Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Massive MIMO Physical Layer Cryptosystem
through Inverse Precoding
Amin Sakzad
Clayton School of IT
Monash University
[email protected]
Joint work with
Ron Steinfeld
October 2015
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
1
Background and Problem Statement
2
Zero-Forcing (ZF) attack and its Advantage Ratio
3
Inverse Precoding
4
Conclusions
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
MIMO Wiretap Channel 1
We consider a slow-fading MIMO wiretap channel model as
follows:
Figure: The block diagram of a MIMO wiretap channel.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
MIMO Wiretap Channel 2
The nr × nt real-valued MIMO channel from user A to user B
is denoted by H.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
MIMO Wiretap Channel 2
The nr × nt real-valued MIMO channel from user A to user B
is denoted by H.
We also denote the channel from A to the adversary E by an
n0r × nt matrix G.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
MIMO Wiretap Channel 2
The nr × nt real-valued MIMO channel from user A to user B
is denoted by H.
We also denote the channel from A to the adversary E by an
n0r × nt matrix G.
The entries of H and G are identically and independently
distributed (i.i.d.) based on a Gaussian distribution N1 . This
model can be written as:
y = Hx + e,
y0 = Gx + e0 .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Dean-Goldsmith Model 1
The entries xi of x ∈ Rnt , for 1 ≤ i ≤ nt , are drawn from a
constellation X = {0, 1, . . . , m − 1} for an integer m.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Dean-Goldsmith Model 1
The entries xi of x ∈ Rnt , for 1 ≤ i ≤ nt , are drawn from a
constellation X = {0, 1, . . . , m − 1} for an integer m.
The components of the noise vectors e and e0 are i.i.d. based
on Gaussian distributions Nm2 α2 and Nm2 β 2 , respectively. We
assume α = β.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Dean-Goldsmith Model 1
The entries xi of x ∈ Rnt , for 1 ≤ i ≤ nt , are drawn from a
constellation X = {0, 1, . . . , m − 1} for an integer m.
The components of the noise vectors e and e0 are i.i.d. based
on Gaussian distributions Nm2 α2 and Nm2 β 2 , respectively. We
assume α = β.
The channel state information (CSI) is available at all the
transmitter and receivers.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Dean-Goldsmith Model 2
To send a message x to B, user A performs a singular value
decomposition (SVD) precoding.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Dean-Goldsmith Model 2
To send a message x to B, user A performs a singular value
decomposition (SVD) precoding.
Let SVD of H be given as H = UΣVt . The user A transmits
Vx instead of x and B applies a filter matrix Ut to the
received vector y.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Dean-Goldsmith Model 2
To send a message x to B, user A performs a singular value
decomposition (SVD) precoding.
Let SVD of H be given as H = UΣVt . The user A transmits
Vx instead of x and B applies a filter matrix Ut to the
received vector y.
With this, the received vectors at B and E are as follows:
ỹ = Σx + ẽ,
y0 = GVx + e0 ,
where ẽ = Ut e.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Correctness Condition for Dean-Goldsmith Cryptosystem
Since Σ = diag(σ1 (H), . . . , σnt (H)) is diagonal, user B
recovers an estimate x̃i of xi as follows:
x̃i = dỹi /σi (H)c = xi + dẽi /σi (H)c .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Correctness Condition for Dean-Goldsmith Cryptosystem
Since Σ = diag(σ1 (H), . . . , σnt (H)) is diagonal, user B
recovers an estimate x̃i of xi as follows:
x̃i = dỹi /σi (H)c = xi + dẽi /σi (H)c .
The decoding process succeeds if |ẽi | < |σi (H)|/2 for all
1 ≤ i ≤ nt .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Correctness Condition for Dean-Goldsmith Cryptosystem
Since Σ = diag(σ1 (H), . . . , σnt (H)) is diagonal, user B
recovers an estimate x̃i of xi as follows:
x̃i = dỹi /σi (H)c = xi + dẽi /σi (H)c .
The decoding process succeeds if |ẽi | < |σi (H)|/2 for all
1 ≤ i ≤ nt .
Let P [B|H] be the probability that B incorrectly decodes x:
P [B|H] ≤ nt Pw←-Nm2 α2 [|w| < |σnt (H)|/2]
= nt Pw←-N1 [|w| < |σnt (H)|/(2mα)]
≤ nt exp (−|σnt (H)|2 )/(8m2 α2 ) ,
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Correctness Condition for Dean-Goldsmith Cryptosystem
Since Σ = diag(σ1 (H), . . . , σnt (H)) is diagonal, user B
recovers an estimate x̃i of xi as follows:
x̃i = dỹi /σi (H)c = xi + dẽi /σi (H)c .
The decoding process succeeds if |ẽi | < |σi (H)|/2 for all
1 ≤ i ≤ nt .
Let P [B|H] be the probability that B incorrectly decodes x:
P [B|H] ≤ nt Pw←-Nm2 α2 [|w| < |σnt (H)|/2]
= nt Pw←-N1 [|w| < |σnt (H)|/(2mα)]
≤ nt exp (−|σnt (H)|2 )/(8m2 α2 ) ,
By choosing parameters like m2 α2≤|σnt (H)|2 /8 log(nt /ε),
one can ensure that B is less than any ε > 0.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Security Condition for Dean-Goldsmith Cryptosystem 1
MIMO − Search problem: Recovering x from y0 = Gv x + e0
and Gv , with non-negligible probability, under certain
parameter settings, upon using massive MIMO systems with
large number of transmit antennas nt .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Security Condition for Dean-Goldsmith Cryptosystem 1
MIMO − Search problem: Recovering x from y0 = Gv x + e0
and Gv , with non-negligible probability, under certain
parameter settings, upon using massive MIMO systems with
large number of transmit antennas nt .
We say that the MIMO − Search problem is hard (secure) if
any attack algorithm against MIMO − Search with run-time
−ω(1)
poly(nt ) has negligible success probability nt
.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Security Condition for Dean-Goldsmith Cryptosystem 2
A polynomial-time complexity reduction is claimed from
worst-case instances of the GapSVPnt /α in lattices of
dimension nt , to the MIMO − Search problem with nt
transmit antennas, noise parameter α and constellation size
m, assuming the following minimum noise level holds:
mα >
Amin Sakzad
√
nt .
(1)
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Security Condition for Dean-Goldsmith Cryptosystem 2
A polynomial-time complexity reduction is claimed from
worst-case instances of the GapSVPnt /α in lattices of
dimension nt , to the MIMO − Search problem with nt
transmit antennas, noise parameter α and constellation size
m, assuming the following minimum noise level holds:
mα >
√
nt .
(1)
The above cryptosystem is called the Massive MIMO Physical
Layer Cryptosystem (MM − PLC).
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Our Contributions
We show that the eavesdropper can decrypt the information
data under the same condition as the legitimate receiver.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Our Contributions
We show that the eavesdropper can decrypt the information
data under the same condition as the legitimate receiver.
We study the signal-to-noise advantage ratio for a more
generalized scheme with an arbitrary precoder and show that
if n0r nt , then there is no such an advantage.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Our Contributions
We show that the eavesdropper can decrypt the information
data under the same condition as the legitimate receiver.
We study the signal-to-noise advantage ratio for a more
generalized scheme with an arbitrary precoder and show that
if n0r nt , then there is no such an advantage.
On the positive side, for the case n0r = nt , we give an O n2
upper bound on the advantage and show that this bound can
be approached using an inverse precoder.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Our Contributions
We show that the eavesdropper can decrypt the information
data under the same condition as the legitimate receiver.
We study the signal-to-noise advantage ratio for a more
generalized scheme with an arbitrary precoder and show that
if n0r nt , then there is no such an advantage.
On the positive side, for the case n0r = nt , we give an O n2
upper bound on the advantage and show that this bound can
be approached using an inverse precoder.
We give a lower bound on the decoding advantage ratio of the
legitimate user over an eavesdropper who is equipped with a
non-linear successive interference cancelation (SIC) stronger
than linear receivers.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Zero-Forcing (ZF) attack
The eavesdropper E receives y0 = Gv x + e0 . Replacing the
SVD, we get y0 = U0 Σ0 (V0 )t x + e0 , where
Σ0 = diag (σ1 (Gv ), . . . , σnt (Gv )) = diag (σ1 (G), . . . , σnt (G)) .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Zero-Forcing (ZF) attack
The eavesdropper E receives y0 = Gv x + e0 . Replacing the
SVD, we get y0 = U0 Σ0 (V0 )t x + e0 , where
Σ0 = diag (σ1 (Gv ), . . . , σnt (Gv )) = diag (σ1 (G), . . . , σnt (G)) .
S(he) computes
ỹ0 = (Gv )−1 y0 = x + ẽ0 ,
(2)
where ẽ0 = V0 (Σ0 )−1 (U0 )t e0 . User E is now able to recover
an estimate x̃0i of xi by rounding:
x̃0i = dỹi0 c = dxi + ẽ0i c = xi + dẽ0i c.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Analysis of ZF attack
Lemma
The components of ẽ0 in (2) are distributed as Nσ2 with
E
σE2 ≤
Amin Sakzad
m2 α2
.
σn2 t (G)
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
The union bound
The above explained ZF attack succeeds if |ẽ0i | < 1/2 for all
1 ≤ i ≤ nt .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
The union bound
The above explained ZF attack succeeds if |ẽ0i | < 1/2 for all
1 ≤ i ≤ nt .
Let PZF [E|G] denotes the decoding error probability that E
incorrectly recovers x using ZF attack. Based on Lemma 1,
we have
1
PZF [E|G] ≤ nt Pw←-Nσ2 |w| <
2
E
|σnt (G)|
≤ nt Pw←-N1 |w| <
.
(3)
2mα
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Distribution of the singular values
Theorem (Edelman89)
Let M be an s × t matrix with i.i.d. entries distributed as N1 . If
s and t tend to infinity in such a way that s/t tends to a limit
y ∈ [1, ∞], then
1 2
σt2 (M)
(4)
→ 1− √
s
y
and
σ12 (M)
→
s
1 2
1+ √
,
y
(5)
almost surely.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Asymptotic probability of error
Theorem
Fix any real ε, ε0 > 0, and y 0 ∈ [1, ∞], and suppose that
n0r /nt → y 0 as nt → ∞. Then, for all sufficiently large nt , the
probability PZF [E] that E incorrectly decodes the message x using
a ZF decoder is upper bounded by ε, if
2
1
0
0
nr
1 − √y 0 − ε
2 2
.
(6)
m α ≤
8 log 2nε t
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Advantage ratio
To analytically investigate the advantage of decoding at B over E,
we define the following advantage ratio.
Definition
For fixed channel matrices H and G, the ratio
advZF ,
σn2 t (H)
,
σn2 t (G)
(7)
is called the advantage of B over E under ZF attack.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Advantage ratio of SVD precoder with ZF attack
Theorem
Let Hnr ×nt be the channel between A and B and Gn0r ×nt be the
channel between A and E, both with i.i.d. elements each with
distribution N1 . Fix real y, y 0 ∈ [1, ∞], and suppose that
nr /nt → y and n0r /nt → y 0 as nt → ∞. Then, using a SVD
precoding technique in MM − PLC, we have
√
advZF → √
y−1
2
y0 − 1
2
almost surely as nt → ∞.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
General Precoder
One may wonder whether a different precoding method
(again, assumed known to E) than used above may provide a
better advantage ratio for B over E.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
General Precoder
One may wonder whether a different precoding method
(again, assumed known to E) than used above may provide a
better advantage ratio for B over E.
Suppose that instead of sending x̃ = Vx, user A precodes
x̃ = P(H)x, where P = P(H) is some other precoding
matrix that depends on the channel matrix H.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
General Precoder
One may wonder whether a different precoding method
(again, assumed known to E) than used above may provide a
better advantage ratio for B over E.
Suppose that instead of sending x̃ = Vx, user A precodes
x̃ = P(H)x, where P = P(H) is some other precoding
matrix that depends on the channel matrix H.
Therefore, in this general case, the advantage ratio of
maximum noise power decodable by B to that decodable by E
under a ZF attack at a given error probability generalizes from
(7) to
σ 2 (HP)
advZF , n2 t
.
(8)
σnt (GP)
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Advantage ratio of general precoder with ZF attack
Theorem
Let H and G be as in Theorem 5. Then we have
advZF ≤ advupZF . Furthermore, fix real y, y 0 ∈ [1, ∞], and suppose
that nr /nt → y and n0r /nt → y 0 as nt → ∞, so that
n0r /nr → y 0 /y , ρ0 . Then, using a general precoding matrix P(H)
in MM − PLC, we have
2
√
y+1
advupZF → √
2
y0 − 1
almost surely as nt → ∞. Hence, in the case n0r = nr and
y 0 = y → ∞, we have advupZF → 1. Moreover, if advupZF → c for
some c ≥ 1, then min(y 0 , ρ0 ) ≤ 9.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Achievable Upper Bound on Advantage Ratio
Theorem (Edelman89)
Let M be a t × t matrix with i.i.d. entries distributed as N1 . The
least singular value of M satisfies
2
h√
i
−x
lim P tσt (M) ≥ x = exp
−x .
(9)
t→∞
2
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
The upper bound
Theorem
Let ε > 0 be fixed, H and G be n × n matrices as in
Proposition 5 with n = nt = nr = n0r . Using a general precoder
P(H) to send the plain text x, the maximum
possible advZF that B
2
can achieve over E, is of order O n , except with probability ≤ ε.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Inverse Precoder Model
We have
ỹ = In x + ẽ,
y0 = GH−1 x + e0 ,
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Inverse Precoder Model
We have
ỹ = In x + ẽ,
y0 = GH−1 x + e0 ,
Note that, for the inverse precoder the advantage ratio (7)
under ZF decoding
algorithm at user E can be written as
2
−1
1/σn GH .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Inverse Precoder Model
We have
ỹ = In x + ẽ,
y0 = GH−1 x + e0 ,
Note that, for the inverse precoder the advantage ratio (7)
under ZF decoding
algorithm at user E can be written as
2
−1
1/σn GH .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Distribution of quotient
Theorem
Let Q = GH−1 , where H and G are two n × n real Gaussian
matrices. The distribution of Q is proportional to
1
.
det (In + QQt )n
Amin Sakzad
(10)
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Inverse Precoder achives maximum advZF
Theorem
Let ε > 0 be fixed, H and G be n × n Gaussian matrices as in
Proposition 5 with n = nt = nr = n0r . Using an inverse precoder
P(H) = H−1 to send the plain text x, the decoding advantage
with respect to zero-forcing
attack advZF , is at least
1
2
2
4 log(1/ε) · n + n = Ω n , except with probability ≤ ε, for
sufficiently large n.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
The exact probability for different orders of n
1
0.9
0.8
0.7
0.6
0.5
G(n) = n
G(n) = n2
0.4
G(n) = 5 ∗ n2
0.3
G(n) = 10 ∗ n2
G(n) = n3
0.2
0.1
0
0
20
40
60
80
100
Figure: The amount of P [advZF < G(n)] for different G(n).
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
advZF for 1000 channel.
8
log10(a dv)
m ea n(lo g10(advup))
m ea n(lo
g10(adv))
log10 n2
7.5
7
lo g10(adv)
6.5
6
5.5
5
4.5
4
3.5
0
100
200
300
400
500
600
Samples
700
800
900
1000
Figure: The advantage ratio (7) for 1000 square channels of size
n = 200 using inverse precoder.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
P n2 σn2 > x for various n
0.8
n = 10
n = 50
n = 100
0.7
Pr n2 σn2 > x
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
x
3
4
5
Figure: The numerical values of P n2 σn2 > x for different dimensions
n = 10, 50, and 100 for 10000 square channels of size n = 100 using
inverse precoder.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Successive Interference Cancellation (SIC) 1
A lattice reduction algorithm is conducted first, and then a
nearest plane algorithm is applied.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Successive Interference Cancellation (SIC) 1
A lattice reduction algorithm is conducted first, and then a
nearest plane algorithm is applied.
Let GH−1 = Q = OR be the QR decomposition of the
equivalent channel. Then the received vector by user E equals
y0 = ORx + e0 .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Successive Interference Cancellation (SIC) 1
A lattice reduction algorithm is conducted first, and then a
nearest plane algorithm is applied.
Let GH−1 = Q = OR be the QR decomposition of the
equivalent channel. Then the received vector by user E equals
y0 = ORx + e0 .
Upon receiving y0 , this user multiplies it by Ot . Hence, we get
ỹ = In x + ẽ,
y00 = Rx + Ot e0 = Rx + e00 ,
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Successive Interference Cancellation (SIC) 2
In SIC decoding framework, the last symbol is decoded first,
i.e.
00 00 yn
en
0
x̃n =
= xn +
rnn
rnn
is an estimate for xn .
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Successive Interference Cancellation (SIC) 2
In SIC decoding framework, the last symbol is decoded first,
i.e.
00 00 yn
en
0
x̃n =
= xn +
rnn
rnn
is an estimate for xn .
The other symbols are approximated iteratively using
$
'
Pn
00 −
0
y
r
x̃
jk
j
k=j+1
k
x̃0j =
,
rjj
for j from n − 1 downward to 1.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Successive Interference Cancellation (SIC) 2
In SIC decoding framework, the last symbol is decoded first,
i.e.
00 00 yn
en
0
x̃n =
= xn +
rnn
rnn
is an estimate for xn .
The other symbols are approximated iteratively using
$
'
Pn
00 −
0
y
r
x̃
jk
j
k=j+1
k
x̃0j =
,
rjj
for j from n − 1 downward to 1.
The above mentioned SIC finds the closest vector if the
distance from input vector to the lattice is less than half the
2
2 , that is rnn .
length of the shortest rjj
2
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Advantage ratio under SIC
We define the following advantage ratio:
advSIC ,
2 (I)
rnn
,
2 (Q)
rnn
(11)
is called the advantage of B over E under SIC attack. Since
2 (I) = 1, the adv
2
rnn
SIC = 1/rnn (Q).
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Distribution of diagonal elements 1
Theorem
2 are
Let the matrices Q, O, and R be as above. Then rjj
independently distributed as B II n−j+1
, 2j , for 1 ≤ j ≤ n.
2
A random variable v is said to have a beta distribution of the
second type (beta prime distribution) B II (a, b) if it has the
following probability density function
1
v a−1 (1 + v)−(a+b) ,
β(a, b)
v > 0,
where both a and b are non-negative and β(a, b) is the beta
function.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Distribution of diagonal elements 2
0.12
1800
1600
0.1
1400
0.08
1200
1000
0.06
800
0.04
600
400
0.02
200
0
0
50
100
150
200
0
0
50
100
150
200
2
Figure: The numerical histogram and the theoretical p.d.f. of rjj
for
j = 10 and 10000 square channels of size n = 100 using inverse precoder.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Distribution of diagonal elements 3
1.5
400
350
300
1
250
200
150
0.5
100
50
0
0
1
2
3
0
0
1
2
3
2
Figure: The numerical histogram and the theoretical p.d.f. of rjj
for
j = 50 and 10000 square channels of size n = 100 using inverse precoder.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Adversary with SIC
Theorem
Let Hn×n be the channel between A and B and Gn×n be the
channel between A and E, both with i.i.d. elements each with
distribution N1 . Then, using an inverse precoding technique in
MM − PLC, we have advSIC = O (n).
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
2
Numerical analysis of P nrnn
(Q) < x
1
0.95
n = 10
n = 50
n = 100
2
Pr nrnn
<x
0.9
0.85
0.8
0.75
0.7
0.65
1
2
3
4
5
x
6
7
8
9
10
2
(Q) < x for different
Figure: The numerical values of P nrnn
dimensions n = 10, 50, and 100 for 10000 square channels of size
n = 100 using inverse precoder.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Conclusions
A Zero-Forcing (ZF) attack has been presented for the
massive multiple-input multiple-output MIMO physical layer
cryptosystem (MM − PLC).
A decoding advantage ratio has been defined and studied for
ZF linear receiver.
It has been shown that this advantage tends to 1 employing a
singular value decomposition (SVD) precoding approach at
the legitimate transmitter and a ZF linear receiver at the
adversary.
An advantage ratio in the order of n2 is achievable if the
legitimate user applies an inverse precoder.
If eavesdropper employs a stronger decoder algorithm such as
a successive interference cancellation (SIC), then the
advantage ratio will be reduced to a constant fraction of n.
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec
Background and Problem Statement
Zero-Forcing (ZF) attack and its Advantage Ratio
Inverse Precoding
Conclusions
Thank you!
Amin Sakzad
Massive MIMO Physical Layer Cryptosystem through Inverse Prec