Real Interference Alignment for the MIMO Multiple Access Wiretap Channel

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Real Interference Alignment for the MIMO Multiple Access Wiretap Channel

Pritam Mukherjee S¸ennur Uluku¸s

University of Maryland, College Park

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The MIMO Multiple Access Wiretap Channel (MAC-WT)

I Consider the two-user N × N × N × K MIMO MAC-WT:

H1

G2

H² G1

N antennas N antennas

N antennas K antennas W1

W2

Wˆ1, ˆW₂

W1W2

I The channel gains arefixed across time-slots.

I All channel gains are known perfectlyat every terminal.

I Question: What is the optimalsum secure degrees of freedom?

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Prior Work: The SISO MAC-WT Channel

I For the case N = K = 1, i.e., the SISO MAC-WT channel:

h1

g2

h² g₁

W1

W2

Wˆ1, ˆW₂

W1W2 I Optimal sum s.d.o.f.^a= ²₃.

I Achievable scheme based onreal interference alignment.

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Prior Work: The SISO MAC-WT Channel (contd.)

I The alignment of signals in this case is as follows:

h1

g2

g1

h2

v1

v2

u1

u2

u1

v1 v2

v2

u2

v1

u1

u₂

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Prior Work: The Fading MIMO MAC-WT Channel

I Thefadingtwo-user N × N × N × K MIMO MAC-WT:

H1(t)

G2(t) H²(t) G1(t)

N antennas N antennas

N antennas K antennas W1

W2

Wˆ1, ˆW₂

W1W2

I The channel gains are i.i.d. across time slots.

I Question: What is the optimalsum secure degrees of freedom?

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The Fading MIMO MAC-WT Channel (contd.)

I Theorem: [Mukherjee, Ulukus, Asilomar 2015]: The optimalsum s.d.o.f. of the N × N × N × K fading MIMO MAC-WT is

d_s =











N, if K ≤ ¹₂N

2

3(2N − K ), if ¹₂N ≤ K ≤ N

2

3N, if N ≤ K ≤⁴₃N 2N − K , if ⁴₃N ≤ K ≤ 2N

0, if K ≥ 2N.

I Note that when N = K = 1, the optimalsum s.d.o.f.^a=²₃.

a[Xie, Ulukus, 2013]

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The Fading MIMO MAC-WT Channel (contd.)

N 2N/3

N2 4N

N 3 2N K

sum s.d.o.f.

multiple access wiretap channel

I Converseproof holds forfixed channel gains as well.

I Question: Is the same s.d.o.f. achievable withfixedchannel gains? Yes!

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The Fading MIMO MAC-WT Channel (contd.)

N 2N/3

N2 4N

N 3 2N K

sum s.d.o.f.

N/2

3N2

wiretap channel with one helper

I Question: Is the same s.d.o.f. achievable withfixedchannel gains? Yes!

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The Fading MIMO MAC-WT Channel (contd.)

N 2N/3

N2 4N

N 3 2N K

sum s.d.o.f.

N/2

3N2

I Question: Is the same s.d.o.f. achievable withfixedchannel gains?

Yes!

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The Fading MIMO MAC-WT Channel (contd.)

N 2N/3

N2 4N

N 3 2N K

sum s.d.o.f.

N/2

3N2

I Question: Is the same s.d.o.f. achievable withfixedchannel gains?

Yes!

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Achievable Scheme for K ≤

^N₂

N 2N /3

N2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I Running example: N = 6. This case: K ≤ 3.

I Optimal sum s.d.o.f.=N.

I Beam-formingis optimal when K ≤ ^N₂.

I Transmitters 1 and 2 sendN − K andK symbols v1and v2, respectively, in the nullspace of the eavesdropper’s channels.

I Note that K ≤ N − K in this regime.

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Achievable Scheme for K ≤

^N₂

N 2N /3

N2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I Running example: N = 6. This case: K ≤ 3.

I Optimal sum s.d.o.f.=N.

I Beam-formingis optimal when K ≤ ^N₂.

I Transmitters 1 and 2 sendN − K andK symbols v1and v2, respectively, in the nullspace of the eavesdropper’s channels.

I Note that K ≤ N − K in this regime.

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Achievable Scheme for

^4N₃

≤ K ≤

^3N₂

N 2N /3

N

2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

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Achievable Scheme for

^4N₃

≤ K ≤

^3N₂

N 2N /3

N

2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I Running example: N = 6. This case: 8 ≤ K ≤ 9.

I Optimal sum s.d.o.f.= 2N − K .

I Transmitter 1:

v₁∈ R^3N−2K, ˜v ∈ R^{3K −4N}

, i.e.,K − N symbols.

I Transmitter 2:

v₂∈ R^3N−2K

, i.e.,3N − 2K symbols.

I Total of2N − K symbols in each time slot.

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Achievable Scheme for

^4N₃

≤ K ≤

^3N₂

(contd.)

I The channel inputs are:

X₁= R1˜v+ P1v₁+ H⁻¹₁ Qu₁ X₂= R₂u˜+ P₂v₂+ H⁻¹₂ Qu₂

I The channel outputs are:

Y =H₁R₁˜v+ H₁P₁v₁+ H₂P₂v₂+ H₂R₂˜u+ Q(u₁+u₂)

Z =G₁R₁˜v+ G2R₂˜u+ G1P₁v₁+ G2H⁻¹₂ Qu₂+ G2P₂v₂+ G1H⁻¹₁ Qu₁

I For security, enforce:

G₁R₁=G₂R₂ G₁P₁=G₂H⁻¹₂ Q G₂P₂=G1H⁻¹₁ Q

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Achievable Scheme for

^4N₃

≤ K ≤

^3N₂

(contd.)

I Feasibility of G₁R₁= G₂R₂with N × (3K − 4N) matrix R_i: [G₁ −G₂]

R₁ R₂

= 0

I This is feasible since 3K − 4N ≤ 2N − K in this regime.

I Choose P_i and Q as solutions of:

G₁ 0_{K ×N} −G₂H⁻¹₂ 0_{K ×N} G₂ −G₁H⁻¹₁



 P₁ P₂ Q



 = 0

I Security: Guaranteed by design.

I Decodability: Number of symbols to decode:

(2N − K )

| {z }

desired symbols

+ (3K − 4N)

| {z }

˜ u

+ (3N − 2K )

| {z }

u1+u2

= N

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Achievable Scheme for

^4N₃

≤ K ≤

^3N₂

(contd.)

I The alignment structure in this case has the following form:

H1

G2 G1

H2

˜

v v1

v2 u1

u2

u1

v2 u2

v1

˜

u ˜u

˜ v

˜

v v1 v2 ˜u

u1

u2

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Achievable Scheme for

^3N₂

≤ K ≤ 2N

N 2N /3

N2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I TheMAC-WTreduces to thewiretap channel with one helper^a.

I Transmitter 1: v ∈ R^2N−K, i.e.,(2N − K )information symbols.

I Transmitter 2 sends only cooperative jammingsignals.

a[Nafea, Yener, 2015]

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Achievable Scheme for

^3N₂

≤ K ≤ 2N

N 2N /3

N2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I TheMAC-WTreduces to thewiretap channel with one helper^a.

I Transmitter 1: v ∈ R^2N−K, i.e.,(2N − K )information symbols.

I Transmitter 2 sends only cooperative jammingsignals.

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Achievable Scheme for

^3N₂

≤ K ≤ 2N (contd.)

I The channel inputs are:

X₁= Pv X₂= Qu

I The received signals are

Y =H₁Pv+ H2Qu Z =G₁Pv+ G₂Qu

I For security, choose P and Q as the solutions to [G1 −G2]

P Q

= 0

I Decodability: Receiver can decode both v and u, since 2(2N − K ) ≤ N

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Achievable Scheme for

^3N₂

≤ K ≤ 2N (contd.)

I The alignment structure in this case has the following form:

u u

v u v v

H1

G2

G1

H2

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Achievable Scheme for

^N₂

≤ K ≤ N

N 2N /3

N

2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I Optimal sum s.d.o.f.=²₃(2N − K ) = 2 N − K + d +₃^l , where d =

2K − N 3

, l = (2N − K ) mod 3

I Examples:

1. When K = 4, sum s.d.o.f.=¹⁶₃, d = 0, l = 2. 2. When K = 5, sum s.d.o.f.=¹⁴₃, d = 1, l = 1.

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Achievable Scheme for

^N₂

≤ K ≤ N

N 2N /3

N

2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I Optimal sum s.d.o.f.=²₃(2N − K ) = 2 N − K + d +₃^l , where d =

2K − N 3

, l = (2N − K ) mod 3

I Examples:

16

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Achievable Scheme for

^N₂

≤ K ≤ N (contd.)

I Main idea: Decompose the channel input at each transmitter into:

1. N − K Gaussiansymbols sent in the nullspace of Eve.

2. Gaussiansymbols carryingd s.d.o.f.

3. structured PAMsymbols carrying ₃^l s.d.o.f.

I Use channel precoding for the Gaussiansymbols.

I Use real alignment schemes for thel × l × l × l MAC-WT achieving

2l

3 sum s.d.o.f. for l = 1, 2.

I For l = 1 : Real alignment scheme for the SISO MAC-WT is known^a.

I Needed: Real alignment scheme for the 2 × 2 × 2 × 2 MAC-WT.

a[Xie, Ulukus, 2013]

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Achievable Scheme for the 2 × 2 × 2 × 2 MAC-WT channel

I Optimal sum s.d.o.f.=⁴₃.

H1

G2

G1

H2

v11 v12

v21 v22

v12 v22

v11 v21

u11 u12

u21 u22

u11

u21

v11

v21

v12

v22

u12

u22

u11

v21

u12

v22

u21

v11

u22

v12

asymptotic alignment

) perfect alignment

)

asymptotic alignment )

I Perfect alignment at the eavesdropper ensures security.

I At receiver, d.o.f. at each antenna = ²₃; total s.d.o.f. =⁴₃.

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The General Scheme for

^N₂

≤ K ≤ N

I Each transmitter wants to send N − K + d + ₃^l s.d.o.f.

I At transmitter i, information bearing symbols: (˜v_i,v⁽¹⁾_i ,v⁽²⁾_i )

I ˜vi :N − K Gaussian symbols that can be sent using Eve’s nullspace

I v⁽¹⁾_i : d Gaussiansymbols each carrying 1 d.o.f.

I v⁽²⁾_i :l structuredsymbols each carrying ¹₃ d.o.f.

I Cooperative jamming signals: u_i= (u⁽¹⁾_i ,u⁽²⁾_i ).

I Let v_i = (v⁽¹⁾_i ,v⁽²⁾_i ). Transmitter i sends:

X_i =G^⊥_i v˜_i+ Piv_i+ H⁻¹_i Qu_i

I The received signals are:

Y =H₁G^⊥₁˜v₁+ H1P₁v₁+ H2P₂v₂+ H2G^⊥₂˜v₂+ Q(u₁+u₂) + N1

Z =G₁P₁v₁+ G₂H⁻¹₂ Qu₂+ G₂P₂v₂+ G₁H⁻¹₁ Qu₁+ N₂

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The General Scheme for

^N₂

≤ K ≤ N (contd.)

I Let Q to be any N × (d + l) matrix with full column rank.

I Set Pi = G^T_i (GiG^T_i )⁻¹(GjH⁻¹_j )Q.

I Eve’s observation is:

Z = G₂H⁻¹₂ Q(v₁+u₂) + G₁H⁻¹₁ Q(v₂+u₁) + N₂

I Perfect alignment ensures security.

I Decoding: Consider B_N×l such that

B^T[H1G^⊥₁ H₂G^⊥₂ H₁P⁽¹⁾₁ H₁P⁽¹⁾₁ Q⁽¹⁾]_N×(N−l)= 0

I Consider ˜Y = (B^TQ⁽²⁾)⁻¹B^TY

Y = DH˜ ₁P⁽²⁾₁ v⁽²⁾₁ + DH₂P⁽²⁾₂ v⁽²⁾₂ + (u⁽²⁾₁ +u⁽²⁾₂ ) + DN₁

I Decode (v⁽²⁾₁ ,v₂⁽²⁾) using the l × l × l × l MAC-WT scheme.

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Achievable Scheme for N ≤ K ≤

^4N₃

N 2N /3

N

2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I Optimal sum s.d.o.f.=^2N₃ .

I The point K = N is achievable using the scheme for ^N₂ ≤K ≤ N.

I The point K = ^4N₃ is achievable using the scheme for ^4N₃ ≤K ≤ ^3N₂ .

I The intermediate points N ≤ K ≤ ^4N₃ are achievable since increasing Eve’s antennas does not increase the sum s.d.o.f.

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Achievable Scheme for N ≤ K ≤

^4N₃

N 2N /3

N

2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I Optimal sum s.d.o.f.=^2N₃ .

I The point K = N is achievable using the scheme for ^N₂ ≤K ≤ N.

I The point K = ^4N₃ is achievable using the scheme for ^4N₃ ≤K ≤ ^3N₂ .

I The intermediate points N ≤ K ≤ ^4N₃ are achievable since increasing

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Achievable Scheme for K ≥ 2N

N 2N /3

N2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I Running example: N = 6. This case: K ≥ 12.

I Optimal sum s.d.o.f.=0.

I Since Eve has more than 2N antennas, the input of both transmitters can be decoded to within noise variance.

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Achievable Scheme for K ≥ 2N

N 2N /3

N2 4N

N 3 2N K

s.d.o.f.

(0, 0) 3N

2

I Running example: N = 6. This case: K ≥ 12.

I Optimal sum s.d.o.f.=0.

I Since Eve has more than 2N antennas, the input of both transmitters can be decoded to within noise variance.

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Conclusions and Future Work

I Provided achievable schemes for the MIMO MAC-WT with fixed channel gains.

I The achievable scheme for the regime ^N₂ < K < N:

I Is based onasymptoticreal interference alignment

I Uses a combination ofGaussianandstructured PAMsymbols.

I Combines channel precoding with real interference alignment

I Open question: What happens if Eve’s CSIT is not available?