# Gravitational wave memory in wormhole spacetimes

Indranil Chakraborty  Department of Physics, Indian Institute of Technology
Kharagpur, Kharagpur 721302, India<EMAIL_ADDRESS>Soumya Bhattacharya
Department of Astrophysics and High Energy Physics, S.N. Bose National Center
for Basic Sciences, Kolkata 700106, India<EMAIL_ADDRESS>Sumanta
Chakraborty  School of Physical Sciences, Indian Association for the
Cultivation of Science, Kolkata 700032, India<EMAIL_ADDRESS>
###### Abstract

Gravitational wave (GW) memory is studied in the context of a certain class of
braneworld wormholes. Unlike other wormhole geometries, this novel class of
wormholes do not require any exotic matter fields for its traversability.
First, we study geodesics in this wormhole spacetime, in the presence of a GW
pulse. The resulting evolution of the geodesic separation shows the presence
of displacement and velocity memory effects. Motivated by the same, we study
the memory effects at null infinity using the Bondi-Sachs formalism, adapted
for braneworld wormhole. Our analysis provides a non-trivial change of the
Bondi mass after the passage of a burst of gravitational radiation and hence
manifests the memory effect at null infinity. In both of these exercises, the
presence of extra dimension and the wormhole nature of the spacetime geometry
gets imprinted in the memory effect. Since future GW detectors will be able to
probe the memory effect, the present work provides another avenue to search
for compact objects other than black holes.

##  I Introduction

The direct detection of Gravitational Waves (GWs) from binary black hole and
binary neutron star merger events [1, 2], as well as the observations of the
shadow of supermassive compact central objects e.g., the M87* and the SgrA*
[3, 4, 5, 6, 7, 8, 9], are the two major observational breakthroughs in the
field of gravitational physics within the last decade. Both of these
observations depend crucially on the strong field behaviour of gravity and, in
principle, can be used to test the robustness of General Relativity (GR) and
also provide crucial pointers to the nature of the compact objects [10, 11,
12, 13, 14, 15, 16, 17, 18]. Despite the fact that — so far GR has passed
these strong field tests without any scar — from a purely theoretical
perspective, the inability of GR to correctly reconcile singularities
occurring at the end point of gravitational collapse, or, as the starting
point of our modern hot big bang cosmology, posits a serious challenge to the
theory itself. This important shortcoming of GR must make room for
investigating alternative possibilities, vis-a-vis modified near-horizon
geometries. Among the alternatives, one can either look for higher
curvature/higher dimensional theories of gravity, possibly emerging from some
quantum gravity scenarios, or, modifications of the black hole paradigm
itself. Such non-black hole compact objects can arise from quantum gravity-
motivated theories e.g., fuzzballs [19] or, compact objects made out of exotic
matter fields, known as ECOs [20, 21, 22, 23, 24]. Both of these classes of
non-black hole objects, behave as a black hole for solar system tests of
gravity, but appears to have distinct phenomenology, in contrast to that of
the black hole, when they are probed using the strong field tests of gravity,
in particular, GWs [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].

However, the present generation of GW detectors are not sensitive enough to
detect the modifications in the GW waveform originating from such non-black
hole objects. In absence of such definitive tests that can either confirm or
nullify the existence of these non-black hole objects, it is necessary to
study the strong-gravity signatures of these exotic objects in order to gain a
better understanding of their physical characteristics. Here we attempt to
study some properties associated with GWs in the background of one of the
oldest as well as interesting class of ECOs, viz. wormholes. These are
spacetimes joining two distinct universes by a throat [37, 38, 39, 40, 41, 42,
43, 44, 45, 46]. If one can travel from one universe to the other, it will be
referred to as a traversable wormhole and this, in general, requires exotic
matter fields. However, there exist one class of wormholes, which do not
require exotic matter for its existence, known as the braneworld wormholes
[47], corresponding to Randall-Sundrum two braneworld scenario [48]. This has
two advantages — (a) the presence of higher dimension is imprinted on this
wormhole solution and hence can possibly be tested using GWs or, black hole
shadow measurements, (b) the wormhole does not require any exotic matter on
the four-dimensional brane, the contributions from higher dimensions take care
of the exotic nature to support traversability of the wormhole. Therefore, it
is worthwhile to study various properties of this wormhole and hence look for
its observational signatures. This is because, it will not only validate the
existence of wormholes, but also of compact extra dimensions, thus providing a
unified testing ground for non-black hole nature of compact objects, vis-a-vis
of theories beyond GR. Several aspects of this braneworld wormhole, e.g., its
stability under perturbations due to various fields, in particular the
ringdown structure [49], as well as implications for black hole shadow
measurements [50], have already been explored. Intriguingly, all of these
explorations have provided quite promising results and in particular,
demonstrates the existence of echoes in the GW signal. In this work, we wish
to explore another direction, which has remained largely unexplored, but holds
immense potential for the future generations of GW detectors, namely the
gravitational memory effect.

With the improvement in the detection prospects for the future ground based GW
detectors and also the launch of the space based-detector LISA in the near
future, may provide us an opportunity to observe GW memory effect [51]. The
memory effect brings in both the strong-field, as well as non-linear aspects
of general relativity, which is yet to be observed. It refers to the lasting
change in the relative distance between test particles when a GW passes
through the ambient spacetime [52, 53]. Being a subtle DC shift to the overall
GW amplitude, it has remained undetected in the past observations taken by
LIGO [54]. There have been proposals of stacking the known GW signals observed
by LIGO-Virgo in order to detect this effect [55]. Initially studied in the
context of hyperbolic scattering [56] and gravitational bremsstrahlung [57],
the memory effect was also sho [58]. Recent works on memory effects involve,
generalization to electrodynamics [59, 60] and Yang-Mills theories [61, 62],
investigating features of extra dimensions [63, 64, 65], distinguishing
modified theories of gravity from GR, e.g., scalar-tensor theories [66, 67,
68, 69] and Chern-Simons gravity [70]. Moreover, there have also been works
generalizing memory effects to the symmetries associated with near horizon
geometries for black holes [71, 72, 73]. In this work, we wish to study the
gravitational memory effect for braneworld wormholes, a class of ECOs, for the
first time. Our aim, in the present context, is to infer the imprints of the
non-black hole nature of this static wormhole spacetime and the presence of
extra dimensions on the memory effect. We attempt this exercise of finding the
memory effects in two distinct ways — (a) we perform a geodesic analysis (as
worked out in [52, 74]) and comment on the presence of displacement and
velocity memory effect in the background wormhole spacetime by introducing an
additional GW perturbation, (b) we study memory effects at null infinity using
the Bondi-Sachs formalism [75].

The organization of the paper is as follows: in II we briefly review the
wormhole geometry in the context of braneworld scenario. III deals with the
study of geodesics and subsequent analysis of displacement and velocity memory
effects and finally in IV, we will study the influence of a perturbing GW on
the wormhole metric in the Bondi-Sachs gauge and the associated null memory
effects. We conclude with a discussion on the results obtained and provide an
outline of the future directions to pursue.

##  II Brief review of wormhole geometry in the braneworld scenario

Let us now discuss briefly the braneworld model under investigation. In the
model, we have a five dimensional spacetime (referred to as the bulk), within
which two four dimensional branes are embedded. The extra dimension is
spacelike in nature and is described by the coordinate $y$. One of the brane
is located at $y=0$, and dubbed as the Planck brane, while the other one is
located at $y=\ell$, referred to as the visible brane. The proper distance
between the two branes is given by the integral of the $g_{yy}$ component over
the extent of the extra dimension, yielding $d(x)=e^{\phi(x)}\ell$, where the
field $\phi(x)$ is referred to as the radion field. Since we are interested in
the measurements done by a four-dimensional observer, it will suffice to
consider the low energy effective gravitational field equations. This can be
achieved by projecting the five dimensional Einstein’s equations on the four
dimensional brane, and then expanding the same in the ratio of the
(bulk/brane) curvature length scale. Thus we finally obtain the following
gravitational field equations on the visible brane [76, 47],

$\begin{split}G_{\mu\nu}&=\frac{\kappa_{5}^{2}}{\ell\Phi}T^{\rm
V}_{\mu\nu}+\frac{\kappa_{5}^{2}(1+\Phi)}{\ell\Phi}T^{\rm
P}_{\mu\nu}+\frac{1}{\Phi}\Big{(}\nabla_{\mu}\nabla_{\nu}\Phi-
g_{\mu\nu}\nabla^{\alpha}\nabla_{\alpha}\Phi\Big{)}-\frac{3}{2\Phi(1+\Phi)}\Big{(}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}\nabla^{\alpha}\nabla_{\alpha}\Phi\Big{)}~{}.\end{split}$
(1)

Here $g_{\mu\nu}$ is the visible brane metric, $\nabla_{\mu}$ is the covariant
derivative with respect to $g_{\mu\nu}$ and $\kappa_{5}^{2}$ is the five
dimensional gravitational coupling constant. Moreover, $T^{\rm P}_{\mu\nu}$
and $T^{\rm V}_{\mu\nu}$ are the energy momentum tensors on the Planck brane
and the visible brane, respectively. The scalar field $\Phi$, appearing in 1
is defined as, $\Phi\equiv\exp[2e^{\phi(x)}]-1$, where $\phi(x)$ is the radion
field. The energy density of the on-brane matter field must be small compared
to the brane tension in order for the low-energy effective theory to hold.
Wormholes, being non-singular solutions, have finite energy density and
pressure for the on-brane matter fields everywhere, thereby ensuring the
validity of this theory in the contexts of wormhole geometry [47].

In addition, it must be noted that the wormholes require exotic matter fields
for their traversability, at least in the context of GR. This is because the
violation in the convergence condition for timelike geodesics, leads to a
violation of the energy conditions for the stress-energy tensor sourcing the
wormhole geometry [37]. However, in the case of braneworld scenario, the total
energy-momentum tensor has contributions from the matter present on the two
3-branes (visible and planck) and a geometric stress due to the radion field
generated from the bulk spacetime. Hence, one can sustain this wormhole
geometry with the on-brane matter satisfying the energy conditions and the
violations of the energy conditions for the total energy-momentum tensor can
be attributed to the bulk spacetime (similar situations may arise in the
context of scalar coupled Gauss-Bonnet gravity, see e.g., [77]). Thus the
resulting wormhole solution will be constructed out of normal matter fields on
the brane, with exotic matter field on the bulk. Since we cannot access the
energy scale of the bulk spacetime, the existence of such exotic matter in the
bulk is not of much concern to our present analysis. Moreover, such braneworld
wormholes have also been shown to be stable under scalar, electromagnetic and
axial gravitational perturbations in [49].

In order to avoid non-locality in the theory, i.e., we do not want the
dynamics of the visible brane to be governed by the energy momentum tensor of
the Planck brane, we will work with $T^{\rm P}_{\mu\nu}=0$ i.e., there is no
matter on the Planck brane. With this choice, the field equation for $\Phi$
takes the following form,

$\nabla^{\alpha}\nabla_{\alpha}\Phi=\frac{\tilde{p}^{2}}{\ell}\frac{T^{\rm
V}}{2\omega+3}-\frac{1}{2\omega+3}\frac{d\omega}{d\Phi}\Big{(}\nabla^{\alpha}\Phi\Big{)}\Big{(}\nabla_{\alpha}\Phi\Big{)}~{},$
(2)

where, $T^{\rm V}$ is the trace of the energy momentum tensor on the visible
brane and the coupling function $\omega(\Phi)$, is defined as,

$\omega(\Phi)=-\frac{3\Phi}{2(1+\Phi)}~{}.$ (3)

Thus, the low energy effective braneworld scenario can be written as a
generalized Brans-Dicke (BD) [78] theory, with a variable BD parameter
$\omega(\Phi)$. Given these, we rewrite the gravitational field equations as,

$G_{\mu\nu}=\frac{\kappa_{5}^{2}}{l\Phi}T_{\mu\nu}^{V}+\frac{1}{\Phi}T_{\mu\nu}^{\Phi}~{}.$
(4)

Here, $T_{\mu\nu}^{\Phi}$ is to be identified with the sum of the third and
the fourth terms of the right hand side of 1, without the $(1/\Phi)$ part.

The above set of equations for the metric functions, as well as for the scalar
field $\Phi$, can be solved assuming a static and spherically symmetric metric
ansatz along with an anisotropic fluid source on the visible brane with
vanishing trace. This simplifies the problem of solving the field equations a
lot and one arrives at a two-parameter family of solutions with $R=0$, written
in the Schwarzschild-like coordinates as [79, 47],

$\displaystyle ds^{2}$
$\displaystyle=-\left(\kappa+\lambda\sqrt{1-\frac{2M}{r}}\right)^{2}dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}$
$\displaystyle\hskip
28.45274pt+r^{2}\Big{(}d\theta^{2}+\sin\theta^{2}d\phi^{2}\Big{)}~{}.$ (5)

Here, in the limit, $\kappa=0$ and $\lambda=1$, we get back the Schwarzschild
geometry. Note that in 5, the $tt$-component of the metric is not given in the
standard asymptotic form, since it does not reduce to unity in the limit of
$r\rightarrow\infty$, rather to $(\kappa+\lambda)^{2}$. Therefore, we rescale
the time coordinate by $t\to t/(\kappa+\lambda)$ and define the ratio
$\frac{\kappa}{\lambda}=p$, since this is the only parameter which
characterises the wormhole geometry. Finally, using the coordinate
transformation $u=t-r_{*}$, where $r_{*}$ is the Tortoise coordinate, defined
as $(dr_{*}/dr)=(1/\sqrt{-g_{tt}g^{rr}})$, the metric in 5 becomes,

$ds^{2}=-f(r)~{}du^{2}-2g(r)~{}dudr+r^{2}~{}d\theta^{2}+r^{2}~{}\sin^{2}\theta\,d\phi^{2}~{}.$
(6)

Here we have denoted the $uu$ and $ur$ components of the metric as $f(r)$ and
$g(r)$, with the following expressions for them,

$\displaystyle f(r)$
$\displaystyle\equiv\bigg{(}\frac{p}{p+1}+\frac{1}{1+p}\sqrt{1-\frac{2M}{r}}\bigg{)}^{2}~{},$
(7) $\displaystyle g(r)$
$\displaystyle\equiv\bigg{(}\frac{p}{p+1}+\frac{1}{1+p}\sqrt{1-\frac{2M}{r}}\bigg{)}\bigg{(}1-\frac{2M}{r}\bigg{)}^{-1/2}~{}.$
(8)

The scalar field, on the other hand, is best written in the isotropic
coordinate $r^{\prime}$, such that

$\Phi(r^{\prime})=\Big{(}\frac{C_{1}}{M}\log\frac{2r^{\prime}q+M}{2r^{\prime}+M}+C_{4}\Big{)}^{2}-1~{},$
(9)

with the isotropic coordinate $r^{\prime}$ being related to the Schwarzschild
coordinate $r$ through the following relation:
$r=r^{\prime}(1+\frac{M}{2r^{\prime}})^{2}$. Note that the two coordinates $r$
and $r^{\prime}$ become identical in the asymptotic limit. Moreover, $C_{1}$
and $C_{4}$, appearing in 9, are positive non-zero constants and
$q=\\{(p+1)/(p-1)\\}$, where $p$ is the wormhole parameter, defined earlier.

Unlike the Schwarzschild spacetime, here the radial coordinate $r$ can be
extended only upto $r=2M$ and it is also evident from 5 that as long as
$\kappa$ is non-zero and positive, $g_{tt}\neq 0$ for all $r\geq 2M$. This
suggests that there is no event horizon in this spacetime. Though, the surface
$r=2M$ is not an event horizon, it is indeed null, as $g^{rr}$ vanishes there
and hence in the above solution as well $r=2M$ is a special surface, and is
referred to as the throat of the wormhole. Physically, the above solution
depicts two separate universes connected together at the throat, located at
$r=2M$, which is traversable. The expression for the anisotropic fluid matter
at the throat, necessary for traversability can be obtained from [47]. We do
not provide it here, since this is not required for the computation of
gravitational memory, which is the prime goal in the present context. The
above provides a broad introduction to the wormhole geometry we will be
working with and we shall apply these results in order to compute the memory
effect, to be discussed below.

##  III Memory of geodesics

In this section, we will present the analysis of the memory effect vis-a-vis
the geodesic deviation between neighbouring geodesics due to a propagating GW,
with the geodesic separation quantifying the amount of displacement memory
effect. Moreover, if the geodesics do not have constant separation, after the
passage of the GW pulse, one can also associate a velocity memory effect with
these geodesics as well.

Such effects have been studied in the recent past [80, 74, 81, 82, 83] by
investigating the evolution of geodesics in exact plane GW spacetimes. By
choosing a Gaussian pulse for the polarization (radiative) term in the line
element of the plane GW spacetime, the geodesic equations can be solved
numerically, and then the change in the separation and velocity (the
displacement and the velocity memory, respectively) can be computed due to the
passage of the GW pulse. Lately, the above formalism has also been generalized
in the context of alternative theories of gravity [84, 85] to look for
signatures of such alternative theories in the displacement and velocity
memory effects. In the present work, we will study the evolution of the
geodesics in the wormhole background presented above, in the presence of a GW
pulse and study how the displacement and velocity memory effects depend on the
wormhole nature of the background geometry. For this purpose, we write down
the spacetime metric as a sum of the background wormhole geometry $g_{\mu\nu}$
and a GW perturbation $h_{\mu\nu}$, such that the line element becomes,

$ds^{2}=(g_{\mu\nu}+h_{\mu\nu})\,dx^{\mu}\,dx^{\nu}~{}.$ (10)

As we have already mentioned, $g_{\mu\nu}$ is the wormhole metric given in 6
and $h_{\mu\nu}$ is the GW perturbation. Using the wormhole geometry presented
in 6 explicitly, The resulting geometry becomes,

$\displaystyle ds^{2}$
$\displaystyle=-f(r)\,du^{2}-2g(r)\,dudr+\left[r^{2}+rH(u)\right]~{}d\theta^{2}$
$\displaystyle\hskip
14.22636pt+\left[r^{2}-rH(u)\right]~{}\sin^{2}\theta\,d\phi^{2}~{},$ (11)

where, $f(r)$ and $g(r)$ are given by 7 and 8, respectively. The function
$H(u)$ corresponds to the GW pulse and we have assumed that $h_{\mu\nu}$ can
be expressed in the transverse-traceless (TT) gauge. In order to generate this
perturbation in the wormhole metric, we need to include in the matter sector
an energy momentum tensor, which can source the GW pulse $H(u)$ in the TT
gauge. Such an energy momentum tensor can arise from an expanding anisotropic
fluid shell. Prior to the perturbation, the fluid was non-dynamical and the
$u$-constant hypersurfaces were spherically symmetric. Due to this expansion,
the GW pulse is generated and it propagates over the wormhole spacetime to
future null infinity. In what follows, we will derive the displacement and the
velocity memory effects as the propagating GW crosses a congruence of timelike
geodesics in the wormhole background, leading to a change in the separation
between these comoving geodesics.

The GW pulse profile described by $H(u)$ is taken to be,
$H(u)=A\operatorname{sech}^{2}(u-u_{0})$, where $A$ denotes the amplitude of
the GW pulse and it is centered around $u=u_{0}$. Since the above wormhole
spacetime along with the GW pulse respects the spherical symmetry, we can
choose the equatorial plane, located at $\theta=(\pi/2)$, to be the plane on
which all the geodesics are located. Therefore, on the equatorial plane, the
geodesic equations for the $u$ coordinate becomes,

$\displaystyle\ddot{u}$
$\displaystyle-\frac{f^{\prime}}{2g}\dot{u}^{2}-\frac{H-2r}{2g}\dot{\phi}^{2}=0~{},$
(12)

Along identical lines one can arrive at the geodesic equations for the other
coordinates on the equatorial plane, in particular, the respective geodesic
equations for the $r$ and the $\phi$ coordinates are given as,

$\displaystyle\ddot{r}$
$\displaystyle+\frac{f}{2g^{2}}f^{\prime}\dot{u}^{2}+\frac{f^{\prime}}{g}\dot{r}\dot{u}+\frac{g^{\prime}}{g}\dot{r}^{2}$
$\displaystyle\hskip 56.9055pt+\frac{fH-2fr-
rgH^{\prime}}{2g^{2}}\dot{\phi}^{2}=0~{},$ (13) $\displaystyle\ddot{\phi}$
$\displaystyle-\frac{H^{\prime}}{r-H}\dot{\phi}\dot{u}+\frac{2r-H}{r(r-H)}\dot{\phi}\dot{r}=0~{}.$
(14)

As mentioned before, all the above equations are written on the
$\theta=\frac{\pi}{2}$ plane and here ‘overdot’ denotes derivative with
respect to the proper time $\tau$ associated with the geodesics, while ‘prime’
denotes derivative with respect to the argument of the respective function.
For example, $H^{\prime}\equiv(dH/du)$ and $f^{\prime}=(df/dr)$.

(a) Variation of the difference between null coordinate of the two timelike
geodesics, denoted by $\Delta u$ with respect to the proper time $\tau$, for
different values of the higher dimensional parameter $p$ has been presented.

(b) Variation of the difference between the radial coordinates of the timelike
geodesics, namely $\Delta r$ has been presented against the proper time $\tau$
for different values of the wormhole parameter $p$.

Figure 1: Variation of the difference of both the coordinates between the
timelike geodesics, namely $\Delta u$ and $\Delta r$ has been plotted with the
proper time for different choices of the wormhole parameters.

(a) Variation of the difference between null coordinate of the two timelike
geodesics, denoted by $\Delta u$, has been presented against the proper time
$\tau$, in the presence of and in the absence of the GW pulse.

(b) Variation of the difference between the radial coordinates of the timelike
geodesics, namely $\Delta r$, has been depicted with the proper time $\tau$ in
the presence as well as in the absence of the GW pulse.

Figure 2: The variation of $\Delta u$ and $\Delta r$ with the proper time of
the geodesics have been presented in the presence and in the absence of GW
pulse. The plots explicitly demonstrates that both $\Delta u$ and $\Delta r$
encodes information about the passage of the GW pulse in the past, thus
depicts the displacement memory effect.

(a) Variation of the difference between the velocity along null direction $u$
of the two timelike geodesics, have been presented against the proper time
$\tau$, in the presence as well as in the absence of the GW pulse.

(b) Variation of the difference between the velocity along the radial
direction $r$ of the timelike geodesics has been presented with the proper
time $\tau$ with and without the GW pulse.

Figure 3: We have explicitly depicted that the expressions for $\Delta\dot{u}$
and $\Delta\dot{r}$ deviate significantly in the presence of a GW pulse. This
immediately suggests the existence of a velocity memory effect.

We have solved the three geodesic equations, presented in 12 \- 14 numerically
in the symbolic manipulation software Mathematica and have analysed the
solutions extensively. In particular, we have started by considering two
neighbouring geodesics in the wormhole background and then have studied the
evolution of their coordinate separation in terms of the proper time $\tau$,
which is also an affine parameter, see 1. Further, we have depicted in 2, how
the evolution of these coordinate separations have been affected because of
the presence of the GW pulse and also by the presence of extra dimension
through non-zero values of $p$. In particular, as 2 demonstrates, even after
the GW pulse has passed, there is a residual effect which manifests as the GW
memory, more specifically, as the displacement memory effect. It turns out
that these geodesics also inhibit velocity memory effects, which can be seen
from 3.

Let us discuss the above scenario in detail and consider the various boundary
conditions imposed on them. First of all, in this work we have considered two
neighbouring timelike geodesics, which in the background wormhole geometry
would satisfy the following condition on the equatorial plane,

$-f(r)\dot{u}^{2}-2g(r)\dot{u}\dot{r}+r^{2}\dot{\phi}^{2}-rH(u)\dot{\phi}^{2}=-1~{}.$
(15)

We choose the initial conditions as follows: both the geodesics are chosen to
have an initial separation in the radial coordinate $r$, as well as in the
null coordinate $u$, however they both start at the same azimuthal coordinate
$\phi$. These geodesics have fixed initial values of $\dot{r}$ and $\dot{u}$,
while the value of $\dot{\phi}$ will depend on the background geometry through
15. Then we define the following quantities $\Delta u$ and $\Delta r$, as
follows,

$\displaystyle\Delta u=u({\rm Geodesic~{}II})-u(\rm Geodesic~{}I)~{},$
$\displaystyle\Delta r=r({\rm Geodesic~{}II})-r(\rm Geodesic~{}I)~{},$

where, $u(\rm Geodesic~{}I)$ and $r(\rm Geodesic~{}I)$ corresponds to the
coordinates associated with the geodesic I and $u(\rm Geodesic~{}II)$ and
$r(\rm Geodesic~{}II)$ are the coordinates associated with the geodesic II.
All of these are obtained by solving the geodesic equations, using appropriate
initial conditions, as discussed before. We choose our pulse profile such as
$A=1$ and $u_{0}=5$. The results arising out of the time evolution of $\Delta
u$ and $\Delta r$ for different values of wormhole parameter $p$ have been
depicted in 1.We consider three cases corresponding to the following values of
$p$ — i) $p=2.0$, ii) $p=1.5$ and iii) $p=1.05$. The value of the mass is set
to $M=3$ for obtaining the plots. It is clearly seen from these plots that
$\Delta u$ is increasing with the decreasing values of $p$ (1(a)), however,
$\Delta r$ is decreasing with the decreasing values of $p$ (1(b)), i.e.,
$\Delta u$ and $\Delta r$ have opposite behaviours with change of $p$. As
evident from these figures, the change $\Delta u$ is more pronounced as
compared to $\Delta r$. For clarity, in 1, the GW pulse is represented by the
filled region. The pulse we have shown in the plot is not the exact profile of
the GW, rather we have used a scaled-up version of the original profile. The
affine parameter interval for which the GW pulse remained significant remains
unaltered. We will now discuss the emergence of displacement and velocity
memory effect in the wormhole geometry.

Existence of displacement memory effect is evident from 2, where we have shown
that the differences $\Delta u$ and $\Delta r$ between two neighbouring
timelike geodesics depends on whether a GW pulse has passed through them or,
not. In other words, the values of $\Delta u$ and $\Delta r$ after the GW has
passed through does not return to that of the wormhole background and hence
provides the displacement memory effect. This memory effect not only depends
on the strength of the GW pulse, but more so on the background spacetime
geometry. Different choices for the parameter $p$ will lead to different
deviations and hence to different memories. In particular, the memory effect
can be a potential candidate to test the existence of additional hairs in the
background spacetime outside a compact object. In the present scenario, it
corresponds to the fact that non-zero $p$ does affect the displacement memory
in neighbouring geodesics.

Finally, in addition to the displacement memory effect, the GW pulse in the
wormhole geometry also describes a velocity memory effect, as clear from 3.
Both $\Delta\dot{u}$ and $\Delta\dot{r}$ are non-zero and differ from their
background values in the presence of the GW pulse. This memory effect also
depends on the choice of the wormhole hair $p$ and possibly a combined study
of displacement and velocity memory effect will lead to an existential proof
of non-zero values of $p$. Therefore, we can conclude that both displacement
and velocity memory effects exist in the case of braneworld wormhole and
depends crucially on the choice of $p$. This provides another avenue to search
for these non-black hole compact objects, which can also hint at the existence
of an extra spatial dimension.

##  IV Bondi-Sachs formalism and memory effect from null infinity

Having discussed the displacement and the velocity memory effects from the
geodesic analysis in the previous section, we now focus our attention in
investigating the memory effect at null infinity [58]. Since the effective
gravitational field equations on the brane, as presented in 1, is equivalent
to the generalized Brans-Dicke theory, with the Brans-Dicke parameter $\omega$
being also a function of the radion field, we try to reformulate the Bondi-
Sachs analysis of the memory effect as prescribed in [67, 68] appropriately.
As in the previous section, here also we assume that the spacetime of interest
corresponds to a background wormhole spacetime with a GW pulse passing through
it, ultimately reaching the future null infinity and as a consequence
modifying the Bondi mass aspect. In particular, we will derive the functional
form of the Bondi mass aspect in terms of the wormhole parameters, which in
turn are related to the physics of the higher spacetime dimensions through the
background wormhole solution. Note that without the GW pulse, there shall be
no dynamics associated with the bondi mass of the background wormhole metric,
since it depicts a static background.

For this purpose, we have to express the background wormhole geometry in the
null coordinates, as presented in 6, in the Bondi-Sachs form by an expansion
of the same in the inverse powers of the radial coordinate $r$. Such an
expansion yields,

$\begin{split}ds^{2}=&-du^{2}-2dudr+r^{2}\gamma_{AB}dx^{A}dx^{B}\\\
&+\frac{2M}{r(p+1)}du^{2}-\frac{2Mp}{r(p+1)}dudr+\mathcal{O}(r^{-2})~{},\end{split}$
(16)

where, $\gamma_{AB}$ is the metric on the unit-two sphere. The terms in the
first line of 16 is the flat metric at future null infinity and the terms in
the second line are the desired $(1/r)$ corrections. As evident, the Bondi
mass aspect is simply given by $\\{M/(p+1)\\}$ and is a constant for the
background spacetime. This provides the behaviour of the background static
wormhole geometry at future null infinity, while the similar result for matter
fields will now follow.

The matter fields consist of two parts, the matter on the four dimensional
brane and the radion field. The dependence of the on-brane matter field on the
radial coordinate $r$ can be found in [47] and it follows that at future null
infinity $\mathscr{I}^{+}$, the matter components fall off faster than
$1/r^{2}$ and hence need not be considered for the calculation. On the other
hand, the fall off behaviour of the radion field111In 9 we had written the
scalar field in terms of an isotropic coordinate $\tilde{r}$. As in the
asymptotic limit the isotropic coordinate $\tilde{r}$ is equal $r$, one can
safely expand the scalar field in terms of $\mathcal{O}(1/r)$. at future null
infinity can be parametrized as: $\Phi_{\rm
b}=\Phi_{0\textrm{(b)}}+\Phi_{1\textrm{(b)}}/r+\Phi_{2\textrm{(b)}}/r^{2}+\cdots$,
where ‘b’ denotes that these are contribution from the background wormhole
geometry. The term $\Phi_{0\textrm{(b)}}$, which is the leading order term in
the asymptotic expansion of the radial field, reads,

$\displaystyle\Phi_{0\textrm{(b)}}$
$\displaystyle=\bigg{(}\frac{C_{1}}{M}\log\frac{p+1}{p-1}+C_{4}\bigg{)}^{2}-1~{}.$
(17)

In an identical manner, we obtain the coefficients of $(1/r)$ and $(1/r^{2})$
terms, in the asymptotic expansion of the radion field $\Phi$ as,

$\displaystyle\Phi_{1\textrm{(b)}}$
$\displaystyle=-\frac{2C_{1}}{p+1}\bigg{(}\frac{C_{1}}{M}\log\frac{p+1}{p-1}+C_{4}\bigg{)}~{},$
(18) $\displaystyle\Phi_{2\textrm{(b)}}$
$\displaystyle=\frac{C_{1}^{2}}{(p+1)^{2}}+\frac{2MpC_{1}}{(p+1)^{2}}\bigg{(}\frac{C_{1}}{M}\log\frac{p+1}{p-1}+C_{4}\bigg{)}~{},$
(19)

where, $C_{1}$ and $C_{4}$ are nonzero constants used to quantify the radion
field [47]. Note that, these expansion coefficients can be expressed in terms
of the parameters of the wormhole spacetime and in particular it depends on
$p$. This fact will be used in the later parts of this work.

The above analysis is about the background spacetime, which is definitely non-
radiative. This is because the metric given in 16 has a constant and non-
dynamical Bondi mass. This is because there is no loss of news in absence of
any dynamics, as fit for a non-radiative geometry. Thus, the memory effect
requires a propagating GW pulse on top of this background geometry, leading to
a finite radiative term. Hence, we introduce a GW component to the above
wormhole metric and the final result is the following axi-symmetric line
element (see [86]),

$\displaystyle ds^{2}=$ $\displaystyle-
du^{2}-2dudr+\bigg{(}\frac{2M}{r(p+1)}+\frac{2M_{\rm
B}(u,\theta)}{r}\bigg{)}du^{2}$
$\displaystyle-\bigg{(}\frac{2Mp}{r(p+1)}+\frac{b(u,\theta)}{r}\bigg{)}du\,dr-2r^{2}U(u,r,\theta)du\,d\theta$
$\displaystyle+r^{2}h_{AB}dx^{A}dx^{B}+\cdots~{}.$ (20)

Note that in the line element presented above, there is a dynamical Bondi mass
term $M_{\rm B}(u,\theta)$. This is due to the presence of the gravitational
radiation in the background of the wormhole spacetime.

Here, $h_{AB}=\gamma_{AB}+c_{AB}r^{-1}+\mathcal{O}(r^{-2})$ and the terms
$b(u,\theta),M_{\rm B}(u,\theta),U(u,r,\theta)$ and $c_{AB}$ arise due to the
presence of the GW pulse and can be considered as perturbations over and above
the braneworld wormhole metric. Moreover, for the above perturbed metric to be
consistent with the gravitational field equations, the radion field should be
perturbed as well and the resultant field becomes $\Phi=\Phi_{\rm b}+\Phi_{\rm
p}$, where $\Phi_{\rm b}$ denotes the background radion scalar field and
$\Phi_{\rm p}$ denotes the perturbed scalar field due to the scalar part of
the GW pulse. We assume that the leading order term in $\Phi_{\rm p}$ is
$\mathcal{O}(1/r)$, such that the Bondi determinant condition can be expressed
as,

$\det(h_{AB})=(\Phi_{0\textrm{(b)}}/\Phi)^{2}\sin^{2}\theta~{}.$ (21)

which yields ,

$c_{AB}=\hat{c}_{AB}-\gamma_{AB}(\Phi_{1}/\Phi_{0(\textrm{b})})~{}.$ (22)

Here, $\hat{c}_{AB}$ is the pure gravitational part and corresponds to the
transverse and traceless degrees of freedom. Also, in the above expression,
$\Phi_{1}=\Phi_{1(\textrm{b})}+\Phi_{1(\textrm{p})}$, is the coefficient of
the $\mathcal{O}(1/r)$ term in the asymptotic expansion of the radion field
$\Phi$. Since there is an additional GW pulse being considered here, it will
be described by a tensorial News $N_{AB}$ as well as a scalar News $N$, which
are given by,

$\displaystyle-\partial_{u}\hat{c}_{AB}$ $\displaystyle=N_{AB}$
$\displaystyle=\mathcal{N}_{1}\operatorname{sech}^{2}u\,(_{-2}Y^{20})\,\,\begin{pmatrix}1&0\\\
0&-\sin^{2}\theta\end{pmatrix}~{},$ (23) $\displaystyle N$
$\displaystyle=\mathcal{N}_{2}\operatorname{sech}^{2}u\,(_{-2}Y^{20})\equiv\partial_{u}\Phi_{1(\textrm{p})}~{}.$
(24)

Note that the $\operatorname{sech}^{2}u$ behaviour is assumed in order to be
consistent with the discussion in the previous section and $N_{AB}$ embodies
the gravitational degrees of freedom, while $N$ encodes the scalar degree of
freedom. The amplitudes $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ are such that
the GW pulse can be considered as perturbations over the wormhole background.
Now, the Bondi determinant condition, along with integration of the above
relations over the null coordinate $u$, yields the following change in the
Bondi shear $c_{AB}$ as,

$\begin{split}\triangle
c_{AB}&=\triangle\hat{c}_{AB}-\gamma_{AB}\frac{\triangle\Phi_{1(\textrm{p})}}{\Phi_{0(\textrm{b})}}=-\mathcal{N}_{1}\,(_{-2}Y^{20})\,\,\begin{pmatrix}1&0\\\
0&-\sin^{2}\theta\end{pmatrix}\int_{-\infty}^{\infty}\operatorname{sech}^{2}u\,du\,\\\
&-\frac{\mathcal{N}_{2}}{\Phi_{0(\textrm{b})}}(_{-2}Y^{20})\begin{pmatrix}1&0\\\
0&\sin^{2}\theta\end{pmatrix}\int_{-\infty}^{\infty}\operatorname{sech}^{2}u\,du=-2\,(_{-2}Y^{20})\,\begin{bmatrix}\mathcal{N}_{1}+\dfrac{\mathcal{N}_{2}}{\Phi_{0(\textrm{b})}}&0\\\
0&\sin^{2}\theta\bigg{(}-\mathcal{N}_{1}+\dfrac{\mathcal{N}_{2}}{\Phi_{0(\textrm{b})}}\bigg{)}\end{bmatrix}\end{split}$
(25)

Here, the term ${}_{-2}Y^{20}$ is the spin-weighted 222Since the system is
axisymmetric, the spherical harmonic is chosen in a way such that there is no
dependence on $\phi$. harmonic, having the expression:
${}_{-2}Y^{20}=(3/4)\sqrt{5/6\pi}~{}\sin^{2}\theta$. The above equation,
namely 25, shows that the change in the Bondi shear, represented by $\triangle
c_{AB}$ is not traceless, with the trace being dependent on the presence of
the scalar memory via $\Phi_{0(\textrm{b})}$. Therefore, the existence of a
non-zero trace for the Bondi shear will signal the presence of an additional
scalar degree of freedom in the system, possibly arising from extra
dimensions. Thus, the total gravitational memory in the spacetime has both a
tensorial part and a scalar part, the later arising from the radion scalar
field of the underlying theory.

(a) We have depicted the variation of the Bondi shear with the wormhole hair
$p$ on the $\theta=(\pi/2)$ plane.

(b) Response of the change in Bondi shear due to the GW pulse has been
presented against $\theta$ for different values of $p$.

Figure 4: Variation of Bondi shear due to the passage of a GW pulse, also
known as the memory tensor has been presented for various choices of the
wormhole parameters.

The effects described above, are clearly depicted in 4, where the behaviour of
the Bondi shear has been presented with variation in the wormhole parameter
$p$.

Finally, we solve the gravitational field equations order by order, to obtain
the change in the Bondi mass aspect of the system, namely $\Delta M_{\rm B}$.
For this purpose, we re-express IV in a convenient form, fit for an
axisymmetric system [86], such that,

$\begin{split}ds^{2}=&-\exp\left[\sigma(u,r,\theta)\right]du^{2}-2\exp\left[2\beta(u,r,\theta)\right]dudr-2r^{2}\exp\left[2\left\\{\gamma(u,r,\theta)-\delta(u,r,\theta)\right\\}\right]U_{1}(u,r,\theta)du\,d\theta\\\
&\hskip
56.9055pt+r^{2}\exp\left[2\left\\{\gamma(u,r,\theta)-\delta(u,r,\theta)\right\\}\right]d\theta^{2}+r^{2}\exp\left[2\left\\{-\gamma(u,r,\theta)-\delta(u,r,\theta)\right\\}\right]d\phi^{2}~{}.\end{split}$
(26)

The metric functions $\sigma$, $\beta$, $U_{1}$, $\gamma$ and $\delta$,
appearing in the above expression, are all expanded in the inverse powers of
the radial coordinate $r$, yielding,

$\displaystyle\sigma(u,r,\theta)=\frac{\sigma_{1}(u,\theta)}{r}+\frac{\sigma_{2}(u,\theta)}{r^{2}}+\mathcal{O}(r^{-3})$
(27)
$\displaystyle\beta(u,r,\theta)=\frac{\beta_{1}(u,\theta)}{r}+\frac{\beta_{2}(u,\theta)}{r^{2}}+\mathcal{O}(r^{-3})$
(28) $\displaystyle
U_{1}(u,r,\theta)=\frac{U_{11}(u,\theta)}{r}+\frac{U_{12}(u,\theta)}{r^{2}}+\mathcal{O}(r^{-3})$
(29)
$\displaystyle\gamma(u,r,\theta)=\frac{\gamma_{1}(u,\theta)}{r}+\frac{\gamma_{2}(u,\theta)}{r^{2}}+\mathcal{O}(r^{-3})$
(30)
$\displaystyle\delta(u,r,\theta)=\frac{\delta_{1}(u,\theta)}{r}+\frac{\delta_{2}(u,\theta)}{r^{2}}+\mathcal{O}(r^{-3})$
(31)

These expansions must be compared with the Bondi-Sachs form of the wormhole
metric with a GW pulse, as presented in IV, from which we find the following
correspondence,

$\displaystyle\sigma_{1}(u,\theta)$ $\displaystyle=$
$\displaystyle-\frac{2M}{p+1}-2M_{\rm B}(u,\theta)~{},$
$\displaystyle\beta_{1}(u,\theta)$ $\displaystyle=$
$\displaystyle\frac{M\,p}{p+1}+\frac{b(u,\theta)}{2}~{},$
$\displaystyle\delta_{1}(u,\theta)$ $\displaystyle=$
$\displaystyle\frac{\Phi_{1}}{2\Phi_{0(\textrm{b})}}~{},$
$\displaystyle\gamma_{1}(u,\theta)$ $\displaystyle=$
$\displaystyle\frac{\hat{c}_{\theta\theta}}{2}\equiv\frac{\hat{c}(u,\theta)}{2}~{},$
$\displaystyle\delta_{2}(u,\theta)$ $\displaystyle=$ $\displaystyle
U(u,r,\theta)$ $\displaystyle=$
$\displaystyle\dfrac{\Phi_{2}U_{1}(u,r,\theta)}{2\Phi_{0(\textrm{b})}}\exp\left[2\left\\{\gamma(u,r,\theta)-\delta(u,r,\theta)\right\\}\right]~{}.$

Figure 5: Variation of the change in the Bondi mass due to GW pulse has been
presented with the null coordinate $u$, for different choices of $p$.

To proceed further, we use the fact that the field equations in Bondi-Sachs
formalism follows a nested pattern. One starts with an initial data,
prescribed in terms of the functions $\hat{c}_{\theta\theta}(u,\theta)$ and
$\Phi_{1(\textrm{p})}(u,\theta)$ at some value of $u$. Subsequently, the
hypersurface equations $G^{u}\,_{r}$ and $G^{u}\,_{\theta}$, respectively
yields, 333The following field equations are solved using the RGTC package in
the symbolic manipulation software Mathematica.

$\displaystyle\mathcal{O}(r^{-1})$ $\displaystyle:\hskip
5.69046ptU_{11}=0~{},$ (33) $\displaystyle\mathcal{O}(r^{-2})$
$\displaystyle:\hskip
5.69046ptU_{12}(u,\theta)=-\frac{\partial_{\theta}\hat{c}(u,\theta)}{2}-\hat{c}(u,\theta)\cot\theta~{},$
(34)

and finally, at order $\mathcal{O}(r^{-3})$, we obtain,

$\beta_{1}=-\frac{\Phi_{1}(u,\theta)}{2\Phi_{0(\textrm{b})}}~{};\qquad
b(u,\theta)=-\frac{\Phi_{1}(u,\theta)}{\Phi_{0(\textrm{b})}}-\frac{2M\,p}{p+1}~{}.$
(35)

Similarly, the hypersurface equation in the $G^{u}\,_{u}$ component of
Einstein’s equations yields an identity at the lowest order of
$\mathcal{O}(r^{-2})$, while the supplementary equation for the component
$G^{r}\,_{u}$ at $\mathcal{O}(r^{-2})$ yields the change in the Bondi mass
aspect of the system,

$\begin{split}\sigma_{1},_{u}=&\frac{\left(\partial_{u}\Phi_{1}\right)^{2}}{2\Phi_{0}^{2}}-\frac{3\left(\partial_{u}\Phi_{1}\right)^{2}}{2\Phi_{0}(1+\Phi_{0})}-\frac{\Phi_{1}\partial_{u}^{2}\Phi_{1}}{\Phi_{0}^{2}}\\\
&-\frac{\partial_{u}\Phi_{1},}{\Phi_{0}}+\frac{\left(\partial_{u}\hat{c}\right)^{2}}{2}+\partial_{u}\hat{c}-\frac{3}{2}\partial_{u}\partial_{\theta}\hat{c}\cot\theta-\frac{1}{2}\partial_{u}\partial_{\theta}^{2}\hat{c}~{}.\end{split}$
(36)

Integrating the above expression, we obtain the effective Bondi mass of the
system to yield,

$\begin{split}M_{\rm
B}=&-\frac{m}{p+1}-\frac{1}{2}\bigg{[}\frac{a^{2}Y^{2}}{2}+b^{2}Y^{2}\bigg{\\{}\frac{1}{2\Phi_{0}^{2}}-\frac{3}{2\Phi_{0}(1+\Phi_{0})}\bigg{\\}}\bigg{]}\times\bigg{(}\frac{2}{3}\tanh
u+\frac{1}{3}\operatorname{sech}^{2}u\tanh u\bigg{)}\\\
&-\frac{1}{3\Phi_{0}^{2}}b^{2}Y^{2}\tanh^{3}u+\frac{1}{2}\tanh
u\bigg{(}\frac{bY}{\Phi_{0}}+aY+\frac{3}{2}aY,_{\theta}\cot\theta+aY,_{\theta\theta}\bigg{)}~{}.\end{split}$
(37)

The evolution of the Bondi mass with variation in $p$ has been depicted in 5.
In the plot we find that the drop in the Bondi mass is higher as the value of
$p$ decreases.

##  V Conclusions

In this article, we have explored certain aspects of the GW memory in a
wormhole background on the brane. The reason for considering the presence of
extra spacetime dimension is two-fold — i) The on-brane gravity theory is a
quasi scalar-tensor theory, with the scalar field capturing the imprints of
the spatial extra dimension and hence any information about the scalar hair
will translate into possible inputs for the extra dimensions. ii) In this
class the wormhole geometry is traversable and can be sustained without
invoking any exotic matter fields. In this manner we arrive at a possible non-
black hole compact object without using exotic matter field and hence it
provides interesting viable alternative to the standard black hole paradigm in
GR. We would like to mention that, besides the wormhole solution considered
here, there are other wormhole solutions, e.g., in the context of Scalar-
coupled Einstein-Gauss-Bonnet Gravity [77], where also the matter field is not
exotic. It will be interesting to study the stability and hence the memory
effect in such wormhole backgrounds as well.

We, at first, have briefly reviewed the geometry of the wormhole spacetime and
how the presence of extra dimension helps in constructing a traversable
wormhole without any exotic matter. Then we have explored the displacement and
velocity memory effects by analysing neighbouring geodesics in the wormhole
background in the presence of a localised GW pulse. We have shown explicitly,
how the geodesic separation evolves before and after the passage of the pulse.
This explicitly establishes the existence of both displacement and velocity
memory effect. In addition, these memory effects depend crucially on the hairs
of this wormhole solution and hence differs from the corresponding memory
effect in Schwarzschild spacetime. Therefore, memory effects are indeed future
pointers towards exploring existence of non-black hole compact objects in our
universe, which in this context can also be related to the existence of extra
spatial dimensions.

Having observed that indeed memory effect exists and depends on the details of
the wormhole geometry, we subsequently proceeded to study the memory effect
using symmetries at null infinity using the Bondi-Sachs formalism. For this
purpose, we have expressed the spacetime metric of the wormhole geometry using
the Bondi coordinate system and have expanded the radion field in inverse
powers of the radial coordinate $r$. Considering a GW perturbation in the
system that satisfies the Bondi gauge conditions, we have computed the Bondi
shear as well as the Bondi mass aspect and hence observed that these also give
rise to the memory effect. Again, the memory depends on the wormhole parameter
($p$) through the leading order contribution from the radial field.

Since the braneworld scenario, considered in the present context, very much
resembles a scalar-tensor theory of gravity, we use the formalism given in
[67], and show how the variation in the Bondi mass aspect, related to the
memory effect, depend explicitly on the wormhole parameters. The same
conclusion holds true for geodesic memory effects as well. This variation of
the Bondi mass aspect is different from the black hole scenario and can
possible be probed using the future GW detectors. Moreover, generalization of
the present result for astrophysically relevant cases of rotating wormholes
will be of significant interest. Besides, this work can also be used as a
prospect to investigate supertranslations and soft hair implants on the throat
of a wormhole geometry (analogous study for black hole horizons can be found
in [87] and Rindler horizons in [88]). These issues we wish to study in the
future.

The memory effect encoded in the Bondi mass aspect is computed by solving the
gravitational field equations of the theory at orders of $\mathcal{O}(1/r)$.
Decomposing the field equations into Hypersurface and Supplementary equations,
we have determined the change in the Bondi mass of the system analytically. We
find that the variation in the value of $p$ produces a change in the evolution
of the Bondi mass. This shows that it will be easier to decipher whether the
central compact object has a non-zero value of $p$, since presence of a non-
zero $p$ will modify the memory significantly. Therefore, as and when the
memory effect can be detected in the future GW detectors, it will possibly
tell us about the existence of non-black hole compact objects and one can
independently verify if the braneworld wormhole is a viable scenario.

There are several possible future extensions of the present work, first of all
a complete analysis of the Bondi-Sachs formalism without assuming axisymmetry
will be an interesting and important extension. Moreover, studying memory
effect for rotating wormholes and other rotating compact objects, which are
more relevant astrophysically, will be another interesting avenue to explore.
Finally, studying GW memory effects for other black hole mimicker spacetimes,
e.g., fuzzballs will be of significant interest.

## Acknowledgements

I.C. acknowledges the University Grants Commission (UGC), Government of India,
for providing financial assistance through a senior research fellowship
(reference ID: 523711). Research of S.C. is funded by the INSPIRE Faculty
fellowship from DST, Government of India (Reg. No. DST/INSPIRE/04/2018/000893)
and by the Start- Up Research Grant from SERB, DST, Government of India (Reg.
No. SRG/2020/000409). S.C. further thanks the Albert-Einstein Institute, where
a part of this work was carried out and the Max-Planck Society for providing
the Max-Planck-India Mobility Grant.

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every pair of servers. Let $\Pi_{\sf OT}^{\sf ij}$ denote an instance of
1-out-of-2 OT with $\mathsf{S}_{\sf i}$ being the sender and $\mathsf{S}_{\sf
j}$ being the receiver. Here, $\mathsf{S}_{\sf i}$ inputs the sender messages
$(x_{0},x_{1})$ while $\mathsf{S}_{\sf j}$ inputs the receiver choice bit
$c\in\mathbb{Z}_{2}$ and obtains $x_{c}$ as the output, for
$x_{0},x_{1}\in\mathbb{Z}_{2^{\ell}}$.

Preprocessing: 1. Execute $\Pi_{\sf BitA}^{\sf
Pre}([{\lambda}_{\mathsf{b}}]^{\bf B})$ to obtain $[{\lambda}_{\mathsf{b}}]$.
2. Locally generate $([r],\langle r\rangle)$ for a random
$r\in\mathbb{Z}_{2^{\ell}}$. Online: 1. $\mathsf{S}_{j}$, for $j\in[\tau]$,
locally computes the following ($\Delta=1$ if $j=1$, else $0$): •
$[(z-r)]_{j}=\Delta\cdot\mathsf{m}_{\mathsf{b}}+(1-2\mathsf{m}_{\mathsf{b}})\cdot[{\lambda}_{\mathsf{b}}]_{j}-[r]_{j}$.
2. $\mathsf{S}_{j}$, for $j\in[\tau]$, sends $[(z-r)]_{j}$ to
$\mathsf{S}_{1}$, who computes $(z-r)$ and sends to all the servers. 3.
Locally compute $\langle z\rangle=\langle(z-r)\rangle+\langle r\rangle$.
Figure 12: Bit-to-arithmetic conversion protocol.

To generate the arithmetic sharing of ${\lambda}_{\mathsf{b}}$ from its
Boolean shares in $[\cdot]$-shared form, a simple method would be to apply a
3-XOR using a daBit-style approach [107], but would result in 12 executions of
1-out-of-2 OTs. However, as pointed out in Prio+ [2], the cost could be
further optimized due to the semi-honest security model being considered in
this work rather than the malicious in [107]. Since Prio+ operates over two
MPC servers, we extend their optimized daBit-generation protocol (cf. [2,
$\mathsf{daBitGen}_{p}$]) to our setting with three servers.

Given two bits $\mathsf{b}_{\sf i},\mathsf{b}_{\sf j}\in\mathbb{Z}_{2}$, the
arithmetic share corresponding to their product can be generated using one
instance of $\Pi_{\sf OT}^{\sf ij}$ with $(x_{0}=r,x_{1}=r+\mathsf{b}_{\sf
i})$ as the OT-sender messages and $\mathsf{b}_{\sf j}$ as the OT-receiver
choice bit. With this observation and using Eq. 3.1, servers can compute
$[\cdot]$-shares corresponding to the bit ${\lambda}_{\mathsf{b}}$ using five
OT invocations. The formal details appear in Fig. B.1.2.

OT Instance - I: $[\mathsf{b}]_{1}[\mathsf{b}]_{2}$ 1. $\mathsf{S}_{1}$
samples random $r_{12}\in\mathbb{Z}_{2^{\ell}}$. 2. $\mathsf{S}_{1}$ and
$\mathsf{S}_{2}$ executes $\Pi_{\sf OT}^{\sf
12}((r_{12},r_{12}+[\mathsf{b}]_{1}),[\mathsf{b}]^{\bf B}_{2})$. 3.
$\mathsf{S}_{1}$ sets $y_{12}^{1}=-r_{12}$ and $\mathsf{S}_{2}$ sets the OT
output as $y_{12}^{2}$. OT Instances - II & III:
$[\mathsf{b}]_{1}[\mathsf{b}]_{3}$, $[\mathsf{b}]_{2}[\mathsf{b}]_{3}$ These
are similar to the computation of $[\mathsf{b}]_{1}[\mathsf{b}]_{2}$ discussed
above. OT Instances - IV & IV:
$[\mathsf{b}]_{1}[\mathsf{b}]_{2}[\mathsf{b}]_{3}$ 1. Computation can be
broken down to
$([\mathsf{b}]_{1}[\mathsf{b}]_{2})\cdot[\mathsf{b}]_{3}=(y_{12}^{1}+y_{12}^{2})\cdot[\mathsf{b}]_{3}$.
2. Execute $\Pi_{\sf OT}^{\sf 13}$ for $y_{12}^{1}\cdot[\mathsf{b}]^{\bf
B}_{3}$. Let $z_{13}^{1}$ and $z_{13}^{2}$ denote the respective shares of
$\mathsf{S}_{1}$ and $\mathsf{S}_{3}$. 3. Execute $\Pi_{\sf OT}^{\sf 23}$ for
$y_{12}^{2}\cdot[\mathsf{b}]^{\bf B}_{3}$. Let $z_{23}^{1}$ and $z_{23}^{2}$
denote the respective shares of $\mathsf{S}_{2}$ and $\mathsf{S}_{3}$.
Computation of final shares $\mathsf{S}_{1}$:
$[\mathsf{b}]_{1}=\mathsf{b}_{1}-2y_{12}^{1}-2y_{13}^{1}+4z_{13}^{1}$.
$\mathsf{S}_{2}$:
$[\mathsf{b}]_{2}=\mathsf{b}_{2}-2y_{12}^{2}-2y_{23}^{1}+4z_{23}^{1}$.
$\mathsf{S}_{3}$:
$[\mathsf{b}]_{3}=\mathsf{b}_{3}-2y_{13}^{2}-2y_{23}^{2}+4z_{13}^{2}+4z_{23}^{2}$.
Figure 13: Bit-to-arithmetic preprocessing.

For the case of approximate bit conversion discussed in §3.1, the number of OT
instances can be further reduced to three following Eq. 9. Concretely, the
conversion involves computation of just
$[\mathsf{b}]_{1}[\mathsf{b}]_{2}[\mathsf{b}]_{3}$ and hence the OT instances
II & III described in Fig. B.1.2 are no longer needed.

Preprocessing: 1. Execute $\Pi_{\sf BitA}^{\sf
Pre}([{\lambda}_{\vec{\mathbf{M}}}]^{\bf B})$ to obtain
$[{\lambda}_{\vec{\mathbf{M}}}]$. 2. Locally generate $([r],\langle r\rangle)$
for a random $r\in\mathbb{Z}_{2^{\ell}}$. Online: 1. $\mathsf{S}_{j}$, for
$j\in[\tau]$, locally computes the following ($\Delta=1$ if $j=1$, else $0$):
• $[(z-r)]_{j}=\Delta\cdot{\sf
Agg\mbox{-}R}(\mathsf{m}_{\vec{\mathbf{M}}})+(1-2\mathsf{m}_{\vec{\mathbf{X}}})\odot[{\lambda}_{\vec{\mathbf{Y}}}]_{j}-[r]_{j}$.
2. $\mathsf{S}_{j}$, for $j\in[\tau]$, sends $[(z-r)]_{j}$ to
$\mathsf{S}_{1}$, who computes $(z-r)$ and sends to all the servers. 3.
Locally compute $\langle z\rangle=\langle(z-r)\rangle+\langle r\rangle$.
Figure 14: Bit-to-arithmetic sum protocol.

When computing the sum of bits directly, the online communication can be
optimized following inner-product protocol and the resulting protocol
$\Pi_{\sf BitA}^{\sf sum}$ is given in Fig. B.1.2.

Bit Injection Protocol. Given a Boolean vector
$\vec{\mathbf{M}}_{\sf\mathsf{d}\times 1}$ and an arithmetic vector
$\vec{\mathbf{N}}_{\sf\mathsf{d}\times 1}$ in the secret-shared form, protocol
$\Pi_{\sf BI}$ computes the inner product of the two vectors, defined as
$z=\vec{\mathbf{M}}\odot\vec{\mathbf{N}}$. This protocol is similar to the
inner product protocol $\Pi_{\sf IP}$ presented in Fig. B.1.2, with the main
difference being that $\vec{\mathbf{M}}$ is a Boolean vector.

During the preprocessing phase, servers first generate the arithmetic shares
of ${\lambda}_{\vec{\mathbf{M}}}$ from its Boolean shares, similar to the bit-
to-arithmetic protocol $\Pi_{\sf BitA}$ in Fig. B.1.2. The remaining steps are
similar to $\Pi_{\sf IP}$ in Fig. B.1.2 and we omit the details.

Preprocessing: 1. Execute $\Pi_{\sf BitA}^{\sf
Pre}([{\lambda}_{\vec{\mathbf{M}}}]^{\bf B})$ to obtain
$[{\lambda}_{\vec{\mathbf{M}}}]$. 2. Execute $\Pi_{\sf BI}^{\sf
Pre}([\vec{\mathbf{{\lambda}_{M}}}],[\vec{\mathbf{{\lambda}_{N}}}])$ to obtain
$[\vec{\mathbf{{\gamma}_{Q}}}]$ with
$\vec{\mathbf{{\gamma}_{Q}}}=\vec{\mathbf{{\lambda}_{M}}}\circ\vec{\mathbf{{\lambda}_{N}}}$.
3. Execute $\Pi_{\sf Tr}()$ to generate $([r],\langle
r/2^{\mathsf{f}}\rangle)$. Online: 1. $\mathsf{S}_{j}$, for $j\in[\tau]$,
locally computes the following ($\Delta=1$ if $j=1$, else $0$): •
$T_{j}^{1}=\Delta\cdot(\mathsf{m}_{\vec{\mathbf{M}}}\odot\mathsf{m}_{\vec{\mathbf{N}}})\
+\ \mathsf{m}_{\vec{\mathbf{M}}}\odot[{\lambda}_{\vec{\mathbf{N}}}]_{j}$. •
$T_{j}^{2}=((1-2\mathsf{m}_{\vec{\mathbf{M}}})\circ\mathsf{m}_{\vec{\mathbf{N}}})\odot[{\lambda}_{\vec{\mathbf{M}}}]_{j}\
+\ (1-2\mathsf{m}_{\vec{\mathbf{M}}})\odot[{\gamma}_{\vec{\mathbf{Q}}}]_{j}$.
• $[(z-r)]_{j}=T_{j}^{1}+T_{j}^{2}-[r]_{j}$. 2. $\mathsf{S}_{j}$, for
$j\in[\tau]$, sends $[(z-r)]_{j}$ to $\mathsf{S}_{1}$, who computes $(z-r)$
and sends to all the servers. 3. Locally compute $\langle
z\rangle=\langle(z-r)/2^{\mathsf{f}}\rangle+\langle r/2^{\mathsf{f}}\rangle$.
Figure 15: Bit injection (sum) protocol.

###  B.2 ScionFL: Additional Details

This section provides addition details of our FL framework ScionFL presented
in §4.

####  B.2.1 Efficiency of Approximate Bit Conversion

In this section, we measure the efficiency gains achieved by our approximation
method, discussed in §3.1, by counting the number of cross terms that must be
computed securely using MPC. Cross terms are terms that compute the product of
two or more shares. While the exact amount of computation and communication
varies depending on the MPC protocol and setting (e.g., honest vs. dishonest
majority or semi-honest vs. malicious security), we believe cross terms can
provide a protocol-independent and realistic assessment of
scalability.121212We acknowledge that the analysis cannot provide an exact
comparison, owing to the presence of the product term in the approximation.
For example, depending on the underlying MPC setup, the product term
($\mathsf{term_{p}}$) may require more communication than the middle terms
($\mathsf{term_{m}}$), and therefore the effect of approximation may be
minimized.

Computation | #cross-terms
---|---
Exact ($\tilde{\mathsf{b}}$) | Approximate ($\hat{\mathsf{b}}$)
Bit-to-Arithmetic | $2^{\mathsf{q}}-\mathsf{q}-1$ | $1$
Bit Injection | $2^{\mathsf{q}}+\mathsf{q}^{2}-2\mathsf{q}-1$ | $\mathsf{q}^{2}-\mathsf{q}+1$

Table 4: Efficiency analysis via approximate bit conversion with respect to
the #cross-terms involved.

Tab. 4 provides the details regarding the number of cross terms involved in
obtaining the arithmetic equivalent of
$\mathsf{b}=\oplus_{i=1}^{\mathsf{q}}\mathsf{b}_{i}$. The gains increase
significantly with a higher number of shares $\mathsf{q}$ due to the
exponential growth in the number of cross terms for the exact computation.
Tab. 4 also provides the details for a bit injection operation in which the
product of a Boolean bit $\mathsf{b}$ and a scale value $\mathsf{s}$ is
securely computed. Given $\mathsf{s}=\sum_{i=1}^{\mathsf{q}}\mathsf{s}_{i}$,
the value $\mathsf{b}\cdot\mathsf{s}$ can be computed by first computing
either $\tilde{\mathsf{b}}$ or $\hat{\mathsf{b}}$ (depending on whether an
exact or approximate value is required) and then multiplying by $\mathsf{s}$.

####  B.2.2 Secure Bit Aggregation with Global Scales

Here, we consider the secure bit aggregation problem in the context of global
scales, as discussed in (cf.§2.2). As shown in Eq. (6), the computation
becomes simpler in the case of global scales since all clients utilize the
same set of public scales, denoted by $\mathsf{s}^{min}$ and
$\mathsf{s}^{max}$, to compute their quantized vector that corresponds to the
rows of $\vec{\mathbf{B}}$. Hence, we just need to compute the column-wise
aggregate of the $\vec{\mathbf{B}}$ matrix and use protocol $\Pi_{\sf
BitA}^{\sf sum}$ (Fig. B.1.2 in §B.1.2) to do so. The resulting protocol
$\Pi_{\sf SecAgg}^{\textsf{Global}}$ appears in Fig. B.2.2.

1. Compute $\langle\vec{\mathbf{W}}^{j}\rangle=\Pi_{\sf BitA}^{\sf sum}(\langle\vec{\mathbf{B}}^{j}\rangle^{\bf B})$, for each $j\in[\mathsf{m}]$. 2. Locally compute $\langle\vec{\mathbf{Y}}_{\sf 1\times\mathsf{m}}\rangle=\mathsf{s}^{min}\oplus\left(\langle\vec{\mathbf{W}}_{\sf 1\times\mathsf{m}}\rangle\cdot(\mathsf{s}^{max}-\mathsf{s}^{min})\right)$. Figure 16: Secure aggregation – Global Scales.

####  B.2.3 Secure Bit Aggregation with Local Scales

In §3.2, we provided an empirical accuracy evaluation for our approximate
secure bit aggregation using global scales.

$10^{-3}$$10^{1}$$10^{5}$x$d=2^{10}$Exactx$d=2^{15}$Approx.x$d=2^{20}$NMSE SQ
---

$10^{0}$$10^{2}$$10^{4}$$10^{-3}$$10^{1}$$10^{5}$number of clients $\mathsf{n}$NMSE HSQ |  $10^{0}$$10^{2}$$10^{4}$number of clients $\mathsf{n}$NMSE KSQ
---|---

Figure 17: NMSE comparison between exact and approximation-based aggregation
for SQ, Hadamard SQ (HSQ), and Kashin SQ (KSQ) for local scales with
$\mathsf{q}=4$ shares, various vector dimensions $d$, and number of clients
$\mathsf{n}$.

Here, in Fig. 17, we additionally provide results considering local scales. In
contrast to global scales, we can observe that for stochastic quantization
without rotation the effect on the NMSE is reduced from three to one order of
magnitude. Also, for rotation-based algorithms there are significant concrete
improvements.

####  B.2.4 Detailed Communication Costs

In this section, we provide more detailed insights into the concrete
communication costs for our secure aggregation protocols described in §4.1.

Approach | Offline | Online
---|---|---
Approach-I | $\mathsf{n}\cdot\mathsf{BitA}^{\mathsf{pre}}+\mathsf{n}\cdot\mathsf{Mult}^{\mathsf{pre}}$ | $\mathsf{n}\cdot\mathsf{BitA}^{\mathsf{on}}+\mathsf{n}\cdot\mathsf{Mult}^{\mathsf{on}}$
Approach-II | $\mathsf{n}\cdot\mathsf{BitA}^{\mathsf{pre}}+\mathsf{n}\cdot\mathsf{Mult}^{\mathsf{pre}}$ | $\mathsf{Mult}^{\mathsf{on}}$
Approach-III | $\mathsf{n}\cdot\mathsf{BitA}^{\mathsf{pre}}+~{}~{}~{}~{}\mathsf{Mult}^{\mathsf{pre}}$ | $\mathsf{Mult}^{\mathsf{on}}$

Table 5: Communication costs aggregating quantized vectors with a single
dimension for $\mathsf{n}$ clients. Protocols $\Pi_{\sf BitA}$ and $\Pi_{\sf
Mult}$ are treated as black-boxes, and their costs are represented as
$\mathsf{BitA}$ and $\mathsf{Mult}$, respectively. The superscript pre in the
costs denotes preprocessing and on denotes the online phase. We compare
Approach-I (cf. Fig. 4.1.1 in §4.1.1), Approach-II (cf. Fig. 4.1.2 in §4.1.2),
and Approach-III (cf. Fig. 4.1.3 in §4.1.3).

First, in Tab. 5 we give a theoretical analysis for the communication cost
when aggregating $\mathsf{n}$ quantized vectors with a single dimension.
Clearly, our Approach-III (cf. Fig. 4.1.3 in §4.1.3) is the most efficient,
with the multiplication-related cost being completely independent of the
number of clients $\mathsf{n}$ due to the integration of SepAgg [14].

|  | Exact | Approx.
---|---|---|---
$\mathsf{n}$ | Method | Offline | Online | Offline | Online
20 | Approach-I | 644.50 | 12.90 | 620.27 | 12.90
Approach-II | 644.50 | 0.59 | 620.27 | 0.59
Approach-III | 89.77 | 0.59 | 65.54 | 0.59
100 | Approach-I | 3222.51 | 64.51 | 3101.36 | 64.51
Approach-II | 3222.51 | 0.59 | 3101.36 | 0.59
Approach-III | 332.08 | 0.59 | 210.93 | 0.59
500 | Approach-I | 16112.56 | 322.56 | 15506.80 | 322.56
Approach-II | 16112.56 | 0.59 | 15506.80 | 0.59
Approach-III | 1543.62 | 0.59 | 937.85 | 0.59

Table 6: Inter-server communication per round in MiB for our MNIST/LeNet
benchmark for different numbers of clients $\mathsf{n}$ per round. Training is
done using $1$-bit SQ with Kashin’s representation (KSQ). We compare
Approach-I (cf. Fig. 4.1.1 in §4.1.1), Approach-II (cf. Fig. 4.1.2 in §4.1.2),
and Approach-III (cf. Fig. 4.1.3 in §4.1.3). Additionally, we distinguish
between using an exact bit-to-arithmetic conversion and our approximation (cf.
§3.1).

Next, in Tab. 6 we provide the detailed communication costs for the secure
aggregation approaches discussed in §4.1 when training the LeNet architecture
for image classification on the MNIST data set [80] using $1$-bit SQ with
Kashin’s representation [28]. We instantiate the OT instances required in the
preprocessing phase, as discussed in §B.1.2, with silent OT [37], following
Prio+ [2]. Here, we can observe the significant impact of including SepAgg
[14] has in practice with performance improvements between Approach-II and
Approach-III of up to $16.6\times$ in the offline phase.

Approach | $\mathsf{n}=10^{2}$ | $\mathsf{n}=10^{3}$ | $\mathsf{n}=10^{4}$ | $\mathsf{n}=10^{5}$
---|---|---|---|---
Prio+ [2] | 9.45 | 94.50 | 945.04 | 9450.44
Approach-III (Exact) | 3.94 | 39.42 | 394.17 | 3941.66
Approach-III (Approx.) | 2.37 | 23.75 | 237.45 | 2374.53

Table 7: Total communication in MiB of our Approach-III (cf. Fig. 4.1.3 in
§4.1.3) compared to Prio+ [2] to calculate the sum of bits for different
numbers of clients $\mathsf{n}$ and dimension $\mathsf{m}=1000$. For our
Approach-III, we distinguish between using an exact bit-to-arithmetic
conversion as in Prio+ [2] and our approximation (cf. §3.1).

Finally, in Tab. 7, we compare the aggregation of bits (i.e., when not
considering quantized inputs that require scale multiplication and hence
without SepAgg [14] being applicable) to Prio+ [2]. For a fair comparison, we
translate the approach Prio+ [2] in our three party dishonest-majority
setting. As we can see, even for exact bit-to-arithmetic conversion, we
improve over Prio+ by factor $2.4\times$ for $\mathsf{n}=10^{5}$. When
applying our approximate bit-to-arithmetic conversion (cf. 3.1), this
improvement increases to a factor of $4\times$.

###  B.3 ScionFL-Aura: Additional Details

Here, we provide additional details ofr ScionFL-Aura.

####  B.3.1 Sub-protocols

In this section, we give the sub-protocols for ScionFL-Aura (cf. §5.1.1). Note
that for sake of simplicity, we do not include optimizations discussed in
§5.1.3.

Alg. 2 computes the aggregation of $\alpha$ quantized vectors. As shown in Eq.
4, the dequantized value of a vector $\vec{Y}$, given its quantized form
$(\vec{\sigma}_{Y},\mathsf{s}_{Y}^{min},\mathsf{s}_{Y}^{max})$, can be
computed as

$\vec{Y}=\mathsf{s}_{Y}^{min}\ \oplus\
\vec{\sigma}_{Y}\circ(\mathsf{s}_{Y}^{max}-\mathsf{s}_{Y}^{min}).$

The above operation essentially places $\mathsf{s}_{Y}^{min}$ in those
positions of the vector $\vec{Y}$ with the corresponding bit in
$\vec{\sigma}_{Y}$ being zero, and the rest with $\mathsf{s}_{Y}^{max}$.

Algorithm 2 Quantized Aggregation

1:procedure
Aggregate($\\{\vec{\sigma}_{Y_{i}},\mathsf{s}_{Y_{i}}^{min},\mathsf{s}_{Y_{i}}^{max}\\}_{i\in\alpha}$)

2: $\vec{Z}\leftarrow\vec{0}$

3: for $k\leftarrow 1$ to $\alpha$ do

4:
$\vec{Z}\leftarrow\vec{Z}+\left(\mathsf{s}_{Y_{k}}^{min}\oplus\vec{\sigma}_{Y_{k}}\circ(\mathsf{s}_{Y_{k}}^{max}-\mathsf{s}_{Y_{k}}^{min})\right)$

5: end for

6: $\vec{Z}\leftarrow\vec{Z}/\alpha$

7: return $\vec{Z}$

8:end procedure

Alg. 3 computes the ${\sf L}_{2}$-norm of a quantized vector. As discussed in
§2.4, a quantized vector $\vec{{Y}}_{\sigma}$ consists of a binary vector
$\vec{\sigma}_{Y}$ and the respective min. and max. scales
$\mathsf{s}_{Y}^{min}/\mathsf{s}_{Y}^{max}$. In this case, we observe that the
squared ${\sf L}_{2}$-norm can be obtained by first counting the number of
zeroes and ones in the vector, denoted by $N_{Z}$ and and $N_{O}$
respectively, followed by multiplying them with the square of the respective
scales and adding the results, i.e.
$N_{Z}\cdot(\mathsf{s}_{Y}^{min})^{2}+N_{O}\cdot(\mathsf{s}_{Y}^{max})^{2}$.
Furthermore, computing the number of ones $N_{O}$ corresponds to the bit-
aggregation of the vector $\vec{Y}$, for which our aggregation methods
discussed in §4.1 can be utilized.

Algorithm 3 L2-Norm Computation (Quantized)

1:procedure
L2-NormQ($\vec{\sigma}_{Y},\mathsf{s}_{Y}^{min},\mathsf{s}_{Y}^{max}$)

2: $\beta\leftarrow\textsc{Len}(\vec{\sigma}_{Y})$ // Dimension of
$\vec{\sigma}_{Y}$

3: $N_{O}\leftarrow\textsc{Sum}(\vec{\sigma}_{Y})$ // Number of ones in
$\vec{\sigma}_{Y}$

4: $N_{Z}\leftarrow\beta-N_{O}$ // Number of zeros in $\vec{\sigma}_{Y}$

5: return
$\sqrt{N_{Z}\cdot(\mathsf{s}_{Y}^{min})^{2}+N_{O}\cdot(\mathsf{s}_{Y}^{max})^{2}}$

6:end procedure

Alg. 4 is used to compute the cosine distance between a quantized vector
$\vec{{Y}}_{\sigma}$ and a reference vector $\vec{S}$. The cosine distance is
given by
$\frac{\vec{{Y}}_{\sigma}\odot\vec{S}}{\lVert\vec{{Y}}_{\sigma}\rVert\cdot\lVert\vec{S}\rVert}$,
where $\lVert\cdot\rVert$ corresponds to the ${\sf L}_{2}$-norm of the input
vector. Using Eq. 4, we can write

$\displaystyle\vec{{Y}}_{\sigma}\odot\vec{S}$
$\displaystyle=(\mathsf{s}_{Y}^{min}\oplus\vec{\sigma}_{Y}\circ(\mathsf{s}_{Y}^{max}-\mathsf{s}_{Y}^{min}))\odot\vec{S}$
$\displaystyle=\mathsf{s}_{Y}^{min}\odot\vec{S}\ +\
(\vec{\sigma}_{Y}\odot\vec{S})\cdot(\mathsf{s}_{Y}^{max}-\mathsf{s}_{Y}^{min}).$

Thus, the inner product computation of $\vec{{Y}}_{\sigma}\odot\vec{S}$
reduces to computing $\vec{\sigma}_{Y}\odot\vec{S}$, followed by two
multiplications.

Algorithm 4 Cosine Distance Calculation

1:procedure
Cosine($(\vec{\sigma}_{Y},\mathsf{s}_{Y}^{min},\mathsf{s}_{Y}^{max})$,
$\vec{S}$)

2: ${\sf
L}_{2}^{Y}\leftarrow\textsc{L2-NormQ}(\vec{\sigma}_{Y},\mathsf{s}_{Y}^{min},\mathsf{s}_{Y}^{max})$

3: ${\sf L}_{2}^{S}\leftarrow\lVert\vec{S}\rVert$ // Computes ${\sf
L}_{2}$-norm

4: $\alpha\leftarrow\textsc{Sum}(\vec{S})$ // Sum of elements of $\vec{S}$

5: $\beta\leftarrow\textsc{Inner\mbox{-}Product}(\vec{\sigma}_{Y},\vec{S})$

6:
$\gamma=\mathsf{s}_{Y}^{min}\cdot\alpha+\beta\cdot(\mathsf{s}_{Y}^{max}-\mathsf{s}_{Y}^{min})$

7: return $\gamma/({\sf L}_{2}^{Y}\cdot{\sf L}_{2}^{S})$

8:end procedure

Figure 18: Effect of  _Min-Max_ attack [112] on training VGG11 with CIFAR10
with and without our defense ScionFL-Aura assuming $20\%$ of $N=50$ clients
being corrupted.

####  B.3.2 Evaluation on VGG11

In addition to our results in §5.1.2, we evaluate the  _Min-Max_ attack on
VGG11 trained with CIFAR10. The experimental setup is identical to §5.1.2. The
results are shown in Fig. 18.

Similarly as for ResNet9 (cf. Fig. 10), the  _Min-Max_ attack substantially
reduces the validation accuracy when training VGG11: We observe drops of up to
$36.8\%$. However, on average, VGG11 is less impacted by the attack.
Concretely, only $15\%$ of the iterations observe a validation accuracy
reduction of about $15\%$ or more when using no compression. One third of the
training rounds are impacted by about $15\%$ or more when using Kashin’s
representation (KSQ) while with the Hadamard transform (HSQ) only very few
training rounds showed a significant accuracy reduction. Thus, HSQ seems to be
inherently more robust against untargeted poisoning.

With ScionFL-Aura, the accuracy reduction is still noticeable smaller for all
variants. With HSQ, on average 0.28 malicious updates are included in global
updated instead of 2.24 without defense. With respect to the validation
accuracy, the difference between having no attack and employing ScionFL-Aura
when under attack is less than $4\%$ in almost all training iterations. When
using KSQ, a global update includes just 0.44 malicious updates on average,
and the attack impact is at least halved in two third of the training
iterations.

####  B.3.3 Comparison to FLAME [97]

Our  _untargeted poisoning_ defense ScionFL-Aura has components with
similarities to the  _backdoor_ defense FLAME [97], which has three steps:
density-based clustering, clipping, and noise addition. However, there are
important differences, which we emphasize in the following.

  1. 1.

Magnitude Boundary. FLAME’s clipping is done with respect to the median
Euclidean distance of the local updates to the previous global model. However,
especially with non-iid data and in early training phases, each training
iteration may exhibit significant differences even for consecutive iterations.
Hence, using the recent average norm (assuming the majority of updates is
benign) as in ScionFL-Aura (cf. Step 1 in §5.1.1 and Lines 3-14 in Alg. 1)
intuitively gives a better estimation for a benign magnitude in the current
training state.

  2. 2.

Filtering. FLAME compares cosine similarity in a pair-wise fashion among
individual updates, i.e., it computes
$\frac{\mathsf{n}\cdot(\mathsf{n}-1)}{2}$ cosine distances per iteration while
ScionFL-Aura does $\mathsf{n}$ (cf. Step 2 in §5.1.1 and Line 16 in Alg. 1).

While ScionFL-Aura sorts local updates based on cosine similarity (cf. Step 2
in §5.1.1 and Line 18 in Alg. 1), FLAME uses secure clustering with low cubic
complexity [25]. FLAME only accepts updates assigned to the largest cluster,
which can lead to an exclusion of benign updates and thus significantly
slowing down the training by removing valuable benign contributions. In
contrast, ScionFL-Aura removes only a fixed number of updates, thereby
enabling an efficient trade-off that reduces the attack’s effect to a
tolerable level (even if a few malicious contributions are not filtered out)
with a low false positive rate.

  3. 3.

Differential Privacy: After the clipping, FLAME aggregates the updates and
adds noise in the cleartext to create a differentially private new global
model. We do not consider differential privacy in our work, however, the noise
addition can trivially be added to our system.

For more details on FLAME, we refer the reader to [97].