Ahmed S. Mansour, Rafael F. Schaefer and Holger Boche. June 9, 2015

Transcription

1 The Individual Secrecy Capacity of Degraded Multi-Receiver Wiretap Broadcast Channels Ahmed S. Mansour, Rafael F. Schaefer and Holger Boche Lehrstuhl für Technische Universität München, Germany Department of Electrical Engineering Princeton University, USA June 9, 2015 Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 1

2 Outline TU München 1 Introduction 2 Joint Secrecy Capacity Region 3 Individual Secrecy Capacity Region 4 Comparison of Secrecy Capacity Region 5 Conclusion & Future Work Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 2

3 Outline TU München 1 Introduction 2 Joint Secrecy Capacity Region 3 Individual Secrecy Capacity Region 4 Comparison of Secrecy Capacity Region 5 Conclusion & Future Work Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 3

4 Information Theoretic Security Secret Key encoding [Shannon 49] M = M K, Secrecy Requirement: I(M; Z n ) = 0. Transmit x n (m ) H(M) H(K) Wiretap Random Coding [Csiszár/Körner 78] Randomization message m r to confuse the eavesdropper. Secrecy Requirement: lim n I(M; Z n ) = 0. Transmit x n (m, m r ) R r I(X; Z). Random Coding + Secret Key encoding [Kang/Liu 10] m plays a role in confusing the eavesdropper. Secrecy Requirement: lim n I(M; Z n ) = 0. Transmit x n (m, m, m r ) R r + R I(X; Z). C. Shannon, Communication theory of secrecy systems, vol. 28, pp , Oct I. Csiszár and J. Körner, Broadcast channels with confidential messages, IEEE Trans. Inf. Theory, vol. 24, no. 3, pp , May 1978 W. Kang and N. Liu, Wiretap channel with shared key, Dublin, Ireland, Sep. 2010, pp. 1 5 Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 4

5 Degraded Multi-Receiver Wiretap BC (M 1,..., M k) X T n x Encoder Channel Yn 1 Y2 n Y3 n Yk n Z R 1 R 2 R n k R Z ˆM 1 ˆM 2 ˆM k Important Feature: X Y 1 Y 2 Y k Z Reliability Requirement: P e (C n ) P [ ˆM1 M 1 or... or ˆM k M k ]. Secrecy Requirement: Joint Secrecy: I(M 1,...,M k ; Z n ) τ n R j 1 n I(M j; Y n j ) + γ j (ɛ n ) Properties?, Which one? and Why??? Individual Secrecy: k j=1 I(M j; Z n ) τ n Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 5

6 Joint Secrecy Vs Individual Secrecy 1 Joint Secrecy: I(M 1,...,M k ; Z n ) τ n Focus of most previous works. Receivers do not trust each other. Conservative and immune against compromised receivers. Vanishing capacity for eavesdropper with statistical advantage. 2 Individual Secrecy: k j=1 I(M j; Z n ) τ n BC with message cognition. [Mansour/Schaefer/Boche 15] Mutual trust and Receiver cooperation. less conservative, secrecy might fails. Higher throughput is expected. How to encode? Use messages as secret keys. A. S. Mansour, R. F. Schaefer, and H. Boche, Capacity regions for broadcast channels with degraded message sets and message cognition under different secrecy constraints, CoRR, vol. abs/ , January 2015, [Online] Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 6

7 Outline TU München 1 Introduction 2 Joint Secrecy Capacity Region 3 Individual Secrecy Capacity Region 4 Comparison of Secrecy Capacity Region 5 Conclusion & Future Work Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 7

8 Joint Secrecy: Achievability and Converse Theorem 1: Capacity Region [Ekrem/Ulukus/ 09] The joint secrecy capacity region of the degraded multi-receiver wiretap BC is given by all rate tuples (R 1,..., R k ) R k + R j I(U j ; Y j U j+1 ) I(U j ; Z U j+1 ) where U 1 = X, U k+1 = and the union is taken over all random variables such that, U k U 2 X Y 1 Y 2 Y k Z. Achievability: Superposition encoding with Random binning. Converse: Standard Techniques + Prop. of Markov chain. For a two-receiver wiretap BC: R 2 I(U; Y 2 ) I(U; Z) R 1 I(X; Y 1 U) I(X; Z U) E. Ekrem and S. Ulukus, Secrecy capacity of a class of broadcast channels with an eavesdropper, EURASIP J. Wirel. Commun. Netw., pp. 1 29, March 2009 Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 8

9 Outline TU München 1 Introduction 2 Joint Secrecy Capacity Region 3 Individual Secrecy Capacity Region 4 Comparison of Secrecy Capacity Region 5 Conclusion & Future Work Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 9

10 Individual Secrecy: Achievability Theorem 2: Degraded Two-Receiver Wiretap BC The individual secrecy capacity region of the degraded two-receiver wiretap BC is given by the union of all rate pairs (R 1, R 2 ) R 2 + that satisfy R 2 I(U; Y 2 ) I(U; Z) R 1 I(X; Y 1 U) + I(U; Z) R 1 I(X; Y 1 U) I(X; Z U) + R 2 where the union is taken over all random variables (U, X), such that U X Y 1 Y 2 Z forms a Markov chain. Achievability: Random coding + Secret Key encoding. Y 1 can decode m 2 : Prop. of degraded BC. m 2 is used a secret key for Y 1. Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 10

11 Individual Secrecy: Multi-Receiver Wiretap BC Theorem 3: Capacity Region The individual secrecy capacity region of the degraded multireceiver wiretap BC is given by the union of all rate tuples (R 1,..., R k ) R k + that satisfy R j I(U j ; Y j U j+1 ) I(U j ; Z U j+1 ) + R j I(U j ; Y j U j+1 ) + I(U j+1 ; Z) k k R l I(U l ; Y l U l+1 ) l=j l=j k l=j+1 where U 1 = X, U k+1 = and the union runs over all random variables such that, U k U 2 X Y 1 Y 2 Y k Z. R l Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 11

12 Bounds Implication and Converse 1 R j I(U j ; Y j U j+1 ) I(U j ; Z U j+1 ) + k l=j+1 R l. Total Rate = Random Coded + Secret Key Encoded. Adapting the Joint converse by k l=j+1 R l. 2 R j I(U j ; Y j U j+1 ) + I(U j+1 ; Z) A User s Rate = Information in his own layer + Randomization Indcies of [ Lower Layers. R j 1 n I(M j ; Yj n M j+1 ) + I( M ] j+1 ; Z n ) + γ j (ɛ n ) 3 k l=j R l k l=j I(U l; Y l U l+1 ) A Randomization index can be used only once. Reliability Bound as in Degraded BC without secrecy. Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 12

13 Outline TU München 1 Introduction 2 Joint Secrecy Capacity Region 3 Individual Secrecy Capacity Region 4 Comparison of Secrecy Capacity Region 5 Conclusion & Future Work Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 13

14 Gaussian Two-Receiver Wiretap BC Theorem 4: Capacity Region: Joint Vs Individual Consider a two-receiver Gaussian wiretap BC, the joint secrecy capacity region is given by ( ) ( ) R 2 f 1 + ᾱp f 1 + ᾱp αp +σ2 2 αp +σ ( ) ( ) Z 2 R 1 f 1 + αp f 1 + αp, σ1 2 σz 2 while the individual secrecy capacity region is given by ( ) ( ) R 2 f 1 + ᾱp f 1 + ᾱp αp +σ2 2 αp +σ ( ) ( ) Z 2 R 1 f 1 + αp + f 1 + ᾱp σ1 2 αp +σ ( ) ( ) Z 2 R 1 f 1 + αp f 1 + αp + R σ1 2 σz 2 2, where f(x) = 1 2 log(x) and ᾱ = 1 α. Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 14

15 Example TU München Joint Secrecy Capacity Individual Secrecy Capacity Sweep over α. σ 2 1 = 0.05, σ 2 2 = 0.1, σ 2 Z = Higher throughput for Individual Secrecy R 1 > 0, although α = 0. R R 1 Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 15

16 Outline TU München 1 Introduction 2 Joint Secrecy Capacity Region 3 Individual Secrecy Capacity Region 4 Comparison of Secrecy Capacity Region 5 Conclusion & Future Work Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 16

17 Conclusion & Future Work Conclusion Investigated secrecy in degraded multi-receiver wiretap BC. Established the individual secrecy capacity region. Bigger capacity region as compared to the joint one. Extensions and Future work Extension to the Gaussian BC. (Done) Extension to the MIMO Gaussian BC. (In Progress) Investigation of the less noisy wiretap BC. (In Progress) Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 17

18 References I TU München C. Shannon, Communication theory of secrecy systems, vol. 28, pp , Oct I. Csiszár and J. Körner, Broadcast channels with confidential messages, IEEE Trans. Inf. Theory, vol. 24, no. 3, pp , May W. Kang and N. Liu, Wiretap channel with shared key, Dublin, Ireland, Sep. 2010, pp A. S. Mansour, R. F. Schaefer, and H. Boche, Capacity regions for broadcast channels with degraded message sets and message cognition under different secrecy constraints, CoRR, vol. abs/ , January 2015, [Online]. E. Ekrem and S. Ulukus, Secrecy capacity of a class of broadcast channels with an eavesdropper, EURASIP J. Wirel. Commun. Netw., pp. 1 29, March Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 18

19 Thank YOU! TU München Mansour, Schaefer and Boche Secrecy in Degraded Wiretap BC 19