# Dendrites with corners

E.S. Nani Corresponding author e-mail address: sumanth-
<EMAIL_ADDRESS>Laboratoire de Physique de la Matière
Condensée, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120
Palaiseau, France  T. Philippe  Laboratoire de Physique de la Matière
Condensée, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120
Palaiseau, France  M. Plapp  Laboratoire de Physique de la Matière Condensée,
CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau,
France

(30/11/2023)

###### Abstract

A phase-field model for diffusion-limited crystal growth is formulated that is
capable of handling highly anisotropic interfaces. It uses a Willmore
regularization that yields corners of finite size. An asymptotic analysis
reveals that Herring’s law is recovered for the advancing surfaces. The model
is validated by conducting simulations of dendritic growth for low anistorpies
and comparing the results to the data from the literature. The model makes it
possible to simulate high anisotropy dendrites for which the standard phase-
field models are ill-posed. In this regime, the interplay between a Herring
instability on the dendrite flanks and the corner regularization creates zig-
zag shaped corrugations and leads to a non-monotonic trend of tip velocity as
a function of anisotropy strength.

##  1 Introduction

Crystal growth from a melt can produce a large variety of complex
morphologies. In solidification, the resulting microstructures determine many
material properties. The phase-field method has been hugely successful in
capturing this pattern formation process in numerical simulations.

Dendrites are one of the most widespread solidification microstructures. Their
morphology is determined by the anisotropy of the solid-liquid interfacial
free energy. Phase-field studies of dendritic growth have mostly been limited
to low and moderate anisotropy strengths. This is the relevant case for metals
which have atomically rough interfaces. However, for substances which have
atomically smooth (facetted) surfaces, the interfacial energy anisotropy is
much stronger [1, 2].

When the interfacial anisotropy becomes so strong that certain surface
orientations are excluded from the equilibrium shape, the equations of motion
need to be regularized both in the sharp-interface and phase-field treatments
[3, 4]. A first regularization technique for phase-field models was proposed
by Eggleston et al. [5], but needs to be adapted to each functional form of
the interfacial anisotropy. More recently, a Willmore regularization was
introduced in phase-field models for surface diffusion [4, 6] and kinetically-
limited crystal growth [7, 8]. This regularization creates corners of fixed
size $\alpha$ which bridge the forbidden orientations. Here, we use the
combination of the latter model with the grand-potential formulation of alloy
solidification[9, 10] to study diffusion-limited dendritic growth with
anisotropy of arbitrary strengths. An asymptotic analysis enables us to make
rigorous connection to the relevant sharp-interface problem.

In the next section, the regularized model is presented and benchmarked.
Simulation results generated using the model are reported and discussed in
section after before concluding the article.

##  2 Model Construction and Benchmarks

###  2.1 Model

The standard formulation of phase-field models is limited to moderate
anisotropies with positive interface stiffness for all orientations. To extend
it to larger anisotropies, we add the Willmore regularization proposed in
Refs. [4, 8]. The core idea of this regularization is to mimic the approach
taken in the sharp-interface theory, namely, to penalize curvatures in the
evolving crystal shape so as to avoid sharp corners. To this end, a new term
is added to the functional of the grand-potential model for isothermal alloy
solidification [9], which becomes

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\int\left[(\omega_{\text{sol}}(\mu)-\omega_{\text{liq}}(\mu))g(\phi)+\omega_{\text{liq}}(\mu)+\frac{\gamma(\theta(\nabla\phi))}{\epsilon}\left(f_{\text{dw}}(\phi)+\frac{\epsilon^{2}}{2}(\nabla\phi)^{2}\right)\right.$
(1)
$\displaystyle\left.\mbox{}+\gamma_{0}\frac{\alpha^{2}}{2\epsilon}\left(\frac{f^{\prime}_{\text{dw}}(\phi)}{\epsilon}-\epsilon\Delta\phi\right)^{2}\right]dV$

where $\omega_{\text{sol}}$ and $\omega_{\text{liq}}$ are the volumetric
grand-potential densities of the crystal and the liquid phases, respectively;
$\mu$ is the diffusion potential; $\phi$ is the phase-indicator marking liquid
and crystal phases with the its values of $0$ and $1$, respectively; $g(\phi)$
is a function interpolating between $0$ and $1$ as $\phi$ ranges over $[0,1]$;
$f_{\text{dw}}(\phi)$ is the double-well potential;
$\gamma(\theta(\nabla\phi))$ is the interfacial energy as a function of
orientation $\theta$ which in turn is specified using $\nabla\phi$;
$\gamma_{0}$ is the average of $\gamma(\theta)$; and $\epsilon$ is the
interface-width specifier. The bracketed expression in the last term of the
functional is the diffuse interface approximation of the local interface
curvature. Hence this term penalizes curvatures of either signs and creates
smoothed corners of fixed size controlled by the Willmore regularization
parameter $\alpha$. In contrast to just adding a $\left(\Delta\phi\right)^{2}$
term [11, 12], this regularization preserves the hyperbolic tangent interface
profile known from the standard phase-field models.

The governing equations derived from this functional are as follows:

$\displaystyle\tau\frac{\partial\phi}{\partial t}=$
$\displaystyle-\frac{1}{\epsilon}\left(\omega_{\text{sol}}(\mu)-\omega_{\text{liq}}(\mu)\right)g^{\prime}(\phi)-\frac{1}{\epsilon^{2}}\gamma(\theta(\nabla\phi))f_{\text{dw}}^{\prime}(\phi)$
$\displaystyle+\nabla\cdot\left[\gamma(\theta(\nabla\phi))\nabla\phi\right]+\frac{1}{\epsilon^{2}}\nabla\cdot\left[\left(f_{\text{dw}}(\phi)+\frac{\epsilon^{2}}{2}|\nabla\phi|^{2}\right)\nabla_{\nabla\phi}\left(\gamma(\theta(\nabla\phi))\right)\right]$
$\displaystyle+\gamma_{0}\frac{\alpha^{2}}{\epsilon^{4}}\left(\epsilon^{2}\nabla^{2}\phi-
f_{\text{dw}}^{\prime}(\phi)\right)f_{\text{dw}}^{\prime\prime}(\phi)-\gamma_{0}\frac{\alpha^{2}}{\epsilon^{2}}\nabla^{2}\left(\epsilon^{2}\nabla^{2}\phi-
f_{\text{dw}}^{\prime}(\phi)\right)$ (2a)
$\displaystyle\frac{\partial\mu}{\partial
t}=\left[\nabla\cdot\left(\left[\left(D_{\text{sol}}-D_{\text{liq}}\right)h(\phi)+D_{\text{liq}}\right]\nabla\mu\right)-\frac{\left(c_{\text{sol}}(\mu)-c_{\text{liq}}(\mu)\right)}{c,_{\mu}}f^{\prime}(\phi)\frac{\partial\phi}{\partial
t}\right]$ (2b)

where $D_{\text{sol}}$ and $D_{\text{liq}}$ are the diffusion coefficients,
and $c_{\text{sol}}$ and $c_{\text{liq}}$ are the compositions of the solid
and the liquid phases, respectively. The symbol $c,_{\mu}$ is used to denote
$\left(\frac{dc_{\text{sol}}}{d\mu}-\frac{dc_{\text{liq}}}{d\mu}\right)g(\phi)+\frac{dc_{\text{liq}}}{d\mu}$.
$h(\phi)$ and $f(\phi)$ are interpolation functions like $g(\phi)$, and $\tau$
is an inverse mobility parameter.

Matched asymptotic analysis demonstrates that the generalized Stefan problem
is recovered by the model of Eq. (2) except for the Gibss-Thomson relation,
which is replaced by Herring’s law:

$\displaystyle\tilde{\tau}v_{n}=\left(\omega_{\text{liq}}(\mu)-\omega_{\text{sol}}(\mu)\right)-\left(\gamma(\theta)+\gamma^{\prime\prime}(\theta)\right)\kappa+\gamma_{0}\alpha^{2}\left(\frac{\kappa^{3}}{2}+\kappa_{ss}\right)$
(3)

where $v_{n}$ and $\kappa$ are respectively the normal velocity and the
curvature of the solid-liquid boundary; $\kappa_{ss}$ is the surface Laplacian
of curvature. $\tilde{\tau}$ is a constant that describes the strength of
interface attachment kinetics.

In the current article, the focus is on diffusion-limited growth. Hence, we
choose the phase-field relaxation parameter $\tau$ such that the interface
attachment kinetics $\tilde{\tau}$ is zero, following methods analogous to
those developed by Almgren [13]. In this preliminary study, we use equal
diffusivities in the liquid and the solid ($D_{\text{sol}}=D_{\text{liq}})$
and denote the common value by $D$.

###  2.2 Dendritic growth for low anisotropies

We tested Eq. (3) by standard benchmarks such as kinetically-limited and
diffusion-limited planar front growth and critical nucleus evolution, with
excellent success. For low anisotropies where the regularization terms are not
required ($\alpha=0$), as a first non-trivial benchmark problem, we simulated
dendritic growth for low anisotropies. An anisotropic crystal seed growing at
the expense of a supersaturated infinite matrix develops into a dendritic
pattern and eventually enters a steady-state. We compared the simulated
operating state with the results of Karma-Rappel [14].

We chose the following forms for the various functions:
$f_{\text{dw}}(\phi)=18\phi^{2}(1-\phi)^{2}$,
$g(\phi)=\phi^{3}(10-15\phi+6\phi^{2})$, $f(\phi)=\phi$,
$\omega_{\text{sol}}(\mu)=-\mu^{2}/4A-\mu c_{\text{sol}}^{\text{eq}}$,
$\omega_{\text{liq}}(\mu)=-\mu^{2}/4A-\mu c_{\text{liq}}^{\text{eq}}$, and

$\gamma(\theta)=\gamma_{0}\left(1+\epsilon_{4}\cos(4\theta)\right).$ (4)

These choices relate the various thermo-physical parameters of the current
model to those of Karma-Rappel’s in the following manner: supersaturation
$\Delta$ matches up with
$\frac{\left(c_{\text{liq}}^{\text{eq}}-c_{0}\right)}{\left(c_{\text{liq}}^{\text{eq}}-c_{\text{sol}}^{\text{eq}}\right)}$
where $c_{0}$ is the liquid’s far-field as well as the initial composition,
the interface width $2\sqrt{2}W_{0}$ scales as $2\epsilon/3$, the capillary
length $d_{0}$ coincides with
$\frac{\gamma_{0}}{2A\left(c_{\text{sol}}^{\text{eq}}-c_{\text{liq}}^{\text{eq}}\right)^{2}}$,
$u$ of Ref. [14] corresponds to
$\frac{\mu}{2A\left(c_{\text{sol}}^{\text{eq}}-c_{\text{liq}}^{\text{eq}}\right)}$,
and finally, the diffusion coefficient $D$ and the anisotropy strength
$\epsilon_{4}$ are the same across both the treatments.

By keeping the set
$\\{c_{\text{liq}}^{\text{eq}},c_{\text{sol}}^{\text{eq}},A\\}$ fixed,
physical situations corresponding to various $\Delta-W_{0}-d_{0}$ combinations
of TABLE II of Karma-Rappel are simulated by choosing the
$c_{0}-\epsilon-\gamma_{0}$ triplets accordingly. The spatial discretizations
and the grid sizes chosen for each simulation are exactly identical to those
listed in the table. The recovered steady-state dendrite tip speeds are within
$2-7\%$ of the expected values for all the cases tested. Growth profiles
simulated for various anisotropy strengths $\epsilon_{4}$ using the parameter
set of Table 1, which corresponds to the dimensionless parameter set of
$\Delta=0.55$, $D=4.0$ and $d_{0}=0.1385$ of Karma and Rappel, are reported in
Fig. 1a). The starting seed size is the same for all the trials. The depicted
interface profiles also correspond to the same but a later timestep. That
means, the trend of dendrite tips becoming faster and sharper as anisotropy
strength is increased can be witnessed to be correctly reproduced by our
simulations. Curves in Fig. 1aa) correspond to concentration profiles along
the dendrite axis. The concentration in the liquid next to the tip decreases
with increasing anisotropy, which is of course consistent with the Gibbs-
Thomson effect.

Table 1: Table showing the material, modeling and simulation parameter set used for benchmarking the model. $c_{\text{liq}}^{\text{eq}}$ | $c_{\text{sol}}^{\text{eq}}$ | $A$ | $\gamma_{0}$ | $D$ | Grid discretization | $\epsilon$ | $\tau$ | $\Delta t$
---|---|---|---|---|---|---|---|---
0.9 | 0.1 | 0.78125 | 0.1385 | 4.0 | $\Delta x=\Delta y=0.2$ | $7.5\sqrt{2}\Delta x$ | $\epsilon\frac{\left(c_{\text{sol}}^{\text{eq}}-c_{\text{liq}}^{\text{eq}}\right)^{2}}{\frac{D}{2A}}\frac{47}{360}$ | $0.2\frac{(\Delta x)^{2}}{D}$

a)b)c)
aa)cc)

Figure 1: Results obtained using the presented model (Eq. (2)). The plots show
the $\phi=0.5$ contours of the simulations for various anisotropy strengths
for a) the traditional dendritic growth, i.e., with $\alpha=0$, b) growth
under constant supersaturation, i.e., by holding $\mu$ constant, and c) non-
zero regularization parameter. Also shown are the concentration profiles along
the central axis of the rightward growing dendrite arms of the cases a) and b)
in sub-figures aa) and cc), respectively.

###  2.3 Growth under constant driving force

In order to benchmark the regularization terms, we switched off the coupling
to transport, started with an initial circular seed of $5$ grid cells, and let
it grow under the drive of a constant supersaturation until it occupies a
certain percentage of the simulation domain, at which point, volume
preservation is imposed for further evolution by a Lagrange multiplier
implementation. The material and simulation parameter-sets used are identical
to those above. Results for various anisotropy strengths and $\alpha$ held
fixed at $0.5$ (in the same units) are reported in Fig. 1b). Except for
corners, these contours were seen to be in excellent agreement with the
respective Wulff shapes.

##  3 Results and Discussion

For $\epsilon_{4}>\nicefrac{{1}}{{15}}$, ranges of interface orientations
around $\theta=0$, $\theta=\pm\pi/2$, and $\theta=\pi$ become unstable, and
the standard phase-field model becomes ill-posed. With our regularized model,
however, dendrites can be simulated for $\epsilon_{4}$ larger than
$\nicefrac{{1}}{{15}}$. This is illustrated in Figs. 1c) and 2 for a value of
$\alpha=3.61\times d_{0}$ and otherwise the same parameter set used for Fig
1a).

Figure 2: Crystal growth morphologies and steady-state tip velocities
predicted by the present model for various anisotropy strengths
$\epsilon_{4}$.

Fig. 1c) shows growth contours of crystals for various anisotropies evolved
for different times to facilitate better visualization. The typical dendritic
shape persists even for large anisotropies, which indicates that growth is
still diffusion-limited. Corrugations of zig-zag shape appear behind the tip
for large enough anisotropy strengths. Fig. 1cc) shows that the concentration
fields remain smooth even for high anisotropies.

For a quantitative evaluation of the tip operating state, one arm is followed
for long times in a moving simulation box. The tip morphologies are
illustrated in the snapshot pictures in Fig 2, and the steady-state tip
velocity is given by the solid curve (with error bars) in the graph. The tip
growth speed first increases as a function of anisotropy strength, then passes
through a maximum, and slowly decreases beyond. Moreover, the tip velocity,
which is constant for low anisotropies, starts to oscillate beyond the
maximum. The error bars in Fig. 2 indicate the minimum and maximum velocities
of the oscillation cycles. These observations can be interpreted as follows.

Three different regimes can be identified. For $\epsilon_{4}$ smaller than
$\nicefrac{{1}}{{15}}$, the dendrite is smooth everywhere, and the
regularization only slightly modifies the anisotropic needle crystal shape.
Therefore, it is not surprising to recover the familiar behavior of increasing
of growth speed with anisotropy strength. This trend continues for a range of
anisotropies beyond $\epsilon_{4}=\nicefrac{{1}}{{15}}$. In this regime, the
dendrite tip is a corner, but the dendrite flanks remain smooth for a
considerable distance behind the tip. Eventually, corrugations start to
develop on the dendrite flank, because the interface enters an unstable
orientation range around $\theta=\pi/2$. Since with increasing anisotropy
strength the unstable ranges get larger, the corrugations develop closer and
closer to the tip until these undulations induce an oscillation of the tip
speed itself. The velocity oscillations can likely be explained by the
modification of the diffusion field created by these zig-zags.

While the corrugations may seem reminiscent of the sidebranches in
solidfication dendrites, they are created by an altogether different
mechanism. Whereas sidebranches result from a selective amplification of
thermal noise along the flanks of a dendrite [15, 14], here the corrugations
are created by a Herring instability that only sets in when the interface
orientation reaches an unstable range. In our simulations, this instability is
triggered by the time-dependent dendrite shape without the explicit addition
of the thermal fluctuations.

For a comprehensive understanding, the dependence of the operating state on
all of the parameters should be investigated. As a first step in this
direction, simulations were repeated for two more values of the surface energy
$\gamma_{0}$, viz. $\gamma_{0}=0.175$ and $\gamma_{0}=0.100$ corresponding to
the red and green data points of Fig. 2, respectively. As expected from the
standard scaling theory of dendritic growth, lowering the surface energy makes
dendrites sharper and faster. However, the tip velocity does not exactly
follow the scaling $v\propto\nicefrac{{1}}{{\gamma_{0}}}$ predicted by
solvability theory [16]. This is due to the presence of the corner size
$\alpha$ as an additional length scale in the problem.

The present simulations have remained limited to the symmetric model
$\left(D_{\text{sol}}=D_{\text{liq}}\right)$. However, solvability theory for
low anisotropy dendrites predicts that unequal diffusivities do only modify
the selection constant, whereas the qualitative behavior remains the same
[17]. There is no reason to assume that it would be different for high
anisotropy dendrites. Hence we expect that our main results, namely the zig-
zagged corrugations and the non-monotonic trend in the tip velocities, should
not be affected by the diffusivity ratio.

##  4 Conclusion

We have demonstrated that our regularized phase-field model can be used to
explore the physics of dendritic growth at large anisotropies. The presence of
a stable corner on the dendrite tip leads to a continuation of needle crystal
growth beyond the limit $\epsilon_{4}=\nicefrac{{1}}{{15}}$. For a better
theoretical understanding of these new solutions, we are presently exploring
an extension of the linearized microscopic solvability theory [17] to the
current problem, namely the generalized Stefan problem with the Gibbs-Thomson
condition replaced by the Herring’s equation.

## Acknowledgement

The authors gratefully acknowledge the financial support provided by Agence
Nationale de la Recherche, France, under the project FACET.

## References

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# White Mirror: Leaking Sensitive Information from Interactive Netflix Movies
using Encrypted Traffic Analysis

Gargi Mitra, Prasanna Karthik Vairam, Patanjali SLPSK, Nitin Chandrachoodan,
Kamakoti V  Indian Institute of Technology Madras
Email: {gargim@cse,pkarthik@cse,slpskp@cse,nitin<EMAIL_ADDRESS>
###### Abstract

Privacy leaks from Netflix videos/movies is well researched. Current state-of-
the-art works have been able to obtain coarse-grained information such as the
genre and the title of videos by passive observation of encrypted traffic.
However, leakage of fine-grained information from encrypted traffic has not
been studied so far. Such information can be used to build behavioral profiles
of viewers.

On 28th December 2018, Netflix released the first mainstream interactive movie
called ‘Black Mirror: Bandersnatch’. In this work, we use this movie as a
case-study to show for the first time that fine-grained information (i.e.,
choices made by users) can be revealed from encrypted traffic. We use the
state information exchanged between the viewer’s browser and Netflix as the
side-channel. To evaluate our proposed technique, we built the first
interactive video traffic dataset of $100$ viewers; which we will be
releasing. Preliminary results indicate that the choices made by a user can be
revealed $96\%$ of the time in the worst case.

##  I Introduction

Recently, Netflix released the first mainstream interactive movie called Black
Mirror: Bandersnatch. In this movie, viewers are allowed to create their own
story-line by choosing one of the on-screen options presented to them. For
instance, the viewers are asked choice-questions such as ‘Frosties or sugar-
puffs?’, ‘visit therapist or follow Colin?’, ‘throw tea over computer or shout
at dad?’. Depending on the viewer’s choice, the corresponding segment of the
movie gets played. Interestingly, the choices made and the path followed can
potentially reveal viewer information that ranges from benign (e.g., their
food and music preferences) to sensitive (e.g., their affinity to violence and
political inclination). Although this information is available to Netflix,
they are bound by legal clauses that prevent them from misusing it. Today, to
prevent this information from leaking to unauthorized parties, Netflix uses
end-to-end encryption. Recent advancements in the domain of encrypted network
traffic analysis make it possible to infer basic information about the
preferences of Netflix viewers. For instance, prior works [1, 2] were able to
infer the title of videos watched by viewers and their preferred genre. In
this work, we raise the following research question: ‘Do interactive movies
leak sensitive fine-grained information about the viewers to passive
eavesdroppers due to the choices that they make while watching the movie?’.

Contributions. (1) We present the first traffic analysis technique for
interactive videos that can leak more information than non-interactive videos.
(2) We present the first dataset for encrypted traffic analysis of interactive
videos.

##  II Related Work

The objectives of existing research works [1, 2, 3] have so far been limited
to discerning video content from other traffic, identifying the titles of
videos and the genre preferred by a viewer from encrypted traffic of
conventional videos. However, the objective of our work is to see if more
fine-grained private information about a viewer can be leaked from encrypted
interactive video traffic.

Existing techniques from the literature are not suitable for encrypted
interactive video traffic analysis due to the following fundamental reason:
inter-video features cannot be used to differentiate between segments from the
same video. For instance, prior works [1] have used bitrate as a feature to
differentiate between two video streams. However, in the context of an
interactive video, the bitrate of chunks pertaining to each choice will be the
same and hence cannot be used to distinguish between two video segments. In
this work, we identify an intra-video side-channel and show that it holds
across various operating conditions.

##  III Proposed Traffic Analysis Technique

Figure 1: The streaming process of Black Mirror: Bandersnatch

In this section, we first describe the streaming process of Bandersnatch. Due
to the non-linearity of the script, the streaming process is check-pointed at
each choice-question in the movie unlike conventional Netflix videos that
stream continuously. The content of Bandersnatch is divided into several
segments, each corresponding to one path segment in the script, with each
segment consisting of multiple chunks. Figure 1 shows an example where a
viewer makes the first two choices. The first segment of the movie (i.e.,
Segment 0) is common for all viewers, and its chunk streaming ends when the
viewer is presented with the first choice-question $Q1$. At this point, the
viewer’s browser sends a JSON file (type-1 JSON file) to the server indicating
that the viewer has encountered $Q1$. The viewers are then given ten seconds
to choose one out of two options. Our experiments revealed that Netflix
considers one of the choices to be the default branch and prefetches chunks
belonging to the default segment. We denote the default choice for question
$Qi$ as $Si$ and the non-default choice as $Si^{\prime}$. If the viewer
chooses $Si$, then the streaming continues uninterrupted. However, if the
choice $Si^{\prime}$ is chosen, the prefetching for $Si$ stops and a request
for $Si^{\prime}$ is sent.

Conditions | Attribute |  Value
---|---|---
Operational | Operating System |  Windows, Linux, Mac
Platform |  Desktop, Laptop
Traffic Conditions |  Morning, Noon and Night
Connection Type |  Wired, Wireless
Browser |  Google-chrome, Firefox
Behavioral | Age-group |  $<$ 20, 20-25,25-30, $>$ 30
Gender |  Male,Female,Undisclosed
Political Alignment |  Liberal, Centrist, Communist, Undisclosed
State of Mind |  Happy, Stressed, Sad, Undisclosed
TABLE I: Attributes of the IITM-Bandersnatch Dataset

In the example from Figure 1, the viewer selects the default choice $S1$ for
$Q1$. The streaming continues uninterrupted until $Q2$ appears on-screen. Like
before, a type-1 JSON file is sent from the viewer’s browser to Netflix at
this point. The viewer selects the non-default option $S2^{\prime}$ for Q2. In
this case, another JSON file (i.e., of type-2) will be sent. Although some
chunks corresponding to $S2$ are pre-fetched, they will be discarded and the
chunks corresponding to $S2^{\prime}$ will be streamed. Therefore, we can
conclude that the number and type of JSON files sent indicate the choice made
by the viewer. However, identifying the two types of JSON files from encrypted
network traffic is challenging.

Our experiments revealed that the packets carrying the encrypted type-1 and
type-2 JSON files can be distinguished from other packets by their SSL record
lengths which are visible even from encrypted traffic. This observation was
found to be consistent across various operating systems, browsers, devices,
connection media, and network conditions. Hence, we use ‘SSL record lengths of
client packets’ as a side-channel to infer the choices made by the viewer
while watching Bandersnatch.

##  IV The Interactive Video Traffic Dataset

In this section, we describe our dataset, called IITM-Bandersnatch, that we
use to evaluate the proposed traffic analysis technique. The dataset comprises
of data points of the form {encrypted traces, ground truth choices}. To
collect each data point, we asked the viewer to watch Bandersnatch from the
beginning and note down the choices made by them. At the same time, we
collected the encrypted network traffic. As of now, our dataset contains
information corresponding to $100$ viewers who volunteered for this study.

The robustness of a traffic analysis technique can only be evaluated if the
dataset has a diverse set of representative data points. In this dataset, we
account for diversity by considering various operating conditions such as
multiple Operating Systems, devices, browsers, connection type, and network
conditions. Table I summarizes the operating conditions considered.

Figure 2: SSL record length distribution for (Desktop, Firefox, Ethernet,
Ubuntu) and (Desktop, Firefox, Ethernet, Windows)

To enable behavioral sciences researchers to evaluate their techniques, we
have also collected the behavioral state of the viewers who participated in
the study.

##  V Results

We conducted our preliminary experiments on the encrypted traffic captured
during $10$ different viewing sessions of the movie ’Black Mirror:
Bandersnatch’. In each case, the movie was viewed by different people under
different combinations of operational and network conditions. Figure 2 shows
the distinguishability of packets carrying type-1 and type-2 JSON files using
the SSL record lengths of client-side application packets obtained from the
encrypted traffic of viewers under two different operational conditions. This
helped us to identify the two types of JSON files with $96\%$ accuracy and
hence the choices made by the viewers.

##  VI Conclusions and Future Directions

In this work, we present a novel side-channel attack on interactive videos. We
show that the identified side-channel holds for various operational and
behavioral conditions. We also present the IITM-Bandersnatch dataset which
contains the behavioural and operational data of 100 viewers.

Countermeasures. An easy fix for the problem would be to either split the JSON
file or to compress it so that it becomes indistinguishable. However, there
could be timing side-channels that may still exist even after this fix.

Extending the Dataset. The set of behavioral attributes in our dataset are
limited. Extending the dataset with information pertaining to more relevant
behavioral traits is an interesting direction.

High-level Implications. Our work is a modest attempt at identifying the
viewer choices from encrypted traffic. We reach out to the research community
to use this information for behavioral studies.

## References

  * [1] A. Reed and M. Kranch, “Identifying https-protected netflix videos in real-time,” in _Proceedings of the Seventh ACM on Conference on Data and Application Security and Privacy_. ACM, 2017, pp. 361–368.
  * [2] R. Schuster, V. Shmatikov, and E. Tromer, “Beauty and the burst: Remote identification of encrypted video streams,” in _USENIX Security_ , 2017.
  * [3] F. Li, J. W. Chung, and M. Claypool, “Silhouette: Identifying youtube video flows from encrypted traffic,” in _Proceedings of the 28th ACM SIGMM Workshop on Network and Operating Systems Support for Digital Audio and Video_. ACM, 2018, pp. 19–24.