Security and Sensitivity of Space Time Communications

Security and Sensitivity of Space Time Communications
Alfred O. Hero
Dept. EECS
University of Michigan - Ann Arbor
[email protected]
http://www.eecs.umich.edu/˜hero
Collaborators:
D. Bliss (MIT-LL), K. Forsythe (MIT-LL), T. Marzetta (Lucent-BL),
M. Godavarti (Altrabroadband, Inc)
Outline
1. Wireless network models
2. Performance metrics: capacity vs security
3. Information security: LPD/LPI-constraints
4. Environmental sensitivity
1
2
HT E = 4
si1
si2
θ11
θ21
θ31
θ12
θ22
θ32
3
5
yi1 ; yi2 ;
si3 ; i = 1; : : : ; T
i = 1; : : : ; T
Eavesdropper
Receiver
Y
Transmitter
2
HT R = 4
h11
h21
h31
T =coherent fade interval
M=number of transmit antennas
N=number of receive antennas
ηr ; ηe = receiver SNR’s
h12
h22
h32
=
pη SH
e
T E + WE , (T
3
5
N)
i = 1; : : : ; T
xi1 ; xi2 ;
Client
Receiver
p
X = η SH
r
T R + WR , (T
N)
Receiver Model
Received signal in l-th frame (t = 1; : : : ; T )
l
l
[xt1 ; : : : ; xtn ] =
pη[sl
t1
2
;:::;
l ] 6
stm
6
6
4
hl11
..
.
hlm1
or, equivalently
X
l
..
.
hl1n
..
.
hlmn
3
7
7
l
l
7 + [wt1 ; : : : ; wtn ];
5
p
l l
l
= ηS H + W
X l : T N received signal matrices
Sl : T M transmitted signal matrices
N
l
H : i.i.d. M N channel matrices Cl N (0 IM IN )
N
l
W : i.i.d. T N noise matrices Cl N (0 IT IN )
;
;
2
Source
T x M matrix
a(t)
A/D
Space
S(t)
Time
Encoder
demux
RF Mod
RF Mod
Transmitter
RF Demod
RF Demod
Codeword
^
S(t)
Estimator
Space
Time
Decoder
D/A
mux
RF Demod
Receiver
T x N matrix
Figure 2. Space-time transmitter/receiver.
3
^a(t)
Space-Time Coding
Block coding: string L codewords over L frames
j S1 j S2 j j SL j
where Sl ’s are selected from a symbol alphabet S Cl T M
Random Block Coding: coder generates Sl at random from S
according to probability distribution P(S) 2 P .
Objective: Find optimal distribution P(S) over P to:
– maximize avg. information rate (achieve capacity)
C = max E [ln P(X jS)=P(X )]
P(S)
– maximize sequentially-decodable rate (achieve cut-off rate)
Ro = max E [expf ND(S1 kS2 )g]
P(S)
Transmitter constraints: average power, peak power, other?
4
Coherent Transmission and Reception − T/R know channel
40
M=32
M=16
M= 4
M= 1
35
30
b/s/hz
25
20
15
10
5
0
−10
−5
0
5
SNR (dB)
10
15
20
Figure 3. Capacity for informed transmitter and receiver (IT-IR).
5
Effect of Incoherent Transmission
1
0.9
0.8
C/C
o
0.7
M=32
M=16
M= 4
M= 1
0.6
0.5
0.4
0.3
−10
−5
0
5
SNR (dB)
10
15
20
Figure 4. Capacity loss due to uninformed transmission (UT-IR).
6
Effect of Training Errors (coherent transmission): T
=128
train
1
0.9
C/C
o
0.8
0.7
M=32
M=16
M= 4
M= 1
0.6
0.5
0.4
−10
−5
0
5
SNR (dB)
10
15
20
Figure 5. Capacity loss due to T/R channel estimation errors.
7
Link Capacity: avg power constraint: tr(E [SS† ]) Po
(1):
Informed transmitter (IT) and informed receiver (IR) capacity:
#
"
C
=
E sup log P(X jS; H )=P(X jH )
PS
"
ln IN + ηHΣH † #
=
TE
=
i
h +
†
T E ln IN + ηHΣpow H = T ∑ E (log µλi )
sup
Σ:trfΣgPo
Capacity achieving source S N (0
;
Σpow = UDU
†
;
λi = eigs ηHH †
IT
NΣ
i
pow )
D = diag (µ
1=λi )
+
µ : tr(Σpow ) = Po
8
IT-IR Link
d1
e1
source
Serial-toParallel
U
eM
Temporal
encoder
dM
Spatial
encoder
Figure 6. Optimal STC for informed-transmitter informed-receiver
9
Waterfilling solution
120
100
1/lambda(i)
80
60
40
mu=34
Power=sigma
20
0
0
5
10
15
20
Mode index i of H HT
25
30
35
Figure 7. Waterpouring solution for power-capacity achieving mode allocation(N = M = 32)
10
(2):
Uninformed transmitter (UT) and IR capacity
C
=
sup E [log P(X jS; H )=P(X jH )]
PS
=
=
where η
0
sup
Σ:trfΣgPo
i
h T E ln IN + ηHΣH † i
h 0
T E ln IN + η HH † = ηPo =M
Capacity achieving source
S
N (0
;
cIT
where c = Po =M
11
OI
M)
UT-IR Link
1
e1
source
Serial-toParallel
eM
Temporal
encoders
1
Figure 8. Optimal STC for uninformed-transmitter informedreceiver
12
(3):
UT-UR: H unknown to either T/R
C3 = max E log PX jS (X jS)=PX (X )
PS
Capacity achieving source
S
VΛ
where
*Λ: non-negative T M block-diagonal matrix
*V : unitary T T matrix
*Λ and V independent
* Λ† Λ = Po
13
UT-UR Link
λ1
e1
source
Serial-toParallel
V
eM
Temporal
encoder
λM
Spatial
encoder
Figure 9. Optimal STC for uninformed-transmitter uninformedreceiver
14
Channel Sensitivity
Rank One Specular Component
Pre-Processing 1
Post-Processing 1
Pre-Processing 2
Post-Processing 2
Received
Information
Information to
be Transmitted
Post-Processing N
Pre-Processing M
M-Transmit Antennas
Independent Paths (Possibly over time)
N-Receive Antennas
Figure 10. Diagram of a multiple antenna communication system
15
Rician Channel Model
Combined Rayleigh and Specular Multipath Fading:
H=
p
p
r G+ r H
1
m
– Gmn are i.i.d. C N (0; 1)
– Hm deterministic matrix such that trfHm Hm† g = NM
– r fraction of channel energy devoted to specular component
– Hm known to both the transmitter and receiver
– G not known to the transmitter
After unitary spatial transformation at T/R: Hm = [D; 0]
16
Rician Capacity: Rank one Hm known to T/R
2
Hm =
p
p
6
6
NM eM eTN = 6
4
3
NM
..
.
:::
.
0
..
.
0
:::
0
..
7
7
7
5
UT-IR Capacity:
CH
= max T E log det[IN + ηHΛ
l ;d
(l ;d )
where
2
Λ
(l ;d )
H †]
=4
3
M
(M
1)d
l1†M 1
l1M
1
dIM
1
5
d is a positive real number such that 0 d M =(M
l is a complex number such that jl j 17
q
M
(M 1
d )d
1)
Optimal UT-IR Rician Link
d
U M-1
e1
source
Serial-toParallel
eM
M-(M-1)d
Temporal
encoder
uM
Spatial
encoder
Figure 11. Optimal STC for Rician uninformed-transmitter
informed-receiver
18
1.4
1.2
dB = 40
1
dB = 20
0.8
d
dB = 0
0.6
dB = −20
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
r (fraction of energy in the specular component)
0.8
0.9
1
Figure 12. Numerical optimization yields l = 0 and values of d
shown as a function of r for different values of ρ.
19
Channel Sensitivity: Physical Scatterers
ray 2
ray 1
L
e2πi[dRx m (n)+dTx m (n)]
=∑
dRx ;m (n)dTx ;m (n)
n
;
Transmitter
Hm;l
;
Figure 13. Physical point scattering model.
20
Receiver
Median Eigenvalues (dB)
IT Capacity Ratio
(a)
0
16x16 Rand
16x16
8x8
4x4
2x2
-10
-20
-30
2
4
6
8
10
12
14
16
(b)
2
1.5
1
0.5
0
-20
-10
2 0
a Po (dB)
10
20
Figure 14. Eigenvalue dsn and capacity ratio (a2 = trfHH † g)
21
Median Eigenvalues (dB)
(a)
0
-10
8x8 Rand
100/L 2
10/L 2
1/L 2
-20
-30
1
2
3
4
6
7
8
(b)
1.5
Capacity Ratio
5
100/L
2
10/L
1/L 2
IT
1
2
UT
0.5
-20
-10
2 0
a Po (dB)
10
20
Figure 15. Median eigenvalue dsn and capacity ratio
22
Channel Sensitivity: Interference
Hypothesis: Strong random interferers
Informed Transmitter (IT) and Informed Receiver (IR)
"
C
=
TE
#
sup
Σ:trfΣgPo
log IM + ηH
1
1
(I + R) 2 Σ(I + R) 2
H†
Uninformed Transmitter (UT) and IR
h
C
=
T
sup
Σ:trfΣgPo
E log I + ηH
1
1
2
2
(I + R)
Σ(I + R)
H†
i
Where R is N N interference spatial covariance matrix at receiver
23
8
(a)
No Interferer
6
4
2
MIMO 8 × 8 to 1 × 8 Capacity Ratio
0
-20
-10
0
10
20
8
(b)
Interferer Power = 20 dB
6
4
2
0
-20
-10
8
6
Random IT
0
Random
UT
Dense IT
Dense UT
Sparse IT
Sparse UT
10
20
(c)
Interferer
Power
= 40 dB
4
2
0
-20
-10
0
2
a Po (dB)
10
20
Figure 16. Spectral efficiency ratio for 8 x 8 system
24
Spectral Efficiency
(b/s/Hz/M)
5
4
3
2
1
0
-10
-5
0
5
10
a 2Po (dB)
15
20
Figure 17. Normalized capacity for no interferers, cooperative interferers, and un-cooperative interferers.
25
I
I
T
I
H TR
I
R
I
H
TE
E
Figure 18. Wireless network with eavesdropper
26
Information Security: Eavesdropper Resistance
Hypotheses:
1. Subscriber links have informed transmitters/receivers (IT-IR):
HT R is known to both parties over a hop
Training generally required to learn channel
Feedback required to inform transmitter of channel
2. Eavesdropper link has uninformed transmitter (UT)
HT E unknown to transmitter
S, HT E may be known or unknown to eavesdropper
Modulation type, signal constellations, source density, may be
known to eavesdropper
27
Eavesdropper Performance Measures
1. Pe eavesdropper error rate for detecting known signal S = s on
link
PF
e
= P(Λ
>
γjS = 0); PM
e
= P(Λ
<
γjS = s)
2. PF , PM = 1 PD : eavesdropper error rates for detecting any
activity on link
PF
e
= P(Λ
>
γjS = 0); PM
e
= P(Λ
<
3. Ce = maxPS I (S;Y ): eavesdropper link capacity
e (K ): eavesdropper symbol intercept error rate
4. Psde
e
e
= P(Sˆ 6= S)
Psde
28
γjS 6= 0)
Computational Cutoff Rates
Ro (H )
=
max
PSjH
ln
ZZ
S1 ;S2 2Cl T M
dPSjH (S1 )dPSjH (S2 ) e
1. T/R Informed cutoff rate: H known to both T/R
η †
D(S1 kS2 ) = tr H (S1 S2 )† (S1
4
2. R informed cutoff rate: H known to R only
η
D(S1 S2 ) = ln IT + (S1
k
4
S2 )(S1
3. Uninformed cutoff rate: H unknown to either T/R
S2 )H
S2 ) †
†
† η
IT + 2 (S1 S1 + S2 S2 )
D(S1 S2 ) = ln r
† †
IT + ηS1 S1 IT + ηS2 S2 k
29
ND(S1 kS2 )
LPI: Uninformed Eavesdropper Lockout Capacity
Lock out condition: Ce = 0
Note: lock out occurs if transmitted signal constellation fSi g satisfies:
Si Si† = A;
8i
Examples:
Doubly unitary codes (T
M):
3
2
Si† Si = IM ;
Si Si† = 4
IM
O
O
O
5
Instances
– Square unitary codes (T
= M):
30
Si Si† = Si† Si = IM
= M = 2):
– Space time QPSK: Quaternion codes (T
S
8 2
<
1
4
=
:
0
3
0
5;
1
2
4
3
j
0
5;
j
0
2
Constant (spatial) modulus (CM) codes (T
4
3
0
1
5;
1 0
= 1):
Si = [S1i ; ; SMi ]
trfSi Si† g = kSi k2 = 1
Note 1: Q. How much subscriber capacity does lockout cost?
A. Dimensionality analysis (T
= M ):
Constraint Si Si† = A reduces coding d.f. by factor
ρ=
M (M + 1)=2
M2
31
1
=
2
2
4
0
j
39
j =
5
0 ;
LPD constraints
The eavesdropper must make a decision between
Xi = Wi ;
H0 :
i = 1; : : : ; L
H1 : Xi = Si Hi + Wi ; i = 1; : : : ; L
His minimum attainable detection error probability has exponential rate
1
lim inf ln Pe
L!∞ L
ρ
=
ρ
=
1
inf lim ln
α2[0;1] L!∞ L
Z
fH11 α (X ) fHα0 (X )dX
ρ is Chernoff error exponent (ρ 0)
ρ is minimal α-divergence between densities fH
Chernoff exponent is achieved for Bayes test
1
32
and fH0
SH-informed Eavesdropper
When eavesdropper knows transmitted sequence S = s = fs1 ; : : : ; sL g and
channel sequence HT E = fH1 ; : : : ; HL g
H0 : S = 0;
H1 :
ρ
=
S=s
1 L
lim ∑ ρ(Hi ; si )
L!∞ L
i=1
where
ρ(Hi ; si ) =
η2e
trfsi Hi Hi† s†i g:
4
33
LPD transmitter strategy: Attain E [maxP(S) ln P(X jHT R ; S)=P(X jHT R )]
subject to constraint on LPD (ρ)
When Hi = HT E are i.i.d. Rayleigh channels:
η2e
E [trfSi Si† g]:
4
ρ=
Relevant LPD constraints on Transmitter are:
Peak power constraint:
trfsi s†i g Popk
Average power constraint:
o
n
tr
E [Si Si† ]
34
Po
S-Informed Eavesdropper
When eavesdropper knows S, but not H, α-divergence is
ln
Z
f1
α
(X
jS = s) fHα (X jS = 0)dX
0
L
=
∑ ln jI
i=1
jIT + ηe si s†i j1
T + ηe (1
α
α) si s†i j
Asymptotic development:
ln
jIT + ηesi s†i j1
jIT + ηe (1
α
α) si s†i j
=
α(1
α)η2e
2
35
trfsi s†i si s†i g + o(η2e ):
Low SNR scenario
Low SNR representation for the Chernoff error exponent
ρ=
η2e 1 L
trfsi s†i si s†i g + o(η2e ):
∑
8 L i=1
Transmitter Strategy:
Attain E [maxP(S) ln P(X jHT R ; S)=P(X jHT R )] subject to either
Peak 4-th moment constraint:
trfsi s†i si s†i g P4pk ;
Average 4-th moment constraint:
trfE [Si Si† Si Si† ]g P4avg ;
36
Uninformed Eavesdropper
When eavesdropper knows neither S nor H
H0 : S = 0;
H1 :
S 6= 0
α-divergence not closed form
Multivariate Edgeworth expansion of f (X jS 6= 0)
ln
= ln Z
f1
α
(X
jS 6= 0) f α (X jS = 0)dY
1 α
IT + ηe SS† IT + ηe (1
+
α)SS† α(1
α)2 η2e
σt ;u κt ;u;v;w (X )σv;w + o(η4e )
8
37
(1)
κr;s;t ;u (X ) is received signal kurtosis and
σt ;u κt ;u;v;w (X )σv;w
2
= ηe 3N
T
M
∑ ∑
cov(skt ; sku )cov(skt sku ; skv skw )cov(skv ; skw )
k=1 t ;u;v;w=1
Observe
Skewness of X is always zero for Gaussian channel
Kurtosis tensor product depends on 4th moment of source:
cov(skt sku ; skv skw ) = E [skt sku skv skw ]
First term in (1) dominates for low SNR
38
E [skt sku ] E [ skv skw ] 0
Uninformed Eavesdropper: Low SNR
ρ
=
=
min
α2[0;1]
α)η2e
α(1
2
trfSS† SS† g + o(η2e )
η2e
trfSS† SS† g + o(η2e )
8
Transmitter strategy:
Attain E [maxP(S) ln P(X jHT R ; S)=P(X jHT R )] subject to
trfSS† SS† g P4avg
Equivalent to constraining S to Gaussian source with
trfSS† SS† g P4avg =3
39
LPD-constrained Capacity
Proposition 1 The LPD-constrained capacity Clpd for the T/R informed
link is
2
0q
i
h Clpd = T E ln IM + ηr HΣlpd H † = T E 4log @
1 + µλ2i
2
13
A5
NΣ
Attained by S N (0 IT lpd )
Σlpd = UDU † , D = diag(σi ),
;
q
σi =
µ
>
1=λ2i + µ
1=λi
2
0 is a parameter such that ∑i σ2i = P4avg .
40
;
(2)
Note:
eigenstructure of Σlpd is matched to modes of H.
power-optimal waterpouring solution is not LPD-optimal
q
n
o
M tr fE [SS† SS† ]g tr E [SS† ]
Conclude: kurtosis constraint also constrains avg power
However: kurtosis constraint produces qualitatively different optimal
source distribution.
41
Optimal Covariance Eigenspectra: SNR =20(dB), M=32
0.06
0.05
spectrum
0.04
0.03
Power−optimal
LPD−opimal
0.02
0.01
0
−0.01
0
5
10
15
20
eigenvalue index
25
30
35
Figure 19. Optimal source spectra: SNR = 20dB; M = N = 32
42
LPD: Tradeoff Study
Define
i
h †
Ic (Σ) = T E ln IM + ηr HΣH 1. IT-IR LPD-Capacity IP4avg (Σlpd )
2. Loss in power-constrained capacity due to LPD constraint
IPo (Σlpd )=IPo (Σpow )
(3)
3. Loss in LPD-constrained capacity due to power constraint
IP4avg (Σpow )=IP4avg (Σlpd )
43
(4)
Informed Transmission and Reception − LPD−Capacity
40
35
M=32
M=16
M= 4
M= 1
30
b/s/hz
25
20
15
10
5
0
−20
−15
−10
−5
0
5
10
mean−squared power (dB)
15
20
Figure 20. IT-IR LPD-constrained capacity (N = M)
44
Loss in Power−Capacity due to MS Power Constraint
1
0.9
0.8
0.7
C/Co
0.6
0.5
0.4
0.3
M=32
M=16
M= 4
M= 1
0.2
0.1
0
−20
−15
−10
−5
0
5
mean power (dB)
10
15
20
Figure 21. Loss in power-capacity due to LPD constraint (N = M)
45
Loss in LPD−Capacity due to Mean−Power Constraint
1
0.9
0.8
0.7
C/Co
0.6
0.5
M=32
M=16
M= 4
M= 1
0.4
0.3
0.2
0.1
0
−20
−15
−10
−5
0
5
10
mean−squared power (dB)
15
20
Figure 22. Loss in LPD-capacity due to Pavg constraint (N = M)
46
Comments
For no transmit diversity (M = 1) there is no loss in capacity
loss increases as more antennas M are deployed by eavesdropper and
client
loss decreases as SNR ηr increases
as ηr decreases to -20 dB loss flattens out.
47
Conclusions
1. For Rician channel T transmits rank-1 component at low SNR
2. Capacity for physical scattering is less optimistic than for Rayleigh
3. High-power interference reduces degrees of freedom (number of
useful channel modes)
4. LPD- and LPI- constrained secure channels are different from open
channels
5. For uninformed eavesdropper 4th moment constraint constrains LPD
6. LPD-constrained information rate advantage increases with M
48

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