Secrecy in the 2-User Symmetric Interference Channel with Transmitter Cooperation: Deterministic View

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1 Secrecy in the 2-User Symmetric Interference Channel with Transmitter Cooperation: Deterministic View P. Mohapatra 9 th March 2013

2 Outline Motivation Problem statement Achievable scheme 1 Weak interference regime 2 Moderate interference regime 3 Very high interference regime (for a specific case) Summary Outer bound (if time permits)

3 Motivation Interference in wireless network Limits the communication rate Allows users to eavesdrop other user s signal Secrecy: important concern in wireless network Cellular network Support high throughput and secure its transmissions Is it possible to get both the benefits? Answer these questions using information theoretic approach

4 Problem statement Users are not completely isolated (e.g.: base stations) Users can cooperate Effectiveness of limited transmitter cooperation in a 2-user symmetric linear deterministic interference channel Interference management Secrecy

5 Why deterministic model Good approximation of Gaussian model at high SNR Give insights into achievable schemes and outer bounds Not feasible in deterministic model: may not be feasible in Gaussian case

6 Deterministic model Tx 1 Rx 1 m C G W1 C G hd hc Ŵ1 Ŵ2 C C Tx 1 n n Rx 1 hc Ŵ2 W2 hd Ŵ1 Tx 2 Rx 2 Tx 2 Rx 2 m Figure: Symmetric GIC Figure: SLDIC m ( log h d 2 ) +, n ( log h c 2 ) + and C C G α n m a i and b i : data bits of transmitter 1 and 2 d i and e i : random bits of transmitter 1 and 2 Data bits and random bits: Bern ( ) 1 2

7 System model Input-output equation y 1 = D q m x 1 D q n x 2 ; y 2 = D q m x 2 D q n x 1 x i and y i : binary vectors of length q max{m,n} D: q q downshift matrix with elements { 1 if 2 j = k +1 q d j,k = 0 otherwise Encoding: x i = f(w i,w r i,v ij) Decoding: solving the set of linear equation Receiver does not require the knowledge of the random bits

8 System model Cooperative links: lossless but of finite capacity Perfect secrecy I(W i ;y j ) = 0, (i j) H(W i ) = H(W i /y j ) Transmitters completely trust each other

9 Weak interference regime (0 α 2 3 ) Achievable scheme Interference cancelation Uncoded bit transmission C = 0 Tx 1 a 2 a 1 a 2 a 1 Rx 1 Tx 2 b 2 b 1 b 2 b 1 Rx 2 Figure: SLDIC with m = 4, n = 2 and C = 0: R S = 2

10 C = 2 a 4 a 4 3 a 3 b 4 a 2 a b 3 a 2 1 a 1 Tx 1 Rx 1 b 4 3 a 4 b 2 a 3 b 1 Tx 2 b 4 b 3 b 2 b 1 Rx 2 Figure: SLDIC with m = 4, n = 2 and C = 4: R S = 4

11 Encoding scheme Encoding for message of transmitter 1 [ ] 0(m (r+c)) x 1 = + 1 a (r+c) 1 [ 0(m C) 1 b c C 1 ] where a [a r+c,a r+c 1,...,a 1 ] T, b c [b r+c,b r+c 1,...,b r+1 ] T and r m n Achievable rate R S = m n }{{} + min{n,c} }{{} uncoded transmission interference cancelation

12 Result Rate (R) without secrecy constraint with secrecy constraint Capacity of the cooperative link (C) Figure: Achievable rate of the SLDIC with m = 4 and n = 2 When 0 α 1 2, one obtains secrecy for free

13 Moderate interference regime ( 2 3 < α < 1) Achievable scheme Interference cancelation Random bit transmission n levels C = 0 a 1 a 1 Tx 1 Rx 1 Tx 2 b1 b 1 Rx 2 Figure: SLDIC with m = 5, n = 4 and C = 0 Possible to transmit at least m n bits securely

14 a 5 d 4 a 5 d 4 b 5 e 4 C = 0 a 1 a 1 Tx 1 Rx 1 b 5 b 5 e 4 e 4 d 4 a 5 Tx 2 b1 b 1 Rx 2 Figure: SLDIC with m = 5, n = 4 and C = 0: R S = 2 Possible to transmit in the upper levels: random bits transmission

15 With cooperation C = 1 a 5 a 5 d 4 d 4 b 5 e 4 a 2 a 2 b 2 a 1 a 1 Tx 1 Rx 1 b 5 b 5 e 4 e 4 d 4 a 5 b 2 a 2 b 1 Tx 2 b 2 b 1 Rx 2 Figure: SLDIC with m = 5, n = 4 and C = 1: R S = 3 Uses combination of interference cancelation and random bit transmission Need to determine the number of data bits that can be transmitted with the help of random bit transmission

16 Encoding scheme Data transmission with the help of random bits transmission Data bits Random bits Zero bits m n m n m n B g 3(m n) g {n (r 2 +C)} + and t g%{3(m n)}, where q min{(t r 2 ) +,r 2 }: number of data bits that can be securely sent on the remaining t levels

17 Encoding scheme Encoding of transmitter 1 message when q = 0 [ ] [ ] ] 0(m (r2 +C)) x 1 = + 1 0(m C) 1 a u [ p 1 0 p 1 a (r2 +C) 1 b c C 1 a u [u 1,d 2,z 3,...,u 3B 2,d 3B 1,z 3B,] T, u l [ a m (l 1)r2,a m (l 1)r2 1,...,a m lr2 +1], d l [ d m (l 1)r2,d m (l 1)r2 1,...,d m lr2 +1], z l is a zero vector of size 1 r 2, p 3B(m n) Achievable rate R S = m n+ B(m n)+q }{{} random bits transmission +min{n,c}

18 Result 6 5 Rate (R) 4 3 without secrecy constraint with secrecy constraint Capacity of the cooperative link (C) Figure: Achievable rate of the SLDIC with m = 6 and n = 5 Gap may be due to the secrecy constraint

19 Very high interference regime (α 2) Scheme for α = 2 and even valued m Achievable scheme Relaying of the other user s data bits Time sharing Techniques used in the moderate interference regime Involves sharing of random bits/data bits or both

20 Achievable scheme When C = 0 Not possible to achieve perfect secrecy We will look at the outer bound (if time permits) When 0 < C m 2 Only random bit sharing When m 2 < C < 3m 2 Data bits and random bits sharing When 3m 2 C n Data bits sharing

21 When 0 < C m : m = 2, n = 4 and C = 1 2 C = 1 a 2 d 1 b 2 e 1 a 1 e 1 b 1 d 1 e 1 a 2 d 1 a 1 Tx 1 Rx 1 C = 1 b 1 a 1 Tx 1 Rx 1 2 e 1 a 2 d 1 b 1 d 1 a 1 e 1 d 1 b 2 e 1 b 1 Tx 2 Rx 2 a 1 Tx 2 b 1 Rx 2 Figure: With random bits sharing Figure: With data bits sharing

22 When m 2 < C < 3m 2 : m = 2, n = 4 and C = 2 Data bits of Tx-1 random bits Data bits of Tx-2 random bits m 2 random bits and C 1 = C m 2 data bits Data bits of Tx-2 random bits Data bits of Tx-2 random bits

23 m = 2, n = 4 and C = 2 b 4 b 2 e 1 a 1 e 1 b 1 d 1 e 1 a 2 d 1 a 1 Tx 1 Rx 1 C = 2 C = 2 a 2 d 1 a 4 a 1 e 1 b 1 d 1 a 4 b 2 a 2 d 1 a 1 Tx 1 Rx 1 b 2 e 1 b 4 b 1 d1 a 1 e 1 b 4 a 2 b 2 e 1 b 1 Tx 2 Rx 2 a 4 b 1 d 1 d 1 e 1 Tx 2 a 2 d 1 a 1 e 1 b 2 b 1 Rx 2 Figure: First time slot Figure: Second time slot Achieves: R S = 2.5

24 Result 4 3 Rate (R) 2 1 without secrecy constraint with secrecy constraint Capacity of the cooperative link (C) Figure: Achievable rate of the SLDIC with m = 2 and n = 4 Possible to achieve non-zero secrecy rate with cooperation

25 Summary When 0 α 1 2 : secrecy comes for free When C = n and α 1: the proposed scheme achieves the maximum possible rate of max{m, n} In all the interference regimes, the proposed scheme always achieves nonzero secrecy rate with cooperation, except for the α = 1 case Very high interference regime α 2: sharing random bits, data bits or both found to be useful

26 Outer bound (α = 1) Based on: Fano s inequality and secrecy constraints C Tx 1 Rx 1 Tx 2 Rx 2 Figure: SLDIC with m = n = 4 y 1 = y 2 = x 1 x 2 nr 1 I(W 1 ;y1 n n = I(W 1 ;y2)+nǫ n n or R 1 = 0

27 α = 2 x 1a x 2a C = 0 x 1b x 1a x 2b Tx 1 Rx 1 x 2a x 2b Tx 2 x 1b x 2a x 1b Rx 2 y 1a = x 2a y 2a = x 1a y 1b = x 1a x 2b y 2b = x 2a x 1b

28 Outline of the proof nr 1 I(W 1,y1 n )+nǫ = I(W 1,y1) I(W n 1,y2)+nǫ n = I(W 1,y1,x n n 2) I(W 1,y2)+nǫ n = I(W 1,y1a,y n 1b n xn 2) I(W 1,y2a,y n 2b n )+nǫ = H(x n 1a ) H(xn 1a W 1) I(W 1 ;y2a n ) I(W 1;y2b n yn 2a )+nǫ I(W 1 ;x n 1a ) I(W 1;x n 1a )+nǫ or R 1 = 0

Outer Bounds on the Secrecy Capacity Region of the 2-user Z Interference Channel With Unidirectional Transmitter Cooperation

Outer Bounds on the Secrecy Capacity Region of the 2-user Z Interference Channel With Unidirectional Transmitter Cooperation Parthajit Mohapatra, Chandra R. Murthy, and Jemin Lee itrust, Centre for Research