# Joint Optimization for Secure and Reliable Communications in Finite
Blocklength Regime

Mintaek Oh, Jeonghun Park, and Jinseok Choi  M. Oh and J. Choi are with the
Department of Electrical Engineering, Ulsan National Institute of Science and
Technology (UNIST), Ulsan, 44919, South Korea (e-mail: {ohmin,
jinseokchoi}@unist.ac.kr). J. Park is with the School of Electronics
Engineering, College of IT Engineering, Kyungpook National University, Daegu,
41566, South Korea (e-mail: jeonghun.park@knu.ac.kr).

###### Abstract

To realize ultra-reliable low latency communications with high spectral
efficiency and security, we investigate a joint optimization problem for
downlink communications with multiple users and eavesdroppers in the finite
blocklength (FBL) regime. We formulate a multi-objective optimization problem
to maximize a sum secrecy rate by developing a secure precoder and to minimize
a maximum error probability and information leakage rate. The main challenges
arise from the complicated multi-objective problem, non-tractable back-off
factors from the FBL assumption, non-convexity and non-smoothness of the
secrecy rate, and the intertwined optimization variables. To address these
challenges, we adopt an alternating optimization approach by decomposing the
problem into two phases: secure precoding design, and maximum error
probability and information leakage rate minimization. In the first phase, we
obtain a lower bound of the secrecy rate and derive a first-order
Karush–Kuhn–Tucker (KKT) condition to identify local optimal solutions with
respect to the precoders. Interpreting the condition as a generalized
eigenvalue problem, we solve the problem by using a power iteration-based
method. In the second phase, we adopt a weighted-sum approach and derive KKT
conditions in terms of the error probabilities and leakage rates for given
precoders. Simulations validate the proposed algorithm.

###### Index Terms:

Physical layer security, finite blocklength, secure precoding, error
probability and information leakage rate minimization, alternating
optimization.

##  I Introduction

Ultra-reliable low latency communications (URLLC) have inevitably become one
of the primary usage scenarios in realizing 5G and 6G communications [1]. For
instance, URLLC is closely related to mission-critical internet-of-things
(IoT) and vehicle-to-everything (V2X) applications oriented toward high
reliability and low latency [2, 3]. To support such delay-sensitive
communications, a short-packet transmission is often taken into account [4, 5,
6, 7]. Unlike conventional communications that mainly consider the infinite
blocklength coding regime, the finite blocklengh (FBL) based communications is
more suitable to leverage the benefit of short data packet transmission [8,
9]. The communication in the FBL regime, however, is limited by a back-off
factor determined by non-negligible decoding error probability and blocklength
[10]. Accordingly, the existing transmission strategies with infinite
blocklength becomes far less optimal in the FBL regime and thus, delay-
constrained applications need to carefully consider the reliability issue as
well as the latency; optimizing the per-user error probability plays a key
role in achieving the high throughput with low latency in the FBL regime [11].

In addition, information security has become a critical issue for future
wireless networks to protect confidential information from eavesdroppers.
Since the wireless communication systems are highly likely to be vulnerable to
eavesdropping due to the broadcast nature of wireless communications [12], the
importance of security has been gaining more attention [13, 14]. Considering
the complexity issues, physical layer security (PLS) has been considered as a
promising solution for secure communications [12, 15]. In [15], a key
mechanism of PLS is to apply a secure channel coding technique to confidential
information so that only the legitimate users are capable of decoding the
information. In [16, 17], the maximal secret communication rate called the
secrecy rate has characterized at which information can be transmitted
securely and reliably over a wiretap channel. In spite of the broad
investigation on the PLS methods, it is inappropriate to directly apply the
existing solutions when communication systems operate in the FBL regime due to
the additional back-off factor from the FBL penalty in PLS [18, 19, 20]. In
this regard, we investigate PLS for downlink FBL-based communications.

###  I-A Prior Work

PLS has been widely studied in wireless communication systems. For the case of
multiple users and a single eavesdropper in multiple-input single-output
(MISO) wireless networks, suboptimal precoding solutions were proposed in
previous works. A secure precoding that maximizes a sum secrecy spectral
efficiency (SE) with suboptimal scheduling was proposed in [21]. In [22], a
secrecy rate optimization was proposed for maximizing the MISO network’s
energy efficiency in terms of users’ quality-of-service (QoS). For a multiple-
input multiple-output (MIMO) system, the secrecy SE with multiple
eavesdroppers equipped with multiple antennas was investigated in [23].
Furthermore, in [24], a secure precoding algorithm was proposed in MIMO
broadcasting channels for a simultaneous wireless information and power
transfer system. For a multi-user multi-eavesdropper network, a secure
transmission strategy was proposed in [25]. Specifically, in [25], an
artificial noise (AN)-aided secure precoding was considered to maximize the
secrecy throughput by addressing the leakage signal from non-colluding
eavesdroppers. Moreover, there exists prior work incorporating a scenario of
colluding eavesdroppers [26].

Based on the analytical finding of the maximal channel coding rate at the FBL
[10], there have been several works that analyzed accurate information-
theoretic approximations of the achievable rates in the FBL regime [27, 28,
29]. Leveraging the results in the FBL regime, in [30, 31], resource
allocation was investigated to maximize the sum rate and to assist the short-
packet communication subject to QoS constraints for URLLC users. In addition,
the decoding error probability also needs to be considered in advanced
wireless communication techniques incorporating reliable and real-time
services. In this regard, the joint optimization of SE and error probability
in the FBL regime is needed to cope with the stringent requirements of URLLC.
The joint optimization problem in [32] was solved by using an alternation
optimization framework between the sum SE maximization and error probability
minimization.

Recent studies have also analyzed PLS in the FBL regime [18, 19]. In [19], the
maximal secret communication rate over a wiretap channel in the FBL regime,
i.e., the secrecy rate, was derived. According to the fundamental results, it
was observed that there exist tradeoffs among delay, reliability, and
security. In this regard, the wireless communication techniques need to be
further investigated for improving the secrecy rate in the FBL regime [33]. To
find the optimal tradeoff, state-of-the-art precoding methods were proposed to
improve the secrecy rate under the reliability and security constraints. For
the case of a single user and eavesdropper, analytical frameworks of the FBL-
based secure communications were investigated in [34]. Specifically, in [34],
the performance of FBL-based communications in the presence of an eavesdropper
was investigated, and the optimal blocklength in maximizing secrecy throughput
was also analyzed by using the AN-aided maximum ratio transmission (MRT)
precoding. For full-duplex MIMO short-packet systems, [35] proposed the secure
precoding method based on zero-forcing (ZF) to maximize the sum secrecy rate.
From [36], an outage probability considering both reliability and security was
proposed according to the characteristics of FBL-based secure communications.
In our previous work [37], we proposed an efficient secure precoding algorithm
for downlink communications with multiple users and a single eavesdropper.

Although the prior works investigated the secure communications in the FBL
regime, the single-eavesdropper scenario is mainly considered. In addition,
the prior work provided theoretical analysis and made an effort of maximizing
the achievable secrecy rate with the conventional linear precoders such as MRT
and ZF. In this regard, thorough optimization of transmission for FBL-based
secure communications under the reliability and security constraints for the
general multi-user multi-eavesdropper MIMO network is still missing. It is,
however, challenging to solve the optimization problem for such sophisticated
networks. In addition, we note that solving the sum secrecy rate optimization
problem is already non-convex and difficult to solve [38], and the back-off
factors entangled with the secrecy rate make the problem more challenging
[19]. Besides, it is infeasible to find direct solutions with the multi-
objectives [32]. Overcoming the aforementioned challenges, we put forth a
novel secure precoding method that brings the optimal tradeoff among rate,
reliability, and security in the FBL regime.

###  I-B Contributions

In this paper, we consider the FBL-based downlink secure communication systems
where an access point (AP) with multiple antennas serves multiple users while
multiple eavesdroppers attempt to wiretap the user messages. Our contributions
are summarized as follows:

  * •

Based on the characteristics of FBL-based communications, we adopt the secrecy
rate as our key performance metric. Using the secrecy rate, we formulate a sum
secrecy rate maximization problem to jointly optimize $(i)$ a precoding
matrix, and $(ii)$ error probability and information leakage rate. However,
there exist several challenges in solving the formulated optimization problem.
First, the problem is inherently non-convex and thus, finding a global optimal
solution is infeasible. Second, the formulated problem is a multi-objective
optimization, and also each user has error probability and information leakage
constraints determined by the system reliability requirements. Third, the
secrecy rate is not tractable due to the back-off factors which are functions
of optimization variables intertwined with each other. Finally, the sum
secrecy rate is non-smooth since the secrecy rate under the presence of
multiple eavesdroppers is determined by the maximum wiretap channel rate.

  * •

To resolve aforementioned difficulties, we first decompose the problem into
two phases adopting alternating optimization: the secrecy rate maximization
precoding design and the maximum error probability and information leakage
rate minimization. In the first phase, for given error probability and leakage
rate, we apply a smooth approximation to a non-smooth objective function.
Then, we make the problem a more tractable form by deriving a lower bound of
the secrecy rate. Due to non-convexity of the problem, we derive a first-order
KKT condition which can be interpreted as a generalized eigenvalue problem and
hence, the best local optimal precoder can be found by finding its principal
eigenvector. It can be solved by a power iteration-based precoding method. In
the second phase, we reformulate the multi-objective optimization problem into
a single-objective optimization problem by using a weighted-sum approach for a
given precoder. Since the problem is still not tractable due to the maximum
function in the secrecy rate, we further consider the lower bound of the
objective function and optimize the maximum error probability and information
leakage rate by solving KKT conditions.

  * •

We note that obtaining the instantaneous channel state information at a
transmitter (CSIT) of eavesdroppers is challenging. Hence, we consider the
case of the partial CSIT of wiretap channels which is regarded as a more
practical scenario. Assuming that only the wiretap channel covariance is
available at the AP, we reformulate the problem by deriving an approximated
lower bound of the secrecy rate which can be handled by using the channel
covariance. Then, we follow similar steps as for the perfect CSIT case and
develop a joint precoder, error probability and information leakage rate
optimization algorithm under the partial CSIT of wiretap channels.

  * •

Via simulations, we verify the rate, reliability, and security performances of
the proposed joint optimization methods. In particular, we demonstrate that
the proposed algorithm achieves the highest secrecy rate while keeping the
lowest maximum error probability and information leakage rate in various
scenarios compared to baseline methods. Therefore, it is concluded that the
proposed algorithms are beneficial to the FBL-based secure communications
under the strict requirement of the reliability and security for future
wireless applications.

Notation: $\bf{A}$ is a matrix and $\bf{a}$ is a column vector. The
superscripts $(\cdot)^{\sf T}$, $(\cdot)^{\sf H}$, and $(\cdot)^{-1}$ denote
the transpose, Hermitian, and matrix inversion, respectively. The blackboard
bold symbols $\mathbb{C}$, $\mbox{$\mathbb{R}$}_{+}$, and
$\mbox{$\mathbb{N}$}_{+}$ denote the complex, nonnegative real, and
nonnegative integer domains, respectively. ${\bf{I}}_{N}$ is the identity
matrix with size $N\times N$. Assuming that
${\bf{A}}_{1},\dots,{\bf{A}}_{N}\in\mathbb{C}^{K\times K}$, ${\bf{A}}={\rm
blkdiag}\left({\bf{A}}_{1},\dots,{\bf{A}}_{N}\right)\in\mbox{$\mathbb{C}$}^{KN\times
KN}$ is a block diagonal matrix. $\|\bf A\|$ represents L2 norm. We use
${\rm{tr}}(\cdot)$ for trace operator, ${\rm{vec}}(\cdot)$ for vectorization,
and ${\mbox{$\mathcal{U}$}}(a,b)$ is a uniform distribution with two
boundaries $a$ and $b$. We also follow MATLAB style notation.

##  II System Model

We consider a downlink network in which the AP equipped with $N$ antennas
serves $K$ single-antenna users. The network includes $M$ single-antenna
eavesdroppers, and they attempt to overhear legitimate user messages from the
AP. In addition, we assume the FBL channel coding, i.e., the coding length is
$L\ll\infty$. We denote an user set and an eavesdropper set as
$\mathcal{K}=\left\\{1,\cdots,K\right\\}$ and
$\mathcal{M}=\left\\{1,\cdots,M\right\\}$, respectively. The data symbol for
user $k$, $s_{k}$, is drawn from a Gaussian distribution with a zero mean and
variance of $\mbox{$\mathbb{E}$}[|s_{k}|^{2}]=P$, $\forall k\in\mathcal{K}$.

The AP broadcasts the data symbols $s_{k},\forall k\in\mathcal{K}$ to each
legitimate user through a linear precoder ${\bf F}=[{\bf f}_{1},\ldots,{\bf
f}_{K}]\in\mbox{$\mathbb{C}$}^{N\times K}$, where ${\bf
f}_{k}\in\mbox{$\mathbb{C}$}^{N}$ indicates a precoding vector for $s_{k}$,
$\forall k\in\mathcal{K}$. A transmitted signal vector ${{\bf
x}}\in\mbox{$\mathbb{C}$}^{N}$ is given by

$\displaystyle{\bf{x}}=\sum_{k=1}^{K}{\bf{f}}_{k}s_{k}={{\bf F}{\bf s}},$ (1)

where ${\bf s}=[s_{1},\dots,s_{K}]^{\sf T}\in\mbox{$\mathbb{C}$}^{K}$. After
transmission, a received signal of user $k$ is given by

$\displaystyle y_{k}={\sqrt{\gamma_{k}}{\bf h}^{\sf H}_{k}{\bf
f}_{k}s_{k}}+{\sum_{\ell\neq k,\ell=1}^{K}\sqrt{\gamma_{k}}{\bf{h}}_{k}^{\sf
H}{\bf f}_{\ell}s_{\ell}+n_{k}},$ (2)

where $\gamma_{k}$ and ${\bf h}_{k}\in\mbox{$\mathbb{C}$}^{N}$ are the large-
scale channel fading term and small-scale channel fading vector between the AP
and user $k$, respectively, and $n_{k}$ is an additive white Gaussian noise
(AWGN) at user $k$ with a zero mean and variance of $\sigma^{2}$. Similarly,
the large-scale channel fading term and small-scale channel fading vector
between the AP and eavesdroppers are represented as $\gamma_{m}^{\sf e}$ and
${\bf g}_{m}\in\mbox{$\mathbb{C}$}^{N}$, respectively. Then, the received
signal at eavesdropper $m$ is given by

$\displaystyle y^{\sf e}_{m}=\sum_{\ell=1}^{K}\sqrt{\gamma_{m}^{\sf
e}}{\bf{g}}_{m}^{\sf H}{\bf f}_{\ell}s_{\ell}+n^{\sf e}_{m},$ (3)

where $n^{\sf e}_{m}$ is the AWGN noise at the eavesdropper $m$ with a zero
mean and variance of $\sigma^{2}_{\sf e}$. We assume the perfect CSIT for both
the legitimate users and eavesdroppers. We then extend our method to a partial
CSIT case.

##  III Problem Formulation

In this section, we introduce performance metrics that incorporate the effect
of FBL in the considered communication system. The achievable rate of user $k$
is

$\displaystyle R_{k}=\log_{2}\left(1+\rho_{k}\right),$ (4)

where $\rho_{k}$ is the signal-to-interference-plus-noise ratio (SINR) of user
$k$ defined as

$\displaystyle\rho_{k}=\frac{\gamma_{k}|{\bf{h}}_{k}^{\sf
H}{\bf{f}}_{k}|^{2}}{\sum_{\ell\neq k,\ell=1}^{K}\gamma_{k}|{\bf{h}}_{k}^{\sf
H}{\bf{f}}_{\ell}|^{2}+\sigma^{2}/P}.$ (5)

The achievable rate of eavesdropper $m$ for $s_{k}$ is

$\displaystyle R_{m,k}^{\sf e}=\log_{2}\left(1+\rho_{m,k}^{\sf e}\right),$ (6)

where $\rho_{m,k}^{\sf e}$ is the SINR of eavesdropper $m$ for $s_{k}$ defined
as

$\displaystyle\rho_{m,k}^{\sf e}=\frac{\gamma^{\sf e}_{m}|{\bf{g}}^{\sf
H}_{m}{\bf{f}}_{k}|^{2}}{\sum_{\ell\neq k,\ell=1}^{K}\gamma^{\sf
e}_{m}|{\bf{g}}^{\sf H}_{m}{\bf{f}}_{\ell}|^{2}+\sigma^{2}_{\sf e}/P}.$ (7)

According to [18, 19], the secrecy rate which measures the maximum
transmission rate of the confidential information that any eavesdropper cannot
decode in the FBL regime is given as

$\displaystyle R_{k}^{\sf sec}({\bf
F},\epsilon_{k},\delta_{m,k};L)=R_{k}-\sqrt{\frac{\mathcal{V}_{k}}{L}}Q^{-1}(\epsilon_{k})-\max_{m\in\mathcal{M}}\left\\{R^{\sf
e}_{m,k}+\sqrt{\frac{\mathcal{V}^{\sf
e}_{m,k}}{L}}Q^{-1}\left(\delta_{m,k}\right)\right\\},$ (8)

where $\mathcal{V}^{\sf e}_{m,k}$ and $\mathcal{V}_{k}$ are channel dispersion
factors that depend on the stochastic variations of the legitimate and the
wiretap channels, respectively [10, 19], $Q^{-1}(\cdot)$ denotes an inverse
Q-function defined as
$Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{\frac{t^{2}}{2}}\,dt$,
$\epsilon_{k}$ is the decoding error probability of user $k$, and
$\delta_{m,k}$ is the secrecy constraint on the information leakage of $s_{k}$
from eavesdropper $m$ [19]. Note that if the blocklength $L$ goes to infinity,
then the secrecy rate in (8) is sufficiently close to the classic secrecy SE.
Thus, back-off factors in (8), i.e.,
$\sqrt{\frac{\mathcal{V}_{k}}{L}}Q^{-1}(\epsilon_{k})$ and
$\sqrt{\frac{\mathcal{V}^{\sf e}_{m,k}}{L}}Q^{-1}\left(\delta_{m,k}\right)$,
play as a drawback which reduces the secrecy rate in the FBL regime.
Considering an interference channel where transmitters uses an independent and
identically distributed (i.i.d.) Gaussian codebook and receivers employ
nearest-neighbor decoding, the channel dispersions factors become [39, 40]

$\displaystyle\mathcal{V}_{k}=\mathcal{V}^{\sf
i.i.d.}(\rho_{k})=\frac{2\rho_{k}}{1+\rho_{k}}\left(\log_{2}e\right)^{2},$ (9)
$\displaystyle\mathcal{V}^{\sf e}_{m,k}=\mathcal{V}^{\sf i.i.d.}(\rho^{\sf
e}_{m,k})=\frac{2\rho^{\sf e}_{m,k}}{1+\rho^{\sf
e}_{m,k}}\left(\log_{2}e\right)^{2}.$ (10)

Now, we let the predetermined maximum error probability and information
leakage constraints as $\hat{\epsilon}_{k}$ and $\hat{\delta}_{m,k},\forall
k\in\mathcal{K},\forall m\in\mathcal{M}$, respectively. Defining
${\boldsymbol{\epsilon}}=[\epsilon_{1},\cdots,\epsilon_{K}]^{\sf
T}\in\mbox{$\mathbb{R}$}^{K}_{+}$ and
${\boldsymbol{\Delta}}=[{\boldsymbol{\delta}}_{1},\cdots,{\boldsymbol{\delta}}_{K}]\in\mbox{$\mathbb{R}$}^{M\times
K}_{+}$, where
${\boldsymbol{\delta}}_{k}=[{\delta}_{1,k},\cdots,{\delta}_{M,k}]^{\sf
T}\in\mbox{$\mathbb{R}$}^{M}_{+},\forall k\in\mathcal{K}$, we formulate a
joint optimization problem to maximize the sum secrecy rate and to minimize
the maximum error probability and information leakage rate as

$\displaystyle\mathop{{\text{maximize}}}_{{\bf
F},{\boldsymbol{\epsilon}},{\boldsymbol{\Delta}}}\;\;\sum_{k=1}^{K}\;{R}_{k}^{\sf
sec}({\bf F},{\boldsymbol{\epsilon}},{\boldsymbol{\Delta}};L),$ (11)
$\displaystyle\mathop{{\text{minimize}}}_{{\boldsymbol{\epsilon}}}\;\;\max\\{\epsilon_{1},\cdots,\epsilon_{K}\\},$
(12)
$\displaystyle\mathop{{\text{minimize}}}_{{\boldsymbol{\Delta}}}\;\;\max\\{\delta_{1,1},\cdots,\delta_{M,K}\\},$
(13) $\displaystyle{\text{subject to}}\;\;{\rm{tr}}\left({\bf F}{\bf F}^{\sf
H}\right)\leq 1,$ (14)
$\displaystyle\qquad\qquad\;\;\epsilon_{k}\leq\hat{\epsilon}_{k},\;\forall
k\in\mathcal{K},$ (15)
$\displaystyle\qquad\qquad\;\;\delta_{m,k}\leq\hat{\delta}_{m,k},\;\forall
m\in\mathcal{M},\forall k\in\mathcal{K},$ (16)

where (12) and (13) are the error probability and information leakage rate
constraints, and (14) is a transmit power constraint at the AP, respectively.
The main challenges in solving the optimization problem are: $1)$ the problem
is the multi-objective optimization, $2)$ the objective function in (11) is
not tractable due to the maximum function which is non-smooth, $3)$ the
problem is inherently non-convex, and $4)$ the error probability and
information leakage need to be considered in the constraints and they are
intertwined with the secrecy rate as parameters of the back-off factors. In
this regard, it is infeasible to find the global optimal solution.

##  IV Proposed Joint Optimization Method

In this section, we develop a novel optimization method for solving the
problem in (11) by employing an alternating optimization approach. We first
begin with obtaining the secure precoder that maximizes the sum secrecy rate
for given error probabilities and information leakage rates, and then solve
the minimization problem in terms of the maximum error probability and
information leakage rate for the obtained precoder.

###  IV-A Phase I: Sum Secrecy Rate Maximization

In this phase, we find the optimal precoder that maximizes the sum secrecy
rate while fixing $\epsilon_{k}$ and $\delta_{m,k},\forall
k\in\mathcal{K},\forall m\in\mathcal{M}$. Since the objective function in (11)
is not smooth, it is necessary to make (8) smooth to find a more tractable
form. To this end, we first adopt a LogSumExp technique [41] to approximate
the maximum function with a parameter $\alpha$ as

$\max_{i=1,\cdots,N}\\{x_{i}\\}\approx\frac{1}{\alpha}\ln\left(\sum_{i=1}^{N}\exp(x_{i}\alpha)\right),$
(17)

where the approximation becomes tight as $\alpha\rightarrow\infty$. Applying
(17) to (8), we have

$\displaystyle\max_{m\in\mathcal{M}}\left\\{R^{\sf
e}_{m,k}+\sqrt{\frac{\mathcal{V}^{\sf
e}_{m,k}}{L}}Q^{-1}(\delta_{m,k})\log_{2}e\right\\}$ (18)
$\displaystyle\approx\;\frac{1}{\alpha}\ln\left[\sum_{m=1}^{M}\exp\left({\alpha
R^{\sf e}_{m,k}}+\alpha\sqrt{\frac{2\rho^{\sf e}_{m,k}}{L(1+\rho^{\sf
e}_{m,k})}}Q^{-1}\left(\delta_{m,k}\right)\log_{2}e\right)\right]$ (19)
$\displaystyle=\;\frac{1}{\alpha}\ln\left[\sum_{m=1}^{M}\left\\{{(1+\rho^{\sf
e}_{m,k})}^{\frac{\alpha}{\ln 2}}\cdot\exp{\left(\frac{\alpha}{\ln
2}\sqrt{\frac{2\rho^{\sf e}_{m,k}}{L(1+\rho^{\sf
e}_{m,k})}}\;Q^{-1}\left(\delta_{m,k}\right)\right)}\right\\}\right]$ (20)
$\displaystyle=\;\tilde{R}^{\sf e}_{k}.$ (21)

Now, we introduce the following lemma [32]:

######  Lemma 1.

For any given $x$, $\tilde{x}>0$, an upper bound of $\sqrt{2x/(1+x)}$ is
obtained as

$\displaystyle\sqrt{\frac{2x}{1+x}}\leq q(\tilde{x})\ln(1+x)+r(\tilde{x}),$
(22)

where $q(\tilde{x})=\frac{1}{\sqrt{2\tilde{x}(1+\tilde{x})}}$ and
$r(\tilde{x})=\sqrt{\frac{2\tilde{x}}{1+\tilde{x}}}-q(\tilde{x})\ln(1+\tilde{x})$.

###### Proof.

Refer to the proof of Lemma 2 in [32]. ∎

Based on Lemma 1 with (21), we can obtain the lower bound of the approximated
secrecy rate. For given $\tilde{\rho}_{k}$ and $\tilde{\rho}^{\sf e}_{m,k}$,
the lower bound of (8) is obtained as

$\displaystyle{R}^{\sf sec}_{k}\overset{(a)}{\approx}$
$\displaystyle\;R_{k}-\sqrt{\frac{2\rho_{k}}{L(1+\rho_{k})}}Q^{-1}(\epsilon_{k})\log_{2}e-\tilde{R}^{\sf
e}_{k}$ (23) $\displaystyle\overset{(b)}{\geq}$
$\displaystyle\;\log_{2}(1+\rho_{k})-\frac{Q^{-1}(\epsilon_{k})}{\sqrt{L}}q(\tilde{\rho}_{k})\ln(1+\rho_{k})\log_{2}e-\psi_{k}$
$\displaystyle-\frac{1}{\alpha}\ln\left\\{\sum_{m=1}^{M}{(1+\rho^{\sf
e}_{m,k})}^{\frac{\alpha}{\ln 2}}\cdot\exp{\left(\ln(1+\rho^{\sf
e}_{m,k})^{\frac{\alpha Q^{-1}(\delta_{m,k})}{\sqrt{L}\ln
2}q(\tilde{\rho}^{\sf e}_{m,k})}+\alpha\psi_{m,k}^{\sf e}\right)}\right\\}$
(24) $\displaystyle=$
$\displaystyle\;\log_{2}(1+\rho_{k})^{\omega_{k}}-\psi_{k}-\frac{1}{\alpha}\ln\left\\{\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}(1+\rho^{\sf
e}_{m,k})^{\omega^{\sf e}_{m,k}}\right\\}$ (25) $\displaystyle=$
$\displaystyle\;{R}^{\sf sec,\sf lb}_{k},$ (26)

where $(a)$ comes from (21), $(b)$ follows from Lemma 1, and

$\displaystyle\psi_{k}=\frac{Q^{-1}(\epsilon_{k})\log_{2}e}{\sqrt{L}}r(\tilde{\rho}_{k}),\;\psi_{m,k}^{\sf
e}=\frac{Q^{-1}(\delta_{m,k})\log_{2}e}{\sqrt{L}}r(\tilde{\rho}_{m,k}^{\sf
e}),\;\beta_{m,k}=\exp{\left(\alpha\psi^{\sf e}_{m,k}\right)},$ (27)
$\displaystyle\omega_{k}=\;1-\frac{Q^{-1}(\epsilon_{k})}{\sqrt{L}}q(\tilde{\rho}_{k}),\text{
and }\omega_{m,k}^{\sf
e}=\;\frac{\alpha}{\ln{2}}\left(1+\frac{Q^{-1}\left(\delta_{m,k}\right)}{\sqrt{L}}q(\tilde{\rho}^{\sf
e}_{m,k})\right).$ (28)

Since we pursue to solve (11) for given $\boldsymbol{\epsilon}$ and
$\boldsymbol{\Delta}$ with the lower bound in (26), our problem is transformed
to the single-objective maximization problem as

$\displaystyle\mathop{{\text{maximize}}}_{{\bf F}}$
$\displaystyle\;\;\sum_{k=1}^{K}\;{R}^{\sf sec,\sf lb}_{k}({\bf F}),$ (29)
subject to $\displaystyle\;\;{\rm{tr}}\left({\bf F}{\bf F}^{\sf H}\right)\leq
1.$ (30)

Next, to further obtain a compact rate expression with respect to the
precoder, we vectorize the precoding matrix as

$\displaystyle\bar{\mathbf{f}}={\rm{vec}}({\bf
F})=\left[\mathbf{f}^{\top}_{1},\mathbf{f}^{\top}_{2},\ldots,\mathbf{f}^{\top}_{K}\right]^{\top}\in\mathbb{C}^{NK}.$
(31)

We set $\|\bar{\bf f}\|^{2}=1$, i.e., transmission with the maximum power,
because the secrecy rate increases with the transmit power. Then, we can
finally reformulate the problem in (29) as the product of Rayleigh quotients:

$\displaystyle\mathop{{\text{maximize}}}_{\bar{\mathbf{f}}}\;\sum_{k=1}^{K}\left[\log_{2}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{A}_{k}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{B}_{k}\bar{\mathbf{f}}}\right)^{\omega_{k}}-\ln\left\\{\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf
e}}\right\\}^{\frac{1}{\alpha}}\right]\\!,$ (32)

where

$\displaystyle\mathbf{A}_{k}=\mathrm{blkdiag}\left(\gamma_{k}\mathbf{h}_{k}\mathbf{h}^{\sf
H}_{k},\cdots,\gamma_{k}\mathbf{h}_{k}\mathbf{h}^{\sf
H}_{k}\right)+\mathbf{I}_{NK}\frac{\sigma^{2}}{P}\in\mathbb{C}^{NK\times NK},$
(33)
$\displaystyle\mathbf{B}_{k}=\mathbf{A}_{k}-\mathrm{blkdiag}(\mathbf{0},\cdots,\gamma_{k}\mathbf{h}_{k}\mathbf{h}^{\sf
H}_{k},\cdots,\mathbf{0})\in\mathbb{C}^{NK\times NK},$ (34)
$\displaystyle\mathbf{C}_{m}=\mathrm{blkdiag}\left(\gamma^{\sf e}_{m}{\bf
g}_{m}{\bf g}_{m}^{\sf H},\cdots,\gamma^{\sf e}_{m}{\bf g}_{m}{\bf g}_{m}^{\sf
H}\right)+\mathbf{I}_{NK}\frac{\sigma^{2}_{\sf e}}{P}\in\mathbb{C}^{NK\times
NK},$ (35)
$\displaystyle\mathbf{D}_{m,k}=\mathbf{C}_{m}-\mathrm{blkdiag}(\mathbf{0},\cdots,\gamma^{\sf
e}_{m}{\bf g}_{m}{\bf g}_{m}^{\sf H},\cdots,\mathbf{0})\in\mathbb{C}^{NK\times
NK}.$ (36)

The second terms in both (34) and (36) have nonzero blocks which are located
at the $k$th diagonal block. We note that the product of Rayleigh quotient
forms in (32) is derived under the assumption $\|\bar{{\bf f}}\|^{2}=1$, and
the problem in (32) is invariant up to the scaling of $\bar{\bf f}$.
Accordingly, the power constraint in (30) is removed in the reformulated
problem.

Now, we focus on identifying the local points of the problem in (32). For
simplicity, we define the objective function in (32) as

$\displaystyle\mbox{$\mathcal{L}$}_{1}(\bar{\bf f})=$
$\displaystyle\;\log_{2}\prod_{k=1}^{K}\left[\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{A}_{k}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{B}_{k}\bar{\mathbf{f}}}\right)^{\omega_{k}}\\!\left\\{\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf
e}}\right\\}^{-\frac{\ln 2}{\alpha}}\right]$ (37) $\displaystyle=$
$\displaystyle\;\log_{2}\lambda(\bar{\bf f}).$ (38)

Then, we derive Lemma 2 to find the condition of stationary points of (37).

######  Lemma 2.

The first-order KKT condition of the problem (32) is satisfied if

$\displaystyle\mathbf{B}^{-1}_{\sf KKT}(\bar{\mathbf{f}})\mathbf{A}_{\sf
KKT}(\bar{\mathbf{f}})\bar{\mathbf{f}}=\lambda(\bar{\mathbf{f}})\bar{\mathbf{f}},$
(39)

where

$\displaystyle\mathbf{A}_{\sf
KKT}(\bar{\mathbf{f}})\\!=\\!\sum_{k=1}^{K}\left[\frac{\omega_{k}}{\ln
2}\left(\frac{\mathbf{A}_{k}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{A}_{k}\bar{\mathbf{f}}}\right)\\!+\\!\frac{1}{\alpha}\left\\{\sum_{m=1}^{M}\left(\frac{\omega^{\sf
e}_{m,k}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf
e}}\frac{\mathbf{D}_{m,k}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}}{\sum_{\ell=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{\ell}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{\ell,k}\bar{\mathbf{f}}}\right)^{\omega_{\ell,k}^{\sf
e}}}\right)\right\\}\right]\lambda_{\sf num}(\bar{\mathbf{f}}),$ (40)
$\displaystyle\mathbf{B}_{\sf
KKT}(\bar{\mathbf{f}})\\!=\\!\sum_{k=1}^{K}\left[\frac{\omega_{k}}{\ln
2}\left(\frac{\mathbf{B}_{k}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{B}_{k}\bar{\mathbf{f}}}\right)\\!+\\!\frac{1}{\alpha}\left\\{\sum_{m=1}^{M}\left(\frac{\omega^{\sf
e}_{m,k}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf
e}}\frac{\mathbf{C}_{m}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}}{\sum_{\ell=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{\ell}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{\ell,k}\bar{\mathbf{f}}}\right)^{\omega_{\ell,k}^{\sf
e}}}\right)\right\\}\right]\lambda_{\sf den}(\bar{\mathbf{f}}),$ (41)
$\displaystyle\lambda_{\sf
num}(\bar{\mathbf{f}})=\prod_{k=1}^{K}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{A}_{k}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{B}_{k}\bar{\mathbf{f}}}\right)^{\omega_{k}},\;\lambda_{\sf
den}(\bar{\mathbf{f}})=\prod_{k=1}^{K}\left\\{\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf
e}}\right\\}^{\frac{\ln 2}{\alpha}}.$ (42)

###### Proof.

See Appendix A. ∎

We note that the first-order KKT condition in (39) can be interpreted as a
generalized eigenvalue problem $\mathbf{B}^{-1}_{\sf
KKT}(\bar{\mathbf{f}})\mathbf{A}_{\sf
KKT}(\bar{\mathbf{f}})\bar{\mathbf{f}}=\lambda(\bar{\mathbf{f}})\bar{\mathbf{f}}$.
Here, $\lambda(\bar{\bf f})$ is as an eigenvalue of ${\bf{B}}_{\sf
KKT}^{-1}(\bar{\bf{f}}){\bf{A}}_{\sf KKT}(\bar{\bf{f}})$ with $\bar{\bf f}$ as
a corresponding eigenvector. As a result, maximizing the objective function
$\mbox{$\mathcal{L}$}_{1}(\bar{\bf f})$ is equivalent to maximizing
$\lambda(\bar{\bf f})$. Therefore, it is desirable to find the principal
eigenvalue of (39) to maximize (38), which is equivalent to finding the best
local optimal solution of (32).

1 initialize: $\bar{\bf{f}}^{(0)}$, $t=1$.

2 while  _$\left\|\bar{\bf{f}}^{(t)}-\bar{\bf{f}}^{(t-1)}\right\|
>\varepsilon$ & $t\leq t_{\max}$_ do

3 Build ${\bf{A}}_{\sf KKT}(\bar{\bf{f}}^{(t-1)})$ and ${\bf{B}}_{\sf
KKT}(\bar{\bf{f}}^{(t-1)})$ according to (40) and (41) for given
$\epsilon_{k}$ and $\delta_{m,k}$.

4 Compute $\bar{\bf{f}}^{(t)}={\bf{B}}^{-1}_{\sf
KKT}(\bar{\bf{f}}^{(t-1)}){\bf{A}}_{\sf
KKT}(\bar{\bf{f}}^{(t-1)})\bar{\bf{f}}^{(t-1)}$.

5 Normalize
$\bar{\bf{f}}^{(t)}=\bar{\bf{f}}^{(t)}/\left\|\bar{\bf{f}}^{(t)}\right\|$.

6 $t\leftarrow t+1$.

7$\bar{\bf{f}}^{\star}\leftarrow\bar{\bf{f}}^{(t)}$

return __ $\bar{{\bf f}}^{\star}=\left[{\bf{f}}_{1}^{\sf T},{\bf{f}}_{2}^{\sf
T},\dots,{\bf{f}}_{K}^{\sf T}\right]^{\sf T}$.

Algorithm 1 Sum Secrecy Rate Maximization

Based on (39), we propose the sum secrecy rate maximization precoding
algorithm by adopting the generalized power iteration (GPI) method [42]. As
described in Algorithm 1, we initialize $\bar{\bf f}^{(0)}$ and update
$\bar{\bf f}^{(t)}$ at each iteration; with the given $L$,
$\boldsymbol{\epsilon}$, and $\boldsymbol{\Delta}$, the algorithm builds
${\bf{A}}_{\sf KKT}(\bar{\bf{f}}^{(t-1)})$ and ${\bf{B}}_{\sf
KKT}(\bar{\bf{f}}^{(t-1)})$ according to (40) and (41). Then, the algorithm
updates $\bar{\bf f}^{(t)}$ by computing
$\bar{\bf{f}}^{(t)}={\bf{B}}^{-1}_{\sf KKT}(\bar{\bf{f}}^{(t-1)}){\bf{A}}_{\sf
KKT}(\bar{\bf{f}}^{(t-1)})\bar{\bf{f}}^{(t-1)}$ and normalizing as
$\bar{\bf{f}}^{(t)}=\bar{\bf{f}}^{(t)}/\left\|\bar{\bf{f}}^{(t)}\right\|$. We
repeat these steps until either $\bar{\bf{f}}^{(t)}$ converges to a tolerance
level $\varepsilon$ or the algorithm reaches $t_{\rm max}$.

The complexity of Algorithm 1 is dominated by the inversion in ${\bf{B}}_{\sf
KKT}^{-1}(\bar{\bf{f}})$. Since ${\bf{B}}_{\sf KKT}(\bar{\bf{f}})$ is a block-
diagonal and symmetric matrix, we implement the inversion by exploiting a
divide-and-conquer method so that we only need a computational complexity
order of $\mathcal{O}\left(\frac{1}{3}KN^{3}\right)$ [42] instead of
$\mathcal{O}\left(K^{3}N^{3}\right)$. In this regard, the total complexity is
$\mathcal{O}\left(\frac{1}{3}TKN^{3}\right)$, where $T$ is the number of
iterations. We note that the complexity of Algorithm 1 is same as a
representative low-complexity framework in sum rate maximization (not even
secrecy rate), namely, the weighted minimum mean square error method [43]. In
[44], a AN-aided transmission method was proposed based on a sequence of semi-
definite programs (SDP) needs a computational complexity order of
$\mathcal{O}\left(N^{6.5}\right)$ for a single confidential message. In [45],
the SDP-based algorithm was also proposed for a single user and multiple
eavesdroppers with a complexity order of $\mathcal{O}\left(N^{8.5}\right)$. In
addition, a jamming noise-aided precoding algorithm was developed for multiple
users and eavesdroppers, which needs a complexity order of
$\mathcal{O}\left((N+K)^{3}\right)$ [46]. Thus, we emphasize that the
complexity of Algorithm 1 is low compared to the existing algorithms.

######  Remark 1 (Secure Precoding in the FBL Regime).

Since Algorithm 1 maximizes the secrecy rate in the FBL regime under given
$\boldsymbol{\epsilon}$ and $\boldsymbol{\Delta}$ with low complexity,
Algorithm 1 is considered to be a computationally efficient secure precoding
algorithm in the FBL-based communications.

###  IV-B Phase II: Maximum Error Probability and Information Leakage Rate
Minimization

We find the optimal ${\boldsymbol{\epsilon}}$ and ${\boldsymbol{\Delta}}$ for
fixed ${\bf F}$. To this end, we transform the multi-objective problem to a
single-objective form and derive a solution for the transformed problem.
Adopting the weighted-sum approach [11], the multi-objective problem in (11)
for given ${\bf F}$ is converted to

$\displaystyle\mathop{{\text{maximize}}}_{{\boldsymbol{\epsilon}},{\boldsymbol{\Delta}}}\;\;\frac{w}{R_{\infty}}\sum_{k=1}^{K}{R}_{k}^{\sf
sec}(\boldsymbol{\epsilon},\boldsymbol{\Delta})\\!+\\!(1-w)\left(\frac{\hat{\epsilon}_{\sf
max}-{\rm max}\\{{\boldsymbol{\epsilon}}\\}}{\hat{\epsilon}_{\sf
max}}+\frac{\hat{\delta}_{\sf max}-{\rm
max}\\{{\boldsymbol{\Delta}}\\}}{\hat{\delta}_{\sf max}}\right),$ (43)
$\displaystyle{\text{subject
to}}\;\;\epsilon_{k}\leq\hat{\epsilon}_{k},\;\forall k\in\mathcal{K},$ (44)
$\displaystyle\qquad\qquad\;\;\delta_{m,k}\leq\hat{\delta}_{m,k},\;\forall
m\in\mathcal{M},\forall k\in\mathcal{K},$ (45)

where $R_{\infty}$ indicates a normalization constant that can be obtained as
the sum secrecy rate in the infinite blocklength regime computed with a state-
of-the-art secure precoder, $\hat{\epsilon}_{\sf max}={\rm
max}\\{\hat{\epsilon}_{1},\cdots,\hat{\epsilon}_{K}\\}$, $\hat{\delta}_{\sf
max}={\rm max}\\{\hat{\delta}_{1,1},\cdots,\hat{\delta}_{M,K}\\}$, and the
weight is $w\in[0,1]$. To solve (43), we need to handle the maximum function
in $R_{k}^{\sf sec}(\epsilon_{k},\delta_{m,k})$. We consider the upper bound
by adding the wiretap rates as $\max_{m\in\mathcal{M}}\left\\{R^{\sf
e}_{m,k}+\sqrt{\frac{\mathcal{V}_{m,k}^{\sf
e}}{L}}Q^{-1}\left(\delta_{m,k}\right)\right\\}\leq\sum_{m=1}^{M}\left(R^{\sf
e}_{m,k}+\sqrt{\frac{\mathcal{V}^{\sf
e}_{m,k}}{L}}Q^{-1}\left(\delta_{m,k}\right)\right)$. We note that although
the bound is not tight and the gap increases with the number of eavesdroppers,
the proposed method based on the bound will show high performance for many
eavesdroppers in Section IV-A. Subsequently, letting $\tau={\rm
max}\\{{\boldsymbol{\epsilon}}\\}$ and $\xi={\rm
max}\\{{\boldsymbol{\Delta}}\\}$ and removing the terms that are not a
function of either $\epsilon_{k}$ or $\delta_{m,k}$, the problem in (43) is
further transformed to the minimization problem as

$\displaystyle\mathop{{\text{minimize}}}_{{\boldsymbol{\epsilon}},{\boldsymbol{\Delta}}}\;\;\frac{w}{R_{\infty}}\sum_{k=1}^{K}\left[\sqrt{\frac{\mathcal{V}_{k}}{L}}Q^{-1}(\epsilon_{k})+\sum_{m=1}^{M}\left(\sqrt{\frac{\mathcal{V}^{\sf
e}_{m,k}}{L}}Q^{-1}\left(\delta_{m,k}\right)\right)\right]+(1-w)\left(\frac{\tau}{\hat{\epsilon}_{\sf
max}}+\frac{\xi}{\hat{\delta}_{\sf max}}\right),$ (46)
$\displaystyle{\text{subject to}}\;\;\epsilon_{k}\leq\tau,$ (47)
$\displaystyle\qquad\qquad\;\;0\leq\epsilon_{k}\leq\hat{\epsilon}_{k},$ (48)
$\displaystyle\qquad\qquad\;\;0\leq\tau\leq\hat{\epsilon}_{\sf max},$ (49)
$\displaystyle\qquad\qquad\;\;\delta_{m,k}\leq\xi,$ (50)
$\displaystyle\qquad\qquad\;\;0\leq\delta_{m,k}\leq\hat{\delta}_{m,k},$ (51)
$\displaystyle\qquad\qquad\;\;0\leq\xi\leq\hat{\delta}_{\sf max},$ (52)

where (48) and (51) are the error probability and information leakage rate
constraints, and (49) and (52) indicate the maximum error probability
assumption and information leakage assumptions. We note that the first term in
(46) is monotonically increasing while the second term in (46) is
monotonically decreasing with decreasing error probability and information
leakage. Therefore, the optimal tradeoff exists among the back-off factors,
error probability, and information leakage rate which is found by solving the
KKT conditions [47] of (46).

######  Lemma 3.

The optimal upper decoding error probability and information leakage rate for
given $\ell,j(k)\in\mbox{$\mathbb{N}$}_{+}$ are derived as

$\displaystyle\tau^{\star}=Q\left(\sqrt{2\ln\left(\frac{\sqrt{L}(1-w)R_{\infty}}{\hat{\epsilon}_{\sf
max}w\sqrt{2\pi}\sum_{k=\ell}^{K}\sqrt{\mathcal{V}_{k}}}\right)}\right),$ (53)
$\displaystyle\xi^{\star}=Q\left(\sqrt{2\ln\left(\frac{\sqrt{L}(1-w)R_{\infty}}{\hat{\delta}_{\sf
max}w\sqrt{2\pi}\sum_{m=j(k)}^{M}\sum_{k=1}^{K}\sqrt{\mathcal{V}_{m,k}^{\sf
e}}}\right)}\right).$ (54)

Then, without loss of generality, we can assume that
$\hat{\epsilon}_{\ell-1}<\tau<\hat{\epsilon}_{\ell}$ for some $\ell$ and
$\hat{\delta}_{j(k)-1,k}<\xi<\hat{\delta}_{j(k),k}$ for some $j(k)$. Then, the
optimal solution of the problem in (46) is

$\displaystyle{\boldsymbol{\epsilon}}^{\star}=[\hat{\epsilon}_{1},\cdots,\hat{\epsilon}_{\ell-1},\tau^{\star},\cdots,\tau^{\star}]^{\sf
T}\in\mbox{$\mathbb{R}$}_{+}^{K},$ (55)
$\displaystyle{\boldsymbol{\Delta}}^{\star}=[{\boldsymbol{\delta}}_{1}^{\star},\cdots,{\boldsymbol{\delta}}_{K}^{\star}]\in\mbox{$\mathbb{R}$}_{+}^{M\times
K},$ (56)

where

$\displaystyle{\boldsymbol{\delta}}_{k}^{\star}=[\hat{\delta}_{1,k},\cdots,\hat{\delta}_{j(k)-1,k},\xi^{\star},\cdots,\xi^{\star}]^{\sf
T}\in\mbox{$\mathbb{R}$}_{+}^{M},\;\forall k\in\mathcal{K}.$ (57)

If $\hat{\epsilon}_{K}<\tau^{\star}$, the optimal error probability becomes

$\displaystyle{\boldsymbol{\epsilon}}^{\star}=[\hat{\epsilon}_{1},\cdots,\hat{\epsilon}_{K}]^{\sf
T}.$ (58)

Similarly, if $\hat{\delta}_{M,k}<\xi^{\star},\forall k\in\mathcal{K}$, the
optimal information leakage rate becomes

$\displaystyle{\boldsymbol{\delta}}^{\star}_{k}=[\hat{\delta}_{1,k},\cdots,\hat{\delta}_{M,k}]^{\sf
T},\;\forall k\in\mathcal{K}.$ (59)

###### Proof.

See Appendix B. ∎

######  Remark 2 (Optimality of Solutions).

The derived solutions in Lemma 3 are the optimal error probability solution
and suboptimal information leakage rate solution for the problem in (43). For
a special case $m=1$, both the derived solutions are optimal.

###  IV-C Joint Secure Precoding Algorithm

1 initialize:
$\bar{\bf{f}}^{(0)},{\boldsymbol{\epsilon}}^{(0)},{\boldsymbol{\delta}}^{(0)}_{k},\forall
k\in\mathcal{K}$, and $t=1$.

2 while  _$increment\;of\;\eqref{eq:multi_obj} >\varepsilon^{\rm out}$ &
$t\leq t_{\rm max}^{\rm out}$_ do

3 $\bar{\bf{f}}^{(t)}\leftarrow\text{Algorithm\leavevmode\nobreak\
\ref{alg:algorithm_1}}\left({\boldsymbol{\epsilon}}^{(t-1)},{\boldsymbol{\Delta}}^{(t-1)}\right)$.

4 Compute $R_{\infty}$ by using Algorithm 1 with $L=\infty$.

5 Find $\tau^{\star}$ and $\xi^{\star}$ according to (53) and (54) for
$\bar{{\bf{f}}}^{(t)}$.

6 Set
${\boldsymbol{\epsilon}}^{(t)}\\!=\\![\hat{\epsilon}_{1},\cdots,\hat{\epsilon}_{\ell-1},\tau^{\star},\cdots,\tau^{\star}]^{\sf
T}$ and
${\boldsymbol{\delta}}_{k}^{(t)}\\!=\\![\hat{\delta}_{1,k},\cdots,\hat{\delta}_{j(k)-1,k},\xi^{\star},\cdots,\xi^{\star}]^{\sf
T}$ $\forall k$.

7 $t\leftarrow t+1$.

8$\bar{\bf{f}}^{\star}\leftarrow\bar{\bf{f}}^{(t)}$,
${\boldsymbol{\epsilon}}^{\star}\leftarrow{\boldsymbol{\epsilon}}^{(t)}$, and
${\boldsymbol{\Delta}}^{\star}\leftarrow{\boldsymbol{\Delta}}^{(t)}$.

return __ $\bar{{\bf f}}^{\star}=\left[{\bf{f}}_{1}^{\sf T},{\bf{f}}_{2}^{\sf
T},\dots,{\bf{f}}_{K}^{\sf T}\right]^{\sf T}$,
${\boldsymbol{\epsilon}}^{\star}$, and ${\boldsymbol{\Delta}}^{\star}$.

Algorithm 2 Joint Optimization based on Alternating Approach

From the results in Section IV-A and Section IV-B, we finally design our
proposed algorithm, which is described in Algorithm 2. First, Algorithm 2
initializes the precoding vector $\bar{\bf f}^{(0)}$,
${\boldsymbol{\epsilon}}^{(0)}$, and ${\boldsymbol{\delta}}^{(0)}_{k},\forall
k\in\mathcal{K}$. To find the best local optimal precoding vector $\bar{\bf
f}^{(t)}$, Algorithm 1 is used for given ${\boldsymbol{\epsilon}}^{(t-1)}$ and
${\boldsymbol{\delta}}^{(t-1)}_{k}$. Then, $R_{\infty}$ is computed by finding
$\bar{\bf f}$ from Algorithm 1 in the infinite blocklength regime. The optimal
$\tau^{\star}$ and $\xi^{\star}$ are computed based on (53) and (54), and we
set ${\boldsymbol{\epsilon}}^{(t)}$ and ${\boldsymbol{\Delta}}^{(t)}$
accordingly. We repeat these steps until either the objective function in (43)
increases smaller than $\varepsilon^{\rm out}$ compared to the previous
iteration, where $\varepsilon^{\rm out}>0$ denotes a tolerance threshold for
the outer loop, or the algorithm reaches a maximum iteration count $t_{\rm
max}^{\rm out}$. Since the solutions in (53) and (54) are closed form, the
complexity order of Algorithm 2 is $\mathcal{O}\left(\frac{1}{3}T_{\rm
tot}KN^{3}\right)$, where $T_{\rm tot}$ is the number of total iterations of
Algorithm 1.

##  V Extension to Partial CSIT of Wiretap Channels

Since it is hard to obtain the perfect channel state information at the
transmitter (CSIT) of eavesdroppers, we consider the case of the partial CSIT
of wiretap channels in which only the long-term channel statistics of
eavesdroppers, i.e., the channel covariance ${\bf R}_{m}^{\sf e}$, is
available. Since the instantaneous CSI of eavesdroppers is not known, it is
infeasible to consider the instantaneous wiretap rate. Accordingly, we
consider the ergodic wiretap rate to exploit the partial CSIT. Then, instead
of the secrecy rate in (8), we have

$\displaystyle R_{k}^{\sf sec,\sf
p}=R_{k}-\sqrt{\frac{\mathcal{V}_{k}}{L}}Q^{-1}(\epsilon_{k})-\max_{m\in\mathcal{M}}\left\\{\mathbb{E}_{{\bf
g}}\left[R^{\sf e}_{m,k}+\sqrt{\frac{\mathcal{V}^{\sf
e}_{m,k}}{L}}Q^{-1}\left(\delta_{m,k}\right)\right]\right\\},$ (60)

where $\mathbb{E}_{{\bf g}}[\cdot]$ indicates the expectation with respect to
wiretap channels. Then, we can rewrite (60) by using the following
proposition.

######  Proposition 1.

The approximated upper bound of the ergodic wiretap secrecy rate in (60) is
obtained as

$\displaystyle\mathbb{E}_{{\bf g}}\left[R_{m,k}^{\sf
e}+\sqrt{\frac{{\mathcal{V}}^{\sf
e}_{m,k}}{L}}Q^{-1}\left(\delta_{m,k}\right)\right]\lessapprox\bar{R}_{m,k}^{\sf
e}+\sqrt{\frac{\bar{\mathcal{V}}^{\sf
e}_{m,k}}{L}}Q^{-1}\left(\delta_{m,k}\right),$ (61)

where

$\displaystyle\bar{R}_{m,k}^{\sf e}=\log_{2}\left(1+\frac{\gamma^{\sf
e}_{m}{\bf f}^{\sf H}_{k}{{\bf R}}^{\sf e}_{m}{\bf f}_{k}}{\sum^{K}_{\ell\neq
k}\gamma^{\sf e}_{m}{{\bf f}}^{\sf H}_{\ell}{{\bf R}}^{\sf e}_{m}{{\bf
f}}_{\ell}+{\sigma}^{2}_{\sf e}/P}\right),$ (62)
$\displaystyle\bar{\mathcal{V}}^{\sf e}_{m,k}=\sqrt{\frac{2\gamma^{\sf
e}_{m}{\bf f}^{\sf H}_{k}{{\bf R}}^{\sf e}_{m}{\bf
f}_{k}}{\sum^{K}_{k=1}\gamma^{\sf e}_{m}{{\bf f}}^{\sf H}_{\ell}{{\bf R}}^{\sf
e}_{m}{{\bf f}}_{\ell}+{\sigma}^{2}_{\sf e}/P}}\log_{2}e.$ (63)

###### Proof.

The ergodic wiretap rate in (6) is approximated as

$\displaystyle\mathbb{E}_{{\bf g}}[R_{m,k}^{\sf e}]=$
$\displaystyle\;\mathbb{E}_{{\bf
g}}\left[\log_{2}\left(1\\!+\\!\frac{\gamma^{\sf e}_{m}|{\bf g}_{m}^{\sf
H}{\bf{f}}_{k}|^{2}}{\sum_{\ell\neq k}^{K}\gamma^{\sf e}_{m}|{\bf g}_{m}^{\sf
H}{\bf{f}}_{\ell}|^{2}+\sigma_{\sf
e}^{2}/P}\right)\right]\overset{(a)}{\approx}\log_{2}\left(1\\!+\\!\frac{\gamma^{\sf
e}_{m}{\bf f}^{\sf H}_{k}{{\bf R}}^{\sf e}_{m}{\bf f}_{k}}{\sum^{K}_{\ell\neq
k}\gamma^{\sf e}_{m}{{\bf f}}^{\sf H}_{\ell}{{\bf R}}^{\sf e}_{m}{{\bf
f}}_{\ell}+{\sigma}^{2}_{\sf e}/P}\right)$ (64) $\displaystyle=$
$\displaystyle\;\bar{R}_{m,k}^{\sf e},$ (65)

where $(a)$ comes from Lemma 1 in [48]. We also have

$\displaystyle\mathbb{E}\left[\sqrt{{\mathcal{V}}^{\sf
e}_{m,k}}\right]\overset{(b)}{\leq}$
$\displaystyle\;\sqrt{\mathbb{E}\left[{\mathcal{V}}^{\sf
e}_{m,k}\right]}\overset{(c)}{\leq}\sqrt{\frac{2}{\frac{1}{\mathbb{E}\left[\rho^{\sf
e}_{m,k}\right]}+1}}\log_{2}e\overset{(d)}{\approx}\sqrt{\frac{2\gamma^{\sf
e}_{m}{\bf f}^{\sf H}_{k}{{\bf R}}^{\sf e}_{m}{\bf
f}_{k}}{\sum^{K}_{k=1}\gamma^{\sf e}_{m}{{\bf f}}^{\sf H}_{\ell}{{\bf R}}^{\sf
e}_{m}{{\bf f}}_{\ell}+{\sigma}^{2}_{\sf e}/P}}\log_{2}e$ (66)
$\displaystyle=$ $\displaystyle\;\sqrt{\bar{\mathcal{V}}^{\sf e}_{m,k}},$ (67)

where $(b)$ and $(c)$ follow from Jensen’s inequality, and $(d)$ comes from
the first-order Taylor series approximation based on statistical linearization
[49]. ∎

Applying (61) to (60), we have the lower bound of secrecy rate approximated as

$\displaystyle{R}^{\sf sec,p}_{k}\gtrapprox$
$\displaystyle\;R_{k}-\sqrt{\frac{\mathcal{V}_{k}}{L}}Q^{-1}(\epsilon_{k})-\max_{m\in\mathcal{M}}\left\\{\bar{R}^{\sf
e}_{m,k}+\sqrt{\frac{\bar{\mathcal{V}}^{\sf
e}_{m,k}}{L}}Q^{-1}\left(\delta_{m,k}\right)\right\\}$ (68) $\displaystyle=$
$\displaystyle\;\bar{R}^{\sf sec,p}_{k}.$ (69)

Replacing ${R}_{k}^{\sf sec}$ in (11) with $\bar{R}_{k}^{\sf sec,p}$, we can
also develop the joint secure precoding algorithm with minimizing the maximum
error probability and information leakage rate. In particular, as in Section
IV-A, we formulate the sum secrecy rate maximization problem with
$\bar{R}_{k}^{\sf sec,p}$ as

$\displaystyle\mathop{{\text{maximize}}}_{{\bf
F}}\;\;\sum_{k=1}^{K}\bar{R}_{k}^{\sf sec,p}({\bf F}),$ (70)
$\displaystyle{\text{subject to}}\;\;{\rm{tr}}\left({\bf F}{\bf F}^{\sf
H}\right)\leq 1.$ (71)

We note that the formulated problem in (70) consists of the eavesdropper’s
channel covariance matrix ${\bf R}_{m}^{\sf e}$. As in Section IV-A, we also
derive the first-order KKT condition for (70) as

$\displaystyle\bar{\mathbf{B}}^{-1}_{\sf
KKT}(\bar{\mathbf{f}})\bar{\mathbf{A}}_{\sf
KKT}(\bar{\mathbf{f}})\bar{\mathbf{f}}=\bar{\lambda}(\bar{\mathbf{f}})\bar{\mathbf{f}},$
(72)

where

$\displaystyle\bar{{\bf A}}_{\sf
KKT}(\bar{\mathbf{f}})=\sum_{k=1}^{K}\left[\frac{\omega_{k}}{\ln
2}\left(\frac{\mathbf{A}_{k}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{A}_{k}\bar{\mathbf{f}}}\right)+\frac{1}{\alpha}\left\\{\sum_{m=1}^{M}\left(\frac{\omega^{\sf
e}_{m,k}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\bar{{\bf C}}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf H}\bar{{\bf
D}}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf e}}\frac{\bar{{\bf
D}}_{m,k}}{\bar{\mathbf{f}}^{\sf H}\bar{{\bf
D}}_{m,k}\bar{\mathbf{f}}}}{\sum_{\ell=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\bar{{\bf C}}_{\ell}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf H}\bar{{\bf
D}}_{\ell,k}\bar{\mathbf{f}}}\right)^{\omega_{\ell,k}^{\sf
e}}}\right)\\!\right\\}\right]\\!\bar{\lambda}_{\sf num}(\bar{\mathbf{f}}),$
(73) $\displaystyle\bar{{\bf B}}_{\sf
KKT}(\bar{\mathbf{f}})=\sum_{k=1}^{K}\left[\frac{\omega_{k}}{\ln
2}\left(\frac{\mathbf{B}_{k}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{B}_{k}\bar{\mathbf{f}}}\right)+\frac{1}{\alpha}\\!\left\\{\sum_{m=1}^{M}\left(\frac{\omega^{\sf
e}_{m,k}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\bar{{\bf C}}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf H}\bar{{\bf
D}}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf e}}\frac{\bar{{\bf
C}}_{m}}{\bar{\mathbf{f}}^{\sf H}\bar{{\bf
C}}_{m}\bar{\mathbf{f}}}}{\sum_{\ell=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\bar{{\bf C}}_{\ell}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf H}\bar{{\bf
D}}_{\ell,k}\bar{\mathbf{f}}}\right)^{\omega_{\ell,k}^{\sf
e}}}\right)\\!\right\\}\right]\\!\bar{\lambda}_{\sf den}(\bar{\mathbf{f}}),$
(74) $\displaystyle\bar{\mathbf{C}}_{m}=\;\mathrm{blkdiag}\left(\gamma^{\sf
e}_{m}\mathbf{R}_{m}^{\sf e},\cdots,\gamma^{\sf e}_{m}\mathbf{R}_{m}^{\sf
e}\right)+\mathbf{I}_{NK}\frac{\sigma^{2}_{\sf e}}{P}\in\mathbb{C}^{NK\times
NK},$ (75) $\displaystyle\bar{{\bf
D}}_{m,k}=\;\bar{\mathbf{C}}_{m}-\mathrm{blkdiag}(0,\cdots,{\gamma^{\sf
e}_{m}{\bf R}_{m}^{\sf e}},\cdots,0)\in\mathbb{C}^{NK\times NK},$ (76)
$\displaystyle\bar{\lambda}(\bar{\mathbf{f}})=\bar{\lambda}_{\sf
num}(\bar{\mathbf{f}})/\bar{\lambda}_{\sf den}(\bar{\mathbf{f}}),$ (77)
$\displaystyle\bar{\lambda}_{\sf
num}(\bar{\mathbf{f}})=\prod_{k=1}^{K}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{A}_{k}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{B}_{k}\bar{\mathbf{f}}}\right)^{\omega_{k}},\;\bar{\lambda}_{\sf
den}(\bar{\mathbf{f}})=\prod_{k=1}^{K}\left\\{\sum_{m=1}^{M}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\beta_{m,k}}\left(\frac{\bar{\mathbf{f}}^{\sf
H}{\bar{{\bf C}}}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf H}\bar{{\bf
D}}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf e}}\right\\}^{\frac{\ln
2}{\alpha}}.$ (78)

Recall that $\mathbf{A}_{k}$ and $\mathbf{B}_{k}$ are derived in (33) and
(34), respectively. Replacing $\mathbf{A}_{\sf KKT}$ and $\mathbf{B}_{\sf
KKT}$ with $\bar{\mathbf{A}}_{\sf KKT}$ and $\bar{\mathbf{B}}_{\sf KKT}$ in
Algorithm 1, we can design the secure precoding algorithm by leveraging the
partial CSIT of the eavesdroppers. The maximum error probability and
information leakage rate can also be optimized by following the same steps in
Section IV-B based on $\bar{R}^{\sf sec,\sf p}_{k}$ as

$\displaystyle\bar{\tau}^{\star}=Q\left(\sqrt{2\ln\left(\frac{\sqrt{L}(1-w)\bar{R}_{\infty}}{\hat{\epsilon}_{\sf
max}w\sqrt{2\pi}\sum_{k=\ell}^{K}\sqrt{{\mathcal{V}}_{k}}}\right)}\right),$
(79)
$\displaystyle\bar{\xi}^{\star}=Q\left(\sqrt{2\ln\left(\frac{\sqrt{L}(1-w)\bar{R}_{\infty}}{\hat{\delta}_{\sf
max}w\sqrt{2\pi}\sum_{m=j(k)}^{M}\sum_{k=1}^{K}\sqrt{\bar{\mathcal{V}}_{m,k}^{\sf
e}}}\right)}\right),$ (80)

where $\bar{R}_{\infty}$ is the normalization constant based on $\bar{R}^{\sf
sec,\sf p}_{k}$ for $L=\infty$. Then, the joint secure precoding algorithm is
proposed with Algorithm 2 by replacing ${\bf A}_{k}$, ${\bf B}_{k}$,
$\tau^{\star}$, and $\xi^{\star}$ with $\bar{{\bf A}}_{k}$, $\bar{{\bf
B}}_{k}$, $\bar{\tau}^{\star}$, and $\bar{\xi}^{\star}$, respectively.

##  VI Numerical Results

In this section, we evaluate the performance of our proposed algorithms along
with other benchmarks and deliver key insights. Considered algorithms are: (1)
Algorithm 1, (2) Algorithm 1 with the channel covariance of wiretaps (Cov) in
Section V, (3) Algorithm 2, (4) Algorithm 2 (Cov), (5) the FBL-based SE
maximization algorithm (FBL-SE-MAX) [32], (6) regularized ZF (RZF), (7) RZF
including the eavesdroppers (RZF-EVE), (8) ZF, (9) ZF-EVE, and (10) MRT.
Specifically, RZF-EVE and ZF-EVE algorithms exploit the wiretap channels by
constructing an effective channel matrix with the $N-K$ strongest wiretap
channels.

To generate the channel vectors ${\bf h}_{k}$ and ${\bf g}_{m}$, we adopt a
one-ring model [50] based on its spatial covariance matrices ${\bf
R}_{k}=\mbox{$\mathbb{E}$}[{\bf h}_{k}{\bf h}_{k}^{\sf H}]$ and ${\bf R}^{\sf
e}_{m}=\mbox{$\mathbb{E}$}[{\bf g}_{m}{\bf g}_{m}^{\sf H}]$. In addition, we
consider that the geometric location of each eavesdropper is correlated with a
particular legitimate user. Accordingly, the channel angle of departure (AoD)
of eavesdropper $m$ follows $\theta^{\sf
e}_{m}\sim\theta_{k}+\mbox{$\mathcal{U}$}(-\Delta\pi,\Delta\pi)$ for randomly
selected user $k$ with a scalar weight $0<\Delta<1$, where $\theta_{k}$
represents the AoD of user $k$. We generate $\hat{\epsilon}_{k}$ and
$\hat{\delta}_{m,k}$ uniformly spaced from $10^{-6}$ to $2\times 10^{-6}$ for
all users and eavesdroppers. For simplicity, we consider
$\hat{\delta}_{m,k}=\hat{\delta}_{m,k^{\prime}},\forall m\in\mathcal{M}$. We
set the scalar weight, coding length, optimization weight, iteration
thresholds, and maximum iteration counts as $\Delta=0.1$, $L=200$, $w=0.01$,
$\varepsilon=\varepsilon^{\rm out}=0.01$, and $t_{\rm max}=15$ and $t_{\rm
max}^{\rm out}=5$ unless mentioned otherwise.

In addition, to generate the large-scale channel fading terms $\gamma_{k}$ and
$\gamma_{m,k}^{\sf e}$, we adopt the ITU-R indoor pathloss model [51] which
considers a non-line-of-sight pathloss environment by setting bandwidth,
carrier frequency, distance power-loss coefficient, and noise figure to be 10
MHz, 5.2 GHz, 31 (equivalent to the pathloss exponent of $3.1$), and 5 dB,
respectively. We assume the noise power spectral density of legitimate users
and eavesdroppers is the same as $-174$ dBm/Hz. The users are randomly
generated around the AP with the maximum distance of $50\;m$ and minimum
distance of $5\;m$ from the AP. The eavesdroppers are randomly distributed
around random users with the maximum distance of $5\;m$ from the users.

###  VI-A Secure Precoding for Given Error and Leakage in FBL Regime

$\begin{array}[]{c
c}{\leavevmode\resizebox{208.13574pt}{}{\includegraphics{Figures/Alg1_PvsRate_N8.pdf}}}&{\leavevmode\resizebox{208.13574pt}{}{\includegraphics{Figures/Alg1_PvsRate_N16.pdf}}}\\\
\mbox{\small(a) $N=8$ }&\mbox{\small(b) $N=16$ }\end{array}$

Figure 1: The sum secrecy rate vs. maximum transmit power with (a) $N=8$ and
(b) $N=16$ AP antennas for $K=4$ users, and $M=4$ eavesdroppers.

We first investigate the precoders that maximize the sum secrecy rate for
given $\epsilon_{k}$ and $\delta_{m,k}$ as $\hat{\epsilon}_{k}$ and
$\hat{\delta}_{m,k}$, respectively. In Fig. 1, we evaluate the sum secrecy
rate of the simulated algorithms with respect to the transmit power $P$ for
(a) $N=8$ and (b) $N=16$ with $K=4$ and $M=4$. Fig. 1 shows that the proposed
algorithms achieve the highest sum secrecy rate compared to the baseline
algorithms because the proposed algorithms consider the effect of back-off
factors due to the FBL as well as the wiretap channels. Algorithm 1 (Cov), in
particular, provides higher performance than the benchmarks without the
perfect wiretap CSIT. We observe that the secure precoders such as the RZF-EVE
and ZF-EVE waste their extra dimension in nullifying the leakage from the
wiretap channels rather than increasing legitimate users’ channel gains,
showing poor performance in Fig. 1(a). However, as shown in Fig. 1(b), when
the AP has the enough antennas, the ZF-based precoders outperform the SE
maximization precoder.

Figure 2: The sum secrecy rate vs. the blocklength $L$ for $N=8$ AP antennas,
$K=4$ users, $M=4$ eavesdroppers, and $P=20$ dBm transmit power.

In Fig. 2, we evaluate the secrecy rate performance versus the blocklength $L$
for $N=8$, $K=4$, $M=4$, and $P=20$ dBm. We note that all algorithms reveal an
increasing trend of the sum secrecy rate with $L$ as theoretically shown in
(8). Fig. 2 shows that the proposed algorithms maintain the highest secrecy
rate performance regardless of $L$. Accordingly, the proposed algorithm is
effective in the general blocklength. Thus, the proposed secure precoder is
preferable in supporting the conventional as well as FBL-based communication
systems.

$\begin{array}[]{c
c}{\leavevmode\resizebox{208.13574pt}{}{\includegraphics{Figures/PvsRate.pdf}}}&{\leavevmode\resizebox{225.48424pt}{}{\includegraphics{Figures/PvsRate_ErrorMin.pdf}}}\\\
\mbox{\small(a) Sum secrecy rate}&\mbox{\small\\!(b) \\!Maximum \\!error
\\!probability \\!and \\!information \\!leakage}\end{array}$

Figure 3: (a) The sum secrecy rate and (b) maximum error probability and
information leakage rate vs. the maximum transmit power $P$ for $N=8$ AP
antennas, $K=4$ users, and $M=8$ eavesdroppers.

###  VI-B Joint Secure Precoding with Error and Leakage Minimization in FBL
Regime

Now, we evaluate the proposed method in Algorithm 2. For comparison, the
baseline algorithms utilize the second phase (Phase II in Section IV-B) of the
proposed optimization framework for minimizing the maximum error probability
and information leakage rate in Section IV-B. Fig. 3 shows that the
performance of the sum secrecy rate, the maximum error probability, and
information leakage rate with respect to the transmit power $P$ for $N=8$,
$K=4$, and $M=8$. The benchmark algorithms with the proposed minimization
method provide the significant reduction in the maximum error probability and
information leakage rate. We note that the proposed algorithms achieve the
highest secrecy rate performance while maintaining the lowest maximum error
probability and information leakage rate for the most signal-to-noise ratio
(SNR). FBL-SE-MAX with Phase II attains the lowest maximum error probability
in most SNR regimes. The security rate of FBL-SE-MAX, however, reveals an
increasing gap from the proposed algorithm due to ignoring the wiretap
channels.

$\begin{array}[]{c
c}{\leavevmode\resizebox{199.4681pt}{}{\includegraphics{Figures/NvsRate.pdf}}}&{\leavevmode\resizebox{225.48424pt}{}{\includegraphics{Figures/NvsRate_ErrorMin.pdf}}}\\\
\mbox{\small(a) Sum secrecy rate}&\mbox{\small\\!(b) \\!Maximum \\!error
\\!probability \\!and \\!information \\!leakage}\end{array}$

Figure 4: (a) The sum secrecy rate and (b) maximum error probability and
information leakage vs. the number AP antennas $N$ for $P=20$ dBm transmit
power, $K=4$ users, and $M=8$ eavesdroppers.

To provide the numerical analysis in terms of the number of AP antennas $N$,
we show the performance of the sum secrecy rate, the maximum error
probability, and information leakage rate with respect to the number of AP
antennas for $P=20$ dBm, $K=4$, and $M=8$ in Fig. 4. It is shown that the
proposed algorithms also achieve the highest secrecy rate performance
regardless of $N$. As observed in Fig. 4(a), RZF-EVE and ZF-EVE reveal an
unique trend of the sum secrecy rate as $N$ increases. Since there exists the
optimal tradeoff between increasing the legitimate channel gain and nullifying
the wiretap channels, the rates do not monotonically increase with $N$ for
RZF-EVE and ZF-EVE. For instance, when $N$ is not large enough, increasing
channel gain should be more suitable to achieve the higher secrecy rate than
nullifying the wiretap channels with extra degree-of-freedom (DoF). Thus, RZF-
EVE and ZF-EVE undergo poor performance in terms of the secrecy rate for small
$N$, and they enhance the secrecy rate with lager $N$. Regarding the
information leakage rate, ZF-EVE that utilizes its extra DoF to fully nullify
the wiretap channels is a near optimal solution with the enough DoF as shown
in Fig. 4(b). Considering the entire performance, however, the proposed
algorithms show the most balanced performance.

$\begin{array}[]{c
c}{\leavevmode\resizebox{208.13574pt}{}{\includegraphics{Figures/KvsRate.pdf}}}&{\leavevmode\resizebox{225.48424pt}{}{\includegraphics{Figures/KvsRate_ErrorMin.pdf}}}\\\
\mbox{\small(a) Sum secrecy rate}&\mbox{\small\\!(b) \\!Maximum \\!error
\\!probability \\!and \\!information \\!leakage}\end{array}$

Figure 5: (a) The sum secrecy rate and (b) maximum error and information
leakage vs. the number of users $K$ for $P=20$ dBm transmit power, $N=8$ AP
antennas, and $M=8$ eavesdroppers.

We further evaluate the proposed algorithms with respect to the number of
legitimate users $K$ for $N=8$, $M=8$, and $P=20$ dBm. Overall, Fig. 5 shows
that the algorithms reveal similar ordering in the sum secrecy rate as in Fig.
3. However, the gaps between the proposed algorithms and FBL-SE-MAX becomes
narrower as $K$ increases. We conjecture that when DoF margin is small
compared to the number of users, focusing on maximizing the sum rate provides
a better tradeoff than allocating resources on mitigating information leakage.
Fig. 5(a) also reveals that there exists the optimal number of users for RZF
in terms of the secrecy rate, which is not the case for the proposed
algorithms. In Fig. 5(b), the proposed algorithm provides the lowest maximum
error probability and information leakage rate regardless of $K$. Therefore,
the proposed algorithm provides improvement in rate, reliability, and security
from the conventional precoding algorithms for the considered system
environment.

$\begin{array}[]{c
c}{\leavevmode\resizebox{208.13574pt}{}{\includegraphics{Figures/MvsRate.pdf}}}&{\leavevmode\resizebox{225.48424pt}{}{\includegraphics{Figures/MvsRate_ErrorMin.pdf}}}\\\
\mbox{\small(a) Sum secrecy rate}&\mbox{\small\\!(b) \\!Maximum \\!error
\\!probability \\!and \\!information \\!leakage}\end{array}$

Figure 6: (a) The sum secrecy rate and (b) maximum error and information
leakage vs. the number of eavesdroppers $M$ for $P=20$ dBm transmit power,
$N=8$ antennas, and $K=4$ users.

In Fig. 6, we assess the sum secrecy rate in terms of the number of
eavesdroppers $M$ for $P=20$ dBm, $N=8$, and $K=4$. Fig. 6(a) shows that the
proposed algorithms still achieve the highest secrecy rates. As $M$ increases,
the relative gaps between the proposed and the other algorithms become larger,
which demonstrate the effective secure precoding performance of the proposed
algorithms. We note that for RZF-EVE and ZF-EVE, decreasing the spatial DoF
margin leads to the poor performance of the secrecy rate. As shown in Fig.
6(b), the proposed algorithms offer good and robust performance in minimizing
the maximum error probability and information leakage rate over $M$. Although
ZF-EVE shows the lowest maximum information leakage rate for $M\leq 4$ by
fully nullifying the wiretap channels, the achieved secrecy rate is
significantly lower and the leakage rate becomes higher than the proposed
algorithms for $M>4$. In this regard, the proposed algorithm is considered to
be a potential PLS solution that is robust to the number of eavesdroppers.

Figure 7: The sum secrecy rate, maximum error probability, and maximum
information leakage evaluated over transmit power $P$ from $0$ to $50$ dBm for
$N=8$ AP antennas, $K=4$ users, and $M=8$ eavesdroppers.

For better visual understanding, we also plot the proposed algorithm by
drawing the secrecy rate, maximum error probability, and maximum information
leakage rate together over $P$ from $0$ to $50$ dBm for $N=8$, $K=4$, and
$M=8$ in Fig. 7. For instance, if the system requirements of the sum secrecy
rate, maximum error probability, and maximum information leakage are
$\sum_{k=1}^{K}R^{\sf sec}_{k}>40$ bits/sec/Hz,
$\max\\{\epsilon_{k}\\}<4.0\times 10^{-11}$, and
$\max\\{\delta_{m,k}\\}<4.0\times 10^{-11}$, the proposed algorithm can
satisfy the requirements with 33.3 dBm transmit power, whereas FBL-SE-MAX with
Phase II requires 44.4 dBm, i.e., more than approximately $12\times$ the
transmit power is needed, and the others cannot meet the requirements in the
feasible transmit power regime. As a result, the proposed algorithm provides
the significant gain of SNR to fulfill the stringent requirements of URLLC in
the FBL regime.

Figure 8: Convergence results in terms of $\log_{2}\lambda(\bar{\bf f})$ for
$N=8$ AP antennas, $K=4$ users, $M=4$ eavesdroppers, and $P=\\{-10,0,10,20\\}$
dBm transmit power.

In Fig. 8, we present the convergence results in terms of the approximated
objective function $\log_{2}\lambda(\bar{\bf f})$ in (38) for
$P\in\\{-10,0,10,20\\}$ dBm, $N=8$, $K=4$, and $M=4$. As shown in Fig. 8, the
proposed algorithm converges within $T=6$ iterations in the practical transmit
power regime. Accordingly, the proposed algorithm demonstrates high potential
for the practical communications compared to other high-complexity precoding
algorithms in the FBL regime. Overall, the proposed algorithms offer
significant improvement in the secrecy rate, error probability, and
information leakage rate, and thus enhance the data rate, security, and
reliability with low complexity in the FBL regime.

##  VII Conclusion

In this paper, we proposed secure precoding algorithms that jointly optimize
rate, reliability, and security in the finite blocklength regime. To this end,
we formulated a joint optimization problem for the sum secrecy rate and error
probability, and information leakage rate according to the characteristics of
finite blocklength channel coding-based communications. To resolve the key
challenges in solving the problem, we first decompose the problem into two
sub-problems. Then, for the first problem, we designed the precoding algorithm
that maximizes the sum secrecy rate for the given error probability and
information leakage rate by identifying stationary points. For the second
problem, we also reformulated the multi-objective optimization problem into
the single-objective optimization problem for a given precoder and solved the
KKT conditions to derive the optimal error probability and information leakage
rate. Through the alternating optimization, we developed the joint
optimization algorithm that provides the significantly improved tradeoff among
the security, the error probability, and information leakage rate. We further
extended the algorithms for the case in which only the partial CSI of wiretap
channels is available. Via simulations, we validated the secrecy rate, error
probability, and information leakage performances of the proposed algorithms.
In particular, we demonstrated that the proposed methods achieve the highest
secrecy performance while satisfying the stringent reliability and security
requirements compared to the existing methods with fast convergence and high
robustness. Therefore, the proposed methods can play a key role in realizing
the URLLC under high information security.

##  Appendix A Proof of Lemma 1

###### Proof.

The Lagrangian function of the problem (32) is

$\displaystyle\mbox{$\mathcal{L}$}_{1}(\bar{\bf
f})=\sum_{k=1}^{K}\left[\log_{2}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{A}_{k}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{B}_{k}\bar{\mathbf{f}}}\right)^{\omega_{k}}\\!-\\!\ln\left\\{\sum_{m=1}^{M}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf
e}}\right\\}^{\frac{1}{\alpha}}\right].$ (81)

According to the first-order KKT condition, the stationary points need to
satisfy zero-gradient, i.e., $\frac{\partial\mbox{$\mathcal{L}$}_{1}(\bar{\bf
f})}{\partial\bar{\bf f}^{\sf H}}=0$. We take the partial derivative of
$\mbox{$\mathcal{L}$}_{1}(\bar{\bf f})$ with respect to ${\bf\bar{f}}$ and set
it to zero. For simplicity, we denote the first and second part of the
Lagrangian function as $\mbox{$\mathcal{L}$}_{1,\sf user}(\bar{\bf f})$ and
$\mbox{$\mathcal{L}$}_{1,\sf eve}(\bar{\bf f})$, i.e.,
$\mbox{$\mathcal{L}$}_{1}(\bar{\bf f})=\mbox{$\mathcal{L}$}_{1,\sf
user}(\bar{\bf f})-\mbox{$\mathcal{L}$}_{1,\sf eve}(\bar{\bf f})$. Then, using

$\displaystyle\frac{\partial\left(\frac{{\bf\bar{f}}^{\sf H}{{\bf
A}}_{k}\bar{\bf f}}{{\bf\bar{f}}^{\sf H}{{\bf B}}_{k}\bar{\bf
f}}\right)}{\partial\bar{\bf f}^{\sf H}}=\left(\frac{{\bf\bar{f}}^{\sf H}{{\bf
A}}_{k}\bar{\bf f}}{{\bf\bar{f}}^{\sf H}{{\bf B}}_{k}\bar{\bf
f}}\right)\left[\frac{{{\bf A}}_{k}\bar{\bf f}}{{\bf\bar{f}}^{\sf H}{{\bf
A}}_{k}\bar{\bf f}}-\frac{{{\bf B}}_{k}\bar{\bf f}}{{\bf\bar{f}}^{\sf H}{{\bf
B}}_{k}\bar{\bf f}}\right],$ (82)

the partial derivative of $\mbox{$\mathcal{L}$}_{1,\sf user}(\bar{\bf f})$ can
be calculated as

$\displaystyle\frac{\partial\mbox{$\mathcal{L}$}_{1,\sf user}(\bar{\bf
f})}{\partial\bar{\bf f}^{\sf H}}=\sum_{k=1}^{K}\frac{1}{\ln
2}\left(\frac{{{\bf A}}_{k}\bar{\bf f}}{{\bf\bar{f}}^{\sf H}{{\bf
A}}_{k}\bar{\bf f}}-\frac{{{\bf B}}_{k}\bar{\bf f}}{{\bf\bar{f}}^{\sf H}{{\bf
B}}_{k}\bar{\bf f}}\right).$ (83)

We also take the partial derivative of $\mbox{$\mathcal{L}$}_{1,\sf
eve}(\bar{\bf f})$ with respect to ${\bf\bar{f}}$ as

$\displaystyle\frac{\partial\mbox{$\mathcal{L}$}_{1,\sf eve}(\bar{\bf
f})}{\partial\bar{\bf f}^{\sf
H}}\\!=\\!\frac{1}{\alpha}\sum_{k=1}^{K}\sum_{m=1}^{M}\left\\{\frac{\omega_{m,k}^{\sf
e}\left(\frac{\bar{\bf f}^{\sf H}{{\bf C}}_{m}\bar{\bf f}}{\bar{\bf f}^{\sf
H}{{\bf D}}_{m,k}\bar{\bf f}}\right)^{\omega_{m,k}^{\sf e}}\left(\frac{{{\bf
C}}_{m}\bar{\bf f}}{{\bf\bar{f}}^{\sf H}{{\bf C}}_{m}\bar{\bf f}}-\frac{{{\bf
D}}_{m,k}\bar{\bf f}}{{\bf\bar{f}}^{\sf H}{{\bf D}}_{m,k}\bar{\bf
f}}\right)}{\sum_{\ell=1}^{M}\left(\frac{\bar{\bf f}^{\sf H}{{\bf
C}}_{\ell}\bar{\bf f}}{\bar{\bf f}^{\sf H}{{\bf D}}_{\ell,k}\bar{\bf
f}}\right)^{\omega_{\ell,k}^{\sf e}}}\right\\}.$ (84)

Then, using (83) and (84), the first-order KKT condition holds if

$\displaystyle\sum_{k=1}^{K}\left[\frac{\omega_{k}}{\ln 2}\frac{{\bf
A}_{k}}{{\bf\bar{f}}^{\sf H}{\bf A}_{k}\bar{\bf
f}}+\frac{1}{\alpha}\left\\{\sum_{m=1}^{M}\left(\frac{\omega^{\sf
e}_{m,k}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf
e}}\frac{\mathbf{D}_{m,k}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}}{\sum_{\ell=1}^{M}\;\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{\ell}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{\ell,k}\bar{\mathbf{f}}}\right)^{\omega_{\ell,k}^{\sf
e}}}\right)\right\\}\right]\bar{\bf f}$
$\displaystyle=\sum_{k=1}^{K}\left[\frac{\omega_{k}}{\ln 2}\frac{{\bf
B}_{k}}{{\bf\bar{f}}^{\sf H}{{\bf B}}_{k}\bar{\bf
f}}+\frac{1}{\alpha}\left\\{\sum_{m=1}^{M}\left(\frac{\omega^{\sf
e}_{m,k}\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{m,k}\bar{\mathbf{f}}}\right)^{\omega_{m,k}^{\sf
e}}\frac{\mathbf{C}_{m}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{m}\bar{\mathbf{f}}}}{\sum_{\ell=1}^{M}\;\left(\frac{\bar{\mathbf{f}}^{\sf
H}\mathbf{C}_{\ell}\bar{\mathbf{f}}}{\bar{\mathbf{f}}^{\sf
H}\mathbf{D}_{\ell,k}\bar{\mathbf{f}}}\right)^{\omega_{\ell,k}^{\sf
e}}}\right)\right\\}\right]\bar{\bf f}.$ (85)

The first-order KKT condition is reorganized as

$\displaystyle{\bf{A}}_{\sf KKT}(\bar{\bf f})\bar{\bf f}=\lambda(\bar{\bf
f}){\bf{B}}_{\sf KKT}(\bar{\bf f})\bar{\bf f}.$ (86)

Since ${\bf{B}}_{\sf KKT}(\bar{\bf f})$ is Hermitian, it is invertible. This
completes the proof. ∎

##  Appendix B Proof of Lemma 3

###### Proof.

We note that $Q^{-1}(x)$ is convex if $x<\frac{1}{2}$, so that the problem in
(46) becomes a convex problem. To solve the problem, we define the Lagrangian
function of the problem in (46) as

$\displaystyle\mbox{$\mathcal{L}$}_{2}=$
$\displaystyle\;\frac{w}{R_{\infty}}\sum_{k=1}^{K}\left[\sqrt{\frac{\mathcal{V}_{k}}{L}}Q^{-1}(\epsilon_{k})+\sum_{m=1}^{M}\left(R^{\sf
e}_{m,k}+\sqrt{\frac{\mathcal{V}_{m,k}^{\sf
e}}{L}}Q^{-1}\left(\delta_{m,k}\right)\right)\right]$
$\displaystyle+(1-w)\left(\frac{\tau}{\hat{\epsilon}_{\sf
max}}\\!+\\!\frac{\xi}{\hat{\delta}_{\sf
max}}\right)\\!-\\!\sum_{k=1}^{K}\lambda_{k}(\tau-\epsilon_{k})-\sum_{k=1}^{K}\nu_{k}(\hat{\epsilon_{k}}-\epsilon_{k})\\!-\\!\mu(\hat{\epsilon_{\sf
max}}-\tau)$ $\displaystyle-\sum_{m=1}^{M}\sum_{k=1}^{K}\lambda_{m,k}^{\sf
e}(\xi-\delta_{m,k})-\sum_{m=1}^{M}\sum_{k=1}^{K}\nu_{m,k}^{\sf
e}(\hat{\delta}_{m,k}-\delta_{m,k})\\!-\\!\mu^{\sf e}(\hat{\delta}_{\sf
max}-\xi),$ (87)

where $\lambda_{k}$, $\nu_{k}$, $\mu$, $\lambda_{m,k}^{\sf e}$,
$\nu_{m,k}^{\sf e}$, and $\mu^{\sf e}$ indicate Lagrangian multipliers. For a
feasible solution, we assume $\lambda_{k}\geq 0$, $\nu_{k}\geq 0,\forall
k\in\mathcal{K}$, $\mu\geq 0$, $\mu^{\sf e}\geq 0$, $\lambda_{m,k}^{\sf e}\geq
0$, and $\nu_{m,k}^{\sf e}\geq 0,\forall m\in\mathcal{M},\forall
k\in\mathcal{K}$. According to the KKT conditions, the optimal solution for
the problem in (46) should satisfy the followings:

$\displaystyle\left.\frac{\partial\mbox{$\mathcal{L}$}_{2}}{\partial\epsilon_{k}}\right|_{\epsilon_{k}=\epsilon^{\star}_{k}}=-\frac{w}{R_{\infty}}\sqrt{\frac{\mathcal{V}_{k}}{L}}\sqrt{2\pi}\exp\left({\frac{\left(Q^{-1}(\epsilon^{\star}_{k})\right)^{2}}{2}}\right)+\lambda^{\star}_{k}+\nu^{\star}_{k}=0,$
(88)
$\displaystyle\left.\frac{\partial\mbox{$\mathcal{L}$}_{2}}{\partial\tau}\right|_{\tau=\tau^{\star}}=\frac{1-w}{\hat{\epsilon}_{\sf
max}}-\sum_{k}\lambda^{\star}_{k}+\mu^{\star}=0,$ (89)
$\displaystyle\left.\frac{\partial\mbox{$\mathcal{L}$}_{2}}{\partial\delta_{m,k}}\right|_{\delta_{m,k}=\delta^{\star}_{m,k}}=-\frac{w}{R_{\infty}}\sqrt{\frac{\mathcal{V}_{m,k}^{\sf
e}}{L}}\sqrt{2\pi}\exp\left({\frac{\left(Q^{-1}(\delta_{m,k}^{\star})\right)^{2}}{2}}\right)+\lambda^{\sf
e,\star}_{m,k}+\nu^{\sf e,\star}_{m,k}=0,$ (90)
$\displaystyle\left.\frac{\partial\mbox{$\mathcal{L}$}_{2}}{\partial\xi}\right|_{\xi=\xi^{\star}}=\frac{1-w}{\hat{\delta}_{\sf
max}}-\sum_{m,k}\lambda^{\sf e,\star}_{m,k}+\mu^{\sf e,\star}=0,$ (91)
$\displaystyle\lambda^{\star}_{k}(\tau^{\star}-\epsilon^{\star}_{k})=0,$ (92)
$\displaystyle\nu^{\star}_{k}(\hat{\epsilon}_{k}-\epsilon^{\star}_{k})=0,$
(93) $\displaystyle\mu^{\star}(\hat{\epsilon}_{\sf max}-\tau^{\star})=0,$ (94)
$\displaystyle\lambda^{\sf
e,\star}_{m,k}(\xi^{\star}-\delta^{\star}_{m,k})=0,$ (95)
$\displaystyle\nu^{\sf
e,\star}_{m,k}(\hat{\delta}_{m,k}-\delta^{\star}_{m,k})=0,$ (96)
$\displaystyle\mu^{\sf e,\star}(\hat{\delta}_{\sf max}-\xi^{\star})=0,$ (97)

where (88) and (90) come from that $\partial Q^{-1}(x)/\partial
x=-\sqrt{2\pi}\exp\left(\left(Q^{-1}(x)\right)^{2}/2\right)$. Now, we obtain
the optimal points $\epsilon^{\star}_{k}$ and $\delta^{\star}_{m,k}$ by using
the derived KKT conditions. From (88), (92), and (93), $\epsilon^{\star}_{k}$
should be equal to $\tau^{\star}$ or $\hat{\epsilon}_{k}$. In the same manner,
$\delta^{\star}_{m,k}$ should be equal to $\xi^{\star}$ or
$\hat{\delta}_{m,k}$ from (90), (95), and (96). Assuming that
$\tau^{\star}<\hat{\epsilon}_{\sf max}$ and $\xi^{\star}<\hat{\delta}_{\sf
max}$, we obtain $\mu^{\star}=\mu^{\sf e,\star}=0$. Since we can assume that
$\hat{\epsilon}_{\ell-1}<\tau^{\star}<\hat{\epsilon}_{\ell}$ for some $\ell$
and $\hat{\delta}_{j(k)-1,k}<\xi^{\star}<\hat{\delta}_{j(k),k}$ for some
$j(k)$, the optimal error probability and information leakage vectors are
obtained as

$\displaystyle{\boldsymbol{\epsilon}}^{\star}=[\hat{\epsilon}_{1},\cdots,\hat{\epsilon}_{\ell-1},\tau^{\star},\cdots,\tau^{\star}]^{\sf
T},$ (98)
$\displaystyle{\boldsymbol{\delta}}_{k}^{\star}=[\hat{\delta}_{1,k},\cdots,\hat{\delta}_{j(k)-1,k},\xi^{\star},\cdots,\xi^{\star}]^{\sf
T}.$ (99)

The information leakage matrix builds upon the derived information leakage
vectors as

$\displaystyle{\boldsymbol{\Delta}}^{\star}=[{\boldsymbol{\delta}}_{1}^{\star},\cdots,{\boldsymbol{\delta}}_{K}^{\star}].$
(100)

Under the assumption that
$\hat{\epsilon}_{\ell-1}<\tau^{\star}<\hat{\epsilon}_{\ell}$ for some $\ell$,
$\nu_{k}^{\star}=0$ and $\mu^{\star}=0$ for $k\geq\ell$. Combining (88) and
(89), we have

$\displaystyle\frac{w}{R_{\infty}}\sum_{k=\ell}^{K}\sqrt{\frac{\mathcal{V}_{k}}{L}}\sqrt{2\pi}\exp\left({\frac{\left(Q^{-1}(\epsilon_{k}^{\star})\right)^{2}}{2}}\right)=\frac{1-w}{\hat{\epsilon}_{\sf
max}},\text{ for }k\geq\ell.$ (101)

We solve (101) with respect to $\epsilon_{k}^{\star}$ and then we have
$\epsilon_{k}^{\star}=\tau^{\star}$ which is in (53) for $k\geq\ell$.
Similarly, we assume that
$\hat{\delta}_{j(k)-1,k}<\xi^{\star}<\hat{\delta}_{j(k),k}$, then $\nu^{\sf
e,\star}_{m,k}=0$ and $\mu^{\sf e,\star}=0$ for $m\geq j(k)$. From (90) and
(91), we have

$\displaystyle\frac{w}{R_{\infty}}\sum_{m=j(k)}^{M}\sum_{k=1}^{K}\left[\sqrt{\frac{\mathcal{V}_{m,k}^{\sf
e}}{L}}\sqrt{2\pi}\exp\left({\frac{\left(Q^{-1}(\delta_{m,k}^{\star})\right)^{2}}{2}}\right)\right]=\frac{1-w}{\hat{\delta}_{\sf
max}},\text{ for }m\geq j(k).$ (102)

Solving (102) with respect to $\delta_{m,k}^{\star}$, we have
$\delta_{m,k}^{\star}=\xi^{\star}$ in (54) for $m\geq j(k)$. ∎

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