On AVCs WITH Quadratic Constraints

ON AVCS WITH QUADRATIC CONSTRAINTS

Farzin HaddadpourJoint work with

Madhi Jafari Siavoshani, Mayank Bakshi and Sidharth Jaggi

Sharif University of Technology, IranISSL, EE Department

2013 ISITJuly 7, 2013

Institute of Network CodingThe Chinese University of Hong Kong

Outline

1/18

•Introduction•System Model•Relation with Prior Works

•Main Result•Proof Steps•Conclusion

Introduction

2/18

Goal: decode message

Goal: interrupt Alice’s information of their movement

Goal: transmitreliably

How can I interrupt this transmission?

Alice

Willie

Bob

System Model

3/18

Enc Deci j( )nx i

ns nV

Power Constraints: 2|| ( ) ||nx i nP2|| ||ns n

nV : i.i.d. Gaussian Vector 2(0, )

ny

Y X s V

Prior Works

4/18

[Hughes and Narayan 1988]

Enc Dec

Jammer

i

Shared common randomness

i

Message Aware Jammer

nV

Capacity Rate:2

1 log(1 )2

P

Prior Works

5/18

[Csizar and Narayan 1991]

Capacity Rate:2

1 log(1 )2

0

P

if P

otherwise

i iEnc Dec

JammernV

ns

Our Model

6/18

Enc Dec

JammernV

ns

Enc Dec

Jammer

i

Shared common randomness

i

Message Aware Jammer

nV

i i

Our Model

7/18

Enc Dec

Jammer

Private randomization

Message- aware Jamming

nV

( , )nx i t

t0n

( )ns i

•Stochastic encoding•Public code•Message-aware jamming•Oblivious adversary

ii

Main Result

8/18

Enc Dec

Jammer

i i

Private randomization

Message- aware Jamming

nV

( , )nx i t

t0n

( )ns i

2

1 log(1 )2

0

P

if

otherwise

P Theorem(Capacity Rate):

Achievability Proof

9/18

•Codebook : 11

2 ... 0ne

2

.

nRe

..

(1,1)x 0(1, )nx e

( ,1)nRx e 0( , )n nRx e e

( , )x i t ( , '')x i t

( , ')x j tError

No Error

•Intuition : Because of our error probability we take average over colored row otherwise Csizar’s approach which has averaging over whole codewords

•Note: Decoder uses ML decoding

( )0i

y

if for

2 2|| || || || ,i jy x y x

:1i i M

i j

if no such exists

Achievability Proof

10/18

•Based on this Criteria error probability is:2 2( , ) [|| ( , ) ( , ') || || ||T Ve s i p p x i T s V x j t s V

for some andi j ']t

[ ( , ) , ( , ') ( , ),T Vp p x i T s V x j t nP x i T s V for some andi j ']t

•Lemma1: fix vector then for every and

*{(,)} Xit

( , )X i t uniformly distributed over (0, )n nP

1 ,nRi e 01 nt e

(0, )ns n 1 00

n 2

1 log(1 )2

PR

for large if

1* 0 1[ ( , ) ] exp( ( log 2 10)exp(( ) ))np e s i Ke K n

Achievability Proof(Lemma1)•Proof of Lemma 1 :

Lemma A1 [Csizar and Narayan 1991] : Let be arbitrary r.v.’s and be arbitrary function with then the condition a,s, implies

1. Using Lemma A1 and taking

we have

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1,..., LX X

1( ,..., )i if X X 0 1, 1,...,if i L 1 1 1[ ( ,..., ) | ,..., ]i i iE f X X X X a 1,...,i L

1

1[ ( ( ,1),..., ( , )) ] exp( ( log 2 ))L

tt

p f X i X i t L aL

( ( ,1),..., ( , )) [ ( , ) , ( , ') ( , ),t Vf X i X i t p x i T s V x j t nP x i T s V

for some andi j 1'] ]nt Ke * [ [ ( , ) , ( , ') ( , ),T Vp p p x i T s V x j t nP x i T s V

0

*0

1

1[ [ [ ( , ) , ( , ') ( , ),ne

Vnt

p p x i t s V x j t nP x i t s Ve

for some andi j ']t

for some andi j 1'] ]nt Ke 0

1*

01

1[ [ ( ( ,1),..., ( , ))] ]ne

ntn

t

p f X i X i t Kee

Achievability Proof(Lemma1)2. So it remains to bound

Where (a) follows by .

12/18

1 1 1[ ( ,..., ) | ,..., ]i i iE f X X X X

*

( , '):

[ [ { ( , '), ( , ) ( , ), }]]Vj t j i

p p X j t X i t s V P X i t s V

*

)

(

(

)

2

1

[ ( , ), ]a

Vp p X i t s V

* 2( , '):

(2)

[ { ( , '), ( , ) ( , ), }, ( , ), ]]Vj t j i

p p X j t X i t s V P X i t s V X i t s V

[ ] [ [ ]] [ ] [ ]p p p p

Achievability Proof(Lemma1)

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Then terms (1) and (2) can be upper bounded using this Lemma.

U

u

1( )2 2[| , | ] 2(1 )n

p U u

1 1

2 n

Lemma [Csizar and Narayan 1991]: u is a fix vector and U is distributed uniformly over sphere and for have

Achievability Proof(Lemma2)

14/18

Lemma 2(Quantizing Adversarial Vector): for a fixed vector , sufficient small and for every there exists a fixed codebook with rate which also does well for every .

Proof of Lemma 2: choosing where is a random vector over unit sphere and , then we can show that

2

1 log(1 )2

PR

' ( , )ns s

's s u u[ , ] 1( ', ) exp( )e s i K n

ssX V

1 00 { ( , )}X i ts

Achievability Proof(Lemma3)

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Lemma 3(Codebook Existence): For every and enough large , there exist a fixed codebook with rate such that forevery vector , and every transmitted message :

Proof of Lemma 3: It’s enough to show that

But using Lemma 2 we don’t need to check for every but only for that covers , therefore we can write

1 00 n{ ( , )}X i t

2

1 log(1 )2

PR

s i

1( , ) ne s i Ke

* * 1lim inf [ , ( , ) exp( )] 0n p s i e i s K n

s net n (0, )n n

* * * *1 1[ , ( , ) exp( )] 1 [ , ( , ) exp( )]n np s i e i s K n p s i e i s K n

* *

( )

11

1 [ ( , ) exp( )]nR

n

ea

si

p e i s K n

Union bound

Achievability Proof(Lemma3)Consider this figure for upper bounding the Cardinality of

16/18

net n

* * 1[ , ( , ) exp( )]np s i e i s K n

0 121 ( ) exp( ) exp( 'exp( ( )))nn nR K n

Conclusion

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Such as Discrete Scenarios Using

Stochastic Encoder won’t Improve Capacity Region

THANKS FOR CONSIDERATIONAny

Questions?