Secret-Key Generation from Channel Reciprocity: A Separation Approach

Transcription

1 Secret-ey Generation from Channel Reciprocity: Separation pproach shish histi Department of Electrical and Computer Engineering University of Toronto Feb 11, 2013

2 Security at PHY-Layer Use PHY Resources for designing security mechanisms. pplication Layer (Semantics of Information) Transport Layer (End to End Connectivity) Encryption, uthentication Secure Socket Layer Wireless Systems Network Layer (Routing and Path Discovery) Virtual Private Networks nonymous Routing Data Link Layer (Error Correction Codes) Physical Layer (Signals, RF hardware) Device level uthentication nti-jamming pplications: Secret-ey Generation Secure Message Transmission Physical Layer uthentication Jamming Resistance Feb 11, / 20

3 Motivation Secret-ey Generation in Wireless Fading Channels m Forward Link y = h x + n y = h x + n Reverse Link m Channel Gain Fading: Reverse Channel Forward Channel y (t) = h (t)x (t)+n (t) Reciprocity: time y (t) = h (t)x (t) + n (t) y (t) = h (t)x (t) + n (t) Feb 11, / 20

4 Motivation Secret-ey Generation in Wireless Fading Channels m Forward Link y = h x + n y = h x + n Reverse Link m z + E = g E x ne z E = g E x + ne Channel Gain Eavesdropper Link Reverse Channel Forward Channel E Spatial Decorrelation: y (t)= h (t)x (t) + n (t) y (t)= h (t)x (t) + n (t) z (t)= g (t)x (t) + n E (t) z (t)= g (t)x (t) + n E (t) time Feb 11, / 20

5 Secret-ey Generation - Systems pproach ey Generation in Wireless Systems UW Systems: Wilson-Tse-Scholz ( 07), M. o ( 07), Madiseh-Neville-McGuire( 12) Narrowband Systems: zimi Sadjadi- iayias-mercado-yener ( 07), Mathur-Trappe-Mandayam -Ye-Reznick ( 10), Patware and asera ( 07) OFDM reciprocity: Haile ( 09), Tsouri and Wulich ( 09) Implementations Experimental UW: Measurements for ey Generation Madiseh ( 12) Software Radio Implementations: Jana et. al. ( 09) MIMO systems: Wallace and Sharma ( 10), Shimizu et al. Zeng-Wu-Mohapatra Signal Processing for Secret-ey Generation ttacks Quantization Techniques: Ye-Reznik-Shah ( 07), Hamida-Pierrot-Castelluccia ( 09), Sun-Zhu-Jiang-Zhao ( 11) daptive Channel Probing: Wei-Zheng-Mohapatra ( 10) Mobility ssisted ey Generation: Gungor-Chen-oksal ( 11) ctive Eavesdroppers: Ebrez et. al ( 11) Zafer-grawal-Srivatsa ( 11), Unauthenticated Channels: Mathur et al. ( 10), Xiao-Greenstein-Mandayam-Trappe ( 07). Feb 11, / 20

6 Secret-ey Generation: Systems pproach II m z E Forward Link g y h E y h x n x n E z x n E Reverse Link g E x n E m Start E Channel Probing N ĥ N ĥ ey Extraction Shared ey Two Phase pproach: Phase I: Channel Probing and Estimation: (ĥn, ĥn ) Phase 2: Source Reconciliation and ey Extraction Secret-ey Generation: Limits Capacity Feb 11, / 20

7 Secret-ey Generation - Source Model Maurer ( 93), hlswede-csiszar ( 93) N X N X Terminal Public Discussion Channel Terminal Eavesdropper DMMS Model: (x N, x N ) N i=1 p x,x (x (i), x (i)) Interactive Public Communication: F ey Generation: k i = F i (xi N, F), i {, }. Reliability: Pr(k k ) ε N, 1 Secrecy: N I(k ; F) ε N Secret-ey Rate: R = 1 N H(k ) Feb 11, / 20

8 Secret-ey Generation - Source Model Maurer ( 93), Csiszar-hlswede ( 93) N X N X Slepian-Wolf Encoder F F Slepian-Wolf Decoder ey-distillation F Eavesdropper N Xˆ ey-distillation Capacity: C = I(x ; x ) One-Round of Communication Capacity Unknown when Eavesdropper also observes a source sequence Feb 11, / 20

9 Problem Setup m Forward Link y = h x + n y = h x + n Reverse Link m z + E = g E x ne z E = g E x + ne E Two-Way Reciprocal Fading Channel y (i) = h (i)x (i) + n (i), y (i) = h (i)x (i) + n (i) z (i) = g (i)x (i) + n E (i), z (i) = g (i)x (i) + n E (i) Feb 11, / 20

10 Problem Setup m Forward Link y = h x + n y = h x + n Reverse Link m z + E = g E x ne z E = g E x + ne Two-Way Reciprocal Fading Channel y (i) = h (i)x (i) + n (i), z (i) = g (i)x (i) + n E (i), E y (i) = h (i)x (i) + n (i) z (i) = g (i)x (i) + n E (i) Channel Model ssumptions: Non-Coherent Model: h (i) and h (i) Perfect Eavesdropper CSI: g (i) & g (i) known to Eve lock-fading Channel with Coherence Period: T. pproximate Reciprocity: (h, h ) p h,h (, ) Independence: (g, g ) (h, h ) Feb 11, / 20

11 Problem Setup m Forward Link y = h x + n y = h x + n Reverse Link m z + E = g E x ne z E = g E x + ne Two-Way Reciprocal Fading Channel y (i) = h (i)x (i) + n (i), z (i) = g (i)x (i) + n E (i), E y (i) = h (i)x (i) + n (i) z (i) = g (i)x (i) + n E (i) Secret-ey greement Protocols: Interactive: x (i) = f (m, y i 1 ), x (i) = f (m, y i 1 verage Power Constraints E[ x 2 ] P, E[ x 2 ] P. k = (y N, m ), k = (y N, m ) Reliability and Secrecy Constraint. Secret-ey Capacity ) Feb 11, / 20

12 Outline Upper ound Lower ound With Public Discussion Lower ound No Public Discussion symptotic Regimes and Numerical Results Feb 11, / 20

13 Secret-ey Capacity Upper ound histi 12 Theorem n upper bound on the secret-key capacity is C R + : R + = 1 [ ( T I(h ; h ) + max E log 1 + P (h ) h 2 )] ) P (h ) P 1 + P (h ) g 2 [ + max E log (1 + P (h ) h 2 )] P (h ) P 1 + P (h ) g 2 ) where P (h ) and P (h ) are power allocation function across the fading states. Feb 11, / 20

14 Secret-ey Capacity Upper ound Genie-ided Channel: h x y y x h Feb 11, / 20

15 Secret-ey Capacity Upper ound Genie-ided Channel: h x y y x h NT R I(m, h, N y NT ; m, h, N y NT z NT, g N ) Feb 11, / 20

16 Secret-ey Capacity Upper ound Genie-ided Channel: h x y y x h NT R I(m, h, N y NT ; m, h, N y NT z NT, g N ) I(x (NT ); y (NT ) h (N), z (NT ), g (N)) + I(x (NT ); y (NT ) h (N), z (NT ), g (N)) + I(m, h N, y NT 1 NT 1 ; m, h, N y z NT 1, g N ) Feb 11, / 20

17 Secret-ey Capacity Upper ound Genie-ided Channel: h x y y x h NT R I(m, h, N y NT ; m, h, N y NT z NT, g N ) NT I(x (n); y (n) h (n), z (n), ḡ (n)) n=1 NT + I(x (n); y (n) h (n), z (n), ḡ (n)) n=1 + NI(h ; h ) Feb 11, / 20

18 Secret-ey Capacity Upper ound Genie-ided Channel: h x y y x h Interpretation of the Upper ound: Channel Reciprocity: 1 T I(h ; h ) Forward Channel: I(y ; x h, z, g ) Reverse Channel: I(y ; x h, z, g ) Feb 11, / 20

19 Secret-ey Capacity Upper ound Genie-ided Channel: h x y y x h Public Discussion Channel Interpretation of the Upper ound: Channel Reciprocity: 1 T I(h ; h ) Forward Channel: I(y ; x h, z, g ) Reverse Channel: I(y ; x h, z, g ) Upper ound also holds if a public discussion channel is available. Feb 11, / 20

20 Lower ound: Separation ased Scheme histi 12 T T P N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) 1 P1 P1 Randomness Sharing Randomness Sharing Training: x (i, 1) = P 1 Randomness Sharing: x (i, t) CN (0, P 2 ) for t = 2,..., T x (i) = [x (i, 2),..., x (i, T )] C T 1. Training: ĥ(i) and ĥ(i) Correlated Sources: Forward Channel: y (i) = h (i)x (i) + n (i) C T 1, Reverse Channel: y (i) = h (i)x (i) + n (i) C T 1. Feb 11, / 20

21 Lower ound: Separation ased Scheme histi 12 T T P N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) 1 P1 P1 Randomness Sharing Randomness Sharing E Channel State ĥ ĥ (g, g ) Forward Channel x y z Reverse Channel y x z Feb 11, / 20

22 Lower ound: Separation ased Scheme histi 12 T T P N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) N(0,P 2 ) 1 P1 P1 Randomness Sharing Randomness Sharing E Channel State ĥ ĥ (g, g ) Forward Channel x y z Reverse Channel y x z Generate a secret-key from these sequences. Feb 11, / 20

23 Lower ound Overview Start Channel Probing Randomness Sharing ĥ ĥ Channel Reconciliation ĥ ĥ ĥ ĥ Secret-ey Extraction T Y T Y Source Reconciliation T T Y Y Shared ey Feb 11, / 20

24 chievable Rate with Public Discussion Theorem (Public Discussion) n achievable rate when a public discussion channel is available is { 1 R key = I(ĥ; ĥ) T }{{} + T 1 T + T 1 T Training [ ] I(y ; x, ĥ) I(y ; z, g, h ) } {{ } Forward Channel [ ] } I(y ; x, ĥ)) I(y ; z, g, h ) } {{ } Reverse Channel Feb 11, / 20

25 chievable Rate with Public Discussion Theorem (Public Discussion) n achievable rate when a public discussion channel is available is { 1 R key = I(ĥ; ĥ) T }{{} + T 1 T + T 1 T Training [ ] I(y ; x, ĥ) I(y ; z, g, h ) } {{ } Forward Channel [ ] } I(y ; x, ĥ)) I(y ; z, g, h ) } {{ } Reverse Channel Feb 11, / 20

26 High SNR Regime Theorem In the high SNR regime our upper and lower bound (with public discussion) coincide: } lim {R + (P ) R PD (P ) c P T where [ c =E log (1 + h 2 )] [ g 2 +E log (1 + h 2 )] g 2 Feb 11, / 20

27 Lower ound Without Public Discussion (T-1) Training Communication Transmission Public Discussion Coherence locks 1 2 Phase Probing + Randomness Sharing Channel-Sequence Reconciliation Source-Sequence Reconciliation Coherence locks ε 1 ε 2 Feb 11, / 20

28 Numerical Plot SNR =35 d, h 1, h 2 CN (0, 1), ρ = Lower ound Upper ound Public Discussion Training Rate (nats/symbol) Coherence Period (T) Feb 11, / 20

29 Symmetric MIMO Extension M. ndersson,. histi and M. Skoglund, 2012 y = H x + n, y = H x + n, z = G E x + n E z = G E x + n E H, H C M M, G E, G E C N E M Independent Rayleigh Fading, pproximate Reciprocity lock Fading with Coherence Period T T M N E Training + Source Emulation achieves degrees of freedom given by: d = max 2(T M )(M N E ) M [1,M] T Feb 11, / 20

30 Conclusions Secret-ey greement in Two-Way fading channels Upper and Lower ounds on Capacity symptotic Optimality Significant Gains over Training ased Schemes Future Work: Upper ounds with Perfect Reciprocity (See lso Lai-Liang-Poor 12) Stationary Fading Channels Low SNR Regime Stronger Eavesdropper Channels Feb 11, / 20

31 Error Reconciliation Public Discussion Channel, Discrete-Valued Sequences Reconciliation of Channel-Estimate Sequences (ĥ, ĥ ) ĥ ĥ ĥ ĥ Discussion Channel ĥ hˆ ( hˆ ĥ, hˆ ) Discussion Channel 1 H(φ ) H(ĥ ĥ), 1 H(φ ) H(ĥ ĥ) Feb 11, / 20

32 Error Reconciliation Public Discussion Channel, Discrete-Valued Sequences Reconciliation of Channel-Estimate Sequences (ĥ, ĥ ) ĥ ĥ ĥ ĥ Discussion Channel ĥ hˆ ( hˆ ĥ, hˆ ) Discussion Channel Reconciliation of (y, y ) y x, hˆ x h ˆ, y Discussion Channel y Common Sequence: ( y y, y, hˆ ) Discussion Channel 1 T H(ψ ) H(y x, ĥ, ĥ), 1 T H(ψ ) H(y x, ĥ, ĥ) Feb 11, / 20