymous ommunications ttacks to mi networks Restricted routes [Dan03] ridging and Fingerprinting [DanSyv08] Social information: Disclosure ttack [Kes03] Statistical Disclosure ttack [Dan03] Perfect Matching Disclosure ttacks [Tron08] Heuristics and specific models 3
4 Mi networks and traffic analysis Determine probability distributions input-output MIX 3 MIX MIX Q S R ( ( ( ( or or or or
5 Mi networks and traffic analysis MIX 3 MIX MIX Q S R ( 0 ( ( ( Non trivial given observation!! onstraintse.g.length= or or
6 The real thing Senders Mies (Threshold = 3 Receivers How to compute probabilities systematically??
7 Mi networks and traffic analysis Find hidden state of the mies M O? M3 Q R S O O M O Prior information O Too large to enumerate!! K
8 Mi networks and traffic analysis hidden state + Observation = M Q R M3 S M P P P 3 M M M3 R M M3 Q M S O O K
9 ayesian Inference ctually we want marginal probabilities ut we cannot obtain them directly j I O Q j Q ( Q S R ( ( ( ( or or or or
10 ayesian Inference - sampling If we obtain samples 3 j ~ O ( Q? 0 0 Q O Markov hain Monte arlo Methods Metropolis Hastings algorithm O I How does look like? Q j ( j
11 Probabilistic model asic onstraints Users decide independently P Length restrictions e.g.uniform ( L min L ma L l L l with any distribution L ma L min Node choice restrictions hoose l out of the N mi node available hoose a set I set ( M M L l P ( N mi l P L l M L l I set ( M
12 Probabilistic model asic onstraints Unknown destinations S S ( ma set L L l M I l L M l L P obs L ma 3
13 Probabilistic model More onstraints ridging Known nodes I bridging ( M Non-compliant clients (with probability p Do not respect length restrictions ( Lmin c p Lma c p hoose l out of the N mi node availableallow repetiti ons M L l I c p ( Path P r ( N c p mi l i P P c p p c p P i I c p ( Pi j P cp ( p c p P j
14 Probabilistic model More constraints Social network information ssuming we know sending profiles Sen Rec P L l M L l I set ( M Sen Rec Other constraints Unknown origin Dummies Other miing strategies.
15 Markov hain Monte arlo Sample from a distribution difficult to sample from directly O O O O K 3 Key advantages: Requires generative model (we know how to compute it! Good estimation of errors Not false positives and negatives Systematic
16 Metropolis Hastings lgorithm onstructs a Markov hain with stationary distribution urrent state Q andidate state Q( candidate current O current Q( current candidate candidate. ompute. If else if candidate current current candidate u ~ U (0 u current candidate else current current Q( Q( candidate current current candidate
17 Our sampler: Q transition Q( candidate current O current Pahts candidate Z candidate current Q( Q( candidate current Transition Q: swap operation current candidate Q( current candidate M M S M3 Q R More complicated transitions for non-compliant clients
18 Iterations Q ( candidate current O Pahts Z current candidate Q ( current candidate onsecutive samples dependant Sufficiently separated i j i i j
19 Error estimation ( Q? P P P 3 P 0 0 Q I Q j Error estimation ernouilli distribution Pr[ 3... Q ] Prior eta( ~ uniform Q Pr[ Q 3...] ~ eta ( I Q ( Path i I Q ( Path i onfidence intervals
20 Evaluation. reate an instance of a network. Run the sampler 3. hoose a target sender and a receiver. Estimate probability Sen Rec 5. heck if actually Sen chose Rec as receiver 6. hoose new network and go to I Sen Rec j ( j I ( SenRec network Events should happen with the estimated probability Sen Rec I Sen Rec j ( j E ( I Sen Rec ( network
21 Results compliant clients E ( Rec I Sen ( network I Sen Rec ( j j
22 Results 50 messages
23 Results 0 messages
24 Results big networks
25 Performance RM usage Nmi t Nmsg Samples RM(Mb Size of network and population Results are kept in memory during simulation Number samples collected increases
26 Performance Running time Nmi t Nmsg iter Full analysis (min One sample(ms Operations should be O( Writing of the results on a file Different number of iterations
27 onclusions Traffic analysis is non trivia l when there are constraints Probabilistic model: incorpor ates most attacks Non-compliant clients Monte arlo Markov hain methods to etract marginal probabilities Future work: SD based on ayesian Inferenc e dded value?
28 Thanks for your attention Microsoft technical report coming soon 8
29 ayes theorem O O O O O O O O O O X X Y Y Y X Y X Joint probability:
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