hiding the rumor source in anonymous messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf ·...
TRANSCRIPT
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Hiding the Rumor Source in Anonymous MessagingSewoong Oh University of Illinois at Urbana-Champaign
Giulia Fanti P. ViswanathPeter Kairouz K. Ramchandran
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Anonymous Social Media provide meta-data privacy
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Existing anonymous messaging apps
10 m
iles
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Threat is real
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Anonymous messaging meets social filtering
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Diffusion of rumor/contagion
Social network/contact network
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Diffusion of rumor/contagion
message author
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Diffusion of rumor/contagion
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Diffusion of rumor/contagion
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Diffusion of rumor/contagion
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Diffusion of rumor/contagion
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Diffusion of rumor/contagion
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Diffusion of rumor/contagion
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can we locate the message author?
Rumor source detection
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1T =
5T = 4T = 4T =
3T =
6T =
6T =
TimingSnapshot
meta-datatime from to control10:12 Alice Spy1 101101
10:25 Bob Spy2 10100111:01 Mary Spy3 100100
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Snapshot-based adversary
16
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Snapshot reveals the source
▪ message author is likely to be in the “center”
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Rumor Centrality [Shah, Zaman ’11]
low likelihood
high likelihood
Similar performance using Jordan centrality [Zhu, Ying ’13]
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Our Goal (on infinite regular trees)
P(Detection) =
1
NT
Adaptive Diffusion
Diffusion
logNT
logP(detection)
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Setting
▪ Contact network is infinite, regular, tree▪ Adversary has the contact network and the snapshot▪ Protocol is allowed to infect any 1-hop neighbors at each time
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Line graph
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Line graph: diffusion
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Line graph: diffusion
p p
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Line graph: diffusion
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Line graph: diffusion
p p
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Line graph: diffusion
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Line graph: diffusion
p p
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Line graph: diffusion
▪ equivalent to two independent random walks
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Adversary with snapshot
nodes with the message
can we locate the message author?
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Maximum likelihood detection
Likelihoods
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Line graph: adaptive diffusion
-1-2 0 1 2
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Line graph: adaptive diffusion
each neighbor is infected w.p. 1/2
-1-2 0 1 2
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Line graph: adaptive diffusion
Node 1 receives message at T = 1
-1-2 0 1 2
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Line graph: adaptive diffusion
-1-2 0 1 2
pinfect =h+ 1
T + 1
▪ Adaptive diffusion prescribes to pass at adaptive rate
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Line graph: adaptive diffusion
▪ node 2 receives the message
-1-2 0 1 2
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Line graph: adaptive diffusion
-1-2 0 1 2
pinfect =h+ 1
T + 1
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Line graph: adaptive diffusion
▪ equivalent to Polya’s urn process▪ adaptive and asymmetric: Nodes that are infected earlier,
spread faster
-1-2 0 1 2
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Given snapshot
can we locate the message author?
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Maximum likelihood detection
Likelihoods
diffusion
adaptive diffusion
probability of detection ⇠ 1
N
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▪ [Shah & Zaman ’11]
Probability of detection using Rumor Centrality
number of nodes with the message
spread with fixed probability p
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Probability of detection using Jordan centrality
number of nodes with the message
spread with probability
p(h, t) =h+ 1
t+ 1
Analytical bound
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▪ Strategy: ▪ Design the infection to be a symmetric ball of depth T/2 ▪ Such that the source is equally likely to be any node at
any given time T
T = 4
Virtual source
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▪ initially, the author is also the virtual source▪ and randomly selects a neighbor to be next virtual source
T = 0
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T = 1
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T = 2
▪ at T=2, the virtual source passes the message to all its neighbors
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keeping the virtual source token passing the virtual source token
▪ at T=2, the protocol has two options
T = 2
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keeping the virtual source token passing the virtual source token
with probability ↵d,T,h with probability 1� ↵d,T,h
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49
Passing the virtual source token
T = 2
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50
T = 3
Passing the virtual source token
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51
Passing the virtual source token
T = 4h = 2
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Keeping the virtual source token
T = 2
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Keeping the virtual source token
T = 3
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54
Keeping the virtual source token
T = 4h = 1
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Adversary with snapshot
can the adversary locate the message author?
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Maximum likelihood estimation
low likelihood
high likelihood
▪ If virtual source is kept with prob.
▪ nodes that receive message faster, spread faster
↵d,T,h =(d� 1)
T2 +1�h � 1
(d� 1)T2 +1 � 1
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57
Theorem. [Fanti, Kairouz, Oh, Viswanath 2015]
On an infinite d-regular tree, 1. adaptive diffusion spreads fast,
2. achieves almost perfect obfuscation
3. The expected distance between the estimated and the true source is T/2
P(Detection) =
1
NT � 1
NT ' (d� 1)T/2
Maximum likelihood estimation
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58
v⇤
T = 0 dv =
⇢3 w.p. 0.55 w.p. 0.5
GT
What if the tree is irregular?
![Page 59: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/59.jpg)
59
v⇤
T = 1 dv =
⇢3 w.p. 0.55 w.p. 0.5
GT
![Page 60: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/60.jpg)
60
v⇤
T = 2 dv =
⇢3 w.p. 0.55 w.p. 0.5
GT
![Page 61: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/61.jpg)
61
v⇤
T = 3 dv =
⇢3 w.p. 0.55 w.p. 0.5
GT
![Page 62: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/62.jpg)
62
v⇤
T = 4 dv =
⇢3 w.p. 0.55 w.p. 0.5
GT
![Page 63: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/63.jpg)
63
T = 4 dv =
⇢3 w.p. 0.55 w.p. 0.5
GT
v̂ML = arg max
v2@GT
1
dvsY
w2�(vs,v)\{vs,v}
(dw � 1)
| {z }F (GT )
![Page 64: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/64.jpg)
64
T = 4 dv =
⇢3 w.p. 0.55 w.p. 0.5
GT
v̂ML = arg max
v2@GT
1
dvsY
w2�(vs,v)\{vs,v}
(dw � 1)
| {z }F (GT )
F (GT ) =1
5⇥ 2
![Page 65: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/65.jpg)
65
Does adaptive diffusion still achieve perfect obfuscation?
100 101 102 10310−3
10−2
10−1
Number of Infected Nodes (N)
Prob
abilit
y of
Det
ectio
n (P
D)
(3,5)=>(0.5,0.5), do=5(3,6)=>(0.5,0.5), do=6(3,7)=>(0.5,0.5), do=7(3,10)=>(0.5,0.5), do=101/N
dv =
⇢3 w.p. 0.55 w.p. 0.5
dv =
⇢3 w.p. 0.510 w.p. 0.5
![Page 66: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/66.jpg)
66
▪Probability of detection:
P(v̂ML = v⇤) =X
GT
P(GT )P(v̂ML = v⇤|GT )| {z }F (GT )
= E(F (GT ))
F (GT ) = max
v2@GT
1
dvsY
w2�(vs,v)\{vs,v}
(dw � 1)
Galton-Watson tree GT
![Page 67: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/67.jpg)
67
If dv
=
(dmin w.p. pmin
.
.
.
.
.
.
, and (dmin � 1)pmin � 1, then
P(v̂ML = v⇤) = EGT
"max
v2@GT
1
dvs
Y
w2�(vs,v)\{vs,v}
(dw
� 1)
#
= (dmin � 1)
�T+o(T )
100 101 102 10310−3
10−2
10−1
Number of Infected Nodes (N)
Prob
abilit
y of
Det
ectio
n (P
D)
(3,5)=>(0.5,0.5), do=5(3,6)=>(0.5,0.5), do=6(3,7)=>(0.5,0.5), do=7(3,10)=>(0.5,0.5), do=101/N
dv =
⇢3 w.p. 0.55 w.p. 0.5
dv =
⇢3 w.p. 0.510 w.p. 0.5
Corollary
![Page 68: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/68.jpg)
68
Proof idea for
dv =
⇢3 w.p. 0.55 w.p. 0.5
dv =
⇢3 w.p. 0.51 w.p. 0.5
X
X
minv2@GT
Y
w2�(vs,v)\{vs,v}
(dw
� 1) = (dmin � 1)T+o(T )
![Page 69: Hiding the Rumor Source in Anonymous Messagingswoh.web.engr.illinois.edu/slide_itw2015.pdf · Theorem. [Fanti, Kairouz, Oh, Viswanath 2015] On an infinite d-regular tree, 1.adaptive](https://reader033.vdocument.in/reader033/viewer/2022050213/5f5f60877b5bd35792329d25/html5/thumbnails/69.jpg)
Messaging App: Wildfire
Alice
anonymous, distributed, secure implementation
Bob
Mary
Alice Faith Saul
Alice Faith Mike Saul
Bob Carol Mary
Wildfire empowers devices by removing central service providersIt also has stronger anonymity properties than Secret, Whisper, and Yik Yak.
Like
Wildfire empowers devices by removing central service providers
It also has stronger anonymity properties than Secret, Whisper, and Yik Yak.
Wildfire empowers devices by removing central service providers
It also has stronger anonymity properties than Secret, Whisper, and Yik Yak.
Like
Like