wits08

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Modeling and Analysis of Anonymous-Communication Systems

Joan Feigenbaumhttp://www.cs.yale.edu/homes/jf

WITS’08; Princeton NJ; June 18, 2008

Acknowledgement: Aaron Johnson

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Outline

• Anonymity: What and why

• Examples of anonymity systems

• Theory: Definition and proof

• Practice: Onion Routing

• Theory meets practice

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Anonymity: What and Why

• The adversary cannot tell who is communicating with whom. Not the same as confidentiality (and hence not solved by encryption).

• Pro: Facilitates communication by whistle blowers, political dissidents, members of 12-step programs, etc.

• Con: Inhibits accountability

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Outline






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Anonymity Systems

• Remailers / Mix Networks– anon.penet.fi– MixMaster– Mixminion

• Low-latency communication– Anonymous proxies, anonymizer.net– Freedom– Tor– JAP

• Data Publishing– FreeNet

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Mix Networks

• First outlined by Chaum in 1981

• Provide anonymous communication– High latency– Message-based (“message-oriented”)– One-way or two-way

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Mix Networks

Users Mixes Destinations

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Mix NetworksAdversary


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Mix Networks


Protocol

Adversary

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Mix Networks

1. User selects a sequence of mixes and a destination.

M1

M2

M3

u d

Protocol

Adversary


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Mix Networks


2. Onion-encrypt the message.

M1

M2

M3

u d

Protocol

Adversary


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Mix Networks



M1

M2

M3

u d

Protocol Onion Encrypt1. Proceed in reverse order

of the user’s path.

2. Encrypt (message, next hop) with the public key of the mix.

Adversary


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Mix Networks



M1

M2

M3

u d




{{{,d}M3,M3}M2

,M2}M1

Adversary


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Mix Networks



3. Send the message, removing a layer of encryption at each mix.

M1

M2

M3

u d




{{{,d}M3,M3}M2

,M2}M1

Adversary


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Mix Networks




M1

M2

M3

u d




{{,d}M3,M3}M2

Adversary


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Mix Networks




M1

M2

M3

u d




{,d}M3

Adversary


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Mix Networks




M1

M2

M3

u d




Adversary


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Mix Networks

u d

Adversary

Anonymity?

1. No one mix knows both source and destination.


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Mix Networks

u d

Adversary

Anonymity?


2. Adversary cannot follow multiple messages through the same mix.

v

f


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Mix Networks

u d

Adversary

Anonymity?


2. Adversary cannot follow multiple messages through the same mix.

3. More users provides more anonymity.

v e

w f


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Outline






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Provable Anonymity in Mix Networks

• N users

• Passive, local adversary

– Adversary observes some of the mixes and the links.

– Fraction f of links are not observed by adversary.

• Users and mixes are roughly synchronized.

• Users choose mixes uniformly at random.

Setting

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• Users should be unlinkable to their destinations.

• Let be a random permutation that maps users to destinations.

• Let C be the traffic matrix observed by the adversary during the protocol. Cei = # of messages on link e in round i

Definition

e1

e2

1 2 3 4 5

1 0

0 1

0

1

1 1

0 0

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• Use information theory to quantify information gain from observing C.

• H(X) = x -Pr[X=x] log(Pr[X=x]) is the entropy of r.v. X

• I(X : Y) is the mutual information between X and Y.

• I(X : Y) = H(X) – H(X | Y) = x,y -Pr[X=xY=y] log(Pr[X=xY=y])


Information-theory background

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Provable Anonymity in Synchronous Protocols

Definition: The protocol is (N)-unlinkable if I(C : ) (N).

Definition: An (N)-unlinkable protocol is efficient if:

1. It takes T(N) = O(polylog(N/(N))) rounds.

2. It uses O(NT(N)) messages.

Theorem (Berman, Fiat, and Ta-Shma, 2004): The basic mixnet protocol is (N)-unlinkable and efficient whenT(N) = (log(N) log2(N/(N))).

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Outline






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Onion Routing [GRS’96]

• Practical design with low latency and overhead

• Connection-oriented, two-way communication

•

• Open source implementation (http://tor.eff.org)

• Over 1000 volunteer routers

• Estimated 200,000 users

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How Onion Routing Works

User u running client Internet destination d

Routers running servers

u d

1 2

3

45

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u d

1 2

3

45

1. u creates 3-hop circuit through routers (u.a.r.).

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u d

1 2

3

45


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u d

1 2

3

45


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2. u opens a stream in the circuit to d.

u d

1 2

3

45

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3. Data are exchanged.

{{{}3}4}1

u d

1 2

3

45

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{{}3}4

u d

1 2

3

45

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{}3

u d

1 2

3

45

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u d

1 2

3

45

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’u d

1 2

3

45

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{’}3

u d

1 2

3

45

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{{’}3}4u d

1 2

3

45

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{{{’}3}4}1

u d

1 2

3

45

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4. Stream is closed.

u d

1 2

3

45

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4. Stream is closed.

5. Circuit is changed every few minutes.

u d

1 2

3

45

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Adversary

u

1 2

3

45

d

Active & Local

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Outline






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Formal Analysis(F., Johnson, and Syverson, 2007)

u 1 2

3

45

d

v

w

e

f

1.

2.

3.

4.

Timing attacks result in four cases:

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1. First router compromised

2.

3.

4.


u 1 2

3

45

d

v

w

e

f


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2. Last router compromised

3.

4.


u 1 2

3

45

d

v

w

e

f


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3. First and last compromised

4.


u 1 2

3

45

d

v

w

e

f


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3. First and last compromised

4. Neither first nor last compromised


u 1 2

3

45

d

v

w

e

f


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Black-Box, Onion-Routing Model

Let U be the set of users.

Let be the set of destinations.

Let the adversary control a fraction b of the routers.

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)]

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Indistinguishabilityu dvw

ef

u dvw

ef

u dvw

ef

u dvw

ef

Indistinguishable configurations

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Indistinguishabilityu dvw

ef

u dvw

ef

u dvw

ef

u dvw

ef

Indistinguishable configurations

Note: Indistinguishable configurations form an equivalence relation.

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Probabilistic Anonymity

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

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

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Note: This is different from the metric of mutual information used to analyze mix nets.

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Exact Bayesian inference

• Adversary after long-term intersection attack

• Worst-case adversary

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Exact Bayesian inference

• Adversary after long-term intersection attack

• Worst-case adversary

Linkability given that u visits d:

E[Y | XD(u)=d]

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Anonymity Bounds

1. Lower bound:E[Y | XD(u)=d] b2 + (1-b2) pu

d

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Anonymity Bounds


d

2. Upper bounds:a. pv

=1 for all vu, where pv pv

e for e d

b. pvd=1 for all vu

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Anonymity Bounds


d

2. Upper bounds:a. pv

=1 for all vu, where pv pv

e for e d

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

b. pvd=1 for all vu

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

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Lower Bound

Theorem 2: E[Y | XD(u)=d] b2 + (1-b2) pud

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Lower Bound


Proof:

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Lower Bound


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

d + (1-b) E[Y | XD(u)=d XI(u)=0]

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Lower Bound



d + (1-b) E[Y | XD(u)=d XI(u)=0]

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Lower Bound


Let {Ci} be the configuration equivalence classes.Let Di be the event Ci XD(u)=d.

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Lower Bound


Let {Ci} be the configuration equivalence classes.Let Di be the event Ci XD(u)=d.E[Y | XD(u)=d XI(u)=0]

= i (Pr[Di])2

Pr[Ci] Pr[XD(u)=d]

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Lower Bound



= i (Pr[Di])2

Pr[Ci] Pr[XD(u)=d]

(i Pr[Di] Pr[Ci] / Pr[Ci])2

Pr[XD(u)=d]by Cauchy-Schwarz

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Lower Bound



= i (Pr[Di])2

Pr[Ci] Pr[XD(u)=d]

(i Pr[Di] Pr[Ci] / Pr[Ci])2

Pr[XD(u)=d]

= pud

by Cauchy-Schwarz

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Lower Bound



d + (1-b) E[Y | XD(u)=d XI(u)=0]

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Lower Bound



d + (1-b) E[Y | XD(u)=d XI(u)=0] b2 + b(1-b) pu

d + (1-b) pud

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Lower Bound



d + (1-b) E[Y | XD(u)=d XI(u)=0] b2 + b(1-b) pu

d + (1-b) pud

= b2 + (1-b2) pud

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Upper Bound

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Upper Bound

Theorem 3: 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

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Upper Bound



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.

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

Upper Bound



2. pvd=1 for all vu

Let pu1 pu

2 pud-1 pu

d+1 … pu


ev = 1 for

some ev.

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

Upper Bound



2. pvd=1 for all vu

Let pu1 pu

2 pud-1 pu

d+1 … pu


ev = 1 for

some ev.

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

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Upper-bound EstimatesLet n be the number of users.

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Upper-bound Estimates

Theorem 4: 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.

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

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

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

Theorem 5: 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)]


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

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

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


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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


For pu small

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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


For pu small

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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


Increased chance of total compromise from b2 to b.

For pu small

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Conclusions

• Many challenges remain in the design, implementation, and analysis of anonymous-communication systems.

• It is hard to prove theorems about real systems – or even to figure out what to prove.

• “Nothing is more practical than a good theory!” (Tanya Berger-Wolfe, UIC)

wits08

Documents

sequence of mixes

reverse order

users path

public key

onion routingtheory

layer of encryption

analysis of anonymous

princeton nj