# Securing the Inter-Spacecraft Links:
Physical Layer Key Generation from Doppler Frequency Shift

Ozan Alp Topal, , Gunes Karabulut Kurt, , Halim Yanikomeroglu  O. A. Topal and
G. Karabulut Kurt are with the Department of Electronics and Communication
Engineering, Istanbul Technical University, Istanbul, 34469 Turkey, (email:
{topalo, gkurt}@itu.edu.tr). G. Karabulut Kurt is also with the Department of
Electrical Engineering, Polytechnique Montréal, Montréal, Canada, (e-mail:
gunes.kurt@polymtl.ca). H. Yanikomeroglu is with the Department of Systems and
Computer Engineering, Carleton University, Ottawa, ON, K1S 5B6, Canada,
(email: halim@sce.carleton.ca).

###### Abstract

In this work, we propose a secret key generation procedure specifically
designed for the inter-spacecraft communication links. As a novel secrecy
source, the spacecrafts utilize Doppler frequency shift based measurements. In
this way, the mobilities of the communication devices are exploited to
generate secret keys, where this resource can be utilized in the environments
that the channel fading based key generation methods are not available. The
mobility of a spacecraft is modeled as the superposition of a pre-determined
component and a dynamic component. We derive the maximum achievable secret key
generation rate from the Doppler frequency shift. The proposed secret key
generation procedure extracts the Doppler frequency shift in the form of
nominal power spectral density samples (NPSDS). We propose a maximum-
likelihood (ML) estimation for the NPSDS at the spacecrafts, then a uniform
quantizer is utilized to obtain secret key bits. The key disagreement rate
(KDR) is analytically obtained for the proposed key generation procedure.
Through numerical studies, the tightness of the provided approximations  is
shown. Both the theoretical and numerical results demonstrate the validity and
the practicality of the presented physical layer key generation procedure
considering the security of the communication links of spacecrafts.

###### Index Terms:

Doppler frequency shift, inter-spacecraft link security, physical layer key
generation, space network.

##  I Introduction

Exploring, understanding, and colonizing other planets  become the future
roadmap for human existence in the space. The availability of low-cost devices
and advanced rocket technology make this roadmap more attainable, where
developing robust communication systems is an integral part for future space
missions. As a starting point, today mostly the small-space satellites are
launched into low-orbital space by top-tier companies like SpaceX, Google,
Facebook, Virgin Galactic[1].

However, exploration of outer space is expected to become the near future
competition for countries and private companies. For example, with the Artemis
program, NASA targets to land the first human on the Moon by 2024[2]. By the
end of the decade, NASA is planning to form sustainable space missions that
eventually sending astronauts to Mars. Another important space mission is
carried out by Chinese National Space Administration, where a relay satellite
is utilized to establish a reliable communication link in the dark side of the
Moon. In order to join the space exploration race, these spacecrafts require
innovative technologies that provide a lower cost of production, launch, and
maintenance. This requirement limits the sensing and the communication
capabilities predicting their limited size, weight, and power. Considering
these characteristics, the spacecrafts that will be utilized in the future
space missions act as space cyber-physical networks, where the control and
connectivity of many low-cost and software-enabled controllable devices are
the main priority. The security of the communication channel and communication
devices becomes the priority of the cyber-physical systems in order to secure
the whole network from the large scale accidents initiated by the security
breaches of low-cost devices [3].

Figure 1: An illustration of an inter-spacecraft link (ISL). Alice, Bob and
Eve represent three different spacecrafts. Alice and Bob try to establish a
secure communication line, while Eve tries to eavesdrop their communication.

The open nature of the wireless communication channel creates security
breaches for the space networks similar to the wireless cyber-physical
systems. Especially considering the critical research and exploration
information harnessed by the space mission, the security breach of an inter-
spacecraft link (ISL) would create the loss of highly profitable information.
Considering the high operational costs and technical difficulties of
restarting a space mission, the security becomes a non-negotiable concept in
space networks. Conventionally, in symmetric key encryption mechanisms, the
security has enabled by cryptographic methods, where the message is encrypted
with a bit sequence at the transmitter and decrypted with the bit sequence at
the receiver. This sequence is named as the secret key, and the sharing of the
secret key becomes a bottleneck for the low-battery power and low chip size
communication devices. Especially to obtain a post-quantum secure
communication systems, the longer secret keys are required to be shared, where
the complexity of these operations become unattainable for the low-cost
devices.

Alternatively, physical layer key generation methods are proposed for
obtaining identical secret keys at the distant nodes without any key sharing
complexities. The distant communication devices exploit the unique
characteristics of the physical layer link that only identical or very similar
for them, and seem random for other observers to extract identical secret
keys. For example, the received signal strengths have been proposed as a
secrecy resource [4] or the reciprocal channel estimates are utilized for
physical layer secret key generation [5]. Also, the spatial correlations of
the channel fadings are analyzed as another resource of secrecy [6]. These
methods provide  a high amount of secrecy under the assumption of  a dynamic
environment and channel fading. However, under the static channel fading
assumptions as in the deep space networks, these methods become inapplicable
since the channel based attenuations become predictable. Therefore, the
utilization of cryptical methods as well as channel based secret key
generation methods becomes unfavorable for the inter-spacecraft communication
networks.

###  I-A Contributions

As a solution to the discussed problem, in [7], we have proposed a mobility
based secret key generation scheme, where the spacecrafts exploits the
reciprocal Doppler frequency shift measurements as a secrecy resource. More
specifically, spacecrafts utilize the measurements of nominal power spectral
density samples (NPSDS), where the effect of Doppler frequency shifts are
fully reserved. As a first step, the communication nodes exchange pilot
signals, and estimate NPSDS with maximum likelihood (ML) estimator. The
estimations are quantized to generate secret key bits.

As an extension of our previous work in [7], the main contributions of this
paper can be enlisted as

  * •

We model the mobility of the spacecrafts as the superposition of a publicly
known, predictable component and a dynamic and randomly modeled component.

  * •

The maximum achievable secret key rate from the Doppler frequency shift
measurements are provided. The dynamic mobility model of the spacecrafts is
modeled with a normalized version of the Brownian motion to obtain the maximum
amount of secrecy.

  * •

The key disagreement rate (KDR) has been analytically modeled considering the
given random mobility model.

  * •

The numerical analysis shows the tightness of the given approximations, and
illustrates the maximum achievable secret key rate and KDR performance.

###  I-B Organization

In the following, we provide related work in physical layer key generation. In
Section II, we provide the considered system model, and the performance of the
Doppler frequency shift as a secrecy resource. In Section III, we explain the
proposed secret key generation procedure. In Section IV, we provide
theoretical analysis on the key disagreement rate of the proposed method. In
Section V, we provide numerical analysis. In Section VI, we conclude the paper
and provide the future work directions.

###  I-C Notation

Scalar variables are denoted by italic symbols, vectors are denoted by
boldface symbols. $\rho(x)$ denotes the probability density function of the
random variable $X$. $\mathcal{N}(\mu,\sigma^{2})$ denotes the normal random
variable with mean $\mu$ and variance $\sigma^{2}$. The non-central chi-
squared distribution with $k$ degrees of freedom and $\delta$ non-centrality
parameter is denoted by ${\chi^{\prime}}_{k}^{2}(\delta)$. $\text{erf}(\cdot)$
denotes the error function as described in [8]. $h(X)$ denotes the
differential entropy of the random variable $X$. $\text{log}(\cdot)$ denotes
the natural logarithm. $\mathcal{I}_{v}(\cdot)$ denotes the $v^{\text{th}}$
order modified Bessel function. $||\mathbf{d}||$ denotes the Euclidean norm of
the vector $\mathbf{d}$. $\mathbb{E}\\{\cdot\\}$ denotes the expectation
operator.

##  II Related Works

TABLE I: Comparison of the related work. Randomness Resource | Reference | Implementation | Channel assumption |  | Applicability
---
to Space Networks
Received signal strengths | [9] | indoor | Non Line Of Sight (NLOS) | No
[10] | outdoor | NLOS and deep fade | No
[11], [12] | indoor | LoS and NLoS | No
Channel amplitude | [5] | indoor | LoS | No
[6] | indoor/outdoor | NLoS, spatial correlation | No
[13] | outdoor | LoS/ spatial correlation | Limited
[14] | outdoor/vehicular | LoS/NLoS - dynamic | No
[12], [15] | indoor | NLoS | No
Channel phase | [16] | outdoor | NLoS | Yes
Power spectral density | [17] | underwater | NLoS - dynamic | Limited
Voltage/current deviations | [18] | power line | n/a | No
Carrier frequency offset | [19] | outdoor/vehicular | NLoS/LoS - dynamic | Yes
Doppler frequency shift | this work | outdoor/ high-mobility | LoS - highly dynamic | Yes

Table I provides a comparison for the state-of-the-art (SOTA) PHY key
generation methods including their applicability for the space communication
networks. In terrestrial networks, especially for the sub 6 GHz systems, the
multi-path fading is the most dominant factor that deteriorates the
communication performance. The multipath fading in terrestrial networks is
generally modeled by Rayleigh or Rician distributions. To ensure reliable
communications, the devices already apply channel estimation and equalization.
The wireless devices have the measurements of channel state information (CSI)
or received signal strengths (RSS) available. One important enabler for
utilizing these estimates for the encryption purposes is that they are
reciprocal for two distant nodes within the coherence time [5]. Therefore,
they have identical secrets without sharing any information about the secrets
itself. The main limiting factor in the space networks is that the fading
channel is dominantly affected by the path loss term, where the main path is
very dominant to other reflections and diffractions. In most of the cases such
as communication between two space shuttles, the EM waves propagate in space
without encountering any blockage. In that case, only fading factor is the
free-space path loss, where it is same for the devices with the same
distances. Hence, CSI or RSS based secret key generation methods cannot be
applied considering the space communication networks.

On the other hand, very high mobility of the spacecrafts result into frequency
and phase shifts for communication devices. The channel phase-based secret key
generation methods are already available in the literature that also given in
Table I. They may also utilized in the space communication networks, but the
estimation process of the channel phases are more complex, and may include
more erroneous results than the rest of the secrecy extraction schemes [20].

As the main novelty, in this work, we propose a Doppler frequency shift based
secret key generation procedure first time in the literature. As described in
the Figure 1, we consider any three ISLs of three distinct spacecrafts. In the
considered scenario, we Alice and Bob exploit the symmetric measurements of
the Doppler frequencies as a secrecy source for the first time in the
literature. Since the mobility of the spacecrafts are updated in each
communication block, the randomness in the mobility reflects on the Doppler
frequency measurements identically for the reciprocal links.

In addition to different key generation schemes, another major contribution is
increasing the efficiency of the proposed key generation mechanisms. The
authors of [21] make use of principle component analysis to decrease the key
disagreement rates at the nodes. In [22], the authors utilize a wavelet-based
pre-processing to eliminate the dissimilarities of the channel observations.
Besides pre-processing after obtaning raw secret keys, information
reconciliation has been applied to eliminate the discrepancies in the
generated keys. In [23], the authors provide a general overview on the
reconciliation performance of the error correcting codes. More recently, the
authors of [24] provide a more efficient key reconciliation protocol. In [25],
the authors propose a polynomial interpolation based information
reconciliation scheme. All of the reconciliation schemes may utilized in
different secret key generation schemes including our work to eliminate
remaining erroneous bits.

##  III System Model

In this paper, we consider point to point communication link between two
spacecrafts. As envisioned in the future space missions, these spacecrafts
might be satellites orbiting around Earth, Moon or Mars, or space shuttles
carrying humans or goods. The spacecrafts for these missions are expected to
have very high mobility that effects the communication links severely.  The
spacecrafts and Doppler frequency shift utilization is given in Figure 2.
Alice and Bob are two distant spacecrafts that  want to communicate with
secrecy. In the meantime, we assume a third spacecraft, Eve, eavesdrops the
communication between Alice and Bob to obtain the same secret of Alice and
Bob. We assume that all spacecrafts have single omni-directional antenna for
message transmission and reception. Alice and Bob are assumed to communicate
in time division duplexing (TDD) fashion. Alice and Bob would require
identical secret keys to encrypt/decrypt their communication messages, where
Eve should not be able to obtain this secret key. Due to the high
computational requirements, commonly utilized secret sharing mechanisms such
as Diffie-Hellman is assumed to be not available in the considered scenario
[26].  As depicted in Figure 2, estimating the Doppler frequency shift is also
an integral part for enabling reliable communication and navigation services
in space [27, 28]. The spacecrafts already utilize Doppler frequency shifts to
compansate the carrier frequency and navigate the spacecraft. Since pilot
sharing and Doppler frequency measurements are already an integral part of the
system, the additional cost of complexity is introduced by the quantization
operation. This complexity is quite low in comparison to the many operations
utilized by the transceivers such as channel estimation and equalization [29].
At least one of the legitimate nodes is assumed to have high mobility. In the
following, we detail the mobility model for the spacecrafts, and provide the
fundamental resource of the secrecy for the communication between Alice and
Bob.

Figure 2: (a) A visiual representation for the relative velocity and position
vectors, (b) Utilization of Doppler frequency shifts in spacecrafts.

###  III-A Spacecraft Mobility Model

In the considered system model, the mobility of Alice and Bob are assumed to
be composed of a static, expected movement component and a random, unexpected
movement component. A communication block is defined for the time instant $t$,
where Alice and Bob respectively share their messages respectively for each
time instant. The velocities and the positions of the spacecrafts are updated
for each time instant.

Let us denote the velocity vector of the $k^{\text{th}}$ spacecraft at time
instant $t$ in Cartesian coordinates by
$\mathbf{v}_{k}(t)=[v^{x}_{k},v^{y}_{k},v^{z}_{k}]^{T}$, where
$k\in\\{a,b,e\\}$ for respectively Alice, Bob and Eve. Similarly, the position
vector of the $k^{\text{th}}$ spacecraft at time instant $t$ in Cartesian
coordinates is denoted by
$\mathbf{p}_{k}(t)=[p^{x}_{k},p^{y}_{k},p^{z}_{k}]^{T}$. In line with the
mission, the spacecrafts are assumed to have a predetermined route and
velocity values that known from all other spacecrafts. In the meantime, due to
the dynamic environmental changes, the spacecrafts have small divergences from
the aimed route and velocity values. We model the total amount of displacement
and velocity as the superposition of these two effects:

$\mathbf{v}_{k}(t)=\mathbf{\hat{v}}_{k}(t)+\mathbf{\tilde{v}}_{k}(t),$ (1)

where $\mathbf{\hat{v}}_{k}(t)$ denotes the velocity of the main route that
publicly known, $\mathbf{\tilde{v}}_{k}(t)$ denotes the velocity introduced by
the displacement due to the dynamic environmental changes. For example, for a
LEO satellite orbiting with a  static route, we can assume
$\mathbf{\tilde{v}}_{k}(t)=0$. However, for a space shuttle the dynamic
environmental would cause higher divergences that result temporal
discrepancies in the main route.  Each spacecraft drafts from its expected
trajectory due to numerous unpredictable reasons [28]. For example, the
pressure of sunlight may catapult to a drift from the main trajectory. After
drifting, the spacecraft starts to follow an arbitrary and random trajectory
that the mobility controllers need to estimate and maneuver the spacecraft
back to its main route. This random displacement is modeled in the manuscript
by the $\tilde{v}_{k}$. Due to the unpredictability and the real-world
randomness, even the space agencies consider these displacements as random
behaviors. In this paper, we model these discrepancies in the velocity with a
stochastic process, where the components in each Cartesian coordinate is
i.i.d,
$\mathbf{\tilde{v}}_{k}(t)=[{\tilde{v}}^{x}_{k},{\tilde{{v}}}^{y}_{k},{\tilde{v}}^{z}_{k}]^{T}$.

As discussed in above, due to the static channel environment in deep space
communication, the legitimate parties may not utilize the channel fading
randomness based secrecy sources to obtain an identical secret keys. The
following subsection light the way for a novel secrecy source for distant
communication parties in this particular challenging environment.

###  III-B Doppler Frequency Shift as a Secrecy Source

Let
$f_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
denote the nominal Doppler frequency, where
$f_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=\frac{c\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}{f_{c}}$,
where
$\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
is the Doppler frequency observed at the $k^{\text{th}}$ spacecraft of the
transmitted signal from the
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}^{\text{th}}$
spacecraft, $c$ is the speed of light in m/s, and $f_{c}$ is the carrier
frequency of the transmitted signal. The nominal Doppler frequency at time
instant $t$ is given by

$\displaystyle
f_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(t)$
$\displaystyle=\frac{\mathbf{v^{T}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)(\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{p_{k}}(t))}{||\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{p_{k}}(t)||}+\frac{\mathbf{v_{k}^{T}}(t)(\mathbf{p_{k}}(t)-\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t))}{||\mathbf{p_{k}}(t)-\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)||},$
(2)
$\displaystyle=\frac{(\mathbf{v_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{v_{k}}(t))^{T}(\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{p_{k}}(t))}{||\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{p_{k}}(t)||}.$

The difference of velocity and position vectors provide us the relative
velocity of the spacecrafts. The following observation is the cornerstone for
the secrecy harnessing from the Doppler frequency shift.

Observation 1. The Doppler frequency of the reciprocal links at a certain time
instant can be uniquely described with
$\omega_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}=-\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$,
while the Doppler frequency of received signal from any other physically
disjoint node $t$ will be different than other ISL’s
$\omega_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}\neq\omega_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n}}\neq\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n}}$.

This observation guarantees that two distant spacecraft, Alice and Bob may
obtain identical Doppler frequency observations resulted from their high
mobilities. In the meantime, Eve also observe a Doppler frequency shift in her
versions of the signal. As stated in the observation, since the relative
velocity and position of Eve with Alice and Bob would be different than the
relative speed of Alice and Bob, her Doppler frequency shift observation would
be different than Alice and Bob.

However, in order to utilize Doppler frequency shift as a secrecy source, two
important properties must be satisfied from the measurements:

  1. 1.

The measurement of Eve should be as uncorrelated as possible with the
measurements at Alice and Bob. In other words, assuming Eve knows the secret
key generation procedure, the generated key at her side must be different than
Alice’s and Bob’s.

  2. 2.

In order to obtain continuous secrecy, Alice and Bob would require update the
generated key. Also, this generated key should have a high amount of entropy
to be utilized in symmetric key encryption. Therefore, Doppler frequency
measurements should embody a random characteristics over a long time period.

With the following proposition and the Observation 1, we guarantee the first
given condition. The second condition is discussed in the following
subsection.

######  Proposition 1.

For the nodes with very high mobility, Doppler frequency shift is a shared
secret between reciprocal nodes, while any other eavesdropping node cannot
retrieve the mobility information of the legitimate nodes.

###### Proof.

Let us denote the nominal Doppler frequency measurements at an arbitrary time
instant at Alice and Bob respectively with $f_{ba}$ and $f_{ab}$. The measured
Doppler frequency shifts at Eve would be denoted by $f_{be}$ and $f_{ae}$.
Considering the definition of the velocity,

$\left((\mathbf{\hat{v}_{b}^{T}}-\mathbf{\hat{v}_{a}^{T}})+(\mathbf{\tilde{v}_{b}^{T}}-\mathbf{\tilde{v}_{a}^{T}})\right)\frac{(\mathbf{p_{b}}-\mathbf{p_{a}})}{||(\mathbf{p_{b}}-\mathbf{p_{a}})||}=f_{ba}.$
(3)

In order to obtain $f_{ba}$ or $f_{ab}$, Eve should obtain the vectors
$\mathbf{\hat{v}_{a}}$, $\mathbf{\hat{v}_{b}}$, $\mathbf{p_{a}}$ and
$\mathbf{p_{b}}$. The observations at Eve are consist of $f_{ae}$, $f_{be}$,
$\mathbf{\hat{v}_{b}}$ and $\mathbf{\hat{v}_{a}}$. Let us rewrite the Doppler
frequency equations at Eve in a set of linear equations form:

$\begin{bmatrix}\mathbf{p}^{T}_{a}-\mathbf{p}^{T}_{e}&-\mathbf{v}^{T}_{e}&0&0\\\
0&0&\mathbf{p}^{T}_{b}-\mathbf{p}^{T}_{e}&-\mathbf{v}^{T}_{e}\end{bmatrix}\begin{bmatrix}\mathbf{v}_{a}\\\
\mathbf{p}_{a}\\\ \mathbf{v}_{b}\\\
\mathbf{p}_{b}\end{bmatrix}=\begin{bmatrix}f_{ae}-\mathbf{v}^{T}_{e}\mathbf{p}_{e}\\\
f_{be}-\mathbf{v}^{T}_{e}\mathbf{p}_{e}\end{bmatrix}.$ (4)

In order to capture the variable vector
$[\mathbf{v}_{a},\mathbf{p}_{a},\mathbf{v}_{b},\mathbf{p}_{b}]^{T}$, Eve needs
to find the inverse of the  left part of the matrix. In order to solve the
equation Eve needs 6 more equations to find the inverse of the matrix given in
the above. Therefore, Eve cannot retrieve the mobility information of Alice
and Bob and cannot obtain their Doppler frequency measurements. ∎

Note that, we do not consider the side channel attacks for the proposed
scenario. Since the mobility of the nodes may obtain in different ways, Eve
may try to utilize side channel attacks. For example, by utilizing multiple
antenna receiver structure, Eve may estimate the position and the velocity of
the spacecrafts. Moreover, by utilizing a radar system, she may track the
legitimate nodes positions and velocities. However, a radar equipped Eve would
be limited by the range of the radar, the number of existing mobile nodes in
the range, and the stealth technology capabilities of Alice and Bob. Based on
the relative positions and velocities of Alice and Bob, they determine the
unambiguous radar range for the eavesdropper as outlined in [30]. If Alice and
Bob are inside the radar range of Eve, than other existing mobile nodes would
force Eve to try $\binom{n_{s}}{2}$ combinations for a single Doppler
measurement, where $n_{s}$ denotes the number of mobile nodes in the radar
range of Eve. By selecting the number of quantization intervals lower than
$\binom{n_{s}}{2}$, brute force attack becomes easier than the radar attack.
For example if there is at least one mobile node existing, they may continue
their secret key generation with a two level quantization process. If there
are no nodes other than Alice and Bob in the radar range, than they may
utilize smart or cognitive jamming [31] or artificial noise methodologies to
confuse the eavesdropper without effecting their own radar measurements [3].
In the following, we discuss how the proposed mechanism satisfies the second
enumerated condition for the secret key generation.

###  III-C Maximum Achievable Secret Key Rate from Doppler Frequency

The superposition of the velocities can also be observed in the Doppler
frequency shifts as
$\omega_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(t)=\hat{\omega}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(t)+\tilde{\omega}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(t)$,
where
$\hat{\omega}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(t)=\frac{f_{c}}{c}\frac{(\mathbf{\hat{v}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{\hat{v}_{k}}(t))^{T}(\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{p_{k}}(t))}{||\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{p_{k}}(t)||}$,
and
$\tilde{\omega}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(t)=\frac{f_{c}}{c}\frac{(\mathbf{\tilde{v}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{\tilde{v}_{k}}(t))^{T}(\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{p_{k}}(t))}{||\mathbf{p_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)-\mathbf{p_{k}}(t)||}$.
Note that, we model
$\hat{\omega}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(t)$
as a deterministic constant, and
$\tilde{\omega}_{k{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}(t)$
as a random i.i.d. process. In the following, we show that, the deterministic
part of the Doppler frequency shift actually do not contribute to the secret
key randomness, whereas the random part determines the maximum achievable
secret key rate.

######  Proposition 2.

The achievable secret key rate when the Doppler frequency shift is utilized as
a secrecy source is equal to $R_{k}=\log({f_{c}}/{c})+h(\tilde{f}_{ab})$.

###### Proof.

The achievable secret key rate is limited by the entropy of the randomness
source as given in [8]. In our case the achievable secret key rate is given by

$R_{k}={h}(\omega_{ab})={h}(\omega_{ba}).$ (5)

Let us rewrite the entropy of the secrecy source as

${h}(\omega_{ab})={h}(\hat{\omega}_{ab}+\tilde{\omega}_{ab}).$ (6)

As given in [8], since the differential entropy is translation invariant, the
constant term does not introduce additional entropy. Hence
${h}(\omega_{ab})=h(\tilde{\omega}_{ab})$. Also considering
$\tilde{\omega}_{ab}=\frac{f_{c}\tilde{f}_{ab}}{c}$, the maximum achievable
key rate is given by

$R_{k}=\log({f_{c}}/{c})+h(\tilde{f}_{ab}).$ (7)

∎

######  Proposition 3.

The achievable secret key rate is maximized when the mobility of the
legitimate nodes follow the Brownian motion as
$\\{(p_{a}(t+1)-p_{a}(t))\sim\mathcal{N}(0,\sigma_{v_{a}}^{2}T)$ and
$(p_{b}(t+1)-p_{b}(t))\\}\sim\mathcal{N}(0,\sigma_{v_{b}}^{2}T)$.

###### Proof.

Let us rephrase $R_{k}=\log({f_{c}}/{c})+h(\tilde{f}_{ab})$. Since the carrier
frequency, speed of light are constants, the achievable secret key rate is
maximized in line with the distribution of the nominal Doppler frequency.
Since a Gaussian random variable has the largest entropy amongst all random
variables of equal variance [8], the maximum achievable secret key rate is
obtained when $\tilde{f}_{ab}\sim\mathcal{N}(0,\sigma^{2})$. $\tilde{f}_{ab}$
can be rephrased by

$\tilde{f}_{ab}=\rho_{x}(\tilde{v}^{x}_{b}-\tilde{v}^{x}_{a})+\rho_{y}(\tilde{v}^{y}_{b}-\tilde{v}^{y}_{a})+\rho_{z}(\tilde{v}^{z}_{b}-\tilde{v}^{z}_{a}),$
(8)

where $\rho_{x}=\frac{p^{x}_{b}-p^{x}_{a}}{||p_{b}-p_{a}||}$,
$\rho_{y}=\frac{p^{y}_{b}-p^{y}_{a}}{||p_{b}-p_{a}||}$, and
$\rho_{z}=\frac{p^{z}_{b}-p^{z}_{a}}{||p_{b}-p_{a}||}$. By assuming each
velocity component is Gaussian distributed, we can get the nominal Doppler
frequency also Gaussian distributed, where
$v^{x}_{a},v^{y}_{a},v^{z}_{a}\sim\mathcal{N}(0,\sigma_{v_{a}}^{2})$, and
$v^{x}_{b},v^{y}_{b},v^{z}_{b}\sim\mathcal{N}(0,\sigma_{v_{b}}^{2})$. The
variance of the nominal Doppler frequency then becomes

$\sigma^{2}_{d}=(\rho^{2}_{x}+\rho^{2}_{y}+\rho^{2}_{z})(\sigma_{v_{a}}^{2}+\sigma_{v_{b}}^{2})=\sigma_{v_{a}}^{2}+\sigma_{v_{b}}^{2}.$
(9)

Since the velocity is constant in the communication period, the displacement
vector of Alice and Bob would be modeled as
$\\{(p_{a}(t+1)-p_{a}(t))\sim\mathcal{N}(0,\sigma_{v_{a}}^{2}T)$ and
$(p_{b}(t+1)-p_{b}(t))\\}\sim\mathcal{N}(0,\sigma_{v_{b}}^{2}T)$, which is the
definition of the Brownian motion [8]. ∎

######  Corollary 1.

The maximum achievable secret key rate can be given by

$R_{k}^{*}=\frac{1}{2}\log(2\pi e\sigma^{2}_{d})+\log(f_{c}/c).$ (10)

###### Proof.

This is a direct result of Proposition 2 and Proposition 3, where

$R^{*}_{k}=\log({f_{c}}/{c})+\max\\{h(\tilde{f}_{ab})\\},$

and the maximization is obtained for
$\tilde{f}_{ab}\sim\mathcal{N}(0,\sigma^{2})$. Since
$h(\tilde{f}_{ab})=\frac{1}{2}\log(2\pi e\sigma^{2}_{d})$, the maximum
achievable secret key rate becomes

$R_{k}^{*}=\frac{1}{2}\log(2\pi e\sigma^{2}_{d})+\log(f_{c}/c).$

∎

In the following, we present the proposed secret key generation procedure to
extract secret keys from the mobility of the spacecrafts.

##  IV Secret Key Generation Procedure

Figure 3: An illustration of secret key generation procedure.

###  IV-A Pilot Transmission & Reception

The secret key generation procedure is described in the Figure 3. As a first
step, Alice and Bob respectively transmit $N$ pilot symbols
$\mathbf{x}=\left[x(1),x(2),\ldots,x(N)\right]$ in a time division duplexing
(TDD) fashion,  where $x(i)=a_{i}e^{j\phi_{i}}$. In the following of this
paper, we consider BPSK modulation for simulations and in Proposition 5. In
this case, $a_{i}=1$, $\phi_{i}\in\\{-\pi,\pi\\}$. The considered spacecrafts
Alice, Bob and Eve are respectively be denoted by $\\{a,b,e\\}$. Considering
the ISLs, the baseband representation of the received symbols at the
spacecraft $k$ from the spacecraft
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}$,
$\mathbf{y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}=\left[y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(1),y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(2),\ldots,y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(N)\right]$
can be modeled as

$y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)=x_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\epsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i),$
(11)

where $i\in\\{1,\ldots,N\\}$,
$x_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)=\zeta
x(i)e^{j\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}iT}$,
$\zeta=\frac{1}{d^{PL}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}$
and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}\in\\{a,b\\}$,
$k\in\\{a,b,e\\}$ and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}\neq
k$ [32]. $\zeta$ denotes the path loss attenuation, where
$d_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
is the distance between spacecraft
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}$
and $k$, and $PL$ is the path loss exponent. $\omega_{jk}$ denotes the Doppler
frequency shift for the link $j-k$, and $T$ is the symbol period.
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\epsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)\sim\mathcal{CN}(0,\sigma_{k}^{2})$
denotes the additive white Gaussian noise at the receiver $k$, where
$\mathcal{CN}(0,\sigma^{2})$ denotes the i.i.d. complex normal distribution
with zero mean and $\sigma^{2}$ variance. Considering the similar quality of
the radio equipments at each spacecraft, we assume that the noise variance is
equal for each receiver as $\sigma_{k}^{2}=\sigma^{2}$.  Note that, we
consider that the carrier frequency information is available at the receiver,
where the received signal can be down-converted to the baseband equivalent
signal. The effect of oscillator-based frequency shifts would be very small in
comparison with the Doppler frequency shift from high mobility. We also assume
that the channel based phase shift can be recovered at receiver by utilizing a
phase locked loop as described in [33]. The spacecrafts are assumed to have
same velocities during the pilot transmission reception scheme. We detail the
timing considerations in the Appendix.

The information contained in
$\mathbf{y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}$
is fully present in its discrete Fourier transform:

$Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)=X_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)+K_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i),$
(12)

where
$\mathbf{Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}=\mathcal{F}\\{y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}\\}=\left[Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(1),Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(2),\ldots,Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(N)\right]$,
$\mathbf{X_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}=\mathcal{F}\\{x_{jk}\\}=\left[X_{jk}(1),X_{jk}(2),\ldots,X_{jk}(N)\right]$
and
$\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}=\mathcal{F}\\{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\epsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}\\}=\left[{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(1),{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(2),\ldots,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(N)\right]$.
Since Gaussian processes are invariant against Fourier transform, the signal
spectrum
$\mathbf{X_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}$,
and the noise spectrum
$\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}$,
are also complex Gaussian, zero-mean, and orthogonal processes. The spectral
samples
${Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}$
are mutually uncorrelated because of the assumed stationarity of
$X_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)$.

Note that, as stated in [34], the phase of
$\mathbf{Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}$
carries no information about the Doppler frequency, since
$\mathbf{Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}$
has been modeled as a stochastic process with the aforementioned properties.
Hence, it is sufficient to consider the power spectrum of the received data as

$\displaystyle\mathbf{S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}$
$\displaystyle=\left[S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(1),S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(2),\ldots,S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(N)\right]$
(13)
$\displaystyle=\left[|Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(1)|^{2},|Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(2)|^{2},\ldots,|Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(N)|^{2}\right].$

Since
$Y_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)$
is a complex Gaussian process, the probability density function of each sample
${S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}(i)$
under the condition of a particular Doppler frequency
$\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
is given by the exponential distribution :

$\rho(S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i);\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k})=\frac{1}{\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}\text{exp}\left(-\frac{S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}{\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}\right),$
(14)

where
$\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)$
denotes the NPSDS, and can be obtained by [34]

$\begin{aligned}
\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)=\mathbb{E}\\{{S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}(i)\\}&=\mathbb{E}\\{|X_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)|^{2}\\}\\\
&=\mathbb{E}\\{|X_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)|^{2}\\}+\mathbb{E}\\{|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varepsilon}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)|^{2}\\}\end{aligned},$
(15)

where
$A^{x}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i\Delta
f-\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k})=\mathbb{E}\\{|X_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)|^{2}\\}$,
$A^{n}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=\mathbb{E}\\{|K_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)|^{2}\\}$.
Note that
$A^{x}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(f)$
is the a priori known nominal power spectral density of the signal; $\Delta f$
is the frequency sampling interval; and
$\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
is the Doppler frequency shift. Considering
$A^{x}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(f)$
is periodic with period $\Delta f$, and
$A^{n}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
is a constant, we can deduce that
$\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)$
is also periodic with $\Delta f$, and consequently we can drop $i$ and denote
the NPSDS as

$\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i),\forall
i.$

######  Proposition 4.

The NPSDSs for reciprocal links will be equal to

$\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=\Theta_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}},$
(16)

while any other spacecraft than
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}$
and $k$ would observe different values.

###### Proof.

We can proof the Proposition 1 in two steps:

  1. 1.

The NPSDS follows the symmetric relation as
$A^{x}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i\Delta
f-\omega_{jk})=A^{x}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i\Delta
f+\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k})$
[35]. As stated in Observation 1,
$\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=-\omega_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}$.
Considering the same pilot sequence is transmitted from both spacecrafts,
$A^{x}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(f)=A^{x}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(f)$.
Consequently, we can state that
$A^{x}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i\Delta
f+\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k})=A^{x}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(i\Delta
f+\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k})$.

  2. 2.

As mentioned above, the variance of the thermal noise at each receiver is
assumed to be equal. Therefore,
$A^{n}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=A^{n}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}$
and

$\underset{\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}{\underbrace{A^{x}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i\Delta
f+\omega_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k})+A^{n}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}}=\underset{\Theta_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}{\underbrace{A^{x}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}(i\Delta
f+\omega_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}})+A^{n}_{k{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}}}$

∎

As indicated in Proposition 4, the ideal NPSDS at Alice and Bob would be equal
for each communication block $\Theta_{ab}(t)=\Theta_{ba}(t)=\Theta_{t}$.
However, considering the randomness in the mobility of the spacecrafts, the
NPSDS are also random variables. Due to the dependency of the power spectral
density to the modulation type, NPSDS distribution would also change with the
modulation type. A general probability distribution model for the NPSDS become
intractable. Therefore, we provide an approximation for NPSDS modeling in the
following proposition.

######  Proposition 5.

Considering BPSK modulation at the transmitter, the NPSDS at Alice and Bob can
be modeled by
$\frac{\Theta_{t}}{\sigma^{2}_{\Theta}T^{2}}\sim{\mathcal{\chi}^{\prime}}^{2}_{1}(\frac{\hat{\omega}^{2}_{ab}}{\sigma^{2}_{v}})$,
where $\sigma^{2}_{\Theta}=\frac{PT^{3}\sigma^{2}_{v}}{2}$. The probability
distribution function of ${\Theta_{t}}$ becomes

$\rho(\Theta_{t})=\frac{1}{2\sigma^{2}_{\Theta}}e^{-\frac{1}{2}\left(\frac{\Theta_{t}}{\sigma^{2}_{\Theta}}+\lambda\right)}\left({\frac{\Theta^{2}_{t}}{\sigma^{2}_{\Theta}\lambda}}\right)^{-\frac{1}{4}}\mathcal{I}_{-1/2}\left(\sqrt{\lambda\frac{\Theta^{2}_{t}}{\sigma^{2}_{\Theta}}}\right),$
(17)

where $\lambda=\frac{\hat{\omega}^{2}_{ab}}{\sigma^{2}_{v}}$.

###### Proof.

As we modeled in Section II, the Doppler frequency shift can be written as the
superposition of two components,
$\omega_{ab}(t)=\hat{\omega}_{ab}(t)+\tilde{\omega}_{ab}(t)$, where
$\tilde{\omega}_{ab}(t)\sim\mathcal{N}(0,\sigma^{2}_{v})$. As given in [35],
considering the BPSK modulation NPSDS becomes

$\Theta_{t}=\frac{PT}{2}\text{sinc}^{2}(1-\omega_{ab}T),$ (18)

where $P$ denotes the average transmission signal power, $T$ denotes the
symbol period. By utilizing the Taylor series expansion of
$\text{sinc}^{2}(x)=(x-1)^{2}+\mathcal{O}(x^{4})$ as $x\rightarrow 1$, the
NPSDS becomes

$\Theta_{t}=\frac{PT^{3}}{2}\left({\omega_{ab}}\right)^{2}.$ (19)

We can model the Doppler frequency shift as
$\omega_{ab}\sim\mathcal{N}(\hat{\omega}_{ab},\sigma_{v}^{2})$. Considering
this, the random variable
$\zeta=\frac{\Theta_{t}}{\sigma^{2}_{v}\frac{PT^{3}}{2}}$ becomes a non-
central chi-squared distribution with 1 degrees of freedom and
$\lambda=\frac{\hat{\omega}^{2}_{ab}}{\sigma^{2}_{v}}$ non-centrality
parameter. Since , ${\Theta_{t}}=\zeta{\sigma^{2}_{v}\frac{PT^{3}}{2}}$, the
probability distribution function of $\Theta_{t}$ can be obtained by the
change of the variables as

$\rho(\Theta_{t})=\frac{1}{2\sigma^{2}_{\Theta}}e^{-\frac{1}{2}\left(\frac{\Theta_{t}}{\sigma^{2}_{\Theta}}+\lambda\right)}\left({\frac{\Theta^{2}_{t}}{\sigma^{2}_{\Theta}\lambda}}\right)^{-\frac{1}{4}}\mathcal{I}_{-1/2}\left(\sqrt{\lambda\frac{\Theta^{2}_{t}}{\sigma^{2}_{\Theta}}}\right).$

∎

Note that, the given distribution in Proposition 5 applies for the BPSK
modulation. With different modulation type, the shape of the power spectral
density would also be different. In Section VI, we provide the tightness of
the assumptions made in the numerical analyses on the Proposition 5. Without
explicitly calculating the Doppler frequency, spacecrafts may individually
estimate the NPSDS from their observation sequences, and they can exploit this
value as a shared secret for key generation mechanism. In the following, we
describe the estimation of the NPSDS process at the spacecrafts.

###  IV-B NPSDS Estimation

Due to the NPSDS change in each communication block, the NPSDS estimation
would be repeated for each communication block separately. Let us denote the
estimated NPSDS at receiving node $k$ in an arbitrary communication block as
${\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$.
Note that considering Alice and Bob, $\Theta_{ab}=\Theta_{ba}=\Theta_{t}$. The
maximum likelihood (ML) estimation of the parameter
${\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
can be given as:

$\displaystyle\hat{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
$\displaystyle=\max_{{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}\left\\{\prod\limits_{i=1}^{N}\frac{1}{{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}\text{exp}\left(-\frac{S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}{\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}\right)\right\\}$
(20)
$\displaystyle=\max_{{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}\left\\{\frac{1}{{\Theta}^{N}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}\text{exp}\left(-\frac{\sum\limits_{i=1}^{N}S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}{\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}\right)\right\\}.$

The log-likelihood function for the estimation problem becomes

$\mathcal{L}(S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i);{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k})=-N\ln({\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k})-\left(\frac{\sum\limits_{i=1}^{N}S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}{\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}\right).$
(21)

We obtain the value of the function when its first derivative is zero to find
the maximum value of the log-likelihood function

$\frac{\partial\mathcal{L}(S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i);{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k})}{\partial\Theta_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}=-\frac{N}{{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}}+\frac{\sum\limits_{i=1}^{N}S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}{\Theta^{2}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(22)

The result of this equation provides us the ML estimator for the
${\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
parameter as

$\hat{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=M^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}_{s}=\frac{\sum\limits_{i=1}^{N}S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}{N},$
(23)

where
$M^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}_{s}$
is the sample mean of the observed power spectral density samples. The steps
followed in NPSDS estimation block can be summarized as in Figure 4.

Figure 4: An illustration of NPSDS estimation at Alice and Bob.

###  IV-C Quantization

After obtaining
$\hat{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}$
values, terminals quantize and encode this value in order to obtain a secret
key sequence. In this paper, we assume uniform quantizer with a step size
$\Delta$ as in [5]. From $N$ observations, we get $Q$ number of quantized
secret key bits as
$q_{b}=\left\lfloor\frac{\hat{\Theta}_{ab}}{\Delta}\right\rfloor$ and
$q_{a}=\left\lfloor\frac{\hat{\Theta}_{ba}}{\Delta}\right\rfloor$, where
$\left\lfloor\cdot\right\rfloor$ denotes the floor function.

Note that, the main focus of this work is producing the raw key from a novel
secrecy source, mainly the Doppler frequency shift. The key reconciliation and
hybrid key generation algorithms in [36] can be applied to the quantized raw
key bits in order to reduce the erroneous elements in the generated key bits.

##  V Key Disagreement Rate

Key disagreement rate (KDR) is the ratio of mismatched bits in the generated
keys. We adopt a similar approach to [7] in order to obtain theoretical
expressions of the KDR considering the proposed key generation system model.
Considering Proposition 4, we can state that
$\Theta_{ba}=\Theta_{ab}=\Theta_{t}$ for the $t^{\text{th}}$ communication
block. Note that, one main difference with our previous work is that, we model
$\Theta_{t}$ with a random variable as given in the Proposition 5. In the
following, we approximate the estimated NPSDS values at Alice and Bob by using
the central limit theorem.

######  Proposition 6.

For large $N$, NPSDS estimates follow the truncated normal distribution with
support $[0,\infty]$ as
$\hat{\Theta}_{ab}\sim\mathcal{N}(\Theta_{t},\frac{\Theta^{2}_{t}}{N})$, and
$\hat{\Theta}_{ba}\sim\mathcal{N}(\Theta_{t},\frac{\Theta^{2}_{t}}{N})$.

###### Proof.

Let us rewrite
$\hat{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=M^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}_{s}=\frac{\sum\limits_{i=1}^{N}S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)}{N}$.
Here,
$S_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(i)$
samples are i.i.d, and each one of them follows the probability distribution
function given in Eq. 14. Considering this, for large $N$, the central limit
theorem states that
$\sqrt{N}\left(\hat{\Theta}_{ab}-\Theta_{t}\right)\sim\mathcal{N}(0,\Theta^{2}_{t})$
[35]. Consequently
$\hat{\Theta}_{ab}\sim\mathcal{N}(\Theta_{t},\frac{\Theta^{2}_{t}}{N})$.
Identical operations can be applied on $\hat{\Theta}_{ba}$, due to
$\Theta_{ab}=\Theta_{ba}=\Theta_{t}$. Note that, the NPSDS should be larger
than equal to zero from its definition, hence their distribution can be given
as the truncated normal distribution with the support of $[0,\infty]$ [37]. ∎

Assuming the uniform quantizer with a step size $\Delta$, an estimated
$\hat{\Theta}_{ab}$ is mapped to the $l^{\text{th}}$ quantization level, where
$l=\left\lfloor\frac{\hat{\Theta}_{ab}}{\Delta}\right\rfloor$ and the
quantization interval is described by $I_{l}=[l\Delta,(l+1)\Delta]$. The
probability that $\hat{\Theta}_{ba}$ locates in $I_{l}$ under given
$\tilde{\Theta}_{ab}$ can be obtained by

$P_{l}=\int\limits_{l\Delta}^{(l+1)\Delta}\rho(\hat{\Theta}_{ba}|\hat{\Theta}_{ab},\Theta_{t})d\hat{\Theta}_{ba},$
(24)

where $\rho(\hat{\Theta}_{ba}|\hat{\Theta}_{ab},\Theta_{t})$ denotes the
probability density function of $\hat{\Theta}_{ba}$ given $\hat{\Theta}_{ab}$
and $\Theta_{t}$. In order to obtain $P_{l}$, let us first obtain
$\rho(\hat{\Theta}_{ba}|\hat{\Theta}_{ab},\Theta_{t})$. For a given
$\hat{\Theta}_{ab}$, the observation signal at Alice can be modeled as
$\hat{\Theta}_{ba}=\hat{\Theta}_{ab}-{K}_{ab}+{K}_{ba}$, where
$\\{{K}_{ab},{K}_{ba}\\}\sim\mathcal{N}(0,\frac{\Theta^{2}_{t}}{N})$. In this
case, we can state that
$\hat{\Theta}_{ba}\sim\mathcal{N}(\hat{\Theta}_{ab},\frac{2\Theta^{2}_{t}}{N})$.
Therefore $P_{l}$ becomes

$P_{l}=\\!\\!\\!\\!\\!\\!\int\limits_{l\Delta}^{(l+1)\Delta}\frac{\sqrt{N}}{\sqrt{8\pi}\Theta_{t}}e^{-\frac{N(\hat{\Theta}_{ba}-\hat{\Theta}_{ab})^{2}}{8\Theta^{2}_{t}}}d\hat{\Theta}_{ba},$
(25)

The closed form expression for the $P_{l}$ can be expressed by

$P_{l}=\text{erf}\left[\frac{(l+1)\Delta-\hat{\Theta}_{ab}}{\sqrt{\frac{8\Theta^{2}_{t}}{N}}}\right]-\text{erf}\left[\frac{(l)\Delta-\hat{\Theta}_{ab}}{\sqrt{\frac{8\Theta^{2}_{t}}{N}}}\right].$
(26)

Considering that $\hat{\Theta}_{ab}$ is also a random variable, the key
matching probability can be obtained by

$P_{c}=\int_{0}^{\infty}\left(\int_{0}^{\infty}P_{l}\rho(\tilde{\Theta}_{ab}|\Theta_{t})d\tilde{\Theta}_{ab}\right)\rho(\Theta_{t})d\Theta_{t}.$
(27)

To the best of authors knowledge, a closed form expression for the key
matching probability is not available. In the most open form, the key matching
probability becomes as in (28). In the following, we provide the numerical
analysis of the proposed key generation mechanism.

$P_{c}=\int_{0}^{\infty}\left(\int_{0}^{\infty}\text{erf}\left[\frac{(l+1)\Delta-\hat{\Theta}_{ab}}{\sqrt{\frac{8\Theta^{2}_{t}}{N}}}\right]-\text{erf}\left[\frac{(l)\Delta-\hat{\Theta}_{ab}}{\sqrt{\frac{8\Theta^{2}_{t}}{N}}}\right]\frac{\sqrt{N}}{\sqrt{2\pi}\Theta_{t}}e^{-\frac{N(\hat{\Theta}_{ab}-{\Theta}_{t})^{2}}{2\Theta^{2}_{t}}}d\hat{\Theta}_{ab}\right)\frac{1}{2\sigma^{2}_{\Theta}}e^{-\frac{1}{2}\left(\frac{\Theta_{t}}{\sigma^{2}_{\Theta}}+\lambda\right)}\left({\frac{\Theta^{2}_{t}}{\sigma^{2}_{\Theta}\lambda}}\right)^{-\frac{1}{4}}\mathcal{I}_{-1/2}\left(\sqrt{\lambda\frac{\Theta^{2}_{t}}{\sigma^{2}_{\Theta}}}\right)d\Theta_{t}.$
(28)

##  VI Numerical Analysis

In this section, we first analyze the secret key rate that can be extracted
from the Doppler frequency measurements in the considered scenario. In the
following, we provide the numerical analysis on the proposed secret key
generation mechanism based on the estimated NPSDSs. We conduct the Monte Carlo
simulations on MATLAB.

###  VI-A Maximum Achievable Key Rate from Doppler Frequency Shift Analysis

In this part, we numerically analyze the results given in Proposition 3 and in
Proposition 2. The velocity model at Alice and Bob follows the model given in
the Section II. In order to illustrate the importance of the random velocity
variable in key generation, we define mobility constant at Alice as
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}_{a}=\frac{\sigma^{2}_{v}}{||\hat{\mathbf{v}}_{a}||}$,
and similarly at Bob as
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}_{b}=\frac{\sigma^{2}_{v}}{||\hat{\mathbf{v}}_{b}||}$.
The parameters for the numerical analyses given in Table II.

Figure 5 illustrates the maximum achievable secret key rate values for
different mobility constants at Alice and Bob. One observation is that the
secret key rate gradually increases as any one of the mobility constants
increases. Since the randomness of the secret key is directly related with the
random movement of Alice and Bob, the secret key rate increases as the
contribution of the random movement increases in any one of the nodes.
Furthermore, the secret key rate gets its maximum value as the random velocity
component at Alice and Bob becomes more dominant to the constant velocity
component. However, this kind of setting is not realizable considering the
space missions requirements. Therefore, we assume that the lower $k$ region
would illustrate more realistically the future space missions. Note that in
line with the designed secret key generation procedure, the secret key rate
changes. The values given here is an upper bound for the maximum amount of
secret key can be extracted from the Doppler frequency shift.

Figure 5: Maximum achievable secret key rate for varying mobility constants of
Alice and Bob.

###  VI-B NPSDS Modeling Analysis

In the following, we present the numerical analysis related with the proposed
secret key generation methodology.

Figure 6: Empirical probability density functions for NPSDSs for three
different $k$ values, (a) $k=0.1$, (b) $k=0.25$ and (c) $k=0.5$, (d) $k=0.8$.

Figure 6 provides a comparison on the probability density functions of NPSDSs
when the value is calculated numerically, and when the value is calculated by
the approximation given in (19).The simulation parameters are given in Table
II. Note that in this simulation we assume
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}_{a}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}_{b}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}$,
where
$0\leq{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}<1$.
One important observation is that, the center point of the distribution shifts
near to zero as the random competent becomes dominant. One reason for this
trend is that as the
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}$
increases $\sigma^{2}_{\Theta}$ also increases. Because of this, the non-
centrality parameter of the $\Theta_{t}$ decreases and the center of the
distribution gets closer to zero. From another perspective, as the
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\kappa}$
increases total amount of movement and Doppler frequency increases. By
increasing these, the samples from the nominal power spectral densities would
be taken from further than their center and maximum values. Considering the
shape of the power spectral density of the modulated waveforms, as the sample
points get farther than the center, the amplitude values decreases. Therefore,
the center of the distribution tends to shift near zero.

###  VI-C NPSDS Estimation Analysis

Figure 7: Histograms for estimated NPSDSs for three different $N$ values, (a)
$N=10$, (b) $N=20$ and (c) $N=50$.

Alice and Bob employ the proposed key generation scheme in Section II in order
to obtain a secret key. In a single key duration, first Alice then Bob
respectively transmits $N$ pilot symbols. Then, they estimate NPSDSs, and feed
them into uniform quantizer. In the meantime, Eve observes the transmitted
pilot symbols from both spacecrafts, and obtain NPSDSs for each transmission.
After each communication block, Alice and Bob update their mobility
information, and consequently their Doppler frequency based observations are
also updated. Each of the legitimate nodes generate single secret key, while
Eve generates two different versions of the secret key.

TABLE II: Simulation parameters Carrier frequency | 1 GHz
---|---
$||\hat{\mathbf{v}}_{a}||$ , $||\hat{\mathbf{v}}_{b}||$ | 6700 m/sc
${v}_{e}$ | 2000 m/sc
Symbol energy | 10 dB
Modulation type | BPSK
Noise variance | 1 dB
Symbol period |  $10^{-6}$ sc.
Figure 8: MSE vs. channel observations considering the different ISLs.

Figure 7 shows the histograms of the estimated NPSDSs at Alice, Bob and Eve
for a given $\Theta_{t}=5$. Since Eve captures both the transmission from
Alice to Bob and from Bob to Alice, she obtains two different versions of the
estimated NPSDSs. From left to right (Figure 7-7-7), the number of
observations increases. One remark is that as the number of observation
increases, the variance of the pdfs become smaller. The distribution at Alice
and Bob are almost identical as defined in (4) and (5). On the other hand, the
pdfs at Eve diverges from Alice and Bob, since their relative velocities and
Doppler frequencies are different.

In order to focus on the relationship between the number of observations and
the estimation disparities, we utilize mean squared error (MSE) as in Figure
8. The MSE for the NPSDS estimator for the
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}-k$
link can be given as

$\text{MSE}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=\frac{1}{N}\sum_{i=0}^{N}|\hat{\Theta}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
As indicated in the figure, the estimations at Alice and Bob converges as the
number of observations increases. Even though estimation error at Eve also
decreases, the error becomes stable after $N=5$. Since the relative velocity
of Alice and Bob differs from the relative velocity of Alice and Eve, and the
relative velocity of Bob and Eve, the Doppler frequency values observed at Eve
is also different than Alice and Bob. Considering the Proposition 1, the only
option for Eve is to utilize her estimation to obtain the secret key generated
by Alice and Bob since she cannot retrieve the mobility information of Alice
and Bob from her observations. The results in both Figure 7 and Figure 8 show
that the NPSDSs or Doppler frequencies can be utilized as a secret source
between two separately moving nodes.

Figure 9: KDR vs. normalized quantization interval considering different
number of channel taps.

###  VI-D KDR Analysis

KDR analysis illustrates the robustness of the secret key scheme. Since even
one bit mismatch in the generated secret key results non-identical secret keys
at Alice and Bob, the error rates should be very low to practically apply the
secret key scheme. Numerically KDR in a single key duration can be described
by

$KDR_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=\begin{cases}1,q_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}}\neq
q_{k}\\\
0,q_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
Considering $D$ number of key durations, KDR for spacecrafts $k$ and $j$
becomes

$\text{KDR}_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}=\frac{\sum\limits_{t=1}^{D}KDR_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}m}k}(t)}{D}.$

Figure 9 provides KDR values of the generated keys at Alice and Bob for
different normalized quantization intervals, where
$\Gamma=\frac{\Delta}{\mathbb{E}\\{\Theta_{t}\\}}$. We consider a two-level
uniform quantizer for the proposed secret key scheme. So the NPSDS estimations
are mapped into two different symbols. Since as the number of observations at
Alice and Bob increases, the error at their estimates decreases. Therefore,
increment in the number of observation samples decreases KDR more
dramatically. Note that, increment in the number of observation samples would
also require transmitting more pilot symbols and reduces the efficiency of the
protocol. Solid lines show the approximate KDR by (28) while the stars
indicate the simulation results. Note that, the performance of the proposed
protocol can further be improved by utilizing hybrid key generation protocol
as proposed in [36].

##  VII Conclusion and Future Work Directions

In this work, we have proposed a security mechanism for the inter-spacecraft
links (ISLs) first time in the literature. We have introduced a novel mobility
model for future space missions, where the dynamic and static mobility
components are modeled separately. By utilizing Doppler frequency shift as a
secrecy source, we have proposed a first example of the mobility-based secret
key generation in the literature, where this novel secrecy source paves the
way for a secret key generation under static channel environments. We have
provided the maximum achievable secret key rate for the Doppler frequency
shift based secret key generation. The proposed key generation procedure
utilizes the nominal power spectral density samples (NSPDSs) as a projection
of Doppler frequency shift measurements. A maximum likelihood (ML) estimation
procedure for the NPSDSs has been introduced, and the corresponding key
disagreement rate (KDR) of the procedure is analytically obtained. The
numerical results highlight the consistency of the analytical bounds and
approximations, and they illustrate the practical applicability of the
proposed secret key generation procedure.

As future work, we believe that utilizing Doppler frequency shift with
oscillator-based frequency offset as a secrecy resource would result into
superposition of two independent random variables. Therefore, the total
randomness obtained would increase, and we obtain higher key entropies in
comparison with only carrier-frequency offset (CFO) or only Doppler frequency
shift based methods. Especially when the mobility of the nodes decreases, the
secret key generation system may switch from the Doppler based secrecy
extraction to CFO based secret key extraction, or vice versa. Another
important future work would be comparing the performance of the proposed
scheme and the fading based key generation schemes under the active physical
layer attacks such as jamming. We believe that the mobility-based secret key
generation would also be more resilient to active attacks since the effect of
noise is negligible for the mobility-based key generation procedures.
Furthermore analyzing the proposed system against an attacker obtaining a
radar system, and the utilization of jamming against this attacker is another
possible future work.

## Appendix: Timing Considerations

The timing distance between Alice and Bob is represented by $\Delta t_{AB}$,
where due to the long distances for spacecrafts, the most important factor in
the timing difference would be the propagation time of the transmitted data.
In this case, we can say that

$\displaystyle\Delta t_{AB}$ $\displaystyle=\frac{||p_{A}(t)-p_{B}(t+\Delta
t_{AB})||}{c}+\frac{||p_{A}(t+\Delta t_{AB}+t^{\prime})}{c}$
$\displaystyle-\frac{p_{B}(t+2\Delta t_{AB}+t^{\prime})||}{c}.$

Let us assume that $||p_{A}(t)-p_{B}(t+\Delta t_{AB})||+||p_{A}(t+\Delta
t_{AB}+t^{\prime})-p_{B}(t+2\Delta
t_{AB}+t^{\prime})||=(1+\alpha)||p_{A}(t)-p_{B}(t)||,$ and

$\Delta t_{AB}=\frac{(1+\alpha)||p_{A}(t)-p_{B}(t)||}{c}.$

We assume that the velocity of the spacecrafts are almost static during the
time difference between the nodes. By saying this, we need to check whether

$\Delta t_{AB}\leq\frac{||v_{max}||}{||a_{A}||+||a_{B}||},$

where $v_{max}$ denotes the maximum velocity change tolerance, and $a_{A}$,
$a_{B}$ are respectively acceleration of Alice and Bob. Considering this, we
would get the following condition for the spacecrafts in order to obtain
identical Doppler frequency shift measurements

$\frac{(1+\alpha)||p_{A}(t)-p_{B}(t)||}{c}\leq\frac{||v_{max}||}{||a_{A}||+||a_{B}||}$
(29)

As a numerical example, a velocity change around 1000 m/s would only create
around 6 kHz Doppler frequency shift for the 2 GHz carrier frequency.
Considering the measured Doppler frequencies would be around 60 KHz for the
secret key generation, we can safely neglect the velocity changes less than
1000 m/s, in this case $||v_{max}||=1000$ m/s. In order to obtain a
communication link between spacecrafts, antenna design is another important
consideration. In order to get signals from the Moon, the radio receivers at
Earth would require a high noise tolerance due to the extremely long
distances. If, we consider that the spacecrafts may can only support the 0.1
times the distance from Earth to Moon, which is $3.810^{8}$ m. By setting
$\alpha=1$, we get $||a_{A}||+||a_{B}||<5000\text{ m/s}^{2}$, which is rather
reasonable considering the acceleration capabilities of the current
spacecrafts.

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