# Novel Secret-Key-Assisted Schemes for Secure MISOME-OFDM Systems

Mohamed Marzban, , Ahmed El Shafie, , Naofal Al-Dhahir,  M. Marzban and N. Al-
Dhahir are with the Electrical and Computer Engineering Department, the
University of Texas at Dallas. A. El Shafie is with the R&D department,
Qualcomm Inc., USA. emails:(mohamed.marzban<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose a new secure transmission scheme for uplink multiple-input single-
output (MISO) orthogonal-frequency multiplexing (OFDM) systems in the presence
of multiple eavesdroppers. Our proposed scheme utilizes the sub-channels
orthogonality of OFDM systems to simultaneously transmit data and secret key
symbols. The base station, Bob, shares secret key symbols with the legitimate
user, Alice, using wiretap coding over a portion of the sub-channels.
Concurrently, Alice uses the accumulated secret keys in her secret-key queue
to encrypt data symbols using a one time pad (OTP) cipher and transmits them
to Bob over the remaining sub-channels. if Alice did not accumulate sufficient
keys in her secret-key queue, she employs wiretap coding to secure her data
transmissions. We propose fixed and dynamic sub-channel allocation schemes to
divide the sub-channels between data and secret keys. We derive the secrecy
outage probability (SOP) and the secure throughput for the proposed scheme. We
quantify the system’s security under practical non-Gaussian transmissions
where discrete signal constellation points are transmitted by the legitimate
source nodes. Numerical results validate our theoretical findings and quantify
the impact of different system design parameters.

###### Index Terms:

PHY security, wiretap channel, OFDM, secret key sharing, OTP encryption

##  I Introduction

Wireless channels are characterized by a broadcast nature which allows
eavesdroppers to overhear exchanged messages between other users co-located in
the same coverage area, introducing serious security challenges to the privacy
of the users and the confidentiality of the transmitted data. Eavesdroppers
usually conceal themselves as regular users and try to eavesdrop the
information transmitted by other users in the same network or coverage area.
Hence, a system designer should expect possible eavesdropping attacks from
some of the potential regular users. None of the users should be taken for
granted as a non-hostile node.

Information security is conventionally attained by deploying cryptography
algorithms at the network layer of the protocol stack. Such algorithms have
shown high resilience against eavesdropping attacks as they are hard to be
broken in a feasible computational time. However, cryptography requires high
storage and computational power resources at the legitimate communicating
nodes. In addition, they are not information-theoretic secure since the
computational power and knowledge of the network parameters are assumed to be
restricted at the eavesdroppers. Recently, physical-layer (PHY) security has
emerged as an additional layer of defense against eavesdroppers at the
waveform level by exploiting the stochastic properties of the wireless
channel. In [1], Wyner proved that information theoretic secure communications
can be attained if the wiretap channel between the transmitter and the
eavesdropper is statistically more degraded than the main channel between the
transmitter and the legitimate receiver. The secrecy capacity, defined as the
maximum data transmission rate with arbitrarily zero information exposure at
the eavesdroppers, is used to quantify the system’s security level.

Nowadays, orthogonal-frequency division multiplexing (OFDM) is being adopted
by most wireless communications standards as it simplifies the channel
equalization process by transforming the frequency-selective channel into a
group of flat fading sub-channels. That is why many research works
investigated the PHY security of OFDM systems, see e.g. [2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14]. Some schemes relied on designing artificial-noise (AN)
precoding schemes to confuse the eavesdropping nodes and reduce their signal-
plus-interference-to-noise ratios (SINRs). These schemes can be divided into
spatial [15, 16], temporal [9, 10, 11], or hybrid spatio-temporal [13] AN
injection schemes. In all scenarios, the AN is injected in the null space of
the legitimate receiver channel to eliminate its effect at the legitimate
receiver. To achieve this goal, spatial AN injection schemes exploit the
degrees of freedom provided by having multiple transmit antennas while the
temporal AN schemes exploit the degrees of freedom due to the cyclic prefix
(CP) insertion in the OFDM waveform. In [9], the authors proposed a temporal
AN injection scheme for multi-user systems and proved that the number of
useful temporal AN streams is dependent on the channel memories of the
transmitter-legitimate receiver and transmitter-eavesdropper links. In [11],
the authors controlled the processing time delays in amplify-and-forward
relay-based communications to increase the AN covariance matrix rank at the
eavesdroppers. In [13], the authors proposed a new hybrid spatio-temporal AN
scheme to enhance the system’s secrecy rate.

Other approaches for PHY security utilized the randomness and reciprocity
properties of the wireless channel to generate identical secret key pairs at
both the legitimate transmitter and receiver ends [17, 18, 19, 20, 21]. The
transmitter encrypts its data symbols using these keys through the one time
pad (OTP) cipher scheme [22]. Different properties of the received signals
were investigated for secret key generation such as the received signal
strength [17], phase differences [18], time delay (in wide-band transmission)
[19, 20], and channel state information (CSI) [21]. OTP encryption is
mathematically unbreakable as long as the the number of secret key symbols is
equal to the number of data symbols. However, the channel-based key generation
rate is typically much lower than the data symbols rate which limits the
secure transmission rate of these schemes.

In [23], the authors proposed the idea of using a secret-key queue in a
single-user system. Different from the data queue at the transmitter, a
secret-key queue is kept at both the legitimate transmitter and the receiver
while being hidden from the eavesdropper. The main idea is to use a portion of
the secrecy rate to send randomly-generated secret key symbols instead of
sending data symbols. These stored key symbols can be used later to achieve
secure communication between the source and its destination when the
transmitter-destination link is not secured. Thus, even at times of _zero_
instantaneous secrecy rate, which is the case when the transmitter-
eavesdropper channel has higher gain than that of the legitimate transmitter-
receiver channel, data symbols can still be transmitted and secured from the
eavesdropper. Based on this idea, Gungor _et al._ [24] showed that a long-term
constant secrecy rate is achievable. The authors of [24] addressed decoding
delays but did not consider the dynamics of the data arrival process at the
transmitter. Mao _et al._ [25] assumed random data arrivals at the source node
and illustrated through simulations that the use of a secret-key queue reduces
the queuing delay for the data packets.

Unlike [23, 24, 25], we propose a secret-key assisted secure scheme that
utilizes the sub-channel orthogonality of OFDM systems, allowing the
simultaneous transmissions of data and secret key symbols in two opposite
directions. In addition, we consider non-Gaussian transmissions, where the
data and keys are generated from discrete signal constellations, which is a
practical hardware constraint for transmitters using a fixed modulation
scheme. Our main contributions in this paper are summarized as follows

  * •

We propose a new secret-key-assisted transmission scheme where a base-station,
Bob, securely transmits secret key packets to a legitimate source node, Alice,
using a wiretap coding scheme. Those packets are accumulated in a secret-key
queue at Alice and Bob. Alice then utilizes her secret keys to encrypt her
data symbols through an OTP cipher scheme and transmits them to Bob. Our
proposed secret-key-assisted scheme is designed based on the connection outage
status of the Alice-Bob link.

  * •

We propose two schemes to divide the OFDM sub-channels between Alice-Bob data
transmissions and Bob-Alice secret key sharing. The first scheme optimizes a
fixed sub-channel allocation and the target data/key rates based on the
channel statistics. The second scheme is dynamic and optimizes the sub-channel
allocation and the target data/key rates every coherence time. We show that
the fixed sub-channel allocation scheme achieves close performance to the
dynamic sub-channel allocation scheme using lower overhead and complexity.
Moreover, we optimize the secret key packet length. We show the impact of all
those system design parameters on the secure throughput.

  * •

We derive the secrecy rate and the secrecy outage probability (SOP)
expressions under gap approximations where non-Gaussian transmissions are
adopted by the legitimate transmitting nodes (i.e., source node when it sends
data packets and the base-station when it sends key packets).

  * •

We analyze the Markov chain of the secret-key queue and show the impact of the
system parameters on the queue dynamics and the amount of arrival and
departure rates.

_Notation:_ Lower- and upper-case bold letters denote column vectors and
matrices, respectively. $\mathbb{C}^{M\times N}$ denotes the set of all
$M\times N$ complex matrices. $(\cdot)^{\top}$ and $(\cdot)^{*}$ denote the
transpose and Hermitian (i.e., complex-conjugate transpose) operations,
respectively, $\|.\|$ denotes the Euclidean norm of a vector.
$[\cdot]^{+}=\max\\{0,\cdot\\}$ returns the maximum between the argument and
zero. The superscripts $(\cdot)^{d}$ and $(\cdot)^{k}$ respectively, refer to
$d$-th data and $k$-th secret key sub-channels quantities. The subscripts
$(\cdot)_{\rm m_{1}-m_{2}}$ denotes a quantity associated with $m_{1}-m_{2}$
link where $m_{1},m_{2}\in\\{{\rm Alice,Bob,\>}m{\rm-th\>Eve}\\}$

##  II System model and proposed secret-key-assisted scheme

We consider an uplink OFDM-based wireless communication system where a
legitimate user, Alice, transmits her data to a base-station (BS), Bob,
through a set of orthogonal OFDM sub-channels in the presence of $M$ other
users who can be treated as potential non-colluding eavesdroppers, Eves. Let
${\rm A}$, ${\rm B}$, ${\rm E}_{m}$ denote Alice, Bob and the $m$-th Eve
($m\in\\{1,2,\dots,M\\}$), respectively. We consider a multiple-input single-
output (MISO) two-way communications system where each transceiver is equipped
with multiple transmit antennas and a single receive antenna. Let the number
of transmit antennas at Alice and Bob be denoted by $N_{\rm A}$ and $N_{\rm
B}$, respectively.

Let $\mathbf{H}_{m_{\rm 1}-m_{\rm 2}}^{\rm time}\in\mathbb{C}^{N_{\rm m_{\rm
1}}\times L}$ denote the multi-path channel coefficients matrix from node
$m_{\rm 1}$ to node $m_{\rm 2}$ where $m_{\rm 1},m_{\rm 2}\in\\{{\rm
A,B,E}_{m}\\}$, the $\nu$-th row represents the channel impulse response (CIR)
with $L$ taps from transmit antenna $\nu$ to the receive antenna. Each CIR tap
follows a complex Gaussian distribution. We assume a quasi-static fading
channel model in which the channel experiences slowly-varying fading and the
entire codeword is experiencing a single fading channel realization. The OFDM
system decouples a frequency-selective channel into a set of orthogonal flat
fading sub-channels. The transmitter performs an inverse discrete Fourier
transform (IDFT) of size $N$ and inserts a CP of size $N_{\rm cp}\geq L$. At
the receivers side, the CP is removed and a DFT operation is performed to
obtain the frequency domain signal. The corresponding frequency-domain
channels are divided into $N$ orthogonal sub-channels. Let the channel
frequency response between a transmitter, $m_{\rm 1}$, and a receiver, $m_{\rm
2}$, over a sub-channel, $k$, be denoted by $\mathbf{h}_{\rm m_{\rm 1}-m_{\rm
2}}^{k}\in\mathbb{C}^{1\times N_{\rm m_{\rm 1}}}$ where $m_{\rm 1},m_{\rm
2}\in\\{{\rm A,B,E}_{m}\\}$. Note that $\mathbf{h}^{k}_{\rm m_{\rm 1}-m_{\rm
2}}$ is not the same as $\mathbf{h}^{k}_{\rm m_{\rm 2}-m_{\rm 1}}$.

We assume that Alice and Eves are potential users in the network communicating
with the BS, Bob, who can estimate his links’ channels with both Alice and
Eves. This assumption is reasonable since Alice and Eves are network users,
feeding back their channel response estimates to Bob. In contrast, the channel
state information (CSI) of the Alice-Eves links are completely unknown at both
Alice and Bob. Hence, Alice cannot design her precoder based on any knowledge
of the Alice-Eves links’ channels. It is noteworthy that Alice could design an
AN signal to hurt Eves using the knowledge of Alice-Bob link where the AN lies
in a subspace orthogonal to the subspace spanned by the Alice-Bob channel
vector [13, 9]. However, we skip this design in this paper since it is not
going to change our scheme’s design or add any insight to the problem under
investigation.

The key idea of our proposed scheme is to exploit the orthogonal structure of
the OFDM sub-channels to simultaneously transmit data and secret keys over
different OFDM sub-channels. We assume two sets of OFDM sub-channels. The set
of data OFDM sub-channels, denoted by ${\mathcal{U}_{\rm data}}$, and has a
cardinality of $N_{\rm data}$, and the set of the secret key OFDM sub-
channels, denoted by ${\mathcal{U}_{\rm key}}$, and has a cardinality of
$N_{\rm key}$. This imposes a constraint that $N_{\rm key}+N_{\rm data}=N$
where $N$ is the total number of OFDM sub-channels111In practice, some sub-
channels are not used (to create guard bands) and others are dedicated for
control functions such as channel estimation and synchronization. In this
case, $N$ will represent the number of active sub-channels dedicated for data
transmissions that does not include the control sub-channels. Alice uses the
$N_{\rm data}$ sub-channels to transmit data to Bob and, simultaneously, Bob
uses the remaining $N_{\rm key}=N-N_{\rm data}$ OFDM sub-channels to transmit
secret-key symbols to Alice as illustrated in Fig. 1. Since the OFDM sub-
channels are orthogonal, there is no interference between the two
transmissions.

Let $d\in\mathcal{U}_{\rm data}$ and $k\in\mathcal{U}_{\rm key}$ represent a
sub-channel assigned for data transmission and secret key sharing,
respectively. Let Alice’s transmitted data symbol over the $d$-th sub-channel
be denoted by $x_{\rm A}^{d}$ and the secret-key symbol shared by Bob over the
$k$-th sub-channel be denoted by $x_{\rm B}^{k}$. Hence, the received data
signal at Bob and the received secret-key symbol at Alice are given,
respectively, by

$y^{d}_{\rm B}=\mathbf{h}_{\rm AB}^{d\top}\>\mathbf{p}_{\rm A}^{d}x_{\rm
A}^{d}+n_{\rm B}^{d},\;\;\;\;y^{k}_{\rm A}=\mathbf{h}_{\rm
BA}^{k\top}\>\mathbf{p}_{\rm B}^{k}x_{\rm B}^{k}+n_{\rm A}^{k}$ (1)

where $\mathbf{h}_{\rm AB}^{d}\in\mathbb{C}^{N_{\rm A}\times 1}$ and
$\mathbf{h}_{\rm BA}^{k}\in\mathbb{C}^{N_{\rm B}\times 1}$ denote the Alice-
Bob and Bob-Alice channel frequency response vectors, at the $d$-th and $k$-th
sub-channels, respectively. Let $\mathbf{p}_{\rm A}^{d}\in\mathbb{C}^{\rm
N_{\rm A}\times 1}$ and $\mathbf{p}_{\rm B}^{k}\in\mathbb{C}^{\rm N_{\rm
B}\times 1}$ denote the $k$-th and $d$-th sub-channel precoding vectors at
Alice and Bob, respectively. Finally, let $n_{\rm A}^{d}$ and $n_{\rm B}^{k}$
denote complex additive white Gaussian noise (AWGN) samples at Alice and Bob,
respectively.

We assume that the target secret-key rate and the target data rate are fixed,
respectively, at $\mathcal{R}_{\rm key}$ and $\mathcal{R}_{\rm data}$
bits/channel use. Generally speaking, $\mathcal{R}_{\rm data}$ is not
necessarily equal to $\mathcal{R}_{\rm key}$. To simplify the relationship
between both quantities, we assume that $\mathcal{R}_{\rm key}$ is divisible
by $\mathcal{R}_{\rm data}$. More specifically, $\mathcal{R}_{\rm
data}/\mathcal{R}_{\rm key}=K$ where $K\in\\{1,2,\dots\\}$ is an integer. Note
that when the direct link from Alice to Bob is in a connection outage, the
link from Bob to Alice is not necessarily in a connection outage since the
channels and precoders of both links are different. In addition,
$\mathcal{R}_{\rm key}$ is typically designed to be less than
$\mathcal{R}_{\rm data}$ which constrains the secret-key packet length to be
relatively smaller than the data packet length.

Figure 1: System model showing Alice sharing data to Bob using $N_{\rm data}$
sub-channels and Bob sharing secret keys to Alice using $N_{\rm keys}$ sub-
channels in the presence of $M$ eavesdroppers who can eavesdrop on both the
data and the secret-key transmissions.

The flow chart of the proposed scheme is depicted in Fig. 2. Let $Q_{\rm
key}(t)$ denote Alice’s secret-key queue state at time $t$ measured by the
number of received secret-key packets from Bob. If the number of accumulated
secret-key packets is greater than a single data packet size (i.e., $Q_{\rm
key}(t)\geq K$), Alice selects the $N_{\rm data}$ sub-channels having highest
channel gains of the Alice-Bob link. Note that the best $N_{\rm data}$ sub-
channels of the Alice-Bob link are not necessarily the best sub-channels of
the Bob-Alice link. Then, Alice utilizes $K$ secret-key packets in her secret-
key queue to encrypt a single data packet using OTP encryption. In this case,
the encrypted data symbols transmitted from Alice to Bob are perfectly secured
and depend only on the Alice-Bob channel link irrespective of the Eve’s
received signals or the links’ qualities. More precisely, the achieved Alice-
Bob secrecy rate, ${R_{\rm s}}_{\rm AB}$, becomes equal to the Alice-Bob link
rate, $R_{\rm AB}$, as follows

${R_{\rm s}}_{\rm AB}=\sum_{d\in\mathcal{U}_{\rm data}}R_{\rm AB}^{d}$ (2)

If the direct link is not in a connection outage (i.e. $R_{\rm
AB}>\mathcal{R}_{\rm data}$), the data transmission is successful and the
secret-key queue moves to state $Q_{\rm key}(t)-K$, i.e., $K$ packets leave
the queue’s head as follows

$Q_{\rm key}(t+1)=Q_{\rm key}(t)-K$ (3)

For secret-key sharing, Bob utilizes knowledge of his channels with Alice and
Eves to adjust the redundancy rate of the wiretap codes to send perfectly-
secured secret-key symbols to Alice over the remaining $N_{\rm key}$ sub-
channels. The details of the wiretap code design are discussed in Appendix -A.
Note that those $N_{\rm key}$ sub-channels are assigned to key sharing only if
the $\rm B-A$ achieved secrecy key rate, ${R_{\rm s}}_{\rm BA}$, is higher
than the target secret-key rate, $\mathcal{R}_{\rm key}$. Otherwise, all sub-
channels are assigned to data transmission (i.e. $N_{\rm data}=N$ and $N_{\rm
key}=0$).

When the number of accumulated secret-key packets at Alice/Bob is less than a
single data packet size (i.e., $Q_{\rm key}(t)<K$), OTP encryption is not
possible, since OTP requires an equal number of secret-key and data symbols.
Hence, data transmissions are secured using wiretap coding, which is
considered secure as long as the achieved secrecy data rate, ${R_{\rm s}}_{\rm
AB}$, is higher than the target data rate, $\mathcal{R}_{\rm data}$.
Otherwise, the system is considered to be in a secrecy outage, as discussed in
Appendix -A. In case of non-sufficient secret key packets in the secret-key
queue ($Q_{\rm key}(t)<K$), Bob is given the priority in selecting the $N_{\rm
key}$ sub-channels used for secret-key transmissions. Bob selects the $N_{\rm
key}$ sub-channels that maximize the difference between the ${\rm B}-{\rm A}$
link rate, ${R}_{\rm BA}^{k}$, and the maximum link rate among the Bob-Eves
links, $\max_{m}R_{{\rm BE}_{m}}^{k}$, as follows

$\small\begin{split}\mathcal{U}_{\rm key}(i)&={\arg}{\max_{k}}\;\left\\{R_{\rm
BA}^{k}-\max_{m}R_{{\rm BE}_{m}}^{k}\right\\},\forall i\in\\{1,2,..,N_{\rm
key}\\},k\neq\left\\{\mathcal{U}_{\rm key}(1),..,\mathcal{U}_{\rm
key}(i-1)\right\\}\end{split}$ (4)

The flow chart of our proposed scheme is depicted in Fig. 2 and the list of
key variables in this paper is summarized in Table I.

TABLE I: List of Key Variables symbol |  Description |  Symbol |  Description
---|---|---|---
$N$ |  total number of sub-channels |  $N_{\rm cp}$ |  CP length
$N_{\rm data}$ |  number of data sub-channels |  $N_{\rm key}$ |  number of secret keys sub-channels
${\mathcal{U}_{\rm data}}$ |  set of data sub-channels |  ${\mathcal{U}_{\rm key}}$ |  set of key sub-channels
$N_{\rm A}$ |  number of transmit antennas at Alice |  $N_{\rm B}$ |  number of transmit antennas at Bob
$M$ |  number of eavesdroppers |  $L$ |  number of CIR taps
$\mathbf{H}_{m_{\rm 1}-m_{\rm 2}}^{\rm time}$ |  multipath CIR of size $N_{\rm m_{\rm 1}}\times L$ from node $m_{\rm 1}$ to node $m_{\rm 2}$ |  $\mathbf{h}_{m_{\rm 1}-m_{\rm 2}}^{k}$ |  frequency domain channel from node $m_{\rm 1}$ to node $m_{\rm 2}$ over sub-channel $k$
$\mathbf{p}_{\rm A}^{d}$ |  $d$-th sub-channel precoding vector at Alice of length $N_{\rm A}$ |  $\mathbf{p}_{\rm B}^{k}$ |  $k$-th sub-channel precoding vector at Bob of length $N_{\rm B}$
$\mathcal{R}_{\rm data}$ |  target data rate |  $\mathcal{R}_{\rm key}$ |  target key rate
$Q_{\rm key}(t)$ |  secret-key queue state at time $t$ |  $Q_{\rm max}$ |  size of the secret-key queue
$K$ |  ratio between $\mathcal{R}_{\rm data}$ and $\mathcal{R}_{\rm key}$ which is equal to the ratio between the data and key packet sizes |  $\lambda$ |  mean arrival rate of the secret-key queue
${R}_{\rm AB}$ |  achieved data rate for ${\rm A}-{\rm B}$ link |  ${R}_{\rm BA}$ |  achieved key-sharing rate for ${\rm B}-{\rm A}$ link
${R_{\rm s}}_{\rm AB}$ |  achieved secrecy data rate for ${\rm A}-{\rm B}$ link |  ${R_{\rm s}}_{\rm BA}$ |  achieved secrecy key rate for ${\rm B}-{\rm A}$ link
$\gamma^{d}_{\rm A}$ |  $d$-th sub-channel received SNR for ${\rm A}-{\rm B}$ transmission |  $\gamma^{k}_{\rm B}$ |  $k$-th sub-channel received SNR for ${\rm B}-{\rm A}$ transmission
$\Gamma_{\rm AB}$ |  SNR gap for ${\rm A}-{\rm B}$ transmissions |  $\Gamma_{\rm BA}$ |  SNR gap for ${\rm B}-{\rm A}$ transmission
$\mathcal{T}_{\rm s}$ |  achieved secure throughput |  $P_{\rm AB}^{\rm OP}$ |  connection outage probability for ${\rm A}-{\rm B}$ data transmission
$P_{\rm AB}^{\rm SOP}$ |  SOP for ${\rm A}-{\rm B}$ data transmissions |  $P_{\rm BA}^{\rm SOP}$ |  SOP for ${\rm B}-{\rm A}$ secret key sharing
Figure 2: Flow chart of our proposed scheme where $\mathcal{P}_{\rm A}=1$
implies $A-B$ link is not in connection outage and thus Alice is allowed to
transmit data and $i_{\rm OTP}=1$ implies Alice accumulated sufficient keys
for OTP encryption.

##  III Secrecy outage probabilities (SOP)

The flow chart in Fig. 2 shows that in some scenarios Alice employs OTP
encryption over its assigned sub-channels while in others, wiretap coding is
adopted. However, Bob always adopts a wiretap coding scheme to share his
generated secret keys with Alice. In this section, we derive the SOP for these
three scenarios: OTP encryption for Alice transmissions to Bob, wiretap coding
for Alice transmissions to Bob, and wiretap coding for Bob transmissions to
Alice.

###  III-A Alice-Bob link SOP using OTP encryption

Alice’s data queue is assumed to be backlogged with data packets. Considering
a channel bandwidth of $W$ and a time slot duration $T$, each packet has
${B}_{\rm data}=\mathcal{R}_{\rm data}WT$ bits where the product $WT$
represents the total number of channel uses and $\mathcal{R}_{\rm data}$ is
the number of bits per channel use. The achievable data rate of the Alice-Bob
link over the $d$-th OFDM sub-channel is given by

$\begin{split}R_{\rm AB}^{d}=\frac{1}{N+N_{\rm
cp}}\log_{2}\left(1+\gamma^{d}_{\rm A}\frac{\mathbf{h}_{\rm
AB}^{d\top}\mathbf{p}_{\rm A}^{d}\left(\mathbf{h}_{\rm
AB}^{d\top}\mathbf{p}_{\rm A}^{d}\right)^{*}}{\Gamma_{\rm
AB}}\right)\end{split}$ (5)

where $\gamma^{d}_{\rm A}$ is the signal-to-noise ratio (SNR) at the $d$-th
sub-channel and $\Gamma_{\rm AB}$ is the SNR gap for the ${\rm A}-{\rm B}$
link. The SNR gap accounts for the mismatch between the theoretical capacity
based on error free transmissions of Gaussian signals and achievable rates
using practical coded quadrature amplitude modulation (QAM) schemes. The SNR
gap is given by[26]

$\displaystyle\Gamma_{\rm
AB}=\frac{\gamma_{m}}{3\gamma_{c}}\left(Q^{-1}\left(\frac{P_{\rm
AB,e}}{4}\right)\right)^{2}$ (6)

where
$Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\text{exp}(-\frac{u^{2}}{2})du$ is
the well-known $Q$-function, $\gamma_{c}$ is the coding gain of any applied
code, $\gamma_{m}$ is the design margin to ensure adequate performance in the
presence of unforeseen system impairments, and $P_{\rm AB,e}$ is the target
error rate. The value of the SNR gap, $\Gamma_{\rm AB}$, defines the required
system quality-of-service (QoS). Therefore, we can define different gap values
for each group to enable different users priority levels.

Alice designs her $d$-th sub-channel precoder, $\mathbf{p}_{\rm A}^{d}$, based
on the CSI of the ${\rm A}-{\rm B}$ link as follows, $\mathbf{p}_{\rm
A}^{d}=\frac{(\mathbf{h}_{\rm AB}^{d})^{*}}{|\mathbf{h}_{\rm AB}^{d}|}$. The
achievable ${\rm A}-{\rm B}$ link rate over the $d$-th OFDM sub-channel is
given by

$\begin{split}R_{\rm AB}^{d}=\frac{1}{N+N_{\rm
cp}}\log_{2}\left(1+\gamma^{d}_{\rm A}\frac{\|\mathbf{h}_{\rm
AB}^{d}\|^{2}}{\Gamma_{\rm AB}}\right)\end{split}$ (7)

When the number of Alice’s transmit antennas is large,
$\|\mathbf{h}_{AB}^{d}\|^{2}$ can be approximated by the expectation of its
elements (i.e. $\|\mathbf{h}_{AB}^{d}\|^{2}\approx N_{\rm A}$). Consequently,

$\begin{split}R_{\rm AB}^{d}\approx\frac{1}{N+N_{\rm
cp}}\log_{2}\left(1+\gamma^{d}_{\rm A}\frac{N_{\rm A}}{\Gamma_{\rm
AB}}\right)\end{split}$ (8)

In OTP encryption, the transmitted symbols are perfectly secured from Eves and
the secrecy rate becomes equal to the data rate (i.e. ${R^{d}_{\rm s}}_{\rm
AB}=R_{\rm AB}^{d}$). Hence, under OTP encryption, the secrecy outage
probability, $P_{\rm AB}^{\rm SOP}$, is equal to the connection outage
probability, $P_{\rm AB}^{\rm OP}$, as follows

$P_{\rm AB}^{\rm OP}=\Pr\left\\{\sum_{d\in\mathcal{U}_{\rm data}}R_{\rm
AB}^{d}\leq\mathcal{R}_{\rm data}\right\\}$ (9)

To ensure that Alice’s transmissions are not in connection outage, the
following condition should be satisfied.

$\begin{split}\sum_{d\in\mathcal{U}_{\rm data}}R_{\rm
AB}^{d}\approx\frac{N_{\rm data}}{N+N_{\rm cp}}\log_{2}\left(1+\gamma^{d}_{\rm
A}\frac{N_{\rm A}}{\Gamma_{\rm AB}}\right)\geq\mathcal{R}_{\rm
data}\end{split}$ (10)

which implies that

$\begin{split}\gamma^{d}_{\rm A}\frac{N_{\rm A}}{\Gamma_{\rm AB}}\geq
2^{\mathcal{R}_{\rm data}\frac{N+N_{\rm cp}}{N_{\rm data}}}-1\end{split}$ (11)

Alice can satisfy this constraint by controlling the number of data OFDM sub-
channels, $N_{\rm data}$, the number of transmit antennas, $N_{\rm A}$, the
transmit power, $\gamma^{k}_{\rm A}$, and the SNR gap, $\Gamma_{\rm AB}$,
which is a function of the required error rate. Under this condition, the
probability of successful (reliable) transmission is almost one. In other
words, the connection outage, which is equal to the SOP in this case, is
almost zero.

###  III-B Alice-Bob link SOP using wiretap coding

If Alice does not have sufficient keys to encrypt one packet of her data
symbols using OTP encryption, she employs wiretap coding over the $N_{\rm
data}$ sub-channels. In this case, the instantaneous secrecy rate of the ${\rm
A}-{\rm B}$ link is given by [27]

$\sum_{d\in\mathcal{U}_{\rm data}}{R_{\rm s}}_{\rm AB}^{d}=\frac{1}{N+N_{\rm
cp}}\min_{m}\left[\sum_{d\in\mathcal{U}_{\rm data}}\left(R_{\rm
AB}^{d}-R_{{\rm A}{\rm E}_{m}}^{d}\right)\right]^{+}$ (12)

where $R_{{\rm A}{\rm E}_{m}}^{d}$ is the achievable rate at the $m$-th Eve
over the $d$-th OFDM sub-channel. Alice does not know ${\rm A}-E_{m}$ channel
links and cannot compute ${R_{\rm s}}_{\rm AB}^{d}$. Hence, perfect secrecy
cannot be attained and a system designer should expect an SOP which can be
given by

$\small\begin{split}P_{\rm AB}^{\rm SOP}&=1-\Pr\\{{R_{\rm s}}_{\rm
AB}\geq\mathcal{R}_{\rm
data}\\}=\Pr\bigg{\\{}\Big{(}\sum_{d\in\mathcal{U}_{\rm data}}R_{\rm
AB}^{d}-\max_{m}\sum_{d\in\mathcal{U}_{\rm data}}R_{{\rm A}{\rm
E}_{m}}^{d}\Big{)}\ <\mathcal{R}_{\rm data}\\!\bigg{\\}}\end{split}$ (13)

The achievable rate of the Alice to $m$-th Eve link, dented by ${\rm A}-{\rm
E}_{m}$, can be given by222We emphasize that the Alice-Eve’s achievable data
rate is arbitrarily small whenever Alice-Bob’s link rate is higher than Alice-
Eve’s link rate as Alice employs wiretap coding. When the secrecy target rate
is lower than the instantaneous secrecy rate, the information leakage at Eve
is zero.

$\small\begin{split}&\sum_{d\in\mathcal{U}_{\rm data}}\\!\\!\\!R_{{\rm A}{\rm
E}_{m}}^{d}=\frac{\sum_{d\in\mathcal{U}_{\rm
data}}\\!\\!\\!\log_{2}\left(1+\gamma^{d}_{\rm A}\mathbf{h}_{{\rm A}{\rm
E}_{m}}^{d\top}\mathbf{p}_{\rm A}^{d}\left(\mathbf{h}_{{\rm A}{\rm
E}_{m}}^{d\top}\mathbf{p}_{\rm A}^{d}\right)^{*}\right)}{N+N_{\rm
cp}}=\frac{\log_{2}\prod_{d\in\mathcal{U}_{\rm data}}\left(1+\gamma^{d}_{\rm
A}\mathbf{h}_{{\rm A}{\rm E}_{m}}^{d\top}\mathbf{p}_{\rm
A}^{d}\left(\mathbf{h}_{{\rm A}{\rm E}_{m}}^{d\top}\mathbf{p}_{\rm
A}^{d}\right)^{*}\right)}{N+N_{\rm cp}}\end{split}$ (14)

Substituting with (8) and (14) into (13), we get

$\begin{split}&P_{\rm AB}^{\rm SOP}=\Pr\bigg{\\{}\frac{1}{N+N_{\rm
cp}}\times\max_{m}\log_{2}\prod_{d\in\mathcal{U}_{\rm
data}}\left(1+\gamma^{d}_{\rm A}\mathbf{h}_{{\rm A}{\rm
E}_{m}}^{d\top}\mathbf{p}_{\rm A}^{d}\left(\mathbf{h}_{{\rm A}{\rm
E}_{m}}^{d\top}\mathbf{p}_{\rm A}^{d}\right)^{*}\right)>\mathcal{R}_{\rm
th}\bigg{\\}}\end{split}$ (15)

where $\mathcal{R}_{\rm th}=\frac{N_{\rm data}}{N+N_{\rm
cp}}\log_{2}\left(1+\gamma^{d}_{\rm A}\frac{N_{\rm A}}{\Gamma_{\rm
AB}}\right)-\mathcal{R}_{\rm data}$. Note that $\mathbf{h}_{{\rm A}{\rm
E}_{m}}^{d}\mathbf{p}_{\rm A}^{d}$ is a complex Gaussian random variable with
zero mean and unit variance since the precoder, $\mathbf{p}_{\rm A}^{d}$, has
a unit norm. Hence, $\mathbf{h}_{{\rm B}{\rm E}_{m}}^{d}\mathbf{p}_{\rm
B}^{d}\left(\mathbf{h}_{{\rm B}{\rm E}_{m}}^{d}\mathbf{p}_{\rm
B}^{d}\right)^{*}$ follows a Chi-square distribution with two degrees of
freedom which is basically an exponential distribution with mean equal to one.
The term, $1+\gamma^{d}_{\rm A}\mathbf{h}_{{\rm A}{\rm
E}_{m}}^{d}\mathbf{p}_{\rm A}^{d}\left(\mathbf{h}_{{\rm A}{\rm
E}_{m}}^{d}\mathbf{p}_{\rm A}^{d}\right)^{*}$ is therefore a shifted
exponential distribution with unit shift and mean $=\gamma_{\rm A}^{d}$. The
probability density function (PDF) of the product of shifted exponential
distributions, $\\\ y=\prod_{d\in\mathcal{U}_{\rm
data}}\left(1+\gamma^{k}_{\rm A}\mathbf{h}_{{\rm A}{\rm
E}_{m}}^{d}\mathbf{p}_{\rm A}^{d}\left(\mathbf{h}_{{\rm A}{\rm
E}_{m}}^{d}\mathbf{p}_{\rm A}^{d}\right)^{*}\right)$ can be expressed as [28]

$\begin{split}\small p_{Y}(y)=\left(\\!\prod_{d\in\mathcal{U}_{\rm
data}}\\!\frac{1}{\gamma^{d}_{\rm A}}\exp\left(\frac{1}{\gamma^{d}_{\rm
A}}\right)\right)\times\mathcal{H}_{0,N_{\rm data}}^{N_{\rm
data},0}\Bigg{[}\frac{y}{\prod_{d\in\mathcal{U}_{\rm data}}\gamma^{d}_{\rm
A}}\;\vline\;\begin{matrix}{(-,-,-)}\\\ {\left(0,1,\frac{1}{\gamma^{1}_{\rm
A}}\right),\dots,\left(0,1,\frac{1}{\gamma^{N_{\rm data}}_{\rm
A}}\right)}\end{matrix}\Bigg{]}\end{split}$ (16)

where $\mathcal{H}$ is the generalized upper incomplete Fox’s H function [28].
The notation $(-,-,-)$ means that these coefficients will be eliminated when
substituted in the $\mathcal{H}$ function. Considering a single Eve and
mutually independent sub-channels, the pdf of Alice-Eve’s rate,
$\sum_{d\in\mathcal{U}_{\rm data}}\\!\\!\\!R_{{\rm A}{\rm E}_{m}}^{d}$ can be
then expressed as

$\small\begin{split}&p_{R_{\rm
AE}}(r)=\frac{1}{\log_{2}\left(\exp(\kappa^{-1})\right)}\times\mathcal{H}_{0,N_{\rm
data}}^{N_{\rm data},0}\Bigg{[}2^{r}\\!\\!\prod_{d\in\mathcal{U}_{\rm
data}}\\!\\!\left(\frac{1}{\gamma^{d}_{\rm
A}}\right)\;\vline\;\begin{matrix}{(-,-,-)}\\\
{\left(1,1,\frac{1}{\gamma^{1}_{\rm
A}}\right),\dots,\left(1,1,\frac{1}{\gamma^{\rm N_{\rm data}}_{\rm
A}}\right)}\end{matrix}\Bigg{]}\end{split}$ (17)

where $\kappa=\left(\\!\prod_{d\in\mathcal{U}_{\rm
data}}\\!\exp^{\frac{1}{\gamma^{d}_{\rm A}}}\right)$. The $A-B$ link SOP can
be expressed as

$\small\begin{split}P_{\rm AB}^{\rm
SOP}&=\Pr\left\\{0\leq\sum_{d\in\mathcal{U}_{\rm data}}{R_{\rm s}}_{\rm
AB}^{d}<\mathcal{R}_{\rm
data}\right\\}\approx\Pr\left\\{\sum_{d\in\mathcal{U}_{\rm data}}R_{\rm
AE}^{d}>\mathcal{R}_{\rm th}\right\\}=\int_{\mathcal{R}_{\rm
th}}^{\infty}p_{R_{\rm AE}}(r)\>dr\end{split}$ (18)

Substituting with (17) in (18) and integrating, we get the SOP expression in
(19) at the top of the next page.

$\begin{split}P_{\rm AB}^{\rm SOP}=\kappa\times\mathcal{H}_{0,N_{\rm
data}+1}^{N_{\rm data}+1,0}\left[2^{\mathcal{R}_{\rm
th}}\\!\\!\prod_{d\in\mathcal{U}_{\rm
data}}\\!\\!\left(\frac{1}{\gamma^{d}_{\rm
A}}\right)\;\vline\;\begin{matrix}{\bigg{(}1,1,0\bigg{)}}\\\
{\bigg{(}0,1,0\bigg{)},\left(1,1,\frac{1}{\gamma^{1}_{\rm
A}}\right),\dots,\left(1,1,\frac{1}{\gamma^{N_{\rm data}}_{\rm
A}}\right)}\end{matrix}\right]\\\ \\\ \hline\cr\end{split}$ (19)

###  III-C Bob-Alice link SOP using wiretap coding

Bob transmits secret key packets to Alice over the $\mathcal{U}_{\rm key}$
sub-channels using wiretap coding. When Bob is given the priority in sub-
channels selection, he selects the $\mathcal{U}_{\rm key}$ sub-channels that
maximize his secrecy rate as shown in Eqn. (4). Designing the precoder at Bob
is similar to Alice with the respective channel parameters, $\mathbf{p}_{\rm
B}^{k}=\frac{(\mathbf{h}_{BA}^{k})^{*}}{|\mathbf{h}_{BA}^{k}|}$. Hence, the
sub-channels assigned to Bob are given by

$\begin{split}\mathcal{U}_{\rm
key}(i)&={\arg}{\max_{k}}\;\left\\{\log_{2}\left(1+\gamma^{k}_{\rm
B}\frac{\|\mathbf{h}_{BA}^{k}\|^{2}}{\Gamma_{\rm
BA}}\right)\right.-\left.\max_{m}\log_{2}\left(1+\gamma^{k}_{\rm
B}\|\mathbf{h}_{BE_{m}}^{k}\|^{2}\right)\right\\},\\\ &\forall
i\in\\{1,2,..,N_{\rm key}\\},k\neq\left\\{\mathcal{U}_{\rm
key}(1),..,\mathcal{U}_{\rm key}(i-1)\right\\}\end{split}$ (20)

At high SNR, Eqn. (20) is approximated as follows

$\small\begin{split}\mathcal{U}_{\rm
key}(i)&={\arg}{\max_{k}}\;\left\\{\frac{\|\mathbf{h}_{BA}^{k}\|^{2}}{\max_{m}\|\mathbf{h}_{BE_{m}}^{k}\|^{2}}\right\\},\forall
i\in\\{1,2,..,N_{\rm key}\\},k\neq\left\\{\mathcal{U}_{\rm
key}(1),..,\mathcal{U}_{\rm key}(i-1)\right\\}\end{split}$ (21)

When the number of Bob’s transmit antennas is large,
$\|\mathbf{h}_{BA}^{k}\|^{2}\approx N_{\rm B}$ and the $B-{\rm A}$ link rate
is given by

$\begin{split}R_{\rm BA}^{k}\approx\frac{N}{N+N_{\rm
cp}}\log_{2}\left(1+\gamma^{k}_{\rm A}\frac{N_{\rm B}}{\Gamma_{\rm
BA}}\right)\end{split}$ (22)

Following the same steps to write down the achievable secrecy rates for the
${\rm B}-{\rm A}$ link secret-key transmissions, the secret keys instantaneous
secrecy rate is given by

$\begin{split}\sum_{k\in\mathcal{U}_{\rm key}}&{R_{\rm s}}_{\rm
BA}^{k}=\min_{m}\sum_{k\in\mathcal{U}_{\rm key}}\left[R_{\rm BA}^{k}-R_{{\rm
B}{\rm E}_{m}}^{k}\right]^{+}\\\ &=\frac{1}{N+N_{\rm
cp}}\Biggr{[}\sum_{k\in\mathcal{U}_{\rm key}}\log_{2}\left(1+\gamma^{k}_{\rm
A}\frac{N_{\rm B}}{\Gamma_{\rm BA}}\right)-\max_{m}\sum_{k\in\mathcal{U}_{\rm
key}}\log_{2}\left(1+\gamma^{k}_{\rm A}\mathbf{h}_{{\rm B}{\rm
E}_{m}}^{k\top}\mathbf{p}_{\rm B}^{k}\left(\mathbf{h}_{{\rm B}{\rm
E}_{m}}^{k\top}\mathbf{p}_{\rm
B}^{k}\right)^{*}\right)\Biggr{]}^{+}\end{split}$ (23)

where $\mathbf{h}_{{\rm B}{\rm E}_{m}}^{k}\mathbf{p}_{\rm B}^{k}$ is a
Gaussian random variable with zero mean and unit variance. Hence, $\\\
\mathbf{h}_{{\rm B}{\rm E}_{m}}^{k}\mathbf{p}_{\rm
B}^{k}\left(\mathbf{h}_{{\rm B}{\rm E}_{m}}^{k}\mathbf{p}_{\rm
B}^{k}\right)^{*}$ follows a Chi-square distribution with two degrees of
freedom. To ensure that Bob’s transmissions are secured, the following
condition should be satisfied

$\begin{split}&\sum_{k\in\mathcal{U}_{\rm key}}{R_{\rm s}}_{\rm
BA}^{k}\geq\mathcal{R}_{\rm key}\end{split}$ (24)

The SOP can be expressed similar to the Alice transmissions SOP wiretap coding
in (19) by replacing the parameters with those associated with the Bob-Alice
link. However, unlike Alice, Bob knows the instantaneous secrecy rate of his
transmissions since he is the intended receiver of all users (i.e., Alice and
Eves). Hence, Bob can determine whether or not a given transmission (from him)
is perfectly secured by ensuring that the instantaneous secrecy rate condition
in (24) is satisfied before transmitting any keys.

##  IV Secure throughput using secret keys

If the secret-key queue accumulates keys equivalent to a single data packet
length, data packet transmissions can be perfectly secured by OTP encryption.
Otherwise, Alice has to employ wiretap coding to secure her transmissions. In
both cases, Alice is assigned $N_{\rm data}$ sub-channels if $B-A$ link is not
in secrecy outage. Otherwise, Alice is assigned $N$ sub-channels. Hence, the
secure throughput of the system in bits/channel use is given by

$\small\begin{split}\mathcal{T}_{\rm s}=&\Bigg{[}\Pr\\{Q_{\rm
key}(t)\geq{{K}}\\}\times\bigg{\\{}\left(1-P_{\rm AB}^{\rm OP}(N_{\rm
data})\right)\left(1-P_{\rm BA}^{\rm SOP}(N_{\rm key})\right)+\left(1-P_{\rm
AB}^{\rm OP}(N)\right)P_{\rm BA}^{\rm SOP}(N_{\rm key})\bigg{\\}}\\\
&+\Pr\\{Q_{\rm key}(t)<{{K}}\\}\times\bigg{\\{}\left(1-P_{\rm AB}^{\rm
SOP}(N_{\rm data})\right)\left(1-P_{\rm BA}^{\rm SOP}(N_{\rm
key})\right)+\left(1-P_{\rm AB}^{\rm SOP}(N)\right)P_{\rm BA}^{\rm SOP}(N_{\rm
key})\bigg{\\}}\Bigg{]}R_{\rm data}\end{split}$ (25)

where $P^{\rm OP}_{\rm m1-m2}(\beta)$, $P^{\rm SOP}_{\rm m1-m2}(\beta)$ denote
respectively, the connection outage probability and the SOP of the $m_{\rm
1}-m_{\rm 2}$ link when $\beta$ sub-channels are used. Note that $P_{\rm
AB}^{\rm OP}(\beta)\leq P_{\rm AB}^{\rm SOP}(\beta)$ where $P_{\rm AB}^{\rm
OP}(\beta)$ is obtained from (9) while $P_{\rm AB}^{\rm SOP}(\beta)$ is
calculated using (19). As can be understood from the expression in (25),
increasing $\Pr\\{Q_{\rm key}(t)\geq{{K}}\\}$ can achieve the perfect security
of the system where the secure throughput is determined by the reliability of
the Alice-Bob link and completely independent of Eve’s presence.

Let $\Pr\\{Q_{\rm key}(t)=\ell\\}$ denote the probability that Alice’s secret-
key queue has accumulated $\ell$ key packets at time $t$. Using the Markov
chain analysis in Appendix -C, the steady state probabilities are given by

$\begin{split}\Pr\\{Q_{\rm
key}(t)=0\\}&=\frac{(1-\lambda)}{K},\;\;\;\;\Pr\\{Q_{\rm
key}(t)=\ell\\}\\!=\\!\frac{1}{K}\forall\ell\in\Sigma,\\\ \Pr\\{Q_{\rm
key}(t)=K\\}\\!&=\\!\frac{\lambda}{K},\;\;\;\;\Pr\\{Q_{\rm
key}(t)=\nu\\}\\!=\\!0\ \forall\nu>K\end{split}$ (26)

where $\lambda$ is the mean arrival rate of the secret-key queue,
$\Sigma=\\{1,2,\dots,K-1\\}$, $K\leq Q_{\max}$ and $Q_{\max}$ is the secret-
key buffer size. From the solution of the Markov chain, we observe that as
long as $K<Q_{\max}$, $Q_{\max}$ has no impact on the security. However,
increasing $Q_{\max}$ can improve the security performance since it increases
the feasible range of the optimization variable, $K$. Substituting with the
steady-state probabilities in (26) into the secure throughput in (25), we get
the secure throughput expression in (28) at the top of the next page, where
the mean arrival rate of the secret-key packets can be expressed as

$\vspace{-0.5 cm}\begin{split}\lambda&=\left(1-P^{\rm SOP}_{\rm BA}(N_{\rm
key})\right)\left(1-P^{\rm OP}_{\rm AB}(N_{\rm data})\right)+\left(1-P^{\rm
SOP}_{\rm BA}(N)\right)P^{\rm OP}_{\rm AB}(N_{\rm data})\end{split}$ (27)

Our ultimate goal is to maximize the secure throughput ($\mathcal{T}_{\rm s}$)
in (28). The expression $1-P^{\rm SOP}_{\rm BA}(\beta)=\Pr\\{{R_{\rm s}}_{\rm
BA}\geq\frac{\mathcal{R}_{\rm data}}{K}\\}$ is a function of $K$ and
monotonically decreasing with it. if one can ensure $\frac{1-P^{\rm SOP}_{\rm
BA}(\beta)}{K}$ is always equal to $1$, we can ensure that all transmissions
can be perfectly secured using secret-keys. The numerator of $\frac{1-P^{\rm
SOP}_{\rm BA}(\beta)}{K}$ is monotonically decreasing with $K$, while the
denominator is monotonically increasing with it. This highlights the
importance of selecting $K$ carefully.

One can further analyze the parameter $K$ as follows. Let ${B}_{\rm
data}=\mathcal{R}_{\rm data}WT$ and ${B}_{\rm key}=B_{\rm
data}/K=\mathcal{R}_{\rm data}WT/K$ denote the number of bits per data and
secret-key packets, respectively. When $K=1$, the secret key packet size is
equal to the data packet size and both have a transmission rate of
$\mathcal{R}_{\rm data}$ bits/channel use. If $K=2$, the secret key packet
size becomes $B_{\rm data}/2$ bits/channel use and becomes easier to achieve.
Increasing $K$, increases the secret-key packets mean arrival rate and
decreases their SOP. However, the steady-state probabilities in (26) decrease
with $K$. This decreases the probability that the secret-key queue accumulates
$K$ secret-key packets which is necessary to secure an entire data packet.
This implies that there is an optimal $K$ such that the secure throughput is
maximized which we optimize in Section V.B.

$\begin{split}\mathcal{T}_{\rm
s}=&\Bigg{[}\frac{\lambda}{K}\times\bigg{\\{}\left(1-P_{\rm AB}^{\rm
OP}(N_{\rm data})\right)\left(1-P_{\rm BA}^{\rm SOP}(N_{\rm
key})\right)+\left(1-P_{\rm AB}^{\rm OP}(N)\right)P_{\rm BA}^{\rm SOP}(N_{\rm
key})\bigg{\\}}\\\ &+\left(1-\frac{\lambda}{k}\right)\bigg{\\{}\left(1-P_{\rm
AB}^{\rm SOP}(N_{\rm data})\right)\left(1-P_{\rm BA}^{\rm SOP}(N_{\rm
key})\right)+\left(1-P_{\rm AB}^{\rm SOP}(N)\right)P_{\rm BA}^{\rm SOP}(N_{\rm
key})\bigg{\\}}\Bigg{]}R_{\rm data}\\\ \\\ \hline\cr\end{split}$ (28)

##  V Analysis and Insights for the proposed scheme

###  V-A Illustrative numerical example when $N_{\rm key}=1$

Assuming independent sub-channels, the sum over sub-channels for the rates can
be approximated as a Gaussian random variable with mean $\mu_{\circ}=N\mu$ and
variance $\sigma_{\circ}^{2}=N\sigma^{2}$. When $N_{\rm A}=N_{\rm B}=1$, using
the gap approximation, the instantaneous secrecy rate of the secret key packet
transmission over sub-channel $k\in\mathcal{U}_{\rm key}$ is given by

$\small\begin{split}\sum_{k\in\mathcal{U}_{\rm key}}{R_{\rm s}}_{\rm
BA}^{k}&=\left[\sum_{k\in\mathcal{U}_{\rm key}}R_{\rm
BA}^{k}-\sum_{k\in\mathcal{U}_{\rm key}}R_{{\rm B}{\rm
E}_{m}}^{k}\right]^{+}=\frac{\left[\sum_{k\in\mathcal{U}_{\rm
key}}\log_{2}\left(\frac{1+\frac{|\mathbf{h}_{\rm BA}^{k}|^{2}\gamma^{k}_{\rm
B}}{\Gamma_{\rm BA}}}{1+{|\mathbf{h}_{{\rm BE}_{m}}^{k}|^{2}\gamma^{k}_{\rm
B}}}\right)\right]^{+}}{N+N_{\rm cp}}\end{split}$ (29)

Assuming a single Eve and that only a single OFDM sub-channel is assigned to
key transmission ($N_{\rm key}=1$), the SOP of the key transmission is given
by [29]

$\begin{split}P_{\rm BA}^{\rm SOP}=1-\frac{1}{1+2^{\mathcal{R}_{\rm
key}(N+N_{\rm cp})}}\exp\left(-\frac{2^{\mathcal{R}_{\rm key}(N+N_{\rm
cp})}-1}{\gamma_{\rm B}^{k}/\Gamma^{k}_{\rm BA}}\right)\end{split}$ (30)

Assuming moderate-to-high SNR levels, we can approximate the secret key
transmission SOP as

$\begin{split}P_{\rm BA}^{\rm SOP}\approx 1-\frac{1}{1+2^{\mathcal{R}_{\rm
key}(N+N_{\rm cp})}}\approx 1-2^{-\mathcal{R}_{\rm key}N}\end{split}$ (31)

where the last approximation holds since we ignored the CP length, $N_{\rm
cp}$, relative to number of OFDM sub-channels, $N$. If $\mathcal{R}_{\rm
key}=2$ bits/channel use, the SOP is almost $1$ regardless of the used
transmit power since we have a factor of $2^{N+N_{\rm cp}}$ in the
denominator. In other words, the SOP is approximately given by

$\begin{split}P_{\rm BA}^{\rm SOP}\approx 1-2^{-2N}\end{split}$ (32)

which is very close to $1$ since $N$ is usually greater than $32$. This is
mainly because we assumed only a single sub-channel assigned for key sharing
($N_{\rm key}=1$) in a single-input single-output (SISO) system. Hence, we are
motivated to optimize the sub-channel assignment between the data and the
secret key symbols.

###  V-B Offline parameters optimization

Our problem is concerned with the secure throughput which is given by

$\underset{\begin{subarray}{c}{{N_{\rm data}}\in\\{1,2,\dots,N\\}}\\\
{K\in\\{1,2,\dots,Q_{\max}\\}}\end{subarray}}{\max:}\ \mathcal{T}_{\rm s}$
(33)

Once $N_{\rm data}$ and $K$ are optimized, we get $N_{\rm key}=N-N_{\rm data}$
and $\mathcal{R}_{\rm key}=\mathcal{R}_{\rm data}/K$. Since both optimization
parameters are integers, a simple approach to obtain the optimal solution that
maximizes the secure throughput is to evaluate the objective function $N\times
Q_{\max}$ times. The optimal solution is the one that yields the highest
objective function in (28). Our proposed approach is summarized in Algorithm
1. Although these computations are performed infrequently and offline, their
complexity can be reduced by using alternative computationally-efficient
methods such as the interior-point method [30, Chapter 11].

Algorithm 1 Optimization Procedure

1: Generate channel realizations using the channel statistics

2: Set $N_{\rm data}=1$

3: while $N_{\rm data}\neq N$ do

4: Compute $N_{\rm key}=N-N_{\rm data}$

5: Set $K=1$

6: while $K\neq Q_{\rm max}$ do

7: For each channel realization, compute the Alice-Bob link rate from (10) and
its secrecy rate outage probability from (19)

8: Compute the Bob-Alice secrecy rate using (23)

9: Compute the secure throughput, $\mathcal{T}_{\rm s}$ using (28)

10: Set $K=K+1$

11: end while

12: Set $N_{\rm data}=N_{\rm data}+1$

13: end while

14: Select $K$ and $N_{\rm data}$ that maximize $\mathcal{T}_{\rm s}$

###  V-C Dynamic parameters optimization

The previous optimization suggests using fixed allocation of data and key sub-
channels which remains the same as long as the channel statistics remain the
unchanged. Another approach to increase the secrecy rate is by relaxing the
fixed allocation requirement of the keys and data sub-channels ($N_{\rm key}$
and $N_{\rm data}$). At the intervals of OTP encryption (i.e., the secret-key
queue has one data packet length of keys), Alice only selects the minimum
number of sub-channels $N_{\rm data}$ with the highest SNR required to ensure
that her link is not in connection outage ($R_{\rm AB}\geq\mathcal{R}_{\rm
data}$). The remaining sub-channels should be used for ${\rm B}-{\rm A}$ key
sharing. Similarly, at the intervals of wiretap coding, Bob, since he knows
the instantaneous secrecy rates, uses the minimum number of sub-channels
required to ensure his key transmission is not in outage (${R_{\rm s}}_{\rm
BA}\geq\mathcal{R}_{\rm key}$). The remaining $N-N_{\rm key}$ OFDM sub-
channels are used by Alice to send data. The flow chart of this scheme is
shown in Fig. 3.

Figure 3: Flow chart of our proposed secret-key-assisted scheme using dynamic
sub-channel allocations

##  VI Simulation results

In this section, we carry out numerical simulations to evaluate the
performance of our proposed schemes in terms of the achieved secure
throughput. The fading channels are assumed to be complex circularly-symmetric
Gaussian random variables with zero mean and unit variances. Two nodes are
assumed to try eavesdropping on the ongoing communications, i.e., $M=2$.
Unless otherwise stated explicitly, we consider an OFDM system with $N=64$,
$N_{\rm cp}=L=8$, $\mathcal{R}_{\rm data}=1.5$ bits/channel use,
${\gamma^{k}_{\rm A}}={\gamma^{d}_{\rm B}}=30$ dB, $\Gamma_{\rm
AB}=\Gamma_{\rm BA}=1.2$, and $Q_{\rm max}=10$. We assume that the base
station, Bob, is equipped with eight transmit antennas while the legitimate
user, Alice, is equipped with two transmit antennas. To simplify the power
allocation, in all transmissions, we consider an equal-power allocation
strategy among all assigned sub-channels. We consider $K=1$ and assume that
the fixed key-sharing uses algorithm 1 to optimize $N_{\rm data}$ only.

Figure 4: Secure throughput versus SNR

Figure 5: Secure throughput versus $N_{\rm B}$

In Fig. 5, we plot the secure throughput performance of our proposed secret-
key-assisted schemes relative to the alignment precoding benchmark, where
Alice uses all sub-channels for data transmissions and aligns her precoder on
each of the Alice-Bob sub-channels. As expected, the secure throughput is
monotonically non-decreasing for all schemes. Our schemes achieved the upper
bound performance where the achievable secure throughput is equal to the
target secrecy rate $\mathcal{R}_{\rm data}=1.5$ bits/channel use. Although
our proposed schemes consume some sub-channel resources for key-sharing, they
achieved almost $9$ times gains relative to the benchmark as the OTP
encryption scheme protects the data packets from eavesdropping. Our proposed
dynamic secret-key sharing scheme achieves the best performance due to its
efficient sub-channels allocation between secret-key symbols and data symbols.
However, the fixed secret-key sharing scheme introduces a simpler and assumes
offline resource allocation with relatively low reduction in performance.

Fig. 5 shows the secure throughput performance versus the number of transmit
antennas at the base station, Bob, at $\mathcal{R}_{\rm data}=1.5$
bits/channel use. The benchmark does not rely on any transmissions from the
base station. Hence, its performance remains constant as we increase the
number of base station transmit antennas. However, increasing $N_{\rm B}$,
enhances Bob’s achievable rate for sharing secret keys with Alice which, in
turn, enhances our scheme’s secure throughput. Increasing the number of
transmit antennas at the base station from $2$ to $8$ enhances the performance
of our proposed scheme by more than $5$ times while the benchmark remains
unchanged.

In Fig. 7, we plot the achieved average secure throughput versus the data
target secrecy rate, $\mathcal{R}_{\rm data}$, when $K=3$. In all scenarios,
the secure throughput keeps increasing until the optimal target secrecy rate
is achieved, then, it decreases again. For the benchmark scheme, the optimal
value of $\mathcal{R}_{\rm data}$ is $0.5$ bits/channel use while the proposed
fixed and dynamic secret-key sharing schemes achieve optimal $\mathcal{R}_{\rm
data}$ of $4$ and $7$ bits/channel use, respectively. This demonstrates the
significant gains introduced by our proposed schemes.

Figure 6: Secure throughput versus $\mathcal{R}_{\rm data}$

Figure 7: Secure throughput versus $A-B$ link SNR gap ($\Gamma_{\rm AB}$)

Fig. 7 depicts the achieved secure throughput versus the SNR gap of the Alice-
Bob link. As we increase the SNR gap (harsher target BER requirements), the
system becomes less secure for all three schemes. However, our proposed
secret-key sharing schemes achieve significantly higher secure throughput
performance even at harsh target BER requirements.

Fig. 9 depicts the achieved average secure throughput versus $N_{\rm data}$
for our proposed key sharing scheme using a fixed sub-channel allocation. For
small $N_{\rm data}$, Alice is not assigned sufficient sub-channels to
transmit data symbols reliably to Bob. However, as $N_{\rm data}$ increases,
significant secure throughput gains are introduced until $N_{\rm data}$
reaches $11$. Then, the secure throughput decreases and saturates at the
benchmark secure throughput. High $N_{\rm data}$ (i.e., low $N_{\rm keys}$)
restricts Bob’s ability in sharing secret keys with Alice which reduces the
secure throughput. Note that the ${\rm B}-{\rm A}$ transmissions rely on
wiretap coding which requires more sub-channels than OTP encryption. This
explains why the optimal $N_{\rm data}$ is $11$ (i.e., optimal $N_{\rm key}$
of $53$).

Figure 8: Secure throughput versus $N_{\rm data}$

Figure 9: Secure throughput versus $K$

Fig. 9 depicts the achieved average secure throughput versus the parameter $K$
which represents the ratio between $\mathcal{R}_{\rm data}$ and
$\mathcal{R}_{\rm key}$. We observe an optimal $K$ of one when
$\mathcal{R}_{\rm data}=1.5$ bits/channel use while the optimal $K$ is two and
three for the dynamic and fixed key sharing schemes, respectively when
$\mathcal{R}_{\rm data}=4$ bits/channel use. At relatively low and medium
target data rates, $\mathcal{R}_{\rm data}$, the optimal value of $K$ is one,
which in turn maximizes $\mathcal{R}_{\rm key}$ (i.e. $\mathcal{R}_{\rm
key}=\mathcal{R}_{\rm data})$ to ensure maximum key sharing from Bob to Alice.
However, at high $\mathcal{R}_{\rm data}$, we have to increase $K$ to ensure
the target key rate is reduced $\mathcal{R}_{\rm key}=\mathcal{R}_{\rm
data}/K$ and becomes achievable by the wiretap coding schemes. Notice that the
OTP encyption-based data transmissions can achieve higher data rates than the
key transmissions which rely on wiretap coding. This explains why a relatively
low $\mathcal{R}_{\rm key}$ is needed when a high $\mathcal{R}_{\rm data}$ is
considered.

##  VII conclusions

We proposed a new secret-key-assisted scheme to secure uplink MISO
transmissions from eavesdropping attacks. Our proposed schemes exploited the
sub-channels orthogonality of OFDM transmissions to simultaneously transmit
data and secret-key symbols in two opposite directions. The base station, Bob,
transmits secret key symbols to Alice and secures his transmissions from
eavesdroppers using wiretap coding. Simultaneously, Alice encrypts her data
symbols using OTP if she accumulates an equivalent number of key symbols.
Otherwise, Alice performs wiretap coding to secure her data transmissions to
Bob. We proposed fixed and dynamic sub-channel allocation schemes to divide
the sub-channels between data transmissions and secret key sharing. The fixed
sub-channel allocation scheme achieves performance close to the dynamic scheme
with lower overhead and complexity. We optimized the key and target data rates
and we showed that the data packet length should be relatively large compared
to the key packet length to enhance the system’s secure throughput. We derived
the SOP and the secure throughput expressions for our proposed scheme. Our
proposed scheme introduced significant secure throughput gains. For example,
at a high SNR of $30$ dB, our proposed scheme achieved more than nine times
secure throughput gains relative to the benchmark due to the introduced gains
of OTP encryption using secret keys.

###  -A Wiretap coding

Consider the case where Alice secures her data packets using wiretap coding.
In a given time slot $t\in\\{1,2,3,...\\}$, Alice adapts her transmission rate
$R_{\rm AB}$ to be arbitrarily equal to the link rate to eliminate the outage
events. Let Alice use a codebook $C(2^{nR_{\rm AB}},2^{n\mathcal{R}_{\rm
data}},n)$ where $n$ denotes the codeword length in channel uses,
$\mathcal{R}_{\rm data}$ denotes the target secrecy rate (i.e. packet size in
bits/channel use), $2^{nR_{\rm AB}}$ represents the codebook size, and
$2^{n\mathcal{R}_{\rm data}}$ is the number of different confidential messages
in the codebook. The $2^{nR_{\rm AB}}$ codewords are randomly and uniformly
grouped into $2^{n\mathcal{R}_{\rm data}}$ bins. To transmit a message
$w\in\\{1,...,2^{n\mathcal{R}_{\rm data}}\\}$, Alice randomly selects a
codeword from bin $w$ and transmits it over the wireless channel. Since Alice
does not know the instantaneous CSI of the Alice-Eves links. We assume that
Alice sets a fixed value for its intended secrecy rate, $\mathcal{R}_{\rm
data}$ (which represents the spectral efficiency of a single packet
transmission). The same wiretap coding scheme is applied when Bob shares his
secret keys with Alice. However, Bob knows the channel of his links with Eves.
Hence, Bob only shares the secret keys to Alice when ${\rm A}-{\rm B}$ link is
not in secrecy outage.

###  -B Markov chain analysis

In this appendix, we analyze the Markov chain secret-key queue. Let $\pi_{i}$
denote the probability that the queue has $i$ packets, $\lambda$ is the
arrival rate to the queue, where it presents the probability that there is an
arrived secret packet at the queue and $f$ is the probability that the queue
is serving $K$ packets under certain conditions. To derive the steady-state
probabilities of the states, we write down and solve the balance equations at
all states. From the balance equation around state $0$, we get

$\begin{split}\pi_{K}=\pi_{0}\frac{\lambda}{(1-\lambda)f}\end{split}$ (34)

The state balance equation at states $1$, $2$, $K-1$ and $K$ are given
respectively, by

$\small\begin{split}\pi_{1}\lambda&=\pi_{0}\lambda+\pi_{K}\lambda
f+\pi_{K+1}(1-\lambda)f\\\ \pi_{2}\lambda&=\pi_{1}\lambda+\pi_{K+1}\lambda
f+\pi_{K+2}(1-\lambda)f\\\
\pi_{K-1}\lambda&=\pi_{K-2}\lambda+\pi_{2K-2}\lambda
f+\pi_{2K-1}(1-\lambda)f\\\ \pi_{K}(1-(1-\lambda)\
\overline{f})&=\pi_{K-1}\lambda+\pi_{2K}\lambda
f+\pi_{2K+1}(1-\lambda)f\end{split}$ (35)

where $\overline{f}=1-f$. The balance equation around state $K+1$ is given by

$\begin{split}\pi_{K+1}(1-(1-\lambda)\
\overline{f})=\pi_{K}\lambda\overline{f}+\pi_{2K+1}\lambda
f+\pi_{2K+2}(1-\lambda)f\end{split}$ (36)

This should continue up to state $Q_{\max}-K+1$. At $Q_{\max}-K$, the balance
equation is

$\begin{split}\pi_{Q_{\max}-K}(1-(1-\lambda)\
\overline{f})&=\pi_{Q_{\max}-K-1}\lambda\overline{f}+\pi_{Q_{\max}-1}\lambda
f+\pi_{Q_{\max}}(1-\lambda)f\end{split}$ (37)

At state $Q_{\max}-K+1$, we have

$\begin{split}\pi_{Q_{\max}-K+1}(1-(1-\lambda)\
\overline{f})=\pi_{Q_{\max}-K}\lambda\overline{f}+\pi_{Q_{\max}}\lambda
f\end{split}$ (38)

After that state, the balance equation of state $Q_{\max}-K+1$ is given by

$\begin{split}\pi_{Q_{\max}-K+2}(1-(1-\lambda)\
\overline{f})=\pi_{Q_{\max}-K+1}\lambda\overline{f}\end{split}$ (39)

This equation is valid until the final state. That is, around state
$Q_{\max}-K+3$, we get

$\begin{split}\pi_{Q_{\max}-K+3}=\pi_{Q_{\max}-K+2}\lambda=\pi_{Q_{\max}-K+1}(\lambda\overline{f})^{2}\end{split}$
(40)

From the balance equation around the final state, i.e., state ${Q_{\max}}$, we
get

$\begin{split}\pi_{Q_{\max}-1}=\pi_{Q_{\max}}\frac{f}{\lambda\overline{f}}\end{split}$
(41)

Going one state before that

$\begin{split}\pi_{Q_{\max}-1}(1-(1-\lambda)\
\overline{f})=\pi_{Q_{\max}-2}(\lambda\overline{f})\end{split}$ (42)

Hence,

$\begin{split}\pi_{Q_{\max}}&=\pi_{Q_{\max}-2}\frac{(\lambda\overline{f})^{2}}{f(1-(1-\lambda)\
\overline{f})}\\\
\pi_{Q_{\max}}&=\pi_{Q_{\max}-k}\frac{(\lambda\overline{f})^{k}}{f(1-(1-\lambda)\
\overline{f})^{(k-1)}}\end{split}$ (43)

where $k\geq K$. Letting $\mathbf{P}_{T}$ denote the the Markov chain
transition probabilities matrix, the steady-state distribution vector
$\mathbf{\Pi}=[\pi_{0},\pi_{1},\dots,\pi_{Q_{\max}}]$ for the given
transitions is obtained from calculating higher powers of the matrix [31] as
follows

$\begin{split}\mathbf{\Pi}=\mathbf{\Pi}^{(0)}\mathbf{P}_{T}^{\infty}\end{split}$
(44)

since the Markov chain is irreducible with ergodic (i.e., aperiodic positive-
recurrent) states. All initial distributions, $\mathbf{\Pi}^{(0)}$, will
eventually lead to the same steady-state [31].

Figure 10: Markov chain model for the energy queues. State-self transitions
are omitted from the graph for visual clarity.

###  -C Approximated Markov Chain Analysis

To obtain closed-form expressions of the Markov chain, we use the assumption
that the queue absorbs $K$ packets once it has them. That is, writing the
balance equations at different states based on that assumption, we get

$\small\begin{split}\pi_{K}=\pi_{0}\frac{\lambda}{(1-\lambda)}\end{split}$
(45)

The state balance equations are given by

$\small\begin{split}\pi_{1}&=\pi_{0}+\pi_{K}=\pi_{0}+\pi_{0}\frac{\lambda}{(1-\lambda)}=\frac{\pi_{0}}{(1-\lambda)}\\\
\pi_{k}&=\pi_{k-1},\;\;\;\;\forall k\in\\{2,\dots,K-1\\}\\\
\pi_{K}&=\pi_{K-1}\lambda\end{split}$ (46)

From the last balance equation, we have

$\begin{split}\pi_{K-1}=\pi_{K-2}=\dots=\pi_{1}=\frac{\pi_{K}}{\lambda}\end{split}$
(47)

Using the identity that the sum over the state probabilities is the unity, we
get

$\small\begin{split}&\pi_{0}+\pi_{1}+\pi_{2}+\dots+\pi_{K-1}+\pi_{K}=1\\\
&\pi_{0}+(K-1)\frac{\pi_{K}}{\lambda}+\pi_{K}=1\\\
&\pi_{0}+(\frac{(K-1)}{\lambda}+1)\pi_{0}\frac{\lambda}{(1-\lambda)}=1\end{split}$
(48)

Thus,

$\begin{split}\pi_{0}=\frac{(1-\lambda)}{K},\;\;\pi_{K}&=\frac{\lambda}{K},\;\;\pi_{k}=\frac{1}{K}\forall
k\neq 0,K\end{split}$ (49)

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