Intelligent Reflecting Surface Assisted Secret Key Generation

Abstract

In secret key generation of physical layer security technology, it is challenging to achieve high key capacity and low secret key inconsistency rate. This paper investigates intelligent reflecting surface (IRS) assisted secret key generation, which aims to maximize the secret key capacity by adjusting the placement of the IRS units. Specifically, we first analyze and deduce the key capacity expression of the IRS-assisted system from the perspective of information theory. Then we investigate how to use the channel state information (CSI) to place the IRS units effectively so as to maximize the secret key capacity. Simulation results show that our scheme could improve the quality of secret key generation significantly.

Full text

1
Intelligent Reflecting Surface Assisted Secret Key
Generation
Xinjin Lu, Jing Lei*, Yuxin Shi, Wei Li
Abstract—In secret key generation of physical layer security
technology, it is challenging to achieve high key capacity and low
secret key inconsistency rate. This paper investigates intelligent
reflecting surface (IRS)-assisted secret key generation, which aims
to maximize the secret key capacity by adjusting the placement
of the IRS units. Specifically, we first analyze and deduce the
key capacity expression of the IRS-assisted system from the
perspective of information theory. Then we investigate how to
use the channel state information (CSI) to place the IRS units
effectively so as to maximize the secret key capacity. Simulation
results show that our scheme could improve the quality of secret
key generation significantly.
Index Terms—secret key generation, intelligent reflecting sur-
face, secret key capacity
I. INTRODUCTION
Since Maurer [1] proposed that both legitimate commu-
nication parties can extract the same key through a related
random source in 1993, the key-based physical layer security
mechanism has drawn significant research attention gradually
[2], [3]. Due to the time-varying, short-term reciprocity and
space-time uniqueness of wireless channels, it can be used as a
natural source to generate secret keys. The sender and receiver
can obtain a secure shared key through channel probing, mea-
surement quantization, information reconciliation and privacy
amplification. Secret key capacity is an important parameter
in the study of secret key generation. Since Ahlswede and
Csiszar [4] et al. derived the theoretical upper limit of the
secret key capacity in source-type model with wiretapper and
channel-type model with wiretapper, the research on the secret
key capacity under more complex system models has been
successively developed [5], [6].
On the other hand, intelligent reflecting surface (IRS)
[7], [8] has emerged as a promising technology to improve
communication qualities through some adjustments. Generally,
signals can be controlled smartly by adjusting the reflection
coefficients of IRS such as the phase, amplitude, frequency,
or even polarization [8], [9]. In essence, IRS is composed of
a large number of reconfigurable and passive reflecting units
whose location can be also adaptively placed. These IRS units
can independently incur some change to the incident signal,
which could help the signal transmissions.
In recent years, more and more studies have applied IRSs to
physical layer security of wireless communications [10], [11].
The intuition of these researchers is that the IRS can be used to
improve the secrecy data rate under wiretap channel, which is
Xinjin Lu, Jing Lei, Yuxin Shi and Wei Li are with College of Electronic
Science and Technology, National University of Defense Technology, Chang-
sha, China. (Email{luxingjin17, leijing, shiyuxin13, weili }@nudt.edu.cn)
called keyless information theory security [12]. However, there
are no research on the IRS-assisted utilization in key-based
physical layer security mechanism. In the key generation of
the physical layer security technology, how to extract the key
effectively and make full use of the channel state information
(CSI) to get more keys has always been an open issue. Since
the IRS is able to configure the wireless channel in real-time
via passive reflection, it has great potential in improving the
secret key capacity
Aiming at the problem of wireless channel key generation,
this paper proposes a scheme based on IRS assisted secret
key generation. By deriving the key capacity expression of
the IRS assisted system, we further optimize the placement of
smart IRS units, or the switch state of IRS units. This scheme
can maximize the key capacity of the system when IRS units
resources are limited. The simulation results show that the
system with location optimization of the IRS units can not
only effectively increase key capacity, but also greatly reduce
the key inconsistency rate.
The main structure of this paper is as follows. Section
II introduces the IRS assisted system model. Section III
derives the key capacity analytical formula for the system and
proposes an optimization scheme for the placement of IRS
units. Section IV gives the simulation results and V concluded
the full paper.
Notations: Throughout our discussions, the distribution of
complex Gaussian random variable with mean 0 and variance
σ2is denoted by ∼CN (0, σ2).CM×Ndenotes norm, the
trace of a matrix and the space of M×Ncomplex-valued
matrices.
II. IRS ASSISTED SYSTEM MODEL
The idea of secret key generation is that the legitimate
communication parties extract the secure shared secret keys
through channel probing, quantizing, information reconcilia-
tion and privacy amplification. The specific process of key
generation based on CSI is shown in Fig. 1 and the steps are
as follows.
•Channel probing [13], [14]: The legitimate communi-
cation parties Alice and Bob successively send channel
sounding signals to each other within the coherence time,
and both parties obtain channel characteristic observation
values based on the received signals.
•Measurement quantization [15], [16]: Both parties of
legitimate communication adopt the same quantization
scheme to quantify the channel feature values obtained
by channel probing to furture get the initial key.

2
TDD Mode
Coherent time
Information
Reconciliation
Privacy
Amplification
Probing of channel
feature
Probing of channel
feature
Measurement
quantization
Key consistency check
Secret keys
Measurement
quantization
Key consistency check
Secret keys
Alice Bob
Wireless channel
with IRS
Fig. 1. The process of secret key generation based on wireless channel.
•Information reconciliation [17], [18]: Due to factors such
as noise, interference, estimation error, half-duplex, etc.,
there may be inconsistent bits in the initial key. The
two parties in legitimate communication complete the
verification of inconsistent key bits through information
exchange on the common channel, and obtain consistent
key bits. The interactive information can be key sequence
number, parity check matrix, etc.
•Privacy amplification [19], [20]: During channel probing
and information reconciliation process, the eavesdropper
Eve may overhear some information about the key, which
poses a potential threat to the security of the key. Privacy
amplification can be used to eliminate the relevant infor-
mation about the key obtained by Eve, which ensures that
Eve can not get any information about the secret key.
The system model of IRS assisted secret key generation is
depicted in Fig.2, Alice and Bob are legitimate communication
nodes who aim to extract the secret key from wireless channel.
IRS is the IRS and Eve is passive eavesdropper. Alice, Bob
and Eve are all equipped with a single antenna, and all
adopt time-division duplex (TDD) working mode and half-
duplex communication style, which can ensure the reciprocity
between the uplink and downlink channels within a coherent
time. Channel coefficients of Alice-Bob link hAB, Bob-Alice
link hBA , Alice-Eve link hAE and Bob-Eve link hBE are
satisfied h∆∼CN (0, σ2
h∆),∆∈(AB, B A, AE, BE ). In
addition, the Alice-IRS-Bob link, the Bob-IRS-Alice link are
denoted by hAIB ∈CN×1and hB IA ∈CN×1, respectively. N
is the number of IRS reflecting units. It is assumed that Eve
can monitor the communication content between Alice and
Bob, but cannot actively interfere with the key establishment
process. Assuming that the distance between Eve and Alice
or Bob is greater than a half wavelength λ/2, the channel
features of the main channel and the eavesdropping channel
are independent of each other [21].
The IRS units receive all multi-path received signals and
reflect the combined signal from via IRS planar array. We
denote Ψ= [β1Ψ1, β2Ψ2, β3Ψ3, ..., βNΨN]Tas the vector
associated with the effective phases shifts Ψiin all IRS units,
where βi∈ {0,1}indicates the switch state of the reflection
BA
h
/2d


AB
h
AE
h
BE
h
/2d


Alice Bob
Eve
IRS
BIA
h
BIA
h
AIB
h
BIA
h
Fig. 2. IRS assisted system model of the secret key generation.
units at the current position. β=1 means that IRS units is power
on, while β=0 means that IRS is power off. It is assumed that
the units of IRS are separate of each other. Thus, the received
signal at Alice and Bob can be respectively written as
yA= (hBA +hBI A)x+nA
= (hBA +
N
P
i=1
hi
BI AβiΨi)x+nA
yB= (hAB +hAIB )x+nB
= (hAB +
N
P
i=1
hi
AIB βiΨi)x+nB
(1)
where nA∼CN (0, σ2
B),nB∼CN (0, σ2
A)denote the noise
at the legitimate users. hi
BI A and hi
AIB are satisfied hi
∆∼
CN (0, σ2
hi
∆),∆∈(BI A, AIB ).
III. SEC RE T KE Y CA PACITY ANALYS IS A ND SCHEME
OPTIMIZATION OF IRS AS SISTED SYSTEM
Secret key capacity is the upper bound of the key generation
rate. As shown in Fig. 3, hAand hBrepresent the main
channel state information (CSI) obtained by Alice and Bob,
respectively. hEis the eavesdropping channel information
obtained by Eve. Therefore, the secret key capacity can be
expressed as a form of mutual information.
C=I(hA;hB|hE)(2)
A
h
B
h
E
h
( ; | )
A B E
I h h h
( ; )
AB
I h h
Fig. 3. The diagram of secret key capacity.
Based on the system model in Fig. 2, Alice and Bob
send known probing signals to each other in turns within the

3
channel coherence time. Alice and Bob’s estimation of CSI
can be expressed as:
hA=hBA +zA
= (hBA +hBI A)x+zA
= (hBA +
N
P
i=1
hi
BI AβiΨi)x+zA
hB=hAB +zB
= (hAB +hAIB )x+zB
= (hAB +
N
P
i=1
hi
AIB βiΨi)x+zB
(3)
where zA∼CN (0, σ2
zA)and zB∼CN (0, σ2
zB)are the
observation noises at the nodes of Alice and Bob. It should be
noted that the noise term includes the noise of direct channel
hBA or hAB and all sub-channels hBI A or hAI B .
The channel estimation values hBA and hAB obey the
following distribution.
hBA ∼CN(0, σ2
hBA )
hAB ∼CN (0, σ2
hAB )
(4)
where σ2
hBA and σ2
hAB can be described as
σ2
hBA =σ2
hBA +
N
P
i=1
β2
iσ2
hi
BIA
σ2
hAB =σ2
hAB +
N
P
i=1
β2
iσ2
hi
AIB
(5)
Since the channel meets reciprocity in the coherent time, we
can further get σ2
hAB =σ2
hBA .
Besides, since the distance between Eve and Alice or Bob is
more than λ/2, the legitimate CSI of hAB and hBA could not
be included in hE. The formula (2) can be further simplified
as C=I(hA;hB|hE)
=I(hA;hB)
=Hd(hA) + Hd(hB)−Hd(hA, hB)
(6)
where Hd(·)represents differential entropy.
Ye et al. pointed out that legitimate communication parties
would establish the following observations in order to get
shared keys from legitimate channel [22].
x=h+na
y=h+nb
(7)
where h∼CN (0, σ2
h),na∼CN (0, σ2
na)and nb∼
CN (0, σ2
nb). And the secret key capacity can be expressed
as
I(x;y) = log2

1 + σ2
h
σ2
na+σ2
nb+σ2
naσ2
nb
σ2
h


(8)
We can take formula (4) into formula (8) to obtain the secret
key capacity of the system:
I(hA;hB) = log2

1 + σ2
hAB
σ2
zA+σ2
zB+σ2
zA
σ2
zB
σ2
hAB


(9)
It is assumed that zAand zBare independent and identically
distributedand σ2
zA=σ2
zB=σ2
z. Formula (9) can be further
simplified as
I(hA;hB) = log2 1 + σ4
hAB /σ4
z
1+2σ2
hAB /σ2
z!(10)
We can take formula (5) into formula (10) to obtain the
final secret key capacity:
I(hA;hB) = log2




1 +
(σ2
hAB +
N
P
i=1
β2
iσ2
hi
AIB )
2
/σ4
z
1 + 2(σ2
hAB +
N
P
i=1
β2
iσ2
hi
AIB
)/σ2
z





(11)
It can be seen from formula (11) that the secret key capacity
of the system can be adjusted by the reflection factor βiof the
IRS units. Alice and Bob take turns to send probing signals in
the channel probing process. The CSI of all sub-channels can
be obtained through channel estimation, which could be used
to further improve the probing of subsequent channel. Alice
and Bob can adjust the smart reflector to obtain more keys by
use of the CSI in the subsequent channel probing process.
If the number of IRS units is limited or the number of
IRS units that can be turned on is limited, i.e.
N
P
i=1
βi6M,
the parameters IRS βineeds to be adjusted effectively to
maximize the secret key capacity. This problem can be further
transformed as
max I(hA;hB)
s.t.
N
P
i=1
βi6M
βi∈ {0,1}, i = 1,2, . . . , N
(12)
In order to find the optimal placement of IRS units, we sort
σ2
hi
AIB
corresponding to different hi
AIB and take the first M
positions with the largest σ2
hi
AIB
to place the IRS units.
IV. SIMULATION RESULTS AND ANALYSIS
We conduct Monte Carlo simulation verification on the pro-
posed scheme. Fig. 4 shows the secret key capacity I(hA;hB)
under different signal-to-noise ratio (SNR) when the number
of IRS units is limited. It can be seen that the I(hA;hB)of
the IRS assisted system has been greatly improved compared
with the system without IRS. The I(hA;hB)of the IRS
assisted system increases as the SNR increases. In addition,
the I(hA;hB)on the scheme of IRS-Optimal placement is
higher than that of IRS-Random placement. The main reason
is that the sub-channels path with better CSI are selected in
the scheme of IRS-Optimal placement.
Fig. 5 compares the I(hA;hB)on the scheme of IRS-
Randam placement and when R= 1/8,1/4,1/2and
N= 128.R=M/N means that MIRS units are selected
from the Navailable IRS units for practical use. It can be
seen that the I(hA;hB)increases with the increase of R.
In addition, the I(hA;hB)on the scheme of IRS-Optimal
placement is better than the scheme of IRS-Randam placement
and this advantage increases with the decrease of R. Fig. 6

4
0 5 10 15 20 25 30
SNR (dB)
10-1
100
101
I(hA;hB)
IRS-Optimal placement
IRS-Random placement
Without IRS
Fig. 4. The comparison of the IRS-assisted system and the system without
IRS on secret key capacity.
0 5 10 15 20 25 30
SNR (dB)
4
6
8
10
12
14
16
18
20
I(hA;hB)
IRS-Randam placement, R = 1/8
IRS-Optimal placement, R = 1/8
IRS-Randam placement, R = 1/4
IRS-Optimal placement, R = 1/4
IRS-Randam placement, R = 1/2
IRS-Optimal placement, R = 1/2
Fig. 5. Comparison of IRS-Randam placement and IRS-Optimal placement
on secret key capacity when R= 1/8,1/4,1/2.
shows the comparison of the secret key capacity on the scheme
of IRS-Randam placement and when N= 32,64,128 and
R= 1/4. We can see that the I(hA;hB)increases with the
increase of Nand the I(hA;hB)on IRS-Optimal placement
is greater. Since the selection space of sub-channels available
increases as the number of IRS units increases, which can
bring higher secret key capacity. In addition, the I(hA;hB)of
the scheme based on the placement optimization of the IRS
units is much higher that of random placement.
The IRS-assisted system can also obtain a lower key incon-
sistency rate. Fig. 7. compares the secret key inconsistency rate
performance of the different schemes which adopt adaptive
single-bit quantization with guard interval [24]. It can be seen
that the key inconsistency rate performance of IRS-Optimal
placement is better than the IRS-Randam placement, which
means that Alice and Bob can obtain the initial secret keys
0 5 10 15 20 25
SNR (dB)
4
6
8
10
12
14
I(hA;hB)
IRS-Randam placement, N=32
IRS-Optimal placement, N=32
IRS-Randam placement, N=64
IRS-Optimal placement, N=64
IRS-Randam placement, N=128
IRS-Optimal placement, N=128
Fig. 6. Comparison of IRS-Randam placement and IRS-Optimal placement
on secret key capacity when N= 32,64,128.
0 5 10 15 20 25 30
SNR (dB)
10-3
10-2
10-1
100
Key inconsistency rate
IRS-Random placement
IRS-Optimal placement
Without IRS
Fig. 7. Comparison of IRS-Randam placement and IRS-Optimal placement
on key inconsistency rate.
with a smaller secret key inconsistency rate. This makes infor-
mation negotiation easier to implement in the key generation
process.
V. CONCLUSION
Aiming at the difficulty of secret key generation in the
wireless channel under single path, this paper proposes an
IRS-assisted secret key generation scheme, which solves the
key generation problem by using IRS units to increase the
number of sub-channels. We give the derivation of the secret
key capacity formula of the IRS assisted system, and further
enhance the secret key capacity of the system by optimizing
the placement of the IRS units. Simulation results shows
that this scheme can not only save IRS resources, but also
effectively improve the performance of secret key generation.

5
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