journalonly conferenceonly intermed

# Spoofing Detection in the Physical Layer with Graph Neural Networks††thanks:
This research has been funded in part by the Research Council of Norway under
IKTPLUSS grant 311994.††thanks: The code and pre-trained models are available
at https://github.com/uiano/gnn_spoofing_detection.

Tien Ngoc Ha  Department of ICT, University of Agder
Grimstad, Norway
<EMAIL_ADDRESS>Daniel Romero  Department of ICT, University of Agder
Grimstad, Norway
<EMAIL_ADDRESS>
###### Abstract

In a spoofing attack, a malicious actor impersonates a legitimate user to
access or manipulate data without authorization. The vulnerability of
cryptographic security mechanisms to compromised user credentials motivates
spoofing attack detection in the physical layer, which traditionally relied on
channel features, such as the received signal strength (RSS) measured by
spatially distributed receivers or access points. However, existing methods
cannot effectively cope with the dynamic nature of channels, which change over
time as a result of user mobility and other factors. To address this
limitation, this work builds upon the intuition that the temporal pattern of
changes in RSS features can be used to detect the presence of concurrent
transmissions from multiple (possibly changing) locations, which in turn
indicates the existence of an attack. Since a localization-based approach
would require costly data collection and would suffer from low spatial
resolution due to multipath, the proposed algorithm employs a deep neural
network to construct a graph embedding of a sequence of RSS features that
reflects changes in the propagation conditions. A graph neural network then
classifies these embeddings to detect spoofing attacks. The effectiveness and
robustness of the proposed scheme are corroborated by experiments with real-
data.

###### Index Terms:

Graph neural networks, spoofing attack, physical layer security, deep
learning, cybersecurity, wireless networks.

##  I Introduction

The prevalence of wireless communications has engendered a panoply of security
threats, including the unauthorized interception of private data, disruptions
to remote services, and user impersonation. Among these pernicious threats,
spoofing attacks pose a particularly troublesome challenge since, in these
attacks, malevolent actors intercept and manipulate data originally intended
for legitimate users [1, 2, 3, 4]. The detection and mitigation of these
attacks are pivotal for data security. Although cryptographic techniques have
traditionally been employed across various communication layers to fortify
security, the potential access of attackers to the credentials of legitimate
users introduces a serious vulnerability. Consequently, the research community
has increasingly focused on detecting spoofing attacks in the physical layer.

For instance, [5, 6, 7, 8] leverage transmitter hardware imperfections, such
as carrier frequency offset (CFO), in-phase and quadrature (I/Q) offset, and
I/Q imbalance, to verify user identity. Regrettably, these methodologies
necessitate knowledge of the communication protocol and may prove ineffective
in the face of environmental changes, such as fluctuations in temperature [6].
These constraints are somehow mitigated in [9, 10, 11], which rely on angle of
arrival (AoA) and time difference of arrival (TDoA) features, and in [12],
where a neural network is trained using signal-to-noise ratio (SNR) traces.
Nonetheless, these approaches still demand synchronization and/or knowledge of
the communication protocol. In contrast, techniques reliant on received signal
strength (RSS) measurements do not require knowledge of the communication
protocol or signal decoding, thus significantly augmenting their generality
and applicability for detecting spoofing attempts [13, 14, 15, 16, 17]. The
predominant approach in this context involves applying clustering primitives
to RSS measurements collected by multiple receivers, such as the access points
of a WiFi network [13, 18, 19]. By exploiting the dependence of RSS signatures
on the transmitter locations, an attack is detected if transmissions with the
same user identifier are found to originate at different locations.
Consequently, this approach results in false alarms when the channel
conditions change, as for example when a legitimate user moves.

To remedy this limitation, the key realization in this work is that it is
possible to tell spoofing from motion and other effects by analyzing the
temporal changes in RSS features. To illustrate this idea, consider a network
that sequentially receives frames from locations denoted as A, B, C, and D. If
all these locations are distinct, it is natural to ascribe these variations to
the movement of the legitimate user. In contrast, if the received
transmissions alternate between points A and B in a pattern such as A, B, A,
B, A, B, etc., it is more likely that one user is transmitting from location A
and another from location B, which indicates the presence of an attack. To the
best of our knowledge, the work at hand is the first to exploit this kind of
information.

To this end, this paper introduces a spoofing attack detection scheme where a
graph embedding is constructed to capture the pattern of changes in RSS
features over a sequence of frames. Then, a _graph neural network_ (GNN)
classifies such graph embeddings as either corresponding to an attack or to
legitimate user activity, which may include user movement. The graph is
constructed by utilizing a _position-change detector_ (PCD) that determines
whether a given pair of frames was transmitted from different locations. Since
changes in the RSS measurements corresponding to different frames may be
caused either by the movement of the transmitter or by the variability due to
the finite number of samples used in the computation of these measurements,
the PCD is designed as a deep neural network that detects position changes by
implicitly learning the distribution of RSS estimates from signal samples. The
proposed scheme can be readily deployed due to the simplicity of the procedure
for collecting the required data set. Specifically, RSS features must be
collected at different locations but those locations need not be recorded.

The rest of the paper is structured as follows. Sec. II formulates the
problem. Sec. III presents the proposed spoofing detection scheme. Sec. IV
presents an extensive performance evaluation using real data. Finally, Sec. V
concludes the paper.

##  II Problem Formulation

Let ${\color[rgb]{0,0,0}\mathcal{X}}\subset\mathbb{R}^{3}$ comprise the
coordinates of all points in the spatial region of interest, where both
legitimate users and attackers are located. A transmitter at
${\color[rgb]{0,0,0}\bm{x}}\in{\color[rgb]{0,0,0}\mathcal{X}}$, which can be
the legitimate user or an attacker, sends a signal ${\color[rgb]{0,0,0}s}(t)$,
where $t$ denotes time. This signal, modeled as an unknown wide-sense
stationary stochastic process, is received by ${\color[rgb]{0,0,0}N}$
receivers, such as the access points or base stations of a wireless network.
Let
${\color[rgb]{0,0,0}h}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)$
denote the impulse response of the channel between the transmitter and the
${\color[rgb]{0,0,0}n}$th receiver, which is assumed to be time-invariant over
the duration of a frame. The received signal at the ${\color[rgb]{0,0,0}n}$th
receiver is given by

$\displaystyle{\color[rgb]{0,0,0}r}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)={\color[rgb]{0,0,0}h}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)\ast{\color[rgb]{0,0,0}s}(t)+{\color[rgb]{0,0,0}z}_{{\color[rgb]{0,0,0}n}}(t),$
(1)

where $\ast$ denotes convolution and
${\color[rgb]{0,0,0}z}_{{\color[rgb]{0,0,0}n}}(t)$ is additive white Gaussian
noise (AWGN) with variance $\sigma^{2}$ and independent of
${\color[rgb]{0,0,0}s}(t)$. Thus, one can define the _received signal
strength_ (RSS)
${\color[rgb]{0,0,0}f}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}}):={\color[rgb]{0,0,0}\mathbb{E}}{|{\color[rgb]{0,0,0}r}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)|^{2}},$
where ${\color[rgb]{0,0,0}\mathbb{E}}$ denotes expectation.

To estimate the RSS of a frame received by the ${\color[rgb]{0,0,0}n}$th
receiver, consider a set of ${\color[rgb]{0,0,0}K}$ samples
${\color[rgb]{0,0,0}\mathcal{K}}_{{\color[rgb]{0,0,0}n}}:=\\{{\color[rgb]{0,0,0}r}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t+{\color[rgb]{0,0,0}k}{\color[rgb]{0,0,0}T})\\}_{{\color[rgb]{0,0,0}k}=1}^{{\color[rgb]{0,0,0}K}}$,
where ${\color[rgb]{0,0,0}T}$ is the sampling period and $t$ is the time when
the frame begins. The RSS can be estimated as

$\displaystyle{\color[rgb]{0,0,0}\hat{f}}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}}):=\frac{1}{{\color[rgb]{0,0,0}K}}\sum_{{\color[rgb]{0,0,0}k}=1}^{{\color[rgb]{0,0,0}K}}|{\color[rgb]{0,0,0}r}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t+{\color[rgb]{0,0,0}k}{\color[rgb]{0,0,0}T})|^{2}.$
(2)

If
${\color[rgb]{0,0,0}r}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},{\color[rgb]{0,0,0}k}{\color[rgb]{0,0,0}T})$
is ergodic for each ${\color[rgb]{0,0,0}\bm{x}}$, it follows that
${\color[rgb]{0,0,0}\hat{f}}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}})$
converges to
${\color[rgb]{0,0,0}f}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}})$ as
${\color[rgb]{0,0,0}K}\rightarrow\infty$.

For notational convenience, the RSS values estimated by all receivers are
collected into a _feature vector_ , generically represented by
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}({\color[rgb]{0,0,0}\bm{x}}):=[{\color[rgb]{0,0,0}\hat{f}}_{1}({\color[rgb]{0,0,0}\bm{x}}),\ldots,{\color[rgb]{0,0,0}\hat{f}}_{{\color[rgb]{0,0,0}N}}({\color[rgb]{0,0,0}\bm{x}})]^{\top}$.
Note that since ${\color[rgb]{0,0,0}K}$ is finite, measuring the RSS
${\color[rgb]{0,0,0}I}$ times for location ${\color[rgb]{0,0,0}\bm{x}}$ yields
${\color[rgb]{0,0,0}I}$ different estimates of
$\bm{{\color[rgb]{0,0,0}f}}({\color[rgb]{0,0,0}\bm{x}}):=[{\color[rgb]{0,0,0}f}_{1}({\color[rgb]{0,0,0}\bm{x}}),\ldots,{\color[rgb]{0,0,0}f}_{{\color[rgb]{0,0,0}N}}({\color[rgb]{0,0,0}\bm{x}})]^{\top}$.
These estimates will be denoted as
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{({{\color[rgb]{0,0,0}i}})}({\color[rgb]{0,0,0}\bm{x}}),{{\color[rgb]{0,0,0}i}}=1,\ldots,{\color[rgb]{0,0,0}I}$.

To introduce the notation for frame sequences, let
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$ denote
the location of the user that transmits the ${\color[rgb]{0,0,0}j}$th frame at
the moment of transmitting that frame and let
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}:={\color[rgb]{0,0,0}{\hat{\bm{f}}}}({\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]})$.
The feature vectors corresponding to a sequence of ${\color[rgb]{0,0,0}J}$
frames are collected into matrix
${\color[rgb]{0,0,0}{\hat{\bm{F}}}}:=[{\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[1]},\ldots,{\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}J}]}]$.
Out of these ${\color[rgb]{0,0,0}J}$ frames, ${\color[rgb]{0,0,0}J}_{L}$
belong to the legitimate user and ${\color[rgb]{0,0,0}J}_{A}$ to the attacker,
where
${\color[rgb]{0,0,0}J}={\color[rgb]{0,0,0}J}_{L}+{\color[rgb]{0,0,0}J}_{A}$.
The set of indices of the frames belonging to the legitimate user is
represented by
${\color[rgb]{0,0,0}\mathcal{J}_{\text{L}}}\subset\\{1,\ldots,{\color[rgb]{0,0,0}J}_{L}\\}$
whereas the set of indices of the frames belonging to the attacker is
represented by
${\color[rgb]{0,0,0}\mathcal{J}_{\text{A}}}\subset\\{1,\ldots,{\color[rgb]{0,0,0}J}_{A}\\}$.

Given ${\color[rgb]{0,0,0}{\hat{\bm{F}}}}$, the problem is to decide between
the following hypotheses:

$\displaystyle\left\\{\begin{matrix}{\color[rgb]{0,0,0}\mathcal{H}}_{0}:{\color[rgb]{0,0,0}\mathcal{J}_{\text{A}}}=\emptyset\\\
{\color[rgb]{0,0,0}\mathcal{H}}_{1}:{\color[rgb]{0,0,0}\mathcal{J}_{\text{A}}}\neq\emptyset.\end{matrix}\right.$
(3)

To this end, a dataset comprising the feature vectors
${\color[rgb]{0,0,0}\mathcal{D}}:=\\{{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{({{\color[rgb]{0,0,0}i}})}({\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}}),{{\color[rgb]{0,0,0}m}}=1,\ldots,{\color[rgb]{0,0,0}M},{{\color[rgb]{0,0,0}i}}=1,\ldots,{\color[rgb]{0,0,0}I}\\}$
is given, where ${\color[rgb]{0,0,0}M}$ is the number of distinct measurement
locations, i.e. ${\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}}\
\neq{\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}^{\prime}}~{}\forall{{\color[rgb]{0,0,0}m}}\neq{{\color[rgb]{0,0,0}m}}^{\prime}$.

##  III Spoofing Detection from RSS Features

Figure 1: RSS measurements along a line vs. their y-coordinate. It is observed
that small variations in the y-coordinate often result in larger RSS changes
than large variations in the y-coordinate. For example, the difference between
$y=24$ and $y=29$ is around 24 dB, whereas the difference between $y=16$ and
$y=40$ is less than 5 dB. This is caused mainly by multipath and suggests that
accurately estimating the position from RSS measurements is not generally
possible.

Evidently, if the time between consecutive frames in
${\color[rgb]{0,0,0}{\hat{\bm{F}}}}$ is too long, then the vectors
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$
may originate at highly distant locations due to user movement, even in the
absence of attacks. As a result,
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$
may be highly different from
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}-1]}$
and
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}+1]}$.
Since this would also be the case in the presence of an attack, solving
problem (3) becomes challenging as the distributions of
${\color[rgb]{0,0,0}{\hat{\bm{F}}}}$ under both hypotheses are highly similar.
Therefore, it becomes imperative to introduce the following assumption:

_Assumption 1: The frame rate is high relative to the speed of the users._

In other words, if
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$
and
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}+1]}$
correspond to the same user, they will be reasonably similar.

In view of Assumption 1, one could consider a strategy to tackle problem (3)
where the locations
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$ are
first estimated based on
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$,
${\color[rgb]{0,0,0}j}=1,\ldots,{\color[rgb]{0,0,0}J}$, and an attack is
declared if multiple transmissions are concurrently received from distant
(possibly moving) users. However, this approach is not viable first because
the data given in the problem formulation of Sec. II does not allow a
reasonably accurate localization of the transmitters. Indeed, if the locations
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$
associated with the vectors in ${\color[rgb]{0,0,0}\mathcal{D}}$ were given,
one could attempt to estimate the locations associated with the frames in
${\color[rgb]{0,0,0}{\hat{\bm{F}}}}$, for instance via fingerprinting-based
localization [20, 21]. However, the error of such approaches is typically in
the order of 10 m in indoor environments (see e.g. [20]), which would hinder
detecting attacks where the attacker is relatively near the legitimate user.
Fig. 1 illustrates why this is the case. Besides, collecting a data set where
the positions of the measurement locations need to be recorded is highly
costly since it would generally require the deployment of an auxiliary
localization system, the use a mobile robot, or to manually measure the
spatial coordinates of all measurement locations.

For this reason, the proposed scheme does not attempt to estimate the
transmitter locations. Instead, it exploits the pattern of dissimilarities
between the feature vectors
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$.
This is accomplished in two steps: First, each pair of vectors
$({\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]},{\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]})$
is compared as described in Sec. III-A. Given these comparisons, a decision is
made on the presence of an attack based on a graph embedding, as described in
Sec. III-B.

###  III-A Position-change Detection

This section presents a PCD, which is a detector that determines whether two
given frames where transmitted from the same location. Specifically, given two
feature vectors
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$
and
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$
respectively corresponding to (possibly equal) locations
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$ and
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$,
the goal is to distinguish between the following hypotheses:

$\displaystyle\left\\{\begin{matrix}{\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{0}^{\text{PCD}}}:{\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}={\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}\\\
{\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{1}^{\text{PCD}}}:{\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}\neq{\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}.\end{matrix}\right.$
(4)

A PCD is a function that maps a pair of feature vectors to a hypothesis, i.e.,
${\color[rgb]{0,0,0}d^{\text{PCD}}}:\mathbb{R}^{{\color[rgb]{0,0,0}N}}\times\mathbb{R}^{{\color[rgb]{0,0,0}N}}\rightarrow\\{{\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{0}^{\text{PCD}}},{\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{1}^{\text{PCD}}}\\}$.

To properly address (4), it is useful to consider the components behind the
dissimilarity between
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$
and
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$.
First, two feature vectors
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$
and
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$
are naturally different because ${\color[rgb]{0,0,0}K}$ is finite, even when
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}={\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$.
Second,
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$
and
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$
will be different when
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}\neq{\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$
because of the different propagation phenomena undergone by the signals
propagating from either location to the receivers. This includes effects such
as path loss, shadowing, and fading. The latter is caused by multipath and
dominates in indoor environments; see for example Fig. 1. The complexity of
these phenomena calls for a PCD that learns to solve (4) in a data-driven
fashion. To this end, in this work, ${\color[rgb]{0,0,0}d^{\text{PCD}}}$ is
implemented using a DNN, as described next.

####  III-A1 Architecture

Following standard practice, the detector is designed to decide
${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{1}^{\text{PCD}}}$ when a
detection statistic
${\color[rgb]{0,0,0}T}^{\text{PCD}}:\mathbb{R}^{{\color[rgb]{0,0,0}N}}\times\mathbb{R}^{{\color[rgb]{0,0,0}N}}\rightarrow\mathbb{R}$
exceeds a predefined threshold, and
${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{0}^{\text{PCD}}}$
otherwise. In this way, the problem of designing
${\color[rgb]{0,0,0}d^{\text{PCD}}}$ becomes that of designing a function
${\color[rgb]{0,0,0}T}^{\text{PCD}}$.

In principle, this function could be directly implemented as a DNN. However,
such a simple approach would result in a non-commutative
${\color[rgb]{0,0,0}T}^{\text{PCD}}$, that is,
${\color[rgb]{0,0,0}T}^{\text{PCD}}({\color[rgb]{0,0,0}{\hat{\bm{f}}}},{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime})$
will generally differ from
${\color[rgb]{0,0,0}T}^{\text{PCD}}({\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime},{\color[rgb]{0,0,0}{\hat{\bm{f}}}})$,
which is clearly undesirable. To remedy this issue, a symmetrization technique
will be adopted. Specifically, ${\color[rgb]{0,0,0}T}^{\text{PCD}}$ will be
implemented based on an auxiliary function
${\color[rgb]{0,0,0}\tilde{T}}^{\text{PCD}}$ by setting
${\color[rgb]{0,0,0}T}^{\text{PCD}}({\color[rgb]{0,0,0}{\hat{\bm{f}}}},{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime}):=({\color[rgb]{0,0,0}\tilde{T}}^{\text{PCD}}({\color[rgb]{0,0,0}{\hat{\bm{f}}}},{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime})+{\color[rgb]{0,0,0}\tilde{T}}^{\text{PCD}}({\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime},{\color[rgb]{0,0,0}{\hat{\bm{f}}}}))/2$.
Observe that this implies that
${\color[rgb]{0,0,0}T}^{\text{PCD}}({\color[rgb]{0,0,0}{\hat{\bm{f}}}},{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime})={\color[rgb]{0,0,0}T}^{\text{PCD}}({\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime},{\color[rgb]{0,0,0}{\hat{\bm{f}}}})$
regardless of ${\color[rgb]{0,0,0}\tilde{T}}^{\text{PCD}}$. Thereby,
${\color[rgb]{0,0,0}\tilde{T}}^{\text{PCD}}$ can be safely implemented as a
DNN.

The architecture of the subnetwork
${\color[rgb]{0,0,0}\tilde{T}}^{\text{PCD}}$ is detailed next. Since
${\color[rgb]{0,0,0}\tilde{T}}^{\text{PCD}}$ is concerned with
dissimilarities, it is natural to include an initial non-trainable layer that
yields
$10\log_{10}[{\color[rgb]{0,0,0}{\hat{\bm{f}}}},{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime},{\color[rgb]{0,0,0}{\hat{\bm{f}}}}-{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime}]$
when the input to the network is
$[{\color[rgb]{0,0,0}{\hat{\bm{f}}}},{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{\prime}]$.
This facilitates learning from moderate-sized datasets. This layer is followed
by three hidden fully-connected layers with $512$ neurons and leaky ReLU
activations [22]. The output layer contains a single neuron with a linear
activation.

####  III-A2 Data set

To train ${\color[rgb]{0,0,0}T}^{\text{PCD}}$, a dataset comprising pairs of
vectors from ${\color[rgb]{0,0,0}\mathcal{D}}$ is constructed. The feature
vectors in ${\color[rgb]{0,0,0}P}$ of these pairs correspond to the same
transmitter location. In the remaining ${\color[rgb]{0,0,0}P}$ pairs, they
correspond to different transmitter locations.

Specifically, the pairs of the first kind are generated for
${{\color[rgb]{0,0,0}p}}=1,\ldots,{\color[rgb]{0,0,0}P}$ by first drawing
${{\color[rgb]{0,0,0}m}}_{{{\color[rgb]{0,0,0}p}}}$ uniformly at random from
the set $\\{1,\ldots,{\color[rgb]{0,0,0}M}\\}$. Then,
${{\color[rgb]{0,0,0}i}}_{{{\color[rgb]{0,0,0}p}}}$ and
${{\color[rgb]{0,0,0}i}}_{{{\color[rgb]{0,0,0}p}}}^{\prime}$ are drawn
uniformly at random without replacement from
$\\{1,\ldots,{\color[rgb]{0,0,0}I}\\}$. This process results in the set
${\color[rgb]{0,0,0}\mathcal{D}_{\text{s}}}:=\\{({\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{({{\color[rgb]{0,0,0}i}}_{{{\color[rgb]{0,0,0}p}}})}({\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}_{{{\color[rgb]{0,0,0}p}}}}),{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{({{\color[rgb]{0,0,0}i}}^{\prime}_{{{\color[rgb]{0,0,0}p}}})}({\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}_{{{\color[rgb]{0,0,0}p}}}})),~{}{{\color[rgb]{0,0,0}p}}=1,\ldots,{\color[rgb]{0,0,0}P}\\}\subset\mathbb{R}^{{\color[rgb]{0,0,0}N}}\times\mathbb{R}^{{\color[rgb]{0,0,0}N}}$.
To generate the pairs of the second kind, draw
${{\color[rgb]{0,0,0}m}}_{{{\color[rgb]{0,0,0}p}}}$ and
${{\color[rgb]{0,0,0}m}}_{{{\color[rgb]{0,0,0}p}}}^{\prime}$ uniformly at
random without replacement from the set $\\{1,\ldots,{\color[rgb]{0,0,0}M}\\}$
for ${{\color[rgb]{0,0,0}p}}=1,\ldots,{\color[rgb]{0,0,0}P}$. Drawing
${{\color[rgb]{0,0,0}i}}_{{{\color[rgb]{0,0,0}p}}}$ and
${{\color[rgb]{0,0,0}i}}_{{{\color[rgb]{0,0,0}p}}}^{\prime}$ as before yields
${\color[rgb]{0,0,0}\mathcal{D}_{\text{d}}}:=\\{({\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{({{\color[rgb]{0,0,0}i}}_{{{\color[rgb]{0,0,0}p}}})}({\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}_{{{\color[rgb]{0,0,0}p}}}}),{\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{({{\color[rgb]{0,0,0}i}}^{\prime}_{{{\color[rgb]{0,0,0}p}}})}({\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}_{{{\color[rgb]{0,0,0}p}}^{\prime}}})),~{}{{\color[rgb]{0,0,0}p}}=1,\ldots,{\color[rgb]{0,0,0}P}\\}\subset\mathbb{R}^{{\color[rgb]{0,0,0}N}}\times\mathbb{R}^{{\color[rgb]{0,0,0}N}}$.
The DNN can then be trained on the dataset
${\color[rgb]{0,0,0}\mathcal{D}_{\text{s}}}\cup{\color[rgb]{0,0,0}\mathcal{D}_{\text{d}}}$.

Recall from Sec. II that obtaining ${\color[rgb]{0,0,0}\mathcal{D}}$ involves
collecting the ${\color[rgb]{0,0,0}I}$ RSS estimates
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{({{\color[rgb]{0,0,0}i}})}({\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}})$,
${{\color[rgb]{0,0,0}i}}=1,\ldots,{\color[rgb]{0,0,0}I}$, for each of the
${\color[rgb]{0,0,0}M}$ locations
${\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}},~{}{{\color[rgb]{0,0,0}m}}=1,\ldots,{\color[rgb]{0,0,0}M}$.
A simpler approach may be to collect a single estimate with a large
${\color[rgb]{0,0,0}K}$ so that it approximately equals
$\bm{{\color[rgb]{0,0,0}f}}({\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}})$
and then generate the
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}^{({{\color[rgb]{0,0,0}i}})}({\color[rgb]{0,0,0}\bm{x}}_{{{\color[rgb]{0,0,0}m}}})$
synthetically. The procedure is described next for the case where
${\color[rgb]{0,0,0}r}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)$
is approximately Gaussian distributed, which would be the case e.g. if
${\color[rgb]{0,0,0}s}(t)$ is an orthogonal frequency division multiplexing
(OFDM) signal; see e.g. [23].

To this end, express
${\color[rgb]{0,0,0}r}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)$
as
${\color[rgb]{0,0,0}r}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)={\color[rgb]{0,0,0}f}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}}){\color[rgb]{0,0,0}\epsilon}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)$,
where
${\color[rgb]{0,0,0}f}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}})$ is
the true RSS and
${\color[rgb]{0,0,0}\epsilon}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)$
is a circularly symmetric zero-mean Gaussian random variable with unit
variance uncorrelated over $t$. Then, (2) becomes

$\displaystyle{\color[rgb]{0,0,0}\hat{f}}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)$
$\displaystyle=\frac{|{\color[rgb]{0,0,0}f}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}})|^{2}}{{\color[rgb]{0,0,0}K}}\sum_{{\color[rgb]{0,0,0}k}=1}^{{\color[rgb]{0,0,0}K}}|{\color[rgb]{0,0,0}\epsilon}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t+{\color[rgb]{0,0,0}k}T)|^{2}$
(5a)
$\displaystyle=\frac{|{\color[rgb]{0,0,0}f}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}})|^{2}}{2{\color[rgb]{0,0,0}K}}\sum_{{\color[rgb]{0,0,0}k}=1}^{{\color[rgb]{0,0,0}K}}\left(\left[\sqrt{2}\text{Re}\\{{\color[rgb]{0,0,0}\epsilon}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t+{\color[rgb]{0,0,0}k}T)\\}\right]\right.^{2}$
$\displaystyle+\left[\sqrt{2}\text{Im}\\{{\color[rgb]{0,0,0}\epsilon}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t+{\color[rgb]{0,0,0}k}T)\\}\right]^{2}\Big{)}.$
(5b)

It follows that
$2{\color[rgb]{0,0,0}K}{\color[rgb]{0,0,0}\hat{f}}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)/|{\color[rgb]{0,0,0}f}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}})|^{2}$
is a $\chi^{2}$ random variable with $2{\color[rgb]{0,0,0}K}$ degrees of
freedom. Thus, ${\color[rgb]{0,0,0}I}$ estimates
${\color[rgb]{0,0,0}\hat{f}}^{({{\color[rgb]{0,0,0}i}})}_{{\color[rgb]{0,0,0}n}}({\color[rgb]{0,0,0}\bm{x}},t)$,
${{\color[rgb]{0,0,0}i}}=1,\ldots,{\color[rgb]{0,0,0}I}$, can be obtained by
generating ${\color[rgb]{0,0,0}I}$ realizations of such a $\chi^{2}$ random
variable.

####  III-A3 Training

The DNN is trained using a binary cross-entropy loss function. Within the
dataset, a subset comprising ${\color[rgb]{0,0,0}M}{{}_{\text{val}}}$
validation points is reserved for validation, while the remainder
${\color[rgb]{0,0,0}M}{{}_{\text{tr}}}={\color[rgb]{0,0,0}M}-{\color[rgb]{0,0,0}M}{{}_{\text{val}}}$
are used for training.

###  III-B Graph Neural Network based Spoofing Detection

Figure 2: Examples of region sequences without an attack ((a)) and with an
attack ((b)-(d)).

The decisions of the PCD for all pairs of frames will be used next to detect
spoofing attacks. To intuitively understand why this is possible, recall from
Sec. III-A that the PCD decides
${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{1}^{\text{PCD}}}$ when
the dissimilarity between the given feature vectors
$\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$ and
$\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$
owes to the difference between the propagation phenomena experienced at
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$ and
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$.
In other words, if
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$ and
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$
are so close that both points see similar propagation conditions to all
receivers, the PCD decides
${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{0}^{\text{PCD}}}$. For
didactical purposes, it is useful to split the space into propagation
_regions_ and assume that
${\color[rgb]{0,0,0}d^{\text{PCD}}}(\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]},\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]})={\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{0}^{\text{PCD}}}$
if ${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$ and
${\color[rgb]{0,0,0}\bm{x}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]}$
belong to the same region and
${\color[rgb]{0,0,0}d^{\text{PCD}}}(\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]},\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]})={\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{1}^{\text{PCD}}}$
otherwise. Using the decisions of the PCD, one can therefore assign each frame
in the given sequence to a region.

This assignment is illustrated in Fig. 2, which sheds light into why it is
possible to solve (3) using the decisions of the PCD. Fig. 2a shows an example
where all frames are generated by a single moving user. Due to Assumption 1,
groups of consecutive frames are declared by the PCD to belong to the same
region. In turn, Fig. 2b depicts the case where two transmissions are
concurrently taking place from different regions, which indicates the presence
of an attack. Finally, Fig. 2c and Fig. 2d respectively correspond to the case
where one or both of the concurrently transmitting users move.

It is important to note that the above considerations are provided to develop
intuition, but in practice do not hold exactly. In particular, the decisions
of the PCD will not generally be transitive, that is, it may hold that
${\color[rgb]{0,0,0}d^{\text{PCD}}}(\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]},\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]})={\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{0}^{\text{PCD}}}$
and
${\color[rgb]{0,0,0}d^{\text{PCD}}}(\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]},\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime\prime}]})={\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{0}^{\text{PCD}}}$
but
${\color[rgb]{0,0,0}d^{\text{PCD}}}(\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]},\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime\prime}]})={\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{1}^{\text{PCD}}}$.
However, this may approximately hold. For this reason, it is useful to
construct a graph ${\color[rgb]{0,0,0}\mathcal{G}}$ where the
${\color[rgb]{0,0,0}j}$-th node corresponds to
$\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]}$ and
there is an edge between nodes ${\color[rgb]{0,0,0}j}$ and
${\color[rgb]{0,0,0}j}^{\prime}$ if
${\color[rgb]{0,0,0}d^{\text{PCD}}}(\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}]},\bm{{\color[rgb]{0,0,0}f}}{\color[rgb]{0,0,0}[{\color[rgb]{0,0,0}j}^{\prime}]})={\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{0}^{\text{PCD}}}$.
Clearly, since ${\color[rgb]{0,0,0}d^{\text{PCD}}}$ is commutative (cf. Sec.
III-A1), this graph is undirected.

Clearly, if the space could be split into propagation regions, as discussed
earlier, then this graph could be partitioned into one component per region
and the presence of an attack would be characterized by an alternating pattern
between components as in Fig. 2b-2d. However, since this is not exactly the
case, it makes sense to train a GNN to detect attacks based on
${\color[rgb]{0,0,0}\mathcal{G}}$ in a data-driven fashion.

####  III-B1 Architecture

A GNN [24] exploits the relation between node features and the graph topology
by performing a sequence of message-passing steps or _layers_ , where the
features associated with a node at layer ${\color[rgb]{0,0,0}l}$ depend on the
features of that node and the neighboring nodes at layer
${\color[rgb]{0,0,0}l}-1$. Specifically, if
$\bm{\phi}_{{\color[rgb]{0,0,0}\nu}}^{({\color[rgb]{0,0,0}l})}$ represents the
features of node ${\color[rgb]{0,0,0}\nu}$ at layer ${\color[rgb]{0,0,0}l}$,
then

$\displaystyle\bm{\phi}_{{\color[rgb]{0,0,0}\nu}}^{({\color[rgb]{0,0,0}l})}=G_{1}^{({\color[rgb]{0,0,0}l})}\left(\bm{\phi}_{{\color[rgb]{0,0,0}\nu}}^{({\color[rgb]{0,0,0}l}-1)},\bigoplus_{{\color[rgb]{0,0,0}\nu}^{\prime}\in{\color[rgb]{0,0,0}\mathcal{N}}_{{\color[rgb]{0,0,0}\nu}}}G_{2}^{({\color[rgb]{0,0,0}l})}(\bm{\phi}_{{\color[rgb]{0,0,0}\nu}}^{({\color[rgb]{0,0,0}l}-1)},\bm{\phi}_{{\color[rgb]{0,0,0}\nu}^{\prime}}^{({\color[rgb]{0,0,0}l}-1)})\right),$
(6)

where $G_{1}^{({\color[rgb]{0,0,0}l})}$ and $G_{2}^{({\color[rgb]{0,0,0}l})}$
are conventional DNNs,
${\color[rgb]{0,0,0}\mathcal{N}}_{{\color[rgb]{0,0,0}\nu}}$ contains the set
of neighbors of node ${\color[rgb]{0,0,0}\nu}$, and $\bigoplus$ is an
aggregation operator such as a summation or maximum operator. The output of
the GNN can be computed by another aggregation operator applied to the
concatenation of the vectors
$\bm{\phi}_{{\color[rgb]{0,0,0}\nu}}^{({\color[rgb]{0,0,0}L})}$ for all
${\color[rgb]{0,0,0}\nu}$, where ${\color[rgb]{0,0,0}L}$ is the number of
layers.

For the problem at hand, a test statistic is obtained with a GNN and then
compared to a threshold to decide between
${\color[rgb]{0,0,0}\mathcal{H}}_{0}$ and
${\color[rgb]{0,0,0}\mathcal{H}}_{1}$. In the adopted architecture,
${\color[rgb]{0,0,0}L}=3$ layers and functions
$G_{1}^{({\color[rgb]{0,0,0}l})}$ and $G_{2}^{({\color[rgb]{0,0,0}l})}$ are
implemented as single-layer fully-connected DNNs with $64$ output neurons and
ReLU activations. The operator $\bigoplus$ is a summation whereas the output
of the GNN is obtained by averaging the features of all nodes at the last
layer and applying a trainable affine transformation. Since the order of the
nodes is relevant (cf. Fig. 2), the features in the first layer are set so
that $\bm{\phi}_{{\color[rgb]{0,0,0}\nu}}^{(1)}$ equals the index of node
${\color[rgb]{0,0,0}\nu}$.

####  III-B2 Dataset

To train the GNN, realizations of ${\color[rgb]{0,0,0}\mathcal{G}}$ must be
generated under both ${\color[rgb]{0,0,0}\mathcal{H}}_{0}$ and
${\color[rgb]{0,0,0}\mathcal{H}}_{1}$. This involves generating frame
sequences ${\color[rgb]{0,0,0}{\hat{\bm{F}}}}$ under both hypotheses. Under
${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{0}}$, the trajectory
${\color[rgb]{0,0,0}\bm{x}}(t)$ of the (single) user is generated as follows.
First, obtain the time duration of the frame sequence, given by
${\color[rgb]{0,0,0}J}{\color[rgb]{0,0,0}R}_{\text{J}}$, where
${\color[rgb]{0,0,0}R}_{\text{J}}$ is the number of frames per second. The
length $\Delta{\color[rgb]{0,0,0}\bm{x}}$ of the trajectory is therefore
$\Delta{\color[rgb]{0,0,0}\bm{x}}={\color[rgb]{0,0,0}J}{\color[rgb]{0,0,0}R}_{\text{J}}{\color[rgb]{0,0,0}v}$,
where ${\color[rgb]{0,0,0}v}$ is the speed of the user. Then, randomly draw a
straight line segment of length $\Delta{\color[rgb]{0,0,0}\bm{x}}$ in
${\color[rgb]{0,0,0}\mathcal{R}}$. The trajectory is therefore
${\color[rgb]{0,0,0}\bm{x}}(t)={\color[rgb]{0,0,0}{\tilde{\bm{x}}}}+{\color[rgb]{0,0,0}\bm{d}}{\color[rgb]{0,0,0}v}t$,
where ${\color[rgb]{0,0,0}{\tilde{\bm{x}}}}$ is the starting point and
${\color[rgb]{0,0,0}\bm{d}}$ is the unit direction vector of the line. For
each ${\color[rgb]{0,0,0}j}$,
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}[{\color[rgb]{0,0,0}j}]$ is obtained by
randomly selecting one of the vectors in ${\color[rgb]{0,0,0}\mathcal{D}}$
that correspond to the location that lies closest to
${\color[rgb]{0,0,0}\bm{x}}({\color[rgb]{0,0,0}j}/{\color[rgb]{0,0,0}R}_{\text{J}})$.

Under ${\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\mathcal{H}}_{1}}$, the frame
sequences of both users are generated following the above procedure. Then, the
frame sequence
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}_{1}[0],\ldots,{\color[rgb]{0,0,0}{\hat{\bm{f}}}}_{1}[{\color[rgb]{0,0,0}J}-1]$
of user-1 is merged with the frame sequence
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}_{2}[0],\ldots,{\color[rgb]{0,0,0}{\hat{\bm{f}}}}_{2}[{\color[rgb]{0,0,0}J}-1]$
of user-2 into a sequence
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}[0],\ldots,{\color[rgb]{0,0,0}{\hat{\bm{f}}}}[{\color[rgb]{0,0,0}J}-1]$
where either
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}[{\color[rgb]{0,0,0}j}]={\color[rgb]{0,0,0}{\hat{\bm{f}}}}_{1}[{\color[rgb]{0,0,0}j}]$
or
${\color[rgb]{0,0,0}{\hat{\bm{f}}}}[{\color[rgb]{0,0,0}j}]={\color[rgb]{0,0,0}{\hat{\bm{f}}}}_{2}[{\color[rgb]{0,0,0}j}]$,
both with probability 1/2 and independently along ${\color[rgb]{0,0,0}j}$.

####  III-B3 Training

The GNN model is trained using a binary cross-entropy loss function.

##  IV Performance Evaluation

This section assesses the performance of the proposed scheme using the dataset
from [20], which contains RSS measurements of $992$ WiFi access points at
$4846$ locations across 4 floors. To ensure a sufficient spatial density, only
the measurements collected at ${\color[rgb]{0,0,0}M}=648$ locations on the
first floor are used. $20\%$ of them are reserved for testing. Since each
measurement location lies out of the range of most of the access points, the
${\color[rgb]{0,0,0}N}=5$ access points that are measured at the greatest
number of the selected locations are considered. The procedure described in
Sec. III-A2 is then used to generate ${\color[rgb]{0,0,0}I}=1000$ feature
vectors for each location. A link to the code is provided on the first page.

The proposed algorithm, referred to as _GNN-based Spoofing Detection_ (GSD),
is compared against four benchmarks which, along the lines of [13, 18, 19],
rely on clustering the feature vectors. This is intuitive as the frames
transmitted from the same location will tend to be clustered together. The
number of clusters is then used as a test statistic and the threshold is
obtained to attain a target probability of false alarm ($P_{\text{FA}}$). The
considered clustering algorithms include _density based spatial clustering of
applications with noise_ (DBSCAN) [25], _hierarchical DBSCAN_ (HDBSCAN) [26],
_ordering points to identify the clustering structure_ (OPTICS) [27], and
_balanced iterative reducing and clustering using hierarchies_ (BIRCH) [28].

Fig. 3 depicts the receiver operating characteristic (ROC) curves [29, Ch. 3]
of GSD and the benchmarks. It is seen that GSD results in a significantly
higher probability of detection ($P_{\text{D}}$) for each $P_{\text{FA}}$.

Figure 3: ROC curves of the proposed algorithm and the benchmarks (10
frames/s, ${\color[rgb]{0,0,0}J}=30$, ${\color[rgb]{0,0,0}K}=150$,
${\color[rgb]{0,0,0}N}=5$).

Fig. 4 analyzes the influence of the speed of the users in the $P_{\text{D}}$
for a given $P_{\text{FA}}=0.1$ for the compared algorithms. Interestingly,
speed seems to positively impact the $P_{\text{D}}$ of all algorithms,
especially those based on clustering. This is because the the number of
clusters per user increases with the speed and, therefore, the test statistic
will tend to be more different between hypotheses.

Figure 4: Pd vs speed of the proposed algorithm and the benchmarks for a fixed
probability of false alarm $P_{\text{FA}}$ (10 frames/s,
${\color[rgb]{0,0,0}J}=30$, ${\color[rgb]{0,0,0}K}=150$,
${\color[rgb]{0,0,0}N}=5$, $P_{\text{FA}}=0.1$).

Fig. 5 investigates the impact of ${\color[rgb]{0,0,0}J}$ on the detection
performance. As expected, $P_{\text{D}}$ tends to increase with
${\color[rgb]{0,0,0}J}$. However, a wiggling effect is observed for the
benchmarks. This does not vanish even if the number of Monte Carlo iterations
is increased. The cause is the discrete nature of the test statistic of the
benchmarks.

Figure 5: Pd vs number of frames of the proposed algorithm and the benchmarks
for a fixed probability of false alarm $P_{\text{FA}}$ (10 frames/s,
${\color[rgb]{0,0,0}K}=150$, ${\color[rgb]{0,0,0}N}=5$, $P_{\text{FA}}=0.1$).

Finally, Fig. 6 analyzes how $P_{\text{D}}$ evolves as a function of
${\color[rgb]{0,0,0}K}$. It is remarkable that GSD attains a very large
$P_{\text{D}}$ even for a small ${\color[rgb]{0,0,0}K}$, which suggests that
the PCD successfully learned to distinguish the two sources of variability in
the feature vectors described in Sec. III-A.

Figure 6: $P_{\text{D}}$ vs number of frames of the proposed algorithm and the
benchmarks for a fixed probability of false alarm $P_{\text{FA}}$ (10
frames/s, ${\color[rgb]{0,0,0}J}=30$, ${\color[rgb]{0,0,0}N}=5$,
$P_{\text{FA}}=0.1$).

##  V Conclusion

This work considered the problem of detecting spoofing attacks in the physical
layer, which is motivated by the vulnerability of cryptographic techniques
when the credentials of the legitimate user are compromised. Unfortunately,
prior schemes based on RSS measurements raise false alarms in the presence of
channel changes or user movement. To remedy this limitation, this work
introduced a deep learning detector robust to these effects. Since
localization-based approaches would suffer from a low spatial resolution due
to multipath effects, a position change detector based on a deep neural
network is used to build a graph embedding of the RSS features. The temporal
pattern of changes in the propagation conditions captured by this embedding is
then learned by a GNN, which then decides on the presence of a spoofing
attack. Empirical evaluation with real-world data showcases the effectiveness
of this scheme as well as its robustness to the mobility of the user and
attacker.

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# Pair Production in time-dependent Electric field at Finite times 111Abstract
of presentation on $30^{th}$ September 2022 in QED Laser Plasmas International
Workshop $26-30$ September 2022 at Max Planck Institute for the Physics of
Complex Systems, Dresden,Germany

Deepak222Corresponding author.
E-mail address<EMAIL_ADDRESS>(Deepak). Manoranjan P. Singh  Theory and
Simulations Lab, Raja Ramanna Centre for Advanced Technology, Indore-452013,
India  Homi Bhaba National Institute, Training School Complex, Anushakti
Nagar, Mumbai 400094, India

##  I Introduction

A strong classical electromagnetic background can lead to vacuum instability
and produce particle-antiparticle pairs. This process of particle creation
from the quantum vacuum was first studied in 1951 by Schwinger under a
constant electric field, and this phenomenon is known as the Schwinger effect
PhysRev.82.664 .This particle creation paradigm has crucial importance for
nonequilibrium processes in heavy-ion collisions24 ; 25 ; 26  as well as
astrophysical phenomena27 and the search for nonlinear and nonperturbative
effects in ultraintense laser systems12 ; 13 ; 14 .
Particle production is the process of evolving a quantum system from an
initial equilibrium configuration to a new final equilibrium configuration via
an intermediate non-equilibrium evolution caused by a strong field background.
Such strong field QED pair production process has been investigated using
scattering calculations Harvey:2009ry , exact solutions Dunne:1998ni , semi-
classical techniques Kim:2000un ; DiPiazza:2004lsj , Monte Carlo simulations
Gies:2005bz  and quantum kinetic equations Alkofer:2001ik ; Blaschke:2005hs
Quantitative description of particle production at all times in time-dependent
electromagnetic field is not possible due to the absence of unique separation
into positive and negative energy states at intermediate times and these
positive and negative states are well-defined only at asymptotically early and
late times where the field vanishes. A common approach is to define particle
numbers in terms of an adiabatic basis using Bogoliubov transformation 2 ; 3 ;
4 ; 5 . In the adiabatic basis, we examine the problem of pair production in a
time-varying spatially uniform Sauter electric field which has been studied by
various authors 6 ; 44 . who calculated the number of particles created at the
asymptotic time but the problem of particle production at the finite time is
not studied for Sauter-Pulse electric field. We looking for the evolution of
the quantum system at some initial time $t_{0}$ in the vacuum state but now
what will be the properties of the quantum system at finite time $t$ ? What
happens to the system properties at that non-asymptotic time? By finding the
exact analytic solution of the mode function, which allows us to study the
behaviour of the quantum vacuum during pair production at any instant of time
for time-dependent Sauter pulsed electric fields in spinor quantum
electrodynamics (QED). We use natural units and set $\hslash=c=m=1$ and
express all variables in terms of the electron mass unit.

##  II Theory

Consider the creation of the electron-positron pair from vacuum by a linearly
polarized time-dependent spatially uniform electric field along the $z$-axis
which is characterized by the four-vector potential $A_{\mu}(t)=(0,0,0,A(t))$
with $E(t)=-\dot{A}(t).$ In our calculations, we choose the Sauter field in
the direction of $z-$axis, which is given by

$E(t)=-\dot{A}(t)=E_{0}sech^{2}(t/\tau)$ (1)

and corresponding vector-potential

$A(t)=-E_{0}\tau tanh(t/\tau).$ (2)

where $E_{0}$ amplitude of the electric field and $\tau$ pulse duration of the
electric field. The one-particle momentum distribution function
$f(\textbf{p},t)$ is an important quantity in the description of the particle
production process in the time-dependent electric field. As it was
demonstrated in 41 ; 42 , f(p,t) can be written in terms of the mode solution
as

$f(\textbf{p},t)=\frac{{(\omega+\Pi)|\dot{\Psi}^{(\pm)}\pm
i\omega\Psi^{(\pm)}|^{2}}}{2\omega}$ (3)

where the mode equation 40 ; 43

$\ddot{\Psi}^{(\pm)}+[\omega^{2}(t)\pm ie\dot{A}(t)]\Psi^{(\pm)}=0$ (4)

with

$\omega(t)^{2}=\Pi^{2}+\epsilon_{\perp}^{2},\quad\Pi=p_{z}+eA(t),\quad\epsilon_{\perp}^{2}=\textbf{p}_{\perp}^{2}+m^{2}.$
(5)

For Sauter-Field, we are able to exactly solve the mode equation 4, which
becomes the hyper-geometric differential equation and its solution is given in
terms of hypergeometric functions. Using this solution, we calculate the one-
particle distribution function in terms of hypergeometric functions. The time
evolution of the particle distribution function $f(\textbf{p},t)$ in momentum
space is studied for $E_{0}=0.2$ and $\tau=10.$

Figure 1: Time evolution of quasi-particle distribution function in the
longitudinal momentum space $f(p_{\parallel},t)$ at different times. The
transverse momentum is considered to zero, and all the units are taken in the
electron mass units.The field parameters are $E_{0}=0.2E_{c}$ and $\tau=10.$

##  III Result

In this section, we discuss about both the longitudinal and transverse
momentum spectrum of the created particle using the solution of the mode
equation in 3 for $E_{0}=0.2$ and $\tau=10$.

###  III.1 Longitudinal momentum spectrum

In Figure 1, we show how the particle-creation proceeds. The produced electron
is accelerated to the negative z-direction by the electric field. At early
times (QEPP region 46 ) when electric field$E(t)$ is increasing, momentum
distribution shows the smooth Gaussian-like structure and after reaching a
maximum value at time $t=0$ electric field starts decaying, and we see the
peak of the momentum distribution function is shifted as shown in figure
1(a),(b),(c). As time progresses, the distribution function tends to be peaked
around the zero value of the longitudinal momentum as seen in figure 1(d). In
this process, it shows a complex behavior of splitting and manifests
oscillation arising at $t=3\tau$ (near REPP region ) where the electric field
nearly vanished see figure 1(e),(f). These complex behavior disappear at the
asymptotic time ( $t=10\tau$), and we see a smooth Gaussian-like structure
1(h).

Figure 2: Time evolution of quasi-particle distribution function in the
transverse momentum space $f(p_{\perp},t)$ at different times. The
longitudinal momentum is considered to be zero, and all the units are taken in
electron mass units.The field parameters are $E_{0}=0.2$ and $\tau=10.$

###  III.2 Transverse momentum spectrum

Time evolution of the transverse momentum spectrum of created quasi-particle
pairs are depicted in the figure 2.The shape of the transverse momentum
distribution is changed when the electric field is switched on. As we see in
the figure 2(c),(d) at the transient stage the peak at $p_{\perp}=0$ splits
into weakly pronounced peaks at $p_{\perp}\approx\pm 0.67m$ and
$p_{\perp}\approx\pm 0.84m.$ At the asymptotic time (REPP stage 46 ), that
splitting disappears and distribution function $f(p_{\perp})$ shows the smooth
structure with a peak at zero transverse momentum as shown in figure 2(h).

##  IV Conclusion

We have analyzed the electron-positron pair creation from the vacuum by a
time-like Sauter pulse electric field. We have seen that the particle
Longitudinal momentum distributions exhibit the oscillating structure at a
finite time where the electric field is nearly zero and this oscillating
structure can be understood in the Dynamical Tunneling picture45 . Dynamical
tunneling can be understood as the process that develops in time inter-band
dynamics in momentum representation for particles. There are different
possibilities of particle tunneling through different channels in momentum
space( or momentum states). As time progresses, different possibilities can
occur: (a) particle can tunnel at time $t$ with $p_{\parallel}^{\prime}-$value
of momentum. (b)At early stage, particle with a lower momentum value
($p_{\parallel}<p_{\parallel}^{\prime}$) can tunnel and then be accelerated to
momentum $p_{\parallel}^{\prime}$, (c)Particle with a higher momentum value
($p_{\parallel}>p_{\parallel}^{\prime}$) can tunnel at time, $t_{1}$, and is
then decelerated to get that momentum $p_{\parallel}^{\prime}$ value. Finally,
when the individual probability amplitudes of this process are added together,
they result in quantum interference at time $t$. At the asymptotic time, those
processes do not share the same phase information and become random due to
averaging over particle paths that do not show quantum interference effects,
which leads to a smooth distribution function. However, multiphoton absorption
is not seen for the value of the Keldysh parameter $\gamma<1,$ but it is
believed to give a smaller contribution in comparison to what we get from
tunneling. Since, in our calculation, it is not much less than one
($\gamma=0.5$). As a result, another process of particle creation with
momentum $p_{\parallel}$, namely the multiphoton process, cannot be ruled out.
These are possibilities, and extensive quantitative analyses of these
processes are in progressDEEPAK:2023 . The transverse momentum distribution
shows only the splitting of smooth structure and the absence of quantum
interference, which is an obvious interference effect that occurs only in the
direction of the electric field.
A more detailed investigation of the momentum distribution of created
particles at the finite time is currently underway DEEPAK:2023 .

##  V Acknowledgments

DEEPAK gratefully acknowledge the financial support from Homi Bhabha National
Institute (HBNI) for carrying out this research work. Deepak also thanks the
organizers of the QED Laser Plasmas International Workshop 2022, Dresden for
the possibility to present our work.

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