The Chinese University of Hong Kong The Institute of Network Coding Pak Hou Che Mayank Bakshi Sidharth Jaggi Alice Reliability Bob Willie the Warden Reliability Deniability ID: 260628
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Slide1
Reliable Deniable Communication: Hiding Messages in Noise
The Chinese University
of Hong Kong
The Institute of
Network Coding
Pak
Hou
Che
Mayank
Bakshi
Sidharth
JaggiSlide2
Alice
Reliability
BobSlide3
Willie
(the Warden)
Reliability
Deniability
Alice
BobSlide4
M
T
t
Alice’s Encoder
Slide5
M
T
Message
Trans. Status
BSC(
p
b
)
Alice’s Encoder
Bob’s Decoder
Slide6
M
T
Message
Trans. Status
BSC(
p
b
)
Alice’s Encoder
Bob’s Decoder
BSC(p
w
)
Willie’s (Best) Estimator
Slide7
Hypothesis Testing
Willie’s Estimate
Alice’s
Transmission Status
Slide8
Hypothesis Testing
Willie’s Estimate
Alice’s
Transmission Status
Want:
Slide9
Hypothesis Testing
Willie’s Estimate
Alice’s
Transmission Status
Want:
Known:
for opt. estimator
Slide10
Hypothesis Testing
Willie’s Estimate
Alice’s
Transmission Status
Want:
Known:
for opt. estimator
,
(
w.h.p.
)
Slide11
Bash, Goeckel & Towsley
[1]
Shared secret
[1] B. A. Bash, D.
Goeckel
and D.
Towsley
,
“Square root law for communication with low probability
of detection
on
AWGN
channels,”
in Proceedings of the IEEE International Symposium on
Information Theory
(ISIT)
, 2012, pp. 448–452.
AWGN channels
Capacity
=
bits
bits
Slide12
This work
No shared secret
BSC(p
b
)
BSC(p
w
)
p
b
< p
wSlide13
Intuition
Slide14
Intuition
Slide15
Main Theorems
Theorem 1Deniability low weight
codewordsTheorem 2 Converse of reliability
Theorem 3Achievability (reliability & deniability)Theorem 4Trade-off between deniability & size of codebookSlide16
Theorem 1 (wt(c.w
.))(high deniability => low weight codewords)
Slide17
Theorem 2 (Converse)
, if
if
Slide18
Theorems 2 &
3(Converse
& achievability for reliable & deniable comm.)Slide19
Theorems 2 & 3
0
1/2
1/2
p
b
>p
wSlide20
Theorems 2 & 3
0
1/2
1/2
Slide21
Theorems 2 & 3
0
1/2
1/2
p
w
=1/2Slide22
Theorems 2 & 3
0
1/2
1/2
(
BSC(p
b
))Slide23
Theorems 2 & 3
0
1/2
1/2
p
b
=0Slide24
Theorems 2 & 3
0
1/2
1/2
Slide25
Theorems 2 & 3
0
1/2
1/2
p
w
>
p
bSlide26
Theorems
2
& 3
0
1/2
1/2
“Standard” IT inequalities
+
wt
(“most
codewords
”)<
(
Thm
1)
Slide27
Theorems 2 &
3
0
1/2
1/2
Achievable region
Main
thm
:Slide28Slide29
Theorem 3 – Reliability
Random codebook ( i.i.d.
) )
minimum
distance decoderFor
,
Slide30
Theorem 3 – Reliability proof sketch
.
.
.
1000001000000000100100000010000000100
0001000000100000010000000010000000001
0010000100000001010010000000100010011
0000100000010000000000010000000010000
Random code
codewords
Weight
Slide31
Theorem 3 – Reliability proof sketch
.
.
.
1000001000010000100100000010000000100
0001000000100000010000000010000000001
0010000100000001010010000000100010011
0000100000010000000000010000000010000
E(Intersection of 2
codewords
) = O(1)
“Most”
codewords
“well-isolated”
Weight
Slide32
Theorem 3 – Reliability proof sketch
.
.
.
1000001000010000100100000010000000100
0001000000100000010000000010000000001
0010000100000001010010000000100010011
0000100000010000000000010000000010000
E(Intersection of 2
codewords
) = O(1)
“Most”
codewords
“well-isolated”
Weight
Slide33
Theorem 3 – dmin
decoding
Pr(x decoded to
x’) <
+
x
x
’Slide34
0
n
l
ogarithm of
# binary vectors
Slide35
0
n
l
og(#
vector
s
)
Slide36
l
og(#
vectors)
Slide37
l
og(#
codewords
)
Slide38
0
n
l
og(#
vectors)
Slide39
Recall: want to show
Theorem 3 –
Den
iability
proof sketchSlide40
Recall: want to show
Theorem 3 –
Den
iability
proof sketchSlide41
0
n
l
og(#
vectors)
Theorem 3 – Deniability proof sketchSlide42
0
n
l
ogarithm of
#
codewords
Theorem 3 –
Den
iability
proof sketchSlide43
!!!
Theorem 3 –
Den
iability
proof sketchSlide44
!!!
Theorem 3 –
Den
iability
proof sketchSlide45
Theorem 3 –
Den
iability
proof sketch
with high probability
Slide46Slide47
0
n
l
ogarithm of
#
vectors
Theorem 3 – Deniability proof sketchSlide48
0
n
l
ogarithm of
#
vectors
Theorem 3 – Deniability proof sketchSlide49
# codewords of “type”
Theorem 3 –
Den
iability
proof sketchSlide50
Theorem 3 –
Den
iability
proof sketch
Slide51
Theorem 3 –
Den
iability
proof sketch
Slide52
Theorem 3 –
Den
iability
proof sketch
Slide53
Theorem 3 –
Den
iability
proof sketch
Slide54
w.p
.
Theorem 3 –
Den
iability
proof sketch
Slide55
w.p
.
close to
w.p.
,
w.h.p
.
Theorem 3 –
Den
iability
proof sketch
Slide56
Theorem 4
0
n
l
ogarithm of
#
codewordsSlide57
0
n
l
ogarithm of
#
codewords
Too
few
codewords
=> Not deniable
Theorem 4Slide58
Summary
0
1/2
1/2
Thm
1 & 2 Converse
(Upper Bound)
Thm
3 Achievability
Thm
4 Lower BoundSlide59
SummarySlide60
Summary