Decomposing the MIMO Wiretap Channel

Decomposing the MIMO Wiretap Channel Anatoly Khina, Tel Aviv University Joint work with: Yuval Kochman, Hebrew University Ashish Khisti, University of Toronto ISIT 2014 Honolulu, Hawai i, USA June 30, 2014

Model Capacity SISO Goal Part I Problem Setting

Model Capacity SISO Goal Channel Model: Gaussian MIMO Wiretap Channel H B Alice H E Bob y B = H B x+z B Eve y E = H E x+z E x N A 1 input vector of power P y B, y E N B 1, N E 1 received vectors H B, H E N B N A, N E N A channel matrices z B CN(0,I NB ), z E CN(0,I NE ) noise vectors Closed loop (full channel knowledge everywhere).

Capacity Model Capacity SISO Goal Gaussian SISO channel capacity [Leung-Yan-Cheong, Hellman 78] I(X;Y B ) I(X;Y E ) [{ ( }} ){{ ( }} )] { C S (h B,h E ) = log 1+ h B 2 P log 1+ h E 2 P Gaussian MIMO channel capacity [Khisti,Wornell 10][Oggier,Hassibi 11] C S (H B,H E ) = max K: trace{k} P I(X;Y B ) I(X;Y E ) [{}}{{}}{ ] log I+H B KH B log I+H E KH E Maximization over all admissible covariance matrices K Power constraint can be replaced with covariance constraint [Liu, Shamai 09] +

Model Capacity SISO Goal Capacity-achieving Codes for Gaussian SISO Wiretap Great effort in constructing practical capacity-achieving codes: LDPC-based [Klinc et al. 11][Andresson, PhD 11] Lattice-based [Oggier et al., submitted 13] Polar-based [Mahdavifar, Vardy 11] Black-box approach [Tyagi, Vardy, ISIT 14] More... What to do for MIMO?

Model Capacity SISO Goal Goal: Practical MIMO Scheme Black box approach Signal processing (SVD-based scheme [Telatar 99], V-BLAST [Foschini 96],...) Any good SISO wiretap codes Achieves capacity Gap-to-capacity dictated by gap-to-capacity of the SISO codes High SNR [Khisti, Wornell 10] Generalized SVD (GSVD) + linear zero-forcing Any good SISO wiretap codes Optimal at SNR B,SNR E Suboptimal at SNR B,SNR E <

SVD V-BLAST (QR) Part II Background: MIMO Without Secrecy

SVD V-BLAST (QR) SVD with water-filling SVD for Given K Practical Schemes for MIMO Without Secrecy (No Eve) Singular-Value Decomposition (SVD) Scheme [Telatar 99] H B = Q B D B V A Q B and V A unitary Alice applies V A and Bob applies Q B d 1 0 0 0 0 d 2 0 0 y 1 = d 1 x 1 +z 1 D B =........ y 2 = d 2 x 2 +z 2. 0 0 d N 1 0. y 0 0 0 d N = d N x N +z N N Results in parallel scalar sub-channels (each sub-channel has a different SNR) Apply water-filling on {x 1,...,x N }: x = V A Wc

SVD V-BLAST (QR) SVD with water-filling SVD for Given K Practical Schemes for MIMO Without Secrecy (No Eve) SVD-based scheme for a given input covariance K H B K 1/2 = Q B D B V A Q B and V A unitary Alice applies K 1/2 V A and Bob applies Q B d 1 0 0 0 0 d 2 0 0 y 1 = d 1 x 1 +z 1 D B =........ y 2 = d 2 x 2 +z 2. 0 0 d N 1 0. y 0 0 0 d N = d N x N +z N N Results in parallel scalar sub-channels (each sub-channel has a different SNR) Apply water-filling on {x 1,...,x n }: x = V A Wc x = K 1/2 V A c

SVD V-BLAST (QR) SVD with water-filling SVD for Given K Practical Schemes for MIMO Without Secrecy (No Eve) SVD scheme with given K achieves : R = log I NA +H B KH B For optimal choice of K attains capacity Can be used to attain capacity for other covariance constraint scenarios (e.g., individual power constraints)

SVD V-BLAST (QR) ZF MMSE Precoded Practical Schemes for MIMO Without Secrecy (No Eve) QR decomp. [Foschini 96][Wolniansky et al. 98] zero-forcing VBLAST H B = Q B T B Q B unitary; T B triangular Bob applies Q B (no SP is required by Alice) t 1 0 t 2 T B =......... 0 0 t N 1 0 0 0 t N Off-diagonal elements are canceled via successive interference cancellation (SIC) y eff 1 = t 1 x 1 +z 1 y eff 2 = t 2 x 2 +z 2. yn eff = t Nx N +z N

SVD V-BLAST (QR) ZF MMSE Precoded Practical Schemes for MIMO Without Secrecy (No Eve) QRD MMSE-VBLAST for a given covariance K [Hassibi 00] [ HB K 1/2 ] = Q B T B I NA Q B unitary; Q B N B N A submatrix of Q B Bob applies Q B (no SP is required by Alice) Q B contains Wiener-filtering ( FFE ) Effective noise has channel noise and ISI components Effective SNRs satisfy: t 2 i = 1+SNR i log(t 2 i ) = log(1+snr i) = I(c i ;y B c N A i+1 ) Off-diagonal elements above diagonal canceled via SIC

SVD V-BLAST (QR) ZF MMSE Precoded Practical Schemes for MIMO Without Secrecy (No Eve) For square invertible H, ZF-VBLAST achieves: R = H B H B ( ) Using K at the transmitter achieves: R = H B KH B MMSE-VBLAST achieves: R = I NB +H B KH B

SVD V-BLAST (QR) ZF MMSE Precoded Practical Schemes for MIMO Without Secrecy (No Eve) MMSE-VBLAST with precoding for a given covariance K [ HB K 1/2 ] V A = Q B T B I NA V A can be used to design diagonal values design SNRs SVD-scheme as MMSE-VBLAST (QR) Choosing V A of the SVD of H B K 1/2 SVD scheme (no SIC needed) Geometric-mean decomposition [Jiang et al. 05]/ QRS [Zhang et al. 05] V A is choosing s.t. all diagonal values (all SNRs) are equal The same codebook can be used over all subchannels No need for bit-loading

Idea SVE@Eve Superposition General scheme Part III Back to Wiretap...

Idea SVE@Eve Superposition General scheme New Scheme for Finite SNR T B { }} { [ HB K 1/2 ] b 1 V A = Q I B. 0.., bi 2 = 1+SNR B i NA 0 0 b N T E { }} { [ HE K 1/2 ] e 1 V A = Q I E. 0.., ei 2 = 1+SNR E i NA 0 0 e N Use any good SISO wiretap codes for SNR-pairs (b 2 i 1,e 2 i 1) V A of Eve s SVD Easy secrecy analysis + strong secrecy

Idea SVE@Eve Superposition General scheme Strong/weak secrecy Orthogonalizing Eve s Channel Take optimal covariance matrix K

Idea SVE@Eve Superposition General scheme Strong/weak secrecy Orthogonalizing Eve s Channel Take optimal covariance matrix K Use optimal SVD-based scheme matched for H E with K: H E K 1/2 = Q E D E V A Alice: x = K 1/2 V A c Eve: ỹ E = Q E y E = Q E H EK 1/2 V A cq E +z E = D E c+ z E

Idea SVE@Eve Superposition General scheme Orthogonalizing Eve s Channel Take optimal covariance matrix K Strong/weak secrecy Use optimal SVD-based scheme matched for H E with K: Alice: x = K 1/2 V A c H E K 1/2 = Q E D E V A Eve: ỹ E = Q E y E = Q E H EK 1/2 V A cq E +z E = D E c+ z E Eve sees parallel subchannels Easy secrecy analysis Bob Uses MMSE-VBLAST: ỹ B = Q B y B = T B c+ z B Use SISO wiretap codes over parallel wiretap subchannels: (SNR B i,snr E i ) ( t B i,d E i )

Idea SVE@Eve Superposition General scheme Orthogonalizing Eve s Channel Strong/weak secrecy SVD of Eve s channel allows easy secrecy-constraints proof Strong/weak secrecy of the SISO codes Strong/weak secrecy of the MIMO scheme Can we use SVD for Bob s channel? Can the same SISO codebook be used over all subchannels?

Idea SVE@Eve Superposition General scheme Superposition Coding for the Memoryless Wiretap Channel Theorem p(y B x) and p(y E x) transition distributions for Bob and Eve x = ϕ(c 1,...,c NA ) c i drawn i.i.d. p ci ( ) Weak-secrecy rates of ( ) ( ) c N R i = I c i ;y A c N B i+1 I c i ;y A E i+1, i = 1,...,N A are achievable Genie-aided secrecy-proof Eve tries to recover sub-messages sequentially (from last to first) For the recovery of sub-message i all previous sub-messages (i = i +1,...,N A ) are revealed

Idea SVE@Eve Superposition General scheme SVD@Bob and 2-GMD General Multi-Stream Scheme for the Gaussian MIMO Wiretap Apply QR decompositions for both Bob and Eve: [ HB K 1/2 ] [ V A HE K = Q B T B, 1/2 ] V A = Q E T E I NA The resulting SNRs satisfy: I NA log(1+snr B;i ) = log(b 2 i ) = I(c i;y B c N A i+1 ) log(1+snr E;i ) = log(e 2 i ) = I(c i;y E c N A i+1 ) Superpoisition theorem Capacity is achieved with SISO codebooks of SNR-pairs (b 2 i 1,e 2 i 1) = Alice can apply any V A!

Idea SVE@Eve Superposition General scheme SVD@Bob and 2-GMD General Multi-Stream Scheme for the Gaussian MIMO Wiretap SVD @ Bob Choose V A such that H B K 1/2 = Q B D B V A Bob: ỹ B = D B c+ z B Same codebook over all subchannels Choose V A s.t. T B and T E have constant diagonals: SNR B 1 = SNRB 2 = = SNRB N SNR E 1 = SNR E 2 = = SNR E N Possible using a space time structure [Khina et al. 11]

High SNR scheme Part IV Supplementary

High SNR scheme MIMO Wiretap Scheme for High SNR [Khisti, Wornell 10] Apply Generalized SVD (GSVD) to H B and H E : H B = Q B D B A H E = Q E D E A

High SNR scheme MIMO Wiretap Scheme for High SNR [Khisti, Wornell 10] Apply Generalized SVD (GSVD) to H B and H E : H B = Q B D B A H E = Q E D E A A invertible matrix; Q B, Q E unitary

High SNR scheme MIMO Wiretap Scheme for High SNR [Khisti, Wornell 10] Apply Generalized SVD (GSVD) to H B and H E : H B = Q B D B A H E = Q E D E A A invertible matrix; Q B, Q E unitary D B, D E diagonal, s.t. D 2 B +D2 E = I Alice uses linear zero-forcing inverts A @ Tx: x = A 1 c Bob and Eve apply Q B and Q E

High SNR scheme MIMO Wiretap Scheme for High SNR [Khisti, Wornell] @ High-SNR (P ) Approaches capacity: N A i=1 log 1+Pd2 B;i 1+Pd 2 E;i N A i=1 log Pd2 B;i Pd 2 E;i Linear zero-forcing attains capacity at high SNR (In contrast to communication without secrecy!) Strong/weak secrecy of the SISO codes Strong/weak secrecy of the MIMO scheme Wasteful at finite SNR!