1. ‘Crowds’ through a PRISM
CS 259
‘Crowds’ through a PRISM

2. Overview Most slides stolen from Vitaly Shmatikov Crowds
Probabilistic model checking
PRISM
PCTL logic
Analyzing Crowds with PRISM

3. Crowds C C C4 C C3 C C C1 C C2 C0 C C [Reiter,Rubin ‘98] pf 1-pf
sender
recipient
After receiving a message, honest router flips a biased coin
With probability Pf randomly selects next router and forwards msg
With probability 1-Pf sends directly to the recipient

4. Probabilistic Notions of Anonymity
Beyond suspicion
The observed source of the message is no more likely to be the true sender than anybody else
Probable innocence
Probability that the observed source of the message is the true sender is less than 50%
Possible innocence
Non-trivial probability that the observed source of the message is not the true sender

5. Probabilistic Model Checking
Participants are finite-state machines
Same as Mur
State transitions are probabilistic
Transitions in Mur are nondeterministic
Standard intruder model
Same as Mur: model cryptography with abstract data types
Mur question:
Is bad state reachable?
Probabilistic model checking question:
What’s the probability of reaching bad state?
0.2
0.3
0.5
...
...
“bad state”

6. Discrete-Time Markov Chains
(S, s0, T, L)
S is a finite set of states
s0 S is an initial state
T :SS[0,1] is the transition relation
s,s’S s’ T(s,s’)=1
L is a labeling function

7. Markov Chain: Simple Example
Probabilities of outgoing
transitions sum up to 1.0
for every state C
0.2
0.5 A
0.1 E s0
0.5
0.8 D
1.0 B
0.9
1.0
Probability of reaching E from s0 is 0.2 0.10.5=0.14
The chain has infinite paths if state graph has loops
Need to solve a system of linear equations to compute probabilities

8. PRISM Probabilistic model checker System specified as a Markov chain
[Kwiatkowska et al., U. of Birmingham]
Probabilistic model checker
System specified as a Markov chain
Parties are finite-state machines w/ local variables
State transitions are associated with probabilities
Can also have nondeterminism (Markov decision processes)
All parameters must be finite
Correctness condition specified as PCTL formula
Computes probabilities for each reachable state
Enumerates reachable states
Solves system of linear equations to find probabilities

9. PRISM Syntax C A E s0 D B module Simple state: [1..5] init 1;
0.2
0.5 A
0.1 E s0
0.5
0.8 D
1.0 B
0.9
1.0
module Simple
state: [1..5] init 1;
[] state=1 -> 0.8: state’= : state’=3;
[] state=2 -> 0.1: state’= : state’=4;
[] state=3 -> 0.5: state’= : state’=5;
endmodule
IF state=3 THEN with prob. 50% assign 4 to state,
with prob. 50% assign 5 to state

10. Modeling Crowds with PRISM
Model probabilistic path construction
Each state of the model corresponds to a particular stage of path construction
1 router chosen, 2 routers chosen, …
Three probabilistic transitions
Honest router chooses next router with probability pf, terminates the path with probability 1-pf
Next router is probabilistically chosen from N candidates
Chosen router is hostile with certain probability
Run path construction protocol several times and look at accumulated observations of the intruder

11. PRISM: Path Construction in Crowds
module crowds
. . .
// N = total # of routers, C = # of corrupt routers
// badC = C/N, goodC = 1-badC
[] (!good & !bad) ->
goodC: (good’=true) & (revealAppSender’=true) +
badC: (badObserve’=true);
// Forward with probability PF, else deliver
[] (good & !deliver) ->
PF: (pIndex’=pIndex+1) & (forward’=true) +
notPF: (deliver’=true);
endmodule
Next router is corrupt with certain probability
Route with probability PF, else deliver

12. PRISM: Intruder Model
module crowds
. . .
// Record the apparent sender and deliver
[] (badObserve & appSender=0) ->
(observe0’=observe0+1) & (deliver’=true);
[] (badObserve & appSender=15) ->
(observe15’=observe15+1) & (deliver’=true);
endmodule
For each observed path, bad routers record apparent sender
Bad routers collaborate, so treat them as a single attacker
No cryptography, only probabilistic inference

13. PCTL Logic Probabilistic Computation Tree Logic
[Hansson, Jonsson ‘94]
Probabilistic Computation Tree Logic
Used for reasoning about probabilistic temporal properties of probabilistic finite state spaces
Can express properties of the form “under any scheduling of processes, the probability that event E occurs is at least p’’
By contrast, Mur can express only properties of the form “does event E ever occur?’’

14. PCTL Syntax State formulas Path formulas
First-order propositions over a single state
 ::= True | a |    |    |  | P>p[]
Path formulas
Properties of chains of states
 ::= X  |  Uk  |  U 
Predicate over state variables
(just like a Mur invariant)
Path formula holds
with probability > p
State formula holds for
every next state in the chain
First state formula holds for every state
in the chain until second becomes true

15. PCTL: State Formulas A state formula is a first-order state predicate
Just like non-probabilistic logic
True
False
X=1
y=2
1.0
True
X=2
y=0
0.2
X=1
y=1
0.5
1.0
X=3
y=0 s0
0.5
0.8
False
 = (y>1) | (x=1)

16. PCTL: Path Formulas
A path formula is a temporal property of a chain of states
1U2 = “1 is true until 2 becomes and stays true”
X=1
y=2
1.0
X=2
y=0
0.2
X=1
y=1
0.5
1.0
X=3
y=0 s0
0.5
0.8
 = (y>0) U (x>y) holds for this chain

17. PCTL: Probabilistic State Formulas
Specify that a certain predicate or path formula holds with probability no less than some bound
True
True
X=1
y=2
1.0
False
X=2
y=0
0.2
X=1
y=1
0.5
1.0
X=3
y=0 s0
0.5
0.8
False
 = P>0.5[(y>0) U (x=2)]

18. Intruder Model Every time a hostile crowd member receives a message
module crowds
. . .
// Record the apparent sender and deliver
[] (badObserve & appSender=0) ->
(observe0’=observe0+1) & (deliver’=true);
[] (badObserve & appSender=15) ->
(observe15’=observe15+1) & (deliver’=true);
endmodule
Every time a hostile crowd member receives a message
from some honest member, he records his observation
(increases the count for that honest member)

19. Negation of Probable Innocence
P >0.5 [true U (observe0>observe1) & done] …
P>0.5[true U (observe0>observe9) & done]
“The probability of reaching a state in which hostile crowd
members completed their observations and observed the
true sender (crowd member #0) more often than any of
the other crowd members (#1 … #9) is greater than 0.5”

20. Dynamic Paths What happens when originator sends new message?
Use same path
Or construct new path
Intruder can correlate messages using content
Originator appears on all paths!
Paths need to be static otherwise probable innocence is broken

21. Path Reformulations Same problem as dynamic paths
When members leave the crowd, paths are reformulated
New paths can be correlated with old paths based on path content
Shmatikov analyzes the effect of path reformulations and crowd size: See case study on PRISM site