justified

††thanks: email<EMAIL_ADDRESS>
# An Innovative Transverse Emittance Cooling Technique using a Laser-Plasma
Wiggler

1,†O. Apsimon  2D. Seipt  1,5,†M. Yadav  1,†A. Perera  3Y. Ma  4,†D.A.
Jaroszynski  3A. G. R. Thomas  6,†G. Xia  1†C. P. Welsch  1The University of
Liverpool, Liverpool L69 3BX, United Kingdom  2Helmholtz-Institut Jena,
Fröbelstieg 3, 07743 Jena, Germany  3Gérard Mourou Center for Ultrafast
Optical Sciences and Department of Nuclear Engineering and Radiological
Sciences, University of Michigan, Ann Arbor, MI 48109, USA  4University of
Strathclyde, Glasgow G11XQ, Scotland  5University of California, Los Angeles,
CA 90095, USA  6 University of Manchester, M13 9PL, Manchester, United Kingdom
† The Cockcroft Institute of Accelerator Science and Technology, Warrington,
WA4 4AD, United Kingdom

###### Abstract

We propose an innovative beam cooling scheme based on laser driven plasma
wakefields to address the challenge of high luminosity generation for a future
linear collider. For linear colliders, beam cooling is realised by means of
damping rings equipped with wiggler magnets and accelerating cavities. This
scheme ensures systematic reduction of phase space volume through synchrotron
radiation emission whilst compensating for longitudinal momentum loss via an
accelerating cavity. In this paper, the concept of a plasma wiggler and its
effective model analogous to a magnetic wiggler are introduced; relation of
plasma wiggler characteristics with damping properties are demonstrated;
underpinning particle-in-cell simulations for laser propagation optimisation
are presented. The oscillation of transverse wakefields and resulting
sinusoidal probe beam trajectory are numerically demonstrated. The formation
of an order of magnitude larger effective wiggler field compared to
conventional wigglers is successfully illustrated. Potential damping ring
designs on the basis of this novel plasma-based technology are presented and
performance in terms of damping times and footprint was compared to an
existing conventional damping ring design.

††preprint: APS/123-QED

##  I Introduction

After the discovery of the Higgs boson at the Large Hadron Collider (LHC) at
CERN [1, 2], the global vision is a successor collider in order to explore the
properties of this newly discovered boson in great detail. The physics of the
Higgs field still holds secrets of fundamental interactions and can only be
addressed in the controlled environment of an electron-positron collider that
allows elementary particle collisions with minimum synchrotron radiation
losses and well defined initial momenta. Over the last few decades, two
different design approaches, using normal conducting (CLIC, [3]) and
superconducting (ILC, [4]) radio frequency structures, revealed the size of
such a machine to be about 30$-$50 km. The electron-electron option of the
Future Circular Collider (FCC) project envisages a 100 km circumference
machine for collisions at centre-of-mass energy of 90 GeV$-$350 TeV [5]. The
limits of conventional technologies for high energy demand of particle physics
exploration persuade us to new frontiers of particle accelerator science.
Paradigm shifting technologies are being developed such as plasma acceleration
[6, 7, 8]. However, there are still many challenges to be addressed before the
maturation of plasma technology for large-scale accelerator applications.
Advanced and novel accelerators community (ICFA- ANAR2017) underlined and
prioritised the following technological challenges; the repetition rate,
efficiency and beam quality of the lasers; scalability of the system; delivery
of high collision luminosity; resolution of diagnostics; comprehensive
simulations and the availability of dedicated test facilities [9]. Driven by
high-power lasers, plasma can generate orders of magnitudes larger electric
fields compared to metallic radio-frequency cavities [6, 10, 11]. In this
paper, an innovative cooling method is presented which shows great promise of
replacing the magnetic wigglers in a damping ring with plasma wigglers and
providing superior beam quality for the future linear collider based on plasma
technology.

In order to achieve high luminosity, a the ILC design relies on emittance
cooling via damping rings followed by a long transfer line from ring to main
linear accelerator followed by beam delivery system to the final interaction
point. Emittance cooling is achieved through radiation damping due to
synchrotron radiation emitted from beam particles moving along curved
trajectories of a circular accelerator (damping ring) that has a circumference
of a few hundred meters to kilometers. Particles emit synchrotron radiation
depending on the local curvature of the orbit within a cone of angle
$\/\gamma$, where $\gamma$ is the relativistic Lorentz factor. Longitudinal
momentum is restored by radio-frequency cavities in the ring, while transverse
momentum is damped on every turn through radiation damping and quantum
excitation are equal and an equilibrium value is reached [12], typically in
milliseconds [13, 14]. A faster damping is achieved by increasing the energy
loss per turn by adding high field periodic magnetic structures (wigglers or
wigglers, depending on the magnetic strength) [15].

As an example, the International Linear Collider (ILC) project proposes a
machine with 250-500 GeV centre-of-mass energy over about a 31 km footprint
[16]. Electrons and positrons emerging from different sources undergo an
initial acceleration up to 5 GeV before they are injected into a their
respective damping rings with a circumference of 3.2 km, housed in the same
tunnel. ILC damping rings are designed in a race track shape to accommodate
two straight sections. A radiative section comprising 54 super-ferric wigglers
is located in one of these straight sections. Each wiggler is 2.1 m long and
generates a 2.16 T peak magnetic field when operating at 4.5 K and radiates 17
kW radiation power [16]. This straight section also houses a superconducting
radio-frequency system to replenish the longitudinal momentum of the beam.

A compact synchrotron radiation insertion device with significantly larger
fields than the current wigglers would limit the footprint and the cost of the
ring by reducing the required number and size of these insertion devices.
There are several methods for plasma assisted radiation generation from a
relativistic particle beam, such as, betatron oscillations, Compton
scattering, bremsstrahlung and transition radiation [17]. We propose to
incorporate the concept of plasma wiggler as radiators in a damping ring to
benefit from their large effective magnetic fields and compactness. There are
various concepts to conceive a plasma wiggler [18, 19, 20, 21]. Following the
one proposed in [18] , a plasma wiggler is formed when a short laser pulse is
injected into plasma off-axis or at an angle that causes the centroid of the
laser pulse to oscillate. Given that the product of the plasma wave number and
the characteristic Rayleigh length of the laser is much larger than one, the
ponderomotively driven plasma wake will follow this centroid. This oscillating
transverse wakefield works as an wiggler forcing particles to follow
sinusoidal trajectories and emit synchrotron radiation. In addition, the
damping time is inversely proportional to the square of the magnetic field of
the damping device. Theoretically, a plasma wiggler can generate order of
magnitude larger effective magnetic fields than conventional wigglers, hence
can reduce the length of the damping units by a factor of hundred while
providing the same damping times.

In this paper, we layout the fundamental design criteria including discussion
on laser-plasma channel coupling and average dephasing length in parabolic
channel to achieve a plasma wiggler in Section II. This is followed by
numerical demonstration of oscillation laser centroid in the channel and
sinusoidal probe beam trajectory in Section III. In this section, we present
the baseline design that achieves 20 T effective magnetic field compared to a
typical 2 T in a superconducting magnetic wiggler, as well as 4 MW average
radiated power by two orders of magnitude improvement from the conventional
approach. This also includes studies discussing correlation of laser injection
offset with laser radius and plasma skin depth laying out interesting
phenomena of radial phase shift in focusing force. In Section IV, we presented
three different damping ring designs that offer compromises between low
damping time or small footprint.

##  II Plasma wiggler

The optimisation criteria for a plasma wiggler include laser and plasma
parametrisation ensuring the matching between the laser spot size and channel
radius; probe beam emittance to the channel width as well as limitations such
as dephasing length in a channel. In this section, characteristics of a plasma
wiggler following the optimisation criteria is numerically demonstrated.

Laser propagation through preformed plasma channels were extensively studied,
previously [22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. A plasma wiggler is
formed when a short laser pulse is injected into a parabolic plasma channel
off-axis or at an angle that causes the centroid of this laser pulse to
oscillate [18] when the normalised vector potential $a_{0}<1$. A Gaussian
laser pulse will be guided in a parabolic channel in Eq.1,

$n(r)=n_{0}(1+(\Delta n/n_{0})r^{2}/w_{0}^{2}),$ (1)

at the matched spot size when the channel depth is equal to the critical
channel depth as in $\Delta n=(\pi r_{e}w_{0}^{2})^{-1}$ [10, 32]. In this
expression $r_{e}=e^{2}/mc^{2}$ is the classical electron radius, $n_{0}$ is
the on-axis plasma density and $w_{0}$ is the laser radius.

For a laser with sufficiently low power to satisfy $P<P_{c}$, the self-
focusing is avoided and the laser spot size is preserved during the
propagation for an initial spot size matched to the channel radius. If the
laser power is higher than the critical power, $P_{c}[GW]\approx
17(k_{L}/k_{p})^{2}$, then relativistic self-focusing will occur [10], where
$k_{L}$ and $k_{p}$ are the wave numbers for the laser and the plasma wave,
respectively.

The period of the laser centroid oscillation is determined by the Rayleigh
length of the laser, $Z_{R}=\pi w_{0}^{2}/\lambda_{L}$, and given by $2\pi
Z_{R}$. According to this, the centroid oscillation follows,

$x=x_{ci}\cos(z/Z_{R}+\phi),$ (2)

where $\phi$ is the initial arbitrary phase and $x_{ci}$ is the initial
transverse offset. The wiggler period, $\lambda_{u}$, is equal to the
wavelength of the laser centroid oscillation,

$\lambda_{u}=2\pi^{2}w_{0}^{2}/\lambda_{L},$ (3)

where $\lambda_{L}$ is the wavelength of the driver laser.

Furthermore, when $k_{p}Z_{R}\gg 1$, the ponderomotively driven wakefields
follow the oscillating laser centroid. In the case of an electron bunch
injected into these oscillating wakefields at an appropriate phase such that
the longitudinal field $E_{x}=0$ and the transverse field $E_{y}$ is nonzero,
then these probe electrons will follow oscillatory trajectories correlated
with the laser centroid.

The strength parameter corresponding to a plasma wiggler is derived in [18]
and given in Eq. 4.

$a_{u}\approx 4\pi a_{0}^{2}Cx_{ci}/\lambda_{L}$ (4)

where $C=\sqrt{\pi/8e}\approx 0.38$ for an optimised laser pulse duration and
$C\rightarrow 2C$ for a circularly polarised laser.

In a magnetic wiggler, assuming that the magnetic field is sinusoidal,
integrating the equation of motion of the electrons in transverse plane,
$\ddot{z}=eB_{y}/p$, yields the maximum horizontal deflection or the wiggler
strength parameter,

$K=\frac{B_{0}}{m_{e}c}\frac{e\lambda_{u}}{2\pi},$ (5)

where $B_{y}$ is the transverse field with a peak value of $B_{0}$, $p$ is the
momentum and $m_{e}$ is the mass of an electron.

Therefore, an effective magnetic field for a plasma wiggler can be defined in
practical units as in

$B_{\textrm{eff}}[\textrm{T}]=a_{u}/(0.0934\lambda_{u}[\textrm{mm}])$ (6)

where $K$ is replaced by the strength of plasma wiggler, $a_{u}$ and
$\lambda_{u}$ now representing the period length of the plasma wiggler. Eq. 6
can be translated into
$B_{\textrm{eff}}[T]=a_{0}^{2}Cx_{ci}[m]/(46.7\pi(w_{0}[m])^{2})$ by
substituting Eq. 3 and Eq. 4 in Eq. 6. Therefore, theoretically, for a
normalised vector potential, $a_{0}=0.36$, and the laser spot size,
$w_{0}=10\mu$m, a plasma wiggler can generate an order of magnitude larger
wiggler field strength than superconducting wigglers designed for ILC. For
$a_{0}$ values close to unity, strong transverse wakefields might cause
ejection of probe electrons outside the wiggler and act as a deflecting magnet
as shown in Section III. Therefore, higher effective wiggler fields may be
achieved by reducing the laser spot size. However, one should note that for
small values of $w_{0}$, fields created behind the laser pulse occupy a narrow
transverse region. Therefore capturing probe electrons can be challenging. The
general rule of thumb is that the transverse extend of both probe electrons
and laser centroid offset should be smaller than the laser spot size.

###  II.1 Coupling into the Channel

#### Laser Matching

For the laser to propagate through the plasma with a constant spot size, the
initial spot size and the channel radius should match. A detailed discussion
is given in [33] by introducing dimensionless variables in units of plasma
frequency in SI units, $\omega_{p}=\sqrt{n_{0}e^{2}/m_{e}\epsilon_{0}}$ and
the wave number, $k_{p}=\omega_{p}/c$. According to this, the evolution of the
laser spot size, $w$, is given by,

$w^{2}=\frac{w_{0}^{2}}{2}+\frac{2R^{2}}{w_{0}^{2}}+\left(\frac{w_{0}^{2}}{2}-\frac{2R^{2}}{w_{0}^{2}}\right)\cos
2\Omega\tau,$ (7)

where $w_{0}$ is the laser spot size, $R$ is the dimensionless channel radius,
$\Omega$ is the characteristic frequency of the laser centroid oscillation and
$\tau$ is dimensionless time. The matched case, where the spot size, $w$, is
constant, requires the second term to be zero. Hence, for a matched case
$R=r_{m}^{2}/2$ where $r_{m}$ is the dimensionless matched laser radius.
Furthermore, $\Omega_{m}=2/(M_{p}r_{m}^{2})$ is the dimensionless frequency of
the centroid oscillation, where $M_{p}=k_{L}/k_{p}$. The coefficient
$M_{p}/2\pi$ transforms the space and time into units measured by plasma
wavelength and period, respectively.

Figure 1: Evolution of the dimensionless laser spot size for the cases where
the plasma channel radius is matched ($w_{0}=r_{m}$) and unmatched
($w_{0}=1.2r_{m}$) to the laser spot size (red and blue curves, respectively).
Laser centroid oscillations in the units of laser wavelength for the matched
case is also presented with the green curve. The plot is generated for
$a_{0}=0.36$, $w_{0}=10\,\mu$m and $n_{0}=7\times 10^{23}$ m-3.

Figure 1 presents the evolution of the laser spot size according to Eq.7, for
matched ($w_{0}=r_{m}$) and unmatched ($w_{0}=1.2r_{m}$) cases in units of
laser wavelength. For the unmatched case (blue curve), laser spot size
oscillates with an amplitude of $\pm$5$\lambda_{L}$. Whereas it is constant at
12.5$\lambda_{L}$ during its propagation for a matched condition. Figure 1
also shows the evolution of the centroid for the matched case which oscillates
at $\Omega_{m}$.

#### Beam Envelope Matching

The beam envelope evolves as a function of the beam emittance and energy as
well as any external focusing or defocusing (such as space charge) and
dispersive effects. Ignoring the adiabatic damping, any dispersive
contribution and space charge, the beam size, $r_{b}$, is given as below.

$\frac{d^{2}r_{b}}{d(ct)^{2}}=\frac{\varepsilon_{n}^{2}}{\gamma^{2}r_{b}^{3}}-k_{\beta}^{2}r_{b}$
(8)

where $\varepsilon_{n}=\gamma r_{b}\theta_{b}$ is the normalised transverse
beam emittance with $\theta_{b}$ is the rms beam angle and $\gamma$ is the
Lorentz factor. To maintain a constant beam size through the channel, i.e.,
$d^{2}r_{b}/dct^{2}=0$, the matched beam size should be
$r_{bm}=(\varepsilon_{n}/\gamma k_{\beta})^{1/2}$ [34]. This matching
condition is also employed for the simulations presented here.

###  II.2 Dephasing Length in a Parabolic Channel

Effective length of propagation for laser-plasma interactions is generally
limited by laser diffraction. Guiding high power laser pulses through a
parabolic plasma channel can circumvent this. Here, the channel acts as
effective radio-frequency cavities that confine and shape the electromagnetic
fields in conventional accelerators. When diffraction is eliminated, the
limiting factor for the effective interaction length is the dephasing length,

$\lambda_{d}=\lambda_{p}^{3}/\lambda_{L}^{2},$ (9)

where $\lambda_{p}=2\pi/k_{p}$ is the on-axis plasma wavelength. This is the
distance over which a trailing electron bunch gains sufficient energy to
outrun the driving laser pulse to cross into decelerating fields. The
dephasing length scales with $n_{0}^{-3/2}$ where $n_{0}$ is the plasma
density on the channel axis. To prevent witness electrons overtaking the
driver laser pulse, the dephasing length should be larger than, or similar to
the length of the plasma wiggler. Therefore, one must optimise the wiggler
target by establishing a compromise between the interaction length and plasma
density [35, 36].

However, in the case of a sinusoidal trajectory of a laser pulse in a channel,
plasma density hence the dephasing length for the laser has a radial
dependence. Therefore, such a laser pulse would experience an average
dephasing length that is shorter than a laser travelling on channel axis
through the channel. The axial dephasing length can be expanded as in Eq. 10,

$\lambda_{d}(r)=\frac{1}{\lambda_{L}^{2}}(2\pi
c)^{3}\bigg{(}\frac{\epsilon_{0}m_{e}}{e^{2}}\bigg{)}^{3/2}n(r)^{-3/2},$ (10)

by substituting the plasma wavelength and isolating the plasma density as a
function of radial coordinate. The average dephasing length is then calculated
as,

$\bar{\lambda}=\frac{1}{y_{i}-y_{f}}\int^{y_{f}}_{y_{i}}\lambda_{d}(y)dy,$
(11)

resulting in,

$\bar{\lambda}_{d}=An_{0}^{-3/2}\bigg{(}1+\frac{\Delta
n}{n_{0}}\frac{y_{f}^{2}}{w_{0}^{2}}\bigg{)}^{-1/2},$ (12)

where the initial injection location $y_{i}=0$, $A=(1/\lambda_{L}^{2})(2\pi
c)^{3}(\epsilon_{0}m_{e}/e^{2})^{3/2}$ and $w_{0}^{2}/(\Delta n/n_{0})$ is
defined as the channel width, $r_{c}$, for practicality. Using a Taylor
expansion on the expression in parenthesis in Eq. 12, it is simplified to Eq.
13,

$\bar{\lambda}_{d}=An_{0}^{-3/2}\bigg{(}1-\frac{y_{f}^{2}}{2r_{c}^{2}}\bigg{)}.$
(13)

where the average dephasing length equals its on-axis value when the radial
extend of the beam, $y_{f}$, is zero.

Given that $\lambda_{u}$ is the wiggler period and $N$ is the number of
periods of the wiggler, to prevent dephasing during the length of the wiggler,
one can introduce the relation $\bar{\lambda}_{d}\geq\lambda_{u}N$ and
substitute $\bar{\lambda}_{d}=\lambda_{u}N$ in Eq. 13 to deduce an average
plasma density, $n_{p,d}$, for a given channel width, $r_{c}$, as given in Eq.
14,

$n_{p,d}=\bigg{(}\frac{4\pi
c^{3}}{w_{0}^{2}\lambda_{L}N}\bigg{)}^{2/3}\bigg{(}\frac{\epsilon_{0}m_{e}}{e^{2}}\bigg{)}\bigg{(}1-\frac{y_{f}^{2}}{3r_{c}^{2}}\bigg{)}$
(14)

The expression will help determine the plasma density to achieve a certain
dephasing length in a channel and entails two terms. The first term is an on-
axis plasma density which can be deduced from Eq. 10 for a $r=0$ and the
second term is that scales with the radial extend of the beam and channel
width.

The other factor effecting the dephasing length is the longitudinal velocity
reduction of the witness particles due to their sinusoidal trajectories which
is approximately equal to $\langle v_{z}\rangle/c\simeq 1-(\langle
p_{\perp}^{2}\rangle+m^{2}c^{2})/2p_{z}^{2}$. However, this effect is not
considered for the results presented in next section.

##  III Simulations and discussions

A plasma wiggler and its interaction with a probe beam are simulated using the
particle-in-cell code, EPOCH [37]. A 200$\,\mu$m long simulation window is
defined with $\pm$70$\,\mu$m transverse acceptance that moves at the 99.9$\%$
of the speed of light.

The domain is set up with 10 cells per laser wavelength in the longitudinal,
and per plasma skin depth for transverse directions, allowing the smallest
features in those directions to be resolved. Electrons in the plasma are
represented with $5$ macroparticles per cell assuming an immobile neutralizing
background. All boundaries are absorbing for particles and fields. Fields are
calculated with a $2^{\text{nd}}$-order Yee solver while currents are smoothed
using a 3-point nearest-neighbour low-pass filter in each direction [38]. QED
photon emission by particles undergoing sinusoidal trajectories is calculated
using an optical-depth-like model through local electromagnetic field
strengths at macroparticle positions and incorporating radiation reaction and
energy straggling effects [39]. The wavelengths of classical fields
represented by such photons are orders of magnitude too small to be resolved
by the grid and are not incorporated into the macroscopic fields of the
simulation.

The physical characteristics of the laser, plasma and probe beam are given in
the following subsection.

###  III.1 Initial simulation settings

#### Plasma

A plasma profile comprising a vacuum section followed by a 10$\,\mu$m up-ramp
section prior to a parabolic distribution was implemented. The zero density
section is provided for initialisation of a laser pulse by ensuring the
consistency of the Maxwell’s equations. The plasma density profile for the
first 200$\,\mu$m of the simulation is presented in Fig. 2.

Figure 2: The initial plasma density configuration including a zero density
and up-ramp region prior to parabolic distribution.

#### Laser

A custom defined laser pulse is used in EPOCH defining 2D Gaussian beam
profiles. Leading order corrections on longitudinal electric field for the
diffraction angle expansion as well as for short pulse duration (first
derivative of pulse envelope in longitudinal field) are implemented. The model
featured control on parameters for focal spot position and temporal pulse
centre as well as longitudinal magnetic fields for out-of-plane laser
polarisation.

The longitudinal pulse profiles are given as,

$E_{\textrm{pulse}}=-E_{0}e^{-(x-x_{t_{0}})^{2}/2(\sigma_{t}c)^{2}},$ (15)
$B_{\textrm{pulse}}=-E_{0}e^{-(x-(x_{t_{0}}-\delta))^{2}/2(\sigma_{t}c)^{2}}/c,$
(16)

where $E_{0}$ is the peak electric field of the laser pulse, $x_{t_{0}}$ is
the spatial position of the temporal pulse centre at $t=0$, $\sigma_{t}$ is
the pulse duration and $\delta$ is the offset for the magnetic field. The
initial distribution of the Gaussian laser field can be seen in Fig. 3 with a
peak electric field of 3.2 TV/m.

Figure 3: Initial laser pulse profile for $a_{0}=0.8$, $\lambda_{L}=800\,$nm
and $w_{0}=30\,\mu$m.

#### Probe beam

A hypothetical 10 pC electron probe beam with a $\gamma=$1000, a matched spot
size of $\sim$1$\,\mu$m and rms length of 1$\,\mu$m. The injection phase is
determined so that no accelerating field acts on the probe beam while the
focusing field is larger than zero and at its maximum value. The transverse
($E_{r}$ and $B_{\theta}$) and longitudinal wakefields ($E_{x}$), that are
related to each other through Panofsky-Wenzel theorem, are presented in Eq. 17
[10].

$\displaystyle
E_{r}-B_{\theta}\sim\left(\frac{4r}{k_{p}w_{0}^{2}}\right)\exp{\left(-\frac{2r^{2}}{w_{0}^{2}}\right)}\sin(k_{p}\xi)$
(17) $\displaystyle
E_{x}\sim\exp{\left(-\frac{2r^{2}}{w_{0}^{2}}\right)}\cos(k_{p}\xi),$

where $\xi=x-ct$. The conditions, $E_{x}=0$ and $E_{r}-B_{\theta}>0$ can be
satisfied simultaneously when $k_{p}\xi=3\pi/2$. According to this, the first
possible location at $\lambda_{p}/4$ behind the head of the laser generally is
located inside the laser therefore is an undesirable injection spot.
Therefore, the next possible location $5\lambda_{p}/4$ behind the head of the
laser is used for the results reported in this paper.

It is non-trivial to exactly determine the start of the wakefields with
respect to the laser pulse front. Therefore, the probe beam injection location
is manually studied the field patterns and determined as approximately
89.5$\,\mu$m behind with respect to the centre of the laser. The field
distribution experienced by the probe beam is presented in Fig. 4.

Figure 4: An example case depicting an optimum injection point with
$E_{x}\approx 0$ and $E_{y}>0$. The colour map presents the transverse (top)
and longitudinal (bottom) field distributions. The injection location is shown
with black circles with respect to the centre of the laser (red circle). Blue
line-outs show the on-axis field values.

###  III.2 Probe beam trajectory and radiated power

Three merits for initial optimisation of the laser-plasma wiggler prior to
injection of a probe beam are effective magnetic field, transverse offset of
laser injection point and dephasing length.

Figure 5: Induced effective magnetic field provided by the plasma wiggler
(black curve) and average emitted radiation by the probe beam (blue curve) as
a function of $a_{0}$. The curves are calculated for $w_{0}=10\,\mu$m and
$x_{ci}=6\,\mu$m.

The effective magnetic field has a strong dependence on the normalised vector
field of the laser, injection offset and radius of the laser. The power
radiated by the probe beam performing lateral acceleration under this field is
given in Eq. 18

$P_{\mathrm{avg}}[\mathrm{W}]=6.336E^{2}\mathrm{[GeV]}B_{\mathrm{eff}}^{2}\mathrm{[kG]}I_{b}L_{u},$
(18)

where $I_{b}$ is the peak beam current and $L_{u}$ is the length of the
radiator (wiggler) which is taken equal to the dephasing length for each given
wiggler solution in this work. Figure 5 shows that an order of magnitude
larger effective magnetic field, compared to current ILC radiators is
theoretically possible at around $a_{0}\approx 0.36$, with two orders of
magnitude larger radiated average power (4 MW).

(a)  Figure 6: Beam trajectory following injection at optimum phase for
increasing $a_{0}$.

In summary, in a parabolic channel, high effective magnetic fields can be
generated by lasers with high $a_{0}$. However, there is an upper limit to
maintain the probe beam trajectory. The probe beam injected into such a
channel might be ejected in one transverse direction before it reaches to the
next polarity of its sinusoidal trajectory. Therefore, for a plasma wiggler
$a_{0}$ should be a fraction of unity. Example trajectories for $a_{0}=\,$0.2,
0.5 and 0.8 are presented in Fig. 6 and the ejection of the probe beam outside
the field regions after a few millimeters of propagation for these three cases
are shown in Fig. 7. Nevertheless, in order to generate bending strengths
significantly larger than the conventional technology, one might consider
smaller laser spot sizes and hence injection offsets, to compensate against
the limit on $a_{0}$.

(a)

(b)

(c)

Figure 7: The ejection of probe beam out of the channel with increasing
$a_{0}$ due to strong transverse focusing field, $eE_{y}+cB_{z}$, driven in
plasma for (a) $a_{0}=0.2$, (b) $a_{0}=0.5$ and (c) $a_{0}=0.8$ (Table 1-(I),
(II) and (III), respectively). All three cases are for an $x_{ci}=5\,\mu$m,
$w_{0}=30\,\mu$m and $n_{e}=1\times 10^{24}\mathrm{m^{-3}}$.

(a)  (b)

(c)  (d)

Figure 8: The radial phase of transverse fields for four different wiggler
solutions providing a) $x_{ci}/w_{0}=0.2$, $x_{ci}/(c/\omega_{p})=0.94$, b)
$x_{ci}/w_{0}=0.6$, $x_{ci}/(c/\omega_{p})=0.94$, c) $x_{ci}/w_{0}=0.3$,
$x_{ci}/(c/\omega_{p})=0.5$ and d) $x_{ci}/w_{0}=0.13$,
$x_{ci}/(c/\omega_{p})=1$ (Table 1-(I), (IV), (V), (VII), respectively). Table
1: Analytical solutions for plasma wigglers yielding different radiation
power, $P_{av}$, and dephasing length, $\lambda_{d}$.

Solution | $a_{0}$ | $w_{0}\,$ | $x_{ci}\,$ | $n_{e}$ | $a_{u}$ | $\lambda_{u}\,$ | $B_{0}\,$ | $\lambda_{d}$ | $P_{av}$
---|---|---|---|---|---|---|---|---|---
|  |  ($\mu$m) |  ($\mu$m) |  (${\color[rgb]{0,1,1}\mathrm{m}}^{-3}$) |  | (mm) | (T) | (cm) | (kW)
(I) | 0.2 | 30 | 5 | $1\times 10^{24}$ | 1.2 | 22 | 0.6 | 3 | 2
(II) | 0.5 | 30 | 5 | $1\times 10^{24}$ | 7.5 | 22 | 3.6 | 3 | 75
(III) | 0.8 | 30 | 5 | $1\times 10^{24}$ | 19 | 22 | 9.2 | 3 | 500
(IV) | 0.3 | 10 | 3 | $0.7\times 10^{24}$ | 1.6 | 2.5 | 7 | 5 | 500
(V) | 0.3 | 10 | 6 | $0.7\times 10^{24}$ | 3.2 | 2.5 | 14 | 5 | 2000
(VI) | 0.36 | 10 | 6 | $0.7\times 10^{24}$ | 4.6 | 2.5 | 20 | 5 | 4000
(VII) | 0.3 | 40 | 5.32 | $1\times 10^{24}$ | 2.8 | 39 | 0.9 | 3 | 3.5

Table 1 summarises a set of numerical solutions for phenomena observed during
the numerical optimisation of a plasma wiggler to study witness ejection from
the channel and radial phase shift of the wakefields. Although, the effective
magnetic field scales with the square of $a_{0}$ and is proportionate to
$x_{ci}$, we showed that $a_{0}\sim 1$ causes the ejection of witness
particles. Hence, angle integrated average power yield through generation of a
large $B_{\mathrm{eff}}$ is achieved by optimising $x_{ci}$. We observed that
a radial phase shift is introduced in the transverse fields if the conditions
of $x_{ci}/(c/\omega_{p})\approx 1$ and $x_{ci}/w_{0}\ll 1$ are not met
simultaneously. This is demonstrated in Fig. 8 using a set of cases
representing different values of the above mentioned ratios. It is also
observed that the ratio of the injection offset and the plasma skin depth has
a more dominant effect in this phenomena.

Counter-intuitively, the radial phase shift seems to accommodate witness
propagation at least throughout the dephasing length as demonstrated in Fig. 9
for the baseline case given in Table 1-(VI). In the figure, the perpendicular
propagation distance in the plasma is 10 mm. However, the effective laser
propagation is almost 10 cm due to the sinusoidal trajectory, achieveing the
laser dephasing length at about 5 mm of perpendicular propagation. Figure also
shows that the witness centroid follows a sinusoidal trajectory as expected.
For this baseline case, the characteristics of the laser, plasma and the probe
beam as well as the performance of the resulting plasma wiggler are summarised
in Table 2. We confirmed the feasibility of generating this parameter space in
a facility such as Scottish Centre for the Application of Plasma-based
Accelerators (SCAPA) [40] and tests will be planned for future.

Figure 9: The trajectories of the rms centroids of the laser pulse (blue
dots), its theoretical prediction (red dashed line) and the witness for
baseline solution given in Table 1-(VI). Table 2: Characteristics of the
initial setup, resulting plasma wiggler and the emitted radiation.

Laser |
---|---
Wavelength (nm), $\lambda_{L}$ | 800
Normalised vector potential, $a_{0}$ | 0.36
Radius at focus($\mu$m), $w_{0}$ | 10
Pulse length (fs), $\sigma_{L}$ | 25
Injection offset ($\mu$m) | 6
Plasma |
Density ($m^{-3}$), $n_{0}$ |  7$\times$1023
Probe beam |
Normalised beam energy $\gamma$, | 1000
Density ($\textrm{m}^{-3}$), $n_{e}$ |  $n_{0}$/1000
Charge (pC), Q | 10
Matched radius ($\mu$m), $\sigma_{y}$ | 0.6
Bunch length ($\mu$m), $\sigma_{x}$ | 1
Dephasing length (cm) | 5
Plasma wiggler |
Period (mm), $\lambda_{u}$ | 2.5
Strength parameter, $a_{u}$ | 4.6
Effective field (T), $B_{\mathrm{eff}}$ | 20
Radiation |
Wavelength (nm), $\lambda_{\gamma}$ | 14
Angle integrated average power (MW) | 4

In this study, we concentrated on numerically demonstrating the feasibility of
a plasma wiggler that is predicted by the theoretical optimisation. We
demonstrated a baseline setting achieving an order of magnitude larger
effective magnetic field (20 T) and two orders of magnitude larger average
radiation power (4 MW) in a single pass compared to those aimed with a
conventional wiggler.

The phase space cooling occurs after tens of hundreds of passages through the
radiator. Hence observation of cooling directly from particle-in-cell
simulations is prohibitively computing-heavy with the current capabilities and
will be feasible in parallel to studies in exascale computing such as the
WARPX project [41]. Therefore, in the next section, we will analytically
demonstrate the impact of the designed baseline plasma wiggler on the cooling
performance of a linear collider damping ring.

##  IV Transverse Emittance Damping

An electron beam circulating in a damping ring losses energy at each turn due
to the synchrotron radiation generated from its movement around the circular
orbit of the ring as well as the radiation generated from lateral acceleration
in the wiggler magnet. The beam emittance decreases exponentially as given in
Eq. 19,

$\varepsilon(t)=(\varepsilon_{\textrm{inj}}-\varepsilon_{\textrm{equ}})e^{-2t/\tau}+\varepsilon_{\textrm{equ}},$
(19)

from its value during injection into damping ring,
$\varepsilon_{\textrm{inj}}$, to an equilibrium value
$\varepsilon_{\textrm{equ}}$ where $\tau$ is the damping time.

The emission of synchrotron radiation is a statistical process involving
quantum fluctuation of beam parameters. An equilibrium emittance is reached
when the quantum excitation is equal to damping [12].

###  IV.1 Contribution from wigglers

The energy loss in wigglers constitutes nearly 95$\%$ of the total energy
loss. This energy loss of a beam during its each passage through a wiggler is
given in Eq.20

$U_{0}=\frac{C_{\gamma}}{2\pi}E^{4}I_{2w}$ (20)

which is proportional to the forth power of its energy $E$ and second
radiation integral for a wiggler $I_{2w}$. This integral can be written as in
Eq.21

$\displaystyle
I_{2w}=\int_{0}^{L_{u}}\frac{1}{\rho^{2}}ds=\frac{1}{(B\rho)^{2}}\int_{0}^{L_{w}}B_{w}^{2}ds=\frac{1}{(B\rho)^{2}}\frac{B_{w}^{2}L_{w}}{2}.$
(21)

for a wiggler with an effective length of $L_{w}$ and a radius of curvature
$\rho$ for the deflected trajectory. To evaluate $I_{2w}$, beam rigidity
$(B\rho=p/e)$ is used to simplify the integral.

From Eq.21, it is straightforward to see that a wiggler contributes to the
energy loss per turn in a damping ring proportionately to the square of its
magnetic field and the effective length, i.e.,

$U_{0}\propto B_{w}^{2}L_{w}.$ (22)

Furthermore, the damping time, which is the time needed for the emittance to
decrease by 1/e of its injected value, is given as,

$\tau=2E_{0}\frac{T_{0}}{U_{0}},$ (23)

where $E_{0}$ is the injection energy of the electrons into the damping ring
and $T_{0}$ is the revolution period of the electrons in the ring. This
implies that an order of magnitude larger field enables the same amount of
damping with a 100 times shorter wiggler. Alternatively, for the same
radiative length, a 100 times faster damping might be achieved with a wiggler
providing an order of magnitude larger wiggler strength.

###  IV.2 Cooling scenarios for an ILC-like machine

The impact of a plasma wiggler on the damping process was studied using
parameters similar to the baseline for ILC damping ring to demonstrate its
potential for future colliders. To study this, an injection emittance of 6 mm
mrad was considered to be damped down to 20 nm ($\mu$m mrad). The ILC-like
baseline case consists of a total radiative length of 113 m including 54
wigglers with 2.16 T (as defined for electrons at ILC in the technical design
report). The evolution of emittance is studied as described in Section IV and
presented in Fig. 10 for different scenarios. For ILC damping rings, the
equilibrium emittance should be reached below 200 ms, between machine pulses,
which is shown in solid black curve in Fig. 10.

Figure 10: Performance of different cooling scenarios using plasma wigglers
for ILC damping ring in comparison to baseline with superferric wigglers.

The first scenario proposes to replace the superferric wigglers by plasma
wigglers with characteristics given in Table 2 over the same 113 m radiative
section. This is presented with hollow dots and provides the largest reduction
in damping time below 1 ms. The second scenario employs the advantage in
significantly increased bending strength and proposes to decrease the
radiative section length an order of magnitude down to 10 m, that achieves
damping within about 5 ms (dashed light curve). Eventually, the third scenario
reduces this down to a one meter-long radiative section (dashed bold curve)
for a larger damping time than the conventional ILC option. Nevertheless, it
still provides an 80 ms damping time which is adequate for the ILC machine
cycle.

A damping ring design with a higher ring energy provides shorter damping
times. However the natural emittance scales with the cube of the energy
deeming a lower energy ring favourable. Keeping in mind that the collective
effects are more severe at low energies, the plasma wiggler offers a solution
where at least two orders of magnitude shorter damping times are possible
while keeping energy low enough to ensure a minimal natural emittance for the
ring. Therefore a next generation damping ring equipped with plasma wigglers
rather than conventional counterparts provides an exciting scenario for the
future linear collider design studies.

###  IV.3 Multi-stage single-pass cooling

The circumference, hence the cost of a damping ring increases with the number
of bunches per linac pulse. The ILC design envisages bunches to be injected
into a ring in a compressed time structure to limit its circumference. After
cooling, these bunches should be decompressed by using state-of-the-art dipole
magnets (kickers) during extraction to provide the desired bunch spacing for
the experiments. Previous studies showed that, in a damping ring, certain
instabilities are introduced during injection/extraction [42] and circulation
until a design orbit is established before cooling.

Furthermore, due to the quantum emission, there is an equilibrium energy
spread of the beam that is established in the ring. Because the ring is
dispersive, this energy spread results in an equilibrium bunch length. This
equilibrium bunch length produced in conventional damping rings (9 mm for
compressor entrance) is much longer than the typical values for operating
conditions of a plasma-based accelerator. Reducing the bunch length is
difficult, and results in intra-beam and Touschek scattering (due to high
current), which are sources of unacceptable emittance growth. Therefore one
must use bunch compression after a damping ring. For example, ILC compresses
about a factor of 30 after the ring. For a plasma accelerator requiring bunch
lengths a fraction of the plasma wavelength, e.g., as small as $\sim$10
microns, this would require bunch compression factors of order $\sim$1000\. It
is difficult to imagine any bunch compression system that could achieve this
while still preserving the beam emittance.

As such, a scheme eliminating the use of a ring altogether would revolutionise
the cooling process for a collider based on plasma technology. This future
work will require extensive computing-heavy numerical work to explore the beam
dynamics of a single-pass linear damping section with adjacent plasma stages
for radiation and acceleration. Staging is already a hot topic in the field,
providing the community with many challanges to tackle [43, 44, 45, 46, 47].

##  V Conclusions

An innovative concept of using a plasma wiggler for beam damping was explored.
Initial results demonstrate an order of magnitude larger effective magnetic
field achievable with realistic laser and plasma parameters. This has direct
implication of several scenarios compromising between orders of magnitude
faster damping or smaller footprint. A comparison to the ILC damping ring
design, as a conventional example, was presented demonstrating, for extreme
scenarios, damping within $<$1 ms or alternatively reduction of footprint of
radiative section down to 1 m from 113 m.

Using plasma wigglers for emittance damping brings an exciting prospect for
future linear colliders especially based on plasma technology. This may
include muon colliders as well, where cooling time might compete against decay
time. This opens up new avenues for exploration of the implementation at
large-scale, numerically, where reduced models or extensive high performance
computing resources will be required as well as studies regarding the multi-
staging.

The reduction in damping time provided by a plasma wiggler can compensate
against the need for large energy in a conventional set-up, allowing to limit
the natural emittance in the ring. That being said, implications on
feasibility of the concept regarding instabilities in a damping ring at low
energies will be explored.

###### Acknowledgements.

This work is supported by the European Union’s Horizon 2020 research and
innovation programme and U.S. NSF grant 1804463 and AFOSR grant
FA9550-19-1-0072. Computing resources provided by STFC Scientific Computing
Department’s SCARF cluster. Authors would like to thank to Jon Roddom and
Derek Ross for their endless support on implementation of EPOCH on SCARF
clusters. We would like to thank to EPOCH community support, especially, Keith
Bennett, Thomas Goffrey, Heather Ratcliffe and Christopher Brady. Authors also
would like to thank to Dr Carl B. Schroder (BELLA Centre, Berkeley Lab), Dr
Hywel Owen (The University of Manchester) and Mr Volodymyr Rodin (The
University of Liverpool) for insightful discussions on the requirements of a
damping system for beams generated by plasma-based accelerators and storage
rings.

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# Homomorphic Encryption-Enabled Federated Learning for Privacy-Preserving
Intrusion Detection in Resource-Constrained IoV Networks

Bui Duc Manh, Chi-Hieu Nguyen, Dinh Thai Hoang and Diep N. Nguyen School of
Electrical and Data Engineering, University of Technology Sydney, Australia.

###### Abstract

This paper aims to propose a novel framework to address the data privacy issue
for Federated Learning (FL)-based Intrusion Detection Systems (IDSs) in
Internet-of-Vehicles (IoVs) with limited computational resources. In
particular, in conventional FL systems, it is usually assumed that the
computing nodes have sufficient computational resources to process the
training tasks. However, in practical IoV systems, vehicles usually have
limited computational resources to process intensive training tasks,
compromising the effectiveness of deploying FL in IDSs. While offloading data
from vehicles to the cloud can mitigate this issue, it introduces significant
privacy concerns for vehicle users (VUs). To resolve this issue, we first
propose a highly-effective framework using homomorphic encryption to secure
data that requires offloading to a centralized server for processing.
Furthermore, we develop an effective training algorithm tailored to handle the
challenges of FL-based systems with encrypted data. This algorithm allows the
centralized server to directly compute on quantum-secure encrypted ciphertexts
without needing decryption. This approach not only safeguards data privacy
during the offloading process from VUs to the centralized server but also
enhances the efficiency of utilizing FL for IDSs in IoV systems. Our
simulation results show that our proposed approach can achieve a performance
that is as close to that of the solution without encryption, with a gap of
less than 0.8%.

###### Index Terms:

Intrusion detection, homomorphic encryption, deep learning, IoV.

##  I Introduction

In recent years, the Internet of Vehicles (IoV) has witnessed as a significant
research field with the development of intelligent transport systems (ITSs),
connected autonomous vehicles (CAV), vehicular ad hoc networks (VANET), and
in-vehicle networks (IVNs) [1]. Particularly, IoV is a combination of VANET
and IoT, which is built based on the communication standards of VANET [2]. In
the IoV network, the vehicles equipped with IoT sensors can transmit and
exchange data via the roadside units (RSUs), thereby providing intelligent
decision-making based on the collected data. However, the massive network
connectivity in IoV leads to substantial concerns in cybersecurity [2].
Regarding cyber threats, the IoV networks are vulnerable to various attacks.
For example, the attacker can impersonate one vehicle in the network stream to
steal or inject false information into a vulnerable vehicle via a spoofing
attack [3]. Additionally, traditional cyberattacks, such as denial of service
(DoS), can severely impact IoV networks by overwhelming them with an intensive
volume of traffic, thereby disrupting network services. Therefore, effective
detection and defence solutions for cybersecurity in IoV networks are urgently
needed to ensure the integrity, reliability, and safety of connected vehicle
systems.

Machine Learning (ML) has emerged as a promising approach that can integrate
with modern networks to form an intelligent intrusion detection system (IDS).
Regarding IoV networks, the authors in [4] propose a Convolutional Long Short
Term Memory Network (ConvLSTM) to detect anomalies in IoT sensors integrated
CAVs. The simulation results show that the deep learning model can detect
various anomalies in sensor data with an F1-score of 97%. Moreover, in [5],
the authors evaluate various deep learning techniques to detect attacks in
vehicular network traffic, which achieve accuracy from 92% to nearly 99%.
However, traditional deep learning approaches operate based on centralized
learning paradigms, which may not be efficient for the decentralized nature of
IoV/IoT networks. Therefore, Federated Learning (FL) is a significant solution
that allows deep learning models to learn with decentralized user-generated
data [6]. Regarding IoT networks, the authors in [7] propose a collaborative
framework that utilizes FL for intrusion detection in IoT networks. Experiment
results using Deep Belief Network (DBN) and Deep Autoencoder (DAE) show the
accuracy of detection from 93% to 98%.

Despite the advantage of FL in IDSs, there are still some major challenges
when deploying it in practical IoV networks. Specifically, in FL-based IDS in
IoV networks, vehicles or RSUs often serve as workers to store and process all
the learning tasks (e.g., training and classification) [1]. However, in
practice, both RSUs and vehicles usually have limited computing and storage
resources, and thus, storing and processing learning tasks at RSUs and VUs are
ineffective. It is important to note that in conventional FL processes, a
delay from one computing node can cause a delay for the whole system [8].
Therefore, several works propose solutions to upload data from RSUs and
vehicles to powerful servers (e.g., centralized servers) for processing [9]
[10]. This approach can be very effective in deploying ML algorithms as all
the data is collected and processed at the centralized servers. However, it
also raises a serious concern regarding the data privacy of VUs, as all the
data is now stored and processed externally.

To overcome the above challenges, we propose a novel privacy-preserving FL
framework for intrusion detection in IoV networks. The proposed framework can
effectively protect VUs’ privacy and detect cyberattacks, given VUs’ limited
computational resources. Specifically, based on the current computing and
storage resource capabilities, VUs can decide the amount of data they need to
upload to the server for processing. To preserve the privacy of the VUs, this
data will be encrypted by employing Homomorphic Encryption (HE) before being
uploaded to the server. While this encryption method enhances data privacy, it
presents significant challenges for the centralized server, which is tasked
with training on the encrypted data offloaded from the VUs. To tackle this
issue, we develop a robust training algorithm leveraging the Single
Instruction Multiple Data (SIMD) and the bootstrapping capabilities of the
underlying HE scheme. This enables direct computation on the quantum-secure
encrypted ciphertexts without the need for decryption. This approach not only
maintains the confidentiality of data during the offloading process from VUs
to the centralized server but also boosts the efficiency of using FL for IDSs
within IoV networks. Simulation results on real-world datasets show that our
proposed framework achieves not only a high accuracy (approximately 91%) in
detecting attacks but also exhibits great performance, closely approaching the
benchmark without using encryption (with a gap of less than 0.8%).

(a) Pre-learning process

(b) Privacy-preserving learning process

Figure 1: The proposed privacy-preserving intrusion detection framework
including pre-learning and privacy-preserving learning.

##  II System Model

The proposed system model is illustrated in Fig. 1. The system consists of a
centralized server (CS), $M$ RSUs and $N$ VUs. Initially, the VUs enter the
pre-learning phase by assessing their computational resources and determining
the optimal amount of data that can be processed locally. The rest of the data
will be offloaded to the centralized server for processing. However, before
offloading the data to the centralized servers, VUs will generate HE key pairs
and use them to encrypt uploading data. The encrypted data will then be
offloaded to the centralized server via RSUs, as illustrated in Fig. 1(a).
Upon receiving the offloaded encrypted data, the centralized server will
compile it into an encrypted dataset. Using this dataset, two learning models
will be developed: the server model and the global model, each serving
distinct purposes. The server model will be utilized to train the encrypted
dataset within the centralized server, whereas the global model will be
distributed to the VUs for local training. Once the global model is sent to
the VUs, the privacy-preserving learning process will commence.

The privacy-preserving learning process will be divided into different
learning periods. During each learning period, each VU will use the global
model to train on its local data. After completing the training, the VU will
encrypt its trained model before sending it to the centralized server.
Concurrently, the centralized server will train its encrypted data using our
proposed CKKS scheme, detailed in Section III-A. Upon receiving all the
encrypted trained models from the VUs, the centralized server will aggregate
them to create a new global model (the aggregation method is detailed in
Section III-B). This updated global model is then sent back to the VUs, and
the next learning period begins. This process repeats until the global model
converges or until a predefined number of learning periods has been completed.

##  III The Privacy-Preserving FL Framework for Encrypted Data

###  III-A The Classification-based Deep Neural Network for Encrypted Data

Figure 2: Encrypt process of a 3x3 matrix W.

To integrate HE with deep neural networks, we propose to use the Cheon-Kim-
Kim-Song (CKKS) scheme. The reason is that it allows the encryption and
calculation of real numbers, which is suitable for deep learning [11]. The
CKKS provides basic HE algorithms as follows [12]:

  * •

$SKGen(n)$: generate random secret key $s_{n}^{sk}$ for user $n$.

  * •

$PKGen(s_{n}^{sk})$: create the public key $s_{n}^{pk}$ for user $n$ based on
the secret key $s_{n}^{sk}$.

  * •

$Enc(s_{n}^{pk},z)$: encrypt vector $z$ into a ciphertext $\hat{z}$ by using
the public key $s_{n}^{pk}$.

  * •

$Dec(s_{n}^{sk},\hat{z})$: decrypt vector $\hat{z}$ into its plain form $z$ by
using the secret key $s_{n}^{sk}$

  * •

$Add(\hat{z},\hat{x})$, $Sub(\hat{z},\hat{x})$, and $Mul(\hat{z},\hat{x})$:
perform element-wise addition, subtraction and multiplication between two
ciphertexts $\hat{z}$ and $\hat{x}$.

Specifically, HE schemes require a ring dimension R, which maintains the
security level, multiplication depth, and noise level [13], thereby allowing
accurate computations over encrypted data. Following that, to design a deep
neural network for encrypted data, we employ the single instruction multiple
data (SIMD) from the CKKS scheme, which packs multiple plaintexts into a
single ciphertext. The size of ciphertext is denoted as B, where
$\textit{B}=\textit{R}/2$. Alternatively, the CKKS can encode and encrypt a
square matrix of size at most $\mu\times\mu$, where
$\mu=\lfloor\sqrt{\textit{B}}\rfloor$ by initially flattening it into a
vector. This thus enables element-wise operation on the plaintext slots
concurrently. For clarity, Fig. 2 describes the implementation of the weight
matrix encryption method. Let $\phi_{i}$ denote the parameter of $i$ linear
layer which $\phi_{i}=(W_{i},b_{i})$. The weight matrix $W^{u\times v}_{i}$
with $u$ and $v$ as the input and output dimensions of the layer is applied to
the encoding process:

$Encode(W_{i})=Flatten\big{(}Pad(W_{i},0,\mu)\big{)},$ (1)

where the matrix $W^{u\times v}_{i}$ is first zero-padded to
$W^{\mu\times\mu}_{i}$ with the size of ($\mu$, $\mu$) to fit within B size.
The weight matrix is then flattened to form an encoded vector. After that,
this encoded vector is padded to ensure its length equals half of the ring
dimension [13]. After that, the encoded vector is encrypted by CKKS, which can
be defined by:

$\hat{W_{i}}=Enc(s^{pk},Encode(W_{i})),$ (2)

where $s^{pk}$ is the public key generated by the users. As a result,
$\hat{W_{i}}$ is the encrypted weight of layer $i$, which can be used to
operate with encrypted training data. Therefore, the output of the forward
propagation over the $i$-th layer can be calculated as:

$\hat{x}_{i+1}=\hat{\sigma}\big{(}Add(Mul(\hat{x}_{i},\hat{W}_{i}),\hat{b}_{i})\big{)},$
(3)

where $\hat{W}_{i}$ and $\hat{b}_{i}$ are the encrypted weight and bias at
layer $i$. Particularly, $\hat{\sigma}$ illustrates the polynomial
approximation of the activation function $\sigma$ using the Chebyshev
polynomial [14]. In the considered deep neural network, the Swish (SiLU)
activation function is chosen due to its advantage in solving the “dying ReLU”
problems [15]. Subsequently, the encrypted output vectors consist of the
distribution of the classes for classification tasks. It is noted that the
Softmax function is not applied in this work due to the exponential and
inverse functions contained, which are non-homomorphic [12].

In the backpropagation process, we apply the Stochastic Gradient Descent (SGD)
for mini-batch regarding the optimization. After calculating the encrypted
gradients for each layer, the SGD update of the encrypted weight can be
formulated as:

$\hat{W_{i}}\leftarrow
BootStrap\Big{(}Sub\Big{(}\hat{W_{i}},Mul\big{(}\eta,\frac{\partial\hat{L}}{\partial\hat{W_{i}}}\big{)}\Big{)}\Big{)}$
(4)

where $\frac{\partial\hat{L}}{\partial\hat{W_{i}}}$ is the calculated
encrypted gradient of $\hat{W_{i}}$, which is computed via the derivative of
encrypted loss function $\hat{L}$. After the SGD update, the encrypted weight
$\hat{W_{i}}$ is applied to the BootStrap method, which renews the ciphertext,
allowing additional computation on $\hat{W_{i}}$ and reduces the magnitude of
accumulated noise [13].

###  III-B The Proposed FL Implementation

In the pre-learning phase, each VU-$n$ evaluates its computing resources and
chooses $p_{n}$% of data to offload. After that, they generate a key pair,
including a secret key $s^{sk}_{n}$ and a public key $s^{pk}_{n}$. In
particular, the VU-$n$ divide its collected dataset $\mathcal{D}_{n}$ into the
local dataset $\mathcal{DR}_{n}$ and offloaded dataset $\mathcal{DS}_{n}$
based on effective computing resources of the vehicles. The $\mathcal{DS}_{n}$
is then encrypted to $\hat{\mathcal{DS}}_{n}$ to protect the user data. The
encrypted data is sent to the CS and combined to form an encrypted dataset
$\hat{\mathcal{DS}}$. After that, the CS initializes the non-encrypt global
model $\mathcal{M}_{g}$ and non-encrypt server model $\mathcal{M}_{s}$, then
distributes $\mathcal{M}_{g}$ to each VU for local training. Subsequently, the
public key is used to initialize the encrypted learning model
$\hat{\mathcal{M}}_{s}$ and encrypted global model $\hat{\mathcal{M}}_{g}$ on
the server.

Regarding the privacy-preserving learning phase, we consider $T$ learning
rounds. At each round $\tau$, privacy is maintained by the non-encrypted
training from the local learning model of VU-$n$ and encrypted training from
the privacy-preserving learning model. After finishing the $\tau^{\text{th}}$
local training round, VU-$n$ encrypts the trained parameters
$\hat{\phi}_{\tau}^{n}$ and sends them to the CS. The CS retrieves the
encrypted parameters from $\hat{\mathcal{M}}_{s}$ along with the local
encrypted parameters and aggregates by the FedAvg algorithm for encrypted
data, which can be defined by:

$\hat{\phi}_{g}^{(\tau+1)}=Mul\Big{(}\frac{1}{N+1},Add\Big{(}\sum_{n=1}^{N}\hat{\phi}_{n}^{\tau},\hat{\phi}_{s}^{\tau}\Big{)}\Big{)}.$
(5)

The global parameters $\hat{\phi}_{g}^{(\tau+1)}$ are then updated to the
encrypted global model $\hat{\mathcal{M}}_{g}$ and sent back to the VUs, which
is then decrypted by the VUs for the next learning round. The learning process
continues until the global model converges and obtains the optimized
parameters.

Algorithm 1 Proposed Privacy-Preserving FL Framework

1:for $\forall n\in N$ do

2: Calculate $p_{n}$% for offloading

3: Generate a secret key and a public key: $s_{n}^{sk}=SKGen(n)$ and
$s_{n}^{pk}=PKGen(s_{n}^{sk})$

4: Split the dataset $\mathcal{D}_{n}$ into $\mathcal{DR}_{n}$ and
$\mathcal{DS}_{n}$ where $\mathcal{DS}_{n}=\mathcal{D}_{n}\times p_{n}$ and
$\mathcal{DR}_{n}=\mathcal{D}_{n}-\mathcal{DS}_{n}$

5: Generate the encrypted data
$\hat{\mathcal{DS}_{n}}=Enc(s_{n}^{pk},\mathcal{DS}_{n})$

6: Send encrypted data $\hat{\mathcal{DS}_{n}}$ to the CS

7:end for

8:CS combines the received encrypted data $\hat{\mathcal{DS}_{n}}$ into
$\hat{\mathcal{DS}}$

9:CS initializes the $\mathcal{M}_{g}$ and $\mathcal{M}_{s}=\mathcal{M}_{g}$

10:Transmit $\mathcal{M}_{n}$ to $N$ VUs which
$\mathcal{M}_{n}=\mathcal{M}_{g}$

11:Generate encrypted model $\hat{\mathcal{M}_{s}}$ and
$\hat{\mathcal{M}_{g}}$ via $s_{n}^{pk}$ where $\phi_{n}=Dec(\hat{\phi_{n}})$

12:while $\tau\leq T_{\max}$ or training process does not converge do

13: $\hat{\mathcal{M}_{s}}$ learns the encrypted data $\hat{\mathcal{DS}}$

14: $\hat{\mathcal{M}_{s}}$ produces encrypted parameters
$\hat{\phi}_{s}^{\tau}$

15: for $n\in N$ do

16: $\mathcal{M}_{n}$ learns the local data $\mathcal{DR}_{n}$

17: Calculate local parameters $\phi_{\tau}^{n}$

18: Encrypt local parameters
$\hat{\phi}_{\tau}^{n}=Enc(s_{n}^{pk},\phi_{\tau}^{n})$

19: Send local encrypted parameters to the CS.

20: end for

21: The CS calculates and produces the encrypted global model
$\hat{\phi}_{g}^{(\tau+1)}$.

22: Send the updated global model $\hat{\phi}_{g}^{(\tau+1)}$ back to $N$ VUs

23: for $\forall n\in N$ do

24: Decrypt the model
$\phi_{g}^{(\tau+1)}=Dec(s_{n}^{sk},\hat{\phi}_{g}^{(\tau+1)})$

25: end for

26:end while

27:Predict $\hat{Y}_{n}$ based on the encrypted training data $\hat{X}_{n}$ at
each VU-n and optimal global model $\phi^{*}$.

In summary, the learning process of the privacy-preserving intrusion detection
framework for IoV is described in Algorithm 1.

##  IV Performance Evaluation

###  IV-A Simulation Setup

In this section, the proposed privacy-preserving model is validated on the
real-world dataset of network traffic attacks on IoT devices, named Edge-IIoT
dataset [16]. The Edge-IIoT dataset includes 20 million raw normal traffic and
attack traffic collected from 13 IoT devices. Attacks can be grouped into the
five most common types, including Distributed Denial of Service (DDoS),
Injection, Man-in-the-Middle (MitM), Malware, and Reconnaissance. After
applying downsample and oversample to overcome the imbalance, the dataset
includes 31,400 samples with details presented in Table I. Subsequently, the
dataset is divided into training and testing sets (80%-20%). The training and
testing sets are then nominalized and scaled within the range of (0,1).
Regarding the neural network, we design a fully connected network consisting
of an input layer, 2 hidden layers, and an output layer. The respective layers
contain 32, 16, 16, and 6 neurons. Apart from the input layer, each layer is
attached to the SiLU activation function.

TABLE I: The distribution of classes of the dataset Class | Number of samples
---|---
Normal | 5,320
DDoS | 5,472
MitM | 4,000
Injection | 5,589
Malware | 5,504
Reconnaissance | 5,515
Total | 31,400

During the simulation, we assume that the collected dataset of each VU has the
same class distribution. Similar to [10], we consider the approach to offload
partial data to the server as the benchmark for our proposed framework.
Nevertheless, it is noted that in such a benchmark, the FL framework does not
consider the privacy of the VUs. In this scenario, we train the non-encrypted
data using the non-encrypted model at both CS and VUs. Consequently, the
trained global model is employed to evaluate the accuracy of our proposed
framework and other benchmarks. In our experiment setup, the proposed
framework consists of 2 VUs and 3 VUs, which can send 10% and 20% of their
local data.

###  IV-B Evaluation Metrics

To evaluate the performance of the detection model, the confusion matrix is
utilized, which is suitable for a machine learning-based classification system
[17]. We denote TP, TN, FP, and FN as “True Positive”, “True Negative”, “False
Positive”, and “False Negative”. Assuming the system consists of $C$ classes,
which include normal and attack traffic, the accuracy can be calculated as:

$\text{$Accuracy$}=\frac{1}{C}\sum_{c=1}^{C}\frac{TP_{c}+TN_{c}}{TP_{c}+TN_{c}+FP_{c}+FN_{c}}.$
(6)

The macro-average precision and recall are utilized in this term. Given $K$ as
the number of classes in the system, the macro-average precision is:

$\text{$Precision$}=\frac{1}{K}\sum_{k=1}^{K}\frac{TP_{k}}{TP_{k}+FP_{k}}.$
(7)

The macro-average recall is calculated as follows:

$\text{$Recall$}=\frac{1}{K}\sum_{k=1}^{K}\frac{TP_{k}}{TP_{k}+FN_{k}}.$ (8)

(a) 2 VUs with 10% offload data

(b) 2 VUs with 20% offload data

(c) 3 VUs with 10% offload data

(d) 3 VUs with 20% offload data

Figure 3: Convergence of privacy-preserving learning with VUs. TABLE II: Simulation results Model | 2 Vehicle Users | 3 Vehicle Users
---|---|---
N-EncFL | EncFL | N-EncFL | EncFL
|  10% data |  20% data |  10% data |  20% data |  10% data |  20% data |  10% data |  20% data
Accuracy |  91.728 |  91.806 |  91.173 |  91.142 |  91.806 |  91.744 |  90.926 |  91.049
Precision |  92.767 |  92.868 |  92.319 |  92.254 |  92.787 |  92.730 |  91.992 |  92.161
Recall |  91.875 |  91.930 |  91.360 |  91.322 |  91.931 |  91.870 |  91.118 |  91.234

(a) Offloading with 10% non-encrypt data.

(b) Offloading with 10% encrypted data.

Figure 4: Classification results of 2 VUs

###  IV-C Simulation Results

(a) Offloading with 20% non-encrypt data.

(b) Offloading with 20% encrypted data.

Figure 5: Classification results of 3 VUs

####  IV-C1 Convergence Analysis

Fig. 3 illustrates the convergence of learning processes from three
approaches, including conventional FL (CFL), FL with non-encrypted offloaded
data (N-EncFL) and the proposed privacy-preserving learning (EncFL). As
observed in Fig. 3(a) and Fig. 3(b) with 2 VUs, the CFL converges after 70
iterations, while the N-EncFL and EncFL require nearly 100 iterations to reach
the convergence. Specifically, due to the different amounts of data handled by
the VU, the CFL demonstrates a slightly better convergence rate compared to
other approaches. Despite the trivial difference in convergence, the accuracy
during the learning process of the three approaches remains nearly identical,
stabilizing at approximately 92%. Additionally, Fig. 3(c) and Fig. 3(d)
describe the convergences in the scenarios with 3 VUs. Although the N-EncFL
and traditional methods converge at nearly the same time, the EncFL require
over 100 iterations to reach the convergence, which is slightly longer than
other approaches. However, the gap in learning rate, which is about 15 to 20
iterations, is trivial. It is worth noting that the accuracy of EncFL remains
consistent with that of the N-EncFL and CFL, regardless of whether the amount
of offloaded data is different. As a result, the proposed framework, which
operates on encrypted data, achieves the same accuracy as those of the other
benchmarks, i.e., N-EncFL and CFL.

####  IV-C2 Performance Evaluation

Table II describes the performance in detecting attacks of two and three VUs
in the IoV network. Overall, the accuracy, precision and recall of the two
scenarios remain nearly identical. Regarding the different amounts of
offloaded data, the results for N-EncFL and EncFL are close to those of other
methods. Specifically, even when the data sent is 10% or 20%, the N-EncFL with
two or three VUs achieves an accuracy of approximately 91.8%. A similar trend
is observed with EncFL, where the accuracy remains consistent regardless of
the varying amounts of data offloaded. When comparing the results of EncFL and
N-EncFL, we can observe that the accuracy, precision and recall of EncFL are
slightly lower than those of the N-EncFL. In detail, the gap between N-EncFL
and EncFL with two VUs is from 0.5% to 0.6%. For instance, with 10% offloaded
data, EncFL achieves an accuracy of 91.173%, which is 0.55% less than
N-EncFL’s 91.728%. Additionally, the results of the three VUs show a similar
pattern, with EncFL performing 0.6% to 0.8% less than RawFL. However, as
observed in Fig. 4 and Fig. 5, the overall accuracy of 6 classes is nearly the
same. The differences primarily lie in the detection of “Injection” and
“Reconnaissance”, which accounts for the small gap between N-EncFL and EncFL.
Although the accuracy of the “Injection” class of EncFL is less than N-EncFL,
EncFL still achieves an 88% detection rate accuracy for the “Injection”
attack. As a result, the small gap between N-EncFL and EncFL is acceptable,
demonstrating that EncFL can classify each class with a high detection rate.

##  V Conclusion

In this paper, we have proposed a novel privacy-preserving FL framework for
intrusion detection in IoVs with limited computing resources. The proposed
framework enables users to offload data to a centralized server, addressing
the computational challenges during local training of the vehicles. To ensure
user privacy, homomorphic encryption (HE) is applied to the data before
offloading it to the server. The encrypted data is then processed by the
training algorithm-based HE, which allows the server to learn from the
encrypted data without knowing its content. The proposed framework can protect
the privacy of users during the learning process, facilitating the efficient
deployment of FL for IDSs in practical IoV networks. The simulation results
show that our proposed framework can accurately detect cyberattacks in IoV
networks. Although the accuracy of the encrypted neural network is slightly
less than that of the raw ones, the gap is acceptable and can be optimized in
future works.

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