A Probabilistic Analysis of Onion Routing in a Black-box

Model10/29/2007

Workshop on Privacy in the Electronic Society

Aaron Johnson (Yale)

with

Joan Feigenbaum (Yale)

Paul Syverson (NRL)

Contributions

Contributions1. Use a black-box abstraction to create a

probabilistic model of onion routing

Contributions1. Use a black-box abstraction to create a

probabilistic model of onion routing

2. Analyze unlinkabilitya. Provide worst-case bounds

b. Examine a typical case

Related Work• A Model of Onion Routing with Provable

AnonymityJ. Feigenbaum, A. Johnson, and P. SyversonFC 2007

• Towards an Analysis of Onion Routing SecurityP. Syverson, G. Tsudik, M. Reed, and C. LandwehrPET 2000

• An Analysis of the Degradation of Anonymous ProtocolsM. Wright, M. Adler, B. Levine, and C. ShieldsNDSS 2002

Anonymous Communication

• Sender anonymity: Adversary can’t determine the sender of a given message

• Receiver anonymity: Adversary can’t determine the receiver of a given message

• Unlinkability: Adversary can’t determine who talks to whom

Anonymous Communication

• Sender anonymity: Adversary can’t determine the sender of a given message

• Receiver anonymity: Adversary can’t determine the receiver of a given message

• Unlinkability: Adversary can’t determine who talks to whom

How Onion Routing Works

User u running client Internet destination d

Routers running servers

u d

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged

{{{m}3}4}1 1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged

{{m}3}4

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged

{m}3

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged

m

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged

m’

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged

{m’}3

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged

{{m’}3}4

1 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged

{{{m’}3}4}11 2

3

45

How Onion Routing Works

u d

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged.

4. Stream is closed.

1 2

3

45

How Onion Routing Works

u

1. u creates 3-hop circuit through routers

2. u opens a stream in the circuit to d

3. Data is exchanged.

4. Stream is closed.

5. Circuit is changed every few minutes.

1 2

3

45

d

Adversary

u

1 2

3

45

d

Active & Local

Anonymity

u 1 2

3

45

d

1.

2.

3.

4.

v

w

e

f

Anonymity

u 1 2

3

45

d

1. First router compromised

2.

3.

4.

v

w

e

f

Anonymity

u 1 2

3

45

d

1. First router compromised

2. Last router compromised

3.

4.

v

w

e

f

Anonymity

u 1 2

3

45

d

1. First router compromised

2. Last router compromised

3. First and last compromised

4.

v

w

e

f

Anonymity

u 1 2

3

45

d

1. First router compromised

2. Last router compromised

3. First and last compromised

4. Neither first nor last compromised

v

w

e

f

Black-box Abstraction

u d

v

w

e

f

Black-box Abstraction

u d

v

w

e

f

1. Users choose a destination

Black-box Abstraction

u d

v

w

e

f

1. Users choose a destination

2. Some inputs are observed

Black-box Abstraction

u d

v

w

e

f

1. Users choose a destination

2. Some inputs are observed

3. Some outputs are observed

Black-box Anonymity

u d

v

w

e

f

• The adversary can link observed inputs and outputs of the same user.

Black-box Anonymity

u d

v

w

e

f

• The adversary can link observed inputs and outputs of the same user.

• Any configuration consistent with these observations is indistinguishable to the adversary.

Black-box Anonymity

u d

v

w

e

f

• The adversary can link observed inputs and outputs of the same user.

• Any configuration consistent with these observations is indistinguishable to the adversary.

Black-box Anonymity

u d

v

w

e

f

• The adversary can link observed inputs and outputs of the same user.

• Any configuration consistent with these observations is indistinguishable to the adversary.

Probabilistic Black-box

u d

v

w

e

f

Probabilistic Black-box

u d

v

w

e

f

• Each user v selects a destination from distribution pv

pu

Probabilistic Black-box

u d

v

w

e

f

• Each user v selects a destination from distribution pv

• Inputs and outputs are observed independently with probability b

pu

Probabilistic Anonymityu dvw

ef

u dvw

ef

u dvw

ef

u dvw

ef

Indistinguishable configurations

Probabilistic Anonymityu dvw

ef

u dvw

ef

u dvw

ef

u dvw

ef

Indistinguishable configurations

Conditional distribution: Pr[ud] = 1

Black Box ModelLet U be the set of users.

Let be the set of destinations.

Configuration C• User destinations CD : U• Observed inputs CI : U{0,1}

• Observed outputs CO : U{0,1}

Let X be a random configuration such that:

Pr[X=C] = u puCD(u)

bCI(u) (1-b)1-CI(u) bCO(u) (1-b)1-CO(u)

Probabilistic Anonymity

The metric Y for the unlinkability of u and d in C is:

Y(C) = Pr[XD(u)=d | XC]

Note: There are several other candidates for a probabilistic anonymity metric, e.g. entropy

Probabilistic Anonymity

The metric Y for the unlinkability of u and d in C is:

Y(C) = Pr[XD(u)=d | XC]

Probabilistic Anonymity

The metric Y for the unlinkability of u and d in C is:

Y(C) = Pr[XD(u)=d | XC]

Exact Bayesian inference

• Adversary after long-term intersection attack

• Worst-case adversary

Probabilistic Anonymity

The metric Y for the unlinkability of u and d in C is:

Y(C) = Pr[XD(u)=d | XC]

Exact Bayesian inference

• Adversary after long-term intersection attack

• Worst-case adversary

Unlinkability given that u visits d:

E[Y | XD(u)=d]

Worst-case Anonymity

Worst-case Anonymity

Theorem 1: The maximum of E[Y | XD(u)=d] over (pv)vu occurs when

1. pv=1 for all vu OR

2. pvd=1 for all vu

Let pu1 pu

2 pud-1 pu

d+1 … pu

Show max. occurs when, for all vu,ev = d orev = .

Worst-case Anonymity

Theorem 1: The maximum of E[Y | XD(u)=d] over (pv)vu occurs when

1. pv=1 for all vu OR

2. pvd=1 for all vu

Let pu1 pu

2 pud-1 pu

d+1 … pu

Show max. occurs when, for all vu, pv

ev = 1 for

some ev.

Show max. occurs when ev=d for all vu, or whenev = for all vu.

Worst-case EstimatesLet n be the number of users.

Worst-case Estimates

Theorem 2: When pv=1 for all vu:

E[Y | XD(u)=d] = b + b(1-b)pud +

(1-b)2 pud [(1-b)/(1-(1- pu

)b)) + O(logn/n)]

Let n be the number of users.

Worst-case Estimates

Theorem 2: When pv=1 for all vu:

E[Y | XD(u)=d] = b + b(1-b)pud +

(1-b)2 pud [(1-b)/(1-(1- pu

)b)) + O(logn/n)]

Theorem 3: When pvd=1 for all vu:

E[Y | XD(u)=d] = b2 + b(1-b)pud +

(1-b) pud/(1-(1- pu

d)b) + O(logn/n)]

Let n be the number of users.

Worst-case Estimates

Theorem 2: When pv=1 for all vu:

E[Y | XD(u)=d] = b + b(1-b)pud +

(1-b)2 pud [(1-b)/(1-(1- pu

)b)) + O(logn/n)]

Let n be the number of users.

Worst-case Estimates

Theorem 2: When pv=1 for all vu:

E[Y | XD(u)=d] = b + b(1-b)pud +

(1-b)2 pud [(1-b)/(1-(1- pu

)b)) + O(logn/n)]

b + (1-b) pud

Let n be the number of users.

Worst-case Estimates

Theorem 2: When pv=1 for all vu:

E[Y | XD(u)=d] = b + b(1-b)pud +

(1-b)2 pud [(1-b)/(1-(1- pu

)b)) + O(logn/n)]

b + (1-b) pud

E[Y | XD(u)=d] b2 + (1-b2) pud

Let n be the number of users.

Worst-case Estimates

Theorem 2: When pv=1 for all vu:

E[Y | XD(u)=d] = b + b(1-b)pud +

(1-b)2 pud [(1-b)/(1-(1- pu

)b)) + O(logn/n)]

b + (1-b) pud

E[Y | XD(u)=d] b2 + (1-b2) pud

Let n be the number of users.

Increased chance of total compromise from b2 to b.

Typical Case

Let each user select from the Zipfian distribution: pdi

= 1/(is)

Theorem 4:E[Y | XD(u)=d] = b2 + (1 − b2)pu

d+ O(1/n)

Typical Case

Let each user select from the Zipfian distribution: pdi

= 1/(is)

Theorem 4:E[Y | XD(u)=d] = b2 + (1 − b2)pu

d+ O(1/n)E[Y | XD(u)=d] b2 + (1 − b2)pu

d

Contributions1. Use a black-box abstraction to create a

probabilistic model of onion routing

2. Analyze unlinkabilitya. Provide worst-case bounds

b. Examine a typical case

Future Work

1. Extend analysis to other types of anonymity and to other systems.

2. Examine how quickly users distribution are learned.

3. Analyze timing attacks.