# Surface-induced ferromagnetism and anomalous Hall transport at Zr2S(001)

Shuyuan Liu1, Yanwei Luo1,2, Chongze Wang1, Hyunsoo Jeon1, Yu Jia3, and Jun-
Hyung Cho1∗ 1Department of Physics and Research Institute for Natural Science,
Hanyang University, 222 Wangsimni-ro, Seongdong-Ku, Seoul 04763, Republic of
Korea
2College of Science, Henan University of Technology, Zhengzhou 450001,
People’s Republic of China
3Key Laboratory for Special Functional Materials of the Ministry of Education,
Henan University, Kaifeng 475004, People’s Republic of China

###### Abstract

Two-dimensional layered electrides possessing anionic excess electrons in the
interstitial spaces between cationic layers have attracted much attention due
to their promising opportunities in both fundamental research and
technological applications. Using first-principles calculations, we predict
that the layered bulk electride Zr2S is nonmagnetic with massive Dirac nodal-
line states arising from Zr-4$d$ cationic and interlayer anionic electrons.
However, the Zr2S(001) surface increases the density of states at the Fermi
level due to the surface potential, thereby inducing a ferromagnetic order at
the outermost Zr layer via the Stoner instability. Consequently, the time-
reversal symmetry breaking at the surface not only generates spin-polarized
topological surface states with intricate helical spin textures but also hosts
an intrinsic anomalous Hall effect originating from the Berry curvature
generated by spin-orbit coupling. Our findings offer a playground to
investigate the emergence of ferromagnetism and anomalous Hall transport at
the surface of nonmagnetic topological electrides.

##  I I. INTRODUCTION

Surface ferromagnetism has been a longstanding issue in condensed matter
physics because of its importance from both the fundamental and technological
points of view [1, 2, 3]. The surface electronic structure often differs from
the bulk one because surface atoms have the reduced coordination number due to
their broken bonds. For example, the (001) surface of a 4$d$ transition metal
Rh increases the density of states (DOS) at the Fermi level $E_{F}$ via
narrowing its associated energy bands, thereby being vulnerable to a surface
ferromagnetic instability [4, 5]. Such surface-induced ferromagnetism in
otherwise nonmagnetic bulk material can be explained by the Stoner model of
itinerant magnetism: i.e., the Stoner criterion [6], where the product of the
exchange integral and the surface DOS at $E_{F}$ in the nonmagnetic state is
greater than 1, is satisfied to induce ferromagnetism.

In the past decade, there have been intense research efforts to explore the
connections between symmetries and topologies of condensed matter [7, 8]. As a
compelling example of symmetry-protected topological states, massless Dirac
fermions with fourfold degenerate band crossings of two doubly degenerate
bands are jointly protected by the time-reversal symmetry (TRS) $T$ and space
inversion symmetry $P$ supplemented by additional crystalline symmetry such as
glide mirror symmetry or screw rotation symmetry [9, 10, 11]. However, when
$T$ or $P$ symmetry is broken, Dirac fermions are transformed into Weyl
fermions with twofold degenerate band crossings of two singly degenerate bands
[11, 12]. Such symmetry-protected topological states having linear band
crossings at nodes or along one-dimensional lines or loops in momentum space
can be classified into Dirac, Weyl, Dirac-nodal-line (DNL), or Weyl-nodal-line
semimetal states [13, 14, 15]. These gapless topological states with the
nontrivial topology of bulk bands host spin-polarized surface states with
helical spin textures. However, without sufficient symmetry protection, the
degeneracies at such band crossings are lifted to form hybridization gaps via
the inclusion of spin-orbit coupling (SOC), which generate the Berry curvature
around gapped crossings [16, 17]. Here, we introduce a nonmagnetic bulk system
that has massive DNL states with SOC-induced gap openings, but its surface
exhibits a ferromagnetic order. The resulting TRS-breaking ferromagnetic
surface hosts the emergence of an intrinsic anomalous Hall effect originating
from the Berry curvature. By utilizing the spin degree of freedom generated
only at surface, it is highly promising to realize anomalous transport
phenomena in future spintronics technologies.

Recently, two-dimensional (2D) layered electrides $A_{2}B$ have aroused great
interest for their exotic electronic properties such as low work function,
high electron mobility, and spin polarization [16, 17, 18, 19, 20]. In such
$A_{2}B$ electrides consisting of a three-atom-thick building block of
$A$-$B$-$A$ stacks [see Figs. 1(a) and 1(b)], anionic excess electrons reside
in the interstitial spaces between positively charged $A$-$B$-$A$ cationic
layers. The global structure search method has been used to predict various
types of 2D layered electrides such as alkaline-earth nitrides [21], rare-
earth carbides [21, 22, 23], rare-earth pnictides [24], rare-earth
chalcogenides [24], rare-earth halides [23, 24], and transition-metal
monochalcogenides [25]. To date, the experimentally synthesized 2D electrides
are relatively few with (i) nonmagnetic Ca2N [18], Hf2S [26], Zr2S [25],
Sr3CrN3 [27], Sr8P5 [28], and Sr5P3 [28], (ii) paramagnetic Y2C [29, 30, 31],
and (iii) ferromagnetic Gd2C [32] and YCl [33]. Interestingly, recent DFT
calculations [34, 35] for the nonmagnetic Hf2S electride predicted the
existence of a ferromagnetic order at the Hf2S(001) surface, where the local
DOS arising from the surface Hf atoms increases at $E_{F}$ and thus induces a
Stoner instability.

In this paper, we investigate the surface-induced ferromagnetism and its
associated anomalous Hall effect in a nonmagnetic bulk electride Zr2S using
first-principles density-functional theory (DFT) calculations. We find that
bulk Zr2S has a topological band structure with massive DNLs consisting of
hybridized Zr-4$d$ cationic and interlayer anionic states, which exhibit a
high DOS distribution between $-$0.3 and $-$1.3 eV below $E_{F}$. However,
such hybridized states associated with the topmost Zr layer at the Zr2S(001)
surface shift toward $E_{F}$ due to the surface potential. The resulting
increase in the DOS at $E_{F}$ induces surface ferromagnetism via a Stoner
instability. Furthermore, we demonstrate that the (001) surface of the
nonmagnetic bulk electride Zr2S not only possesses highly spin-polarized
topological surface states with intricate helical spin textures but also
exhibits a surface anomalous Hall effect originating from the Berry curvature.
The present results offer a playground to investigate the intriguing interplay
between electride materials, surface ferromagnetism, and anomalous transport
phenomena.

Figure 1: Side (left panels) and top (right panels) views of the optimized
structures of the (a) 1T and (b) 2H phases of bulk Zr2S. The green and orange
balls in panel (b) represent Zr atoms locating in neighboring layers. The
lattice parameters $a$, $b$, and $c$ are drawn in each unit cell (indicated by
the dashed lines). Here, the 1T phase has $a$ = $b$ = 3.548 Å and $c$ = 5.600
Å, while the 2H phase has $a$ = $b$ = 3.443 Å and $c$ = 11.935 Å.

##  II II. CALCULATIONAL METHODS

Our first-principles DFT calculations were performed using the Vienna $ab$
$initio$ simulation package (VASP) with the projector-augmented wave method
[36, 37, 38]. The exchange-correlation energy was treated with the
generalized-gradient approximation functional of Perdew-Burke-Ernzerhof [39].
The plane wave basis was employed with a kinetic energy cutoff of 550 eV, and
the $k$-space integration was done with 18${\times}$18${\times}$12,
18${\times}$18${\times}$6, and 18${\times}$18${\times}$1 meshes for the 1T
bulk, 2H bulk, and (001) surface, respectively. All atoms were allowed to
relax along the calculated forces until all the residual force components were
less than 0.005 eV/Å. The phonon spectrum calculation of the 1T bulk was
carried out by using the QUANTUM ESPRESSO package [40], with the
6${\times}$6${\times}$4 $q$ points. The $ab$ $initio$ molecular dynamics
simulations were performed by using a 3${\times}$3${\times}$2 supercell. The
Zr2S(001) surface was simulated using a periodic slab of twelve Zr-S-Zr stacks
with ${\sim}$25 Å vacuum in-between adjacent slabs.

##  III III. RESULTS AND DISCUSSION

We begin by optimizing the 1T and 2H phases of bulk Zr2S using DFT
calculations. Here, the 1T phase crystallizes in an octahedral geometry with
the space group $P\overline{3}m1$ (No. 164), while the 2H phase crystallizes
in a trigonal prismatic geometry with the space group $P6_{3}/mmc$ (No. 194).
Figures 1(a) and 1(b) show the optimized structures of the 1T and 2H phases,
respectively. Our spin-polarized calculations for the two phases show that any
initial ferromagnetic or antiferromagnetic configuration converges to a
nonmagnetic one, indicating that bulk Zr2S is nonmagnetic. We find that the 1T
phase is more energetically favored over the 2H phase by 55.5 meV per 1T unit
cell. Interestingly, the preferred 1T phase in Zr2S contrasts with bulk Hf2S
where the 2H phase has been experimentally synthesized [26]. Hereafter, we
focus on the bulk and surface properties of the 1T phase of Zr2S.

Figure 2: (a) Calculated phonon spectrum of bulk Zr2S. The total energy versus
time for bulk Zr2S, computed from $ab$ $initio$ molecular dynamics simulations
at different temperatures, is displayed in panel (b). The inset in panel (b)
shows the top and side views of the simulated structure at 1000 K after 10 ps.
Figure 3: (a) Calculated ELF of bulk Zr2S on the (1$\overline{1}$0) plane with
a contour spacing of 0.05. Here, the dashed circles represent $X_{1}$ and
$X_{2}$ anions, the muffin-tin radii of which are chosen as 1 Å. The
calculated band structure and partial DOS of bulk Zr2S are displayed in panel
(b), where the projected bands on Zr-4$d$ and $X_{1}$-, $X_{2}$-$s$-like
orbitals are represented by circles whose radii are proportional to the
weights of the corresponding orbitals. In panel (b), the red arrows indicate
various nodal lines NL1, NL2, NL3, and NL4 (see text) around $E_{F}$ whose
dispersions over the Brillouin zone are drawn in panel (c). The right panels
in (c) show the top views of nodal lines.

To examine the dynamic stability of bulk Zr2S, we calculate its phonon
spectrum using the density-functional perturbation theory [41, 42]. As shown
in Fig. 2(a), there are no imaginary phonon frequencies over the whole
Brillouin zone, indicating that bulk Zr2S is dynamically stable. We further
perform $ab$ $initio$ molecular dynamics simulations to ensure the
thermodynamic stability of bulk Zr2S. Figure 2(b) shows the time evolution of
total energies at different temperatures of 300, 600, and 1000 K. We find that
the total energy at each temperature is well converged without large
deviations. Specifically, the top and side views of the simulated structure at
1000 K after 10 ps show that the layered structure is preserved without any
bond breakage. It is thus likely that bulk Zr2S would be thermodynamically
stable even at a high temperature of ${\sim}$1000 K. Recently, Zr2S was
experimentally synthesized with a polymorphic phase with the $P\overline{3}m1$
(equivalent to the 1T phase) and $Pnnm$ structures [25]. Further experimental
synthetic works will be demanded for the formation of a single 1T phase in the
future.

Figure 3(a) shows the electron localization function (ELF) of bulk Zr2S. It is
seen that anionic excess electrons are well localized at the positions marked
as $X_{1}$ and $X_{2}$ in the interlayer space, demonstrating that bulk Zr2S
is characterized as a 2D layered electride. In Fig. 3(b), the calculated band
structure of bulk Zr2S shows that Zr-4$d$ cationic and interstitial anionic
states are strongly hybridized in the energy range between ${\approx}$$-$0.3
and ${\approx}$$-$1.3 eV below $E_{F}$, giving rise to several common peaks in
their partial densities of states. We note that S-3$s$ and S-3$p$ orbitals are
located below $-$3 eV (see Fig. S1 in the Supplemental Material [43]),
indicating that S atoms hardly participate in hybridization with interstitial
anionic states. Interestingly, at the Zr2S(001) surface, the hybridized
Zr-4$d$ cationic and interstitial anionic states associated with the topmost
Zr layer are shifted toward $E_{F}$, thereby inducing a surface ferromagnetic
instability, as discussed below.

Using the tight-binding (TB) Hamiltonian with a basis of maximally localized
Wannier functions [45, 46], we investigate the topological properties of bulk
Zr2S. The TB Wannier bands of bulk Zr2S agree well with the corresponding DFT
ones calculated using the VASP code (see Fig. S2 in the Supplemental Material
[43]). As shown in Figs. 3(b) and 3(c), we find fourfold degenerate band
touching points around $E_{F}$, forming nodal lines NL1 and NL3 along the
${\Gamma}-A$ and $K-H$ paths, respectively, and nodal loops NL2 and NL4. It is
noted that the NL2 and NL4 DNLs are respected by the $P$ symmetry, whereas the
NL1 (NL3) DNL is respected by the $C_{3z}$ ($C_{3z}$ and $P$) symmetry [47]
(see Fig. S3 in the Supplemental Material [43]). The nontrivial topological
characterization of DNLs is identified by calculating the topological $Z_{2}$
index [48], defined as ${\zeta}_{1}$ = ${\frac{1}{\pi}}$ ${\oint}$c
$dk$${\cdot}$A($k$), along a closed loop encircling each DNL. Here, A($k$) =
$-i$$<$$u_{k}$$\mid$$\partial$k$\mid$$u_{k}$$>$ is the Berry connection for
the related Bloch states. We obtain ${\zeta}_{1}$ = ${\pm}$1 for the DNLs,
indicating that they are stable against perturbations without breaking
involved symmetries. However, by taking into account SOC, the fourfold
degeneracy of DNLs is lifted to open gaps, thereby leading to massive DNLs
around $E_{F}$.

Next, we investigate the electronic structure of the Zr2S(001) surface. Figure
4(a) shows the band structure of the nonmagnetic Zr2S(001) surface in the
absence of SOC. We find that there are two surface states $SS_{1}$ and
$SS_{2}$ along the ${\Gamma}$-$K$-$M$ path near $E_{F}$, which are composed of
the hybridized Zr-1 4$d$ cationic and interstitial $X_{1}$($X_{2}$)-$s$-like
anionic states. Here, Zr-1, $X_{1}$, and $X_{2}$ represent Zr atom and
interstitial anions at the topmost surface layer [see Fig. 4(b)]. It is noted
that the number of electrons within the muffin-tin sphere of the surface
$X_{1}$ ($X_{2}$) anion is 0.321 (0.204) electrons [see Fig. 4(b)], different
from the corresponding bulk value of 0.570 (0.394) electrons [see Fig. 3(a)].
Figure 4(c) shows the local DOS (LDOS) projected on the Zr-1 and Zr-2 atoms
[see Fig. 4(b)]. We find that the LDOS of Zr-1 at $E_{F}$ increases
significantly compared to those of Zr-2 and bulk Zr [see Fig. 3(b)]. This
dramatic change of Zr-4$d$ states together with the rearrangement of
interstitial anionic electrons at Zr2S(001) manifests strong surface effects.
Due to the high LDOS of Zr-1 at $E_{F}$, the Stoner criterion may be satisfied
to drive a ferromagnetic instability at the Zr2S(001) surface. Indeed, the
ferromagnetic phase is found to be energetically favored over the nonmagnetic
one by 5.6 meV per Zr surface atom. For the ferromagnetic phase, the LDOS of
Zr-1 shows that the spin-up and -down states are separated by ${\approx}$0.261
eV (see Fig. S4 in the Supplemental Material [43]). By dividing this exchange
splitting by the magnetic moment of Zr-1, we can estimate the Stoner parameter
$I$, which in turn satisfies the Stoner criterion $I{\cdot}D(E_{F})>1$ (see
Fig. S4 [43]). Here, $D(E_{F})$ is the LDOS of Zr-1 at $E_{F}$, obtained from
the nonmagnetic surface. Thus, we can say that surface ferromagnetism emerging
at the Zr2S(001) surfaces is driven by the Stoner instability due to an
increase in the surface DOS at $E_{F}$. To estimate the Curie temperature
$T_{\rm c}$, we perform spin-polarized DFT calculations for various
antiferromagnetic surface configurations (see Fig. S5 [43]). We find that the
lowest antiferromagnetic configuration is less stable than the ferromagnetic
one by 11.1 meV per Zr surface atom. Using the mean field approximation [44],
we estimate a $T_{\rm c}$ of ${\approx}$85 K at the Zr2S(001) surface.

Figure 4: (a) Calculated band structure of the nonmagnetic phase of the
Zr2S(001) surface. Here, the surface states $SS_{1}$ and $SS_{2}$ are
projected onto the Zr-1 $d$, $X_{1}$ $s$-, and $X_{2}$ $s$-like orbitals where
the radii of circles are proportional to the weights of the corresponding
orbitals. The grey shaded region indicates the projection of bulk states. The
ELF of the Zr2S(001) surface with a contour spacing of 0.05 is displayed in
panel (b), where the dashed circles represent $X_{1}$ and $X_{2}$ anions
locating at the topmost surface layer. The calculated LDOS of Zr-1 and Zr-2
atoms at the Zr2S(001) surface are given in panel (c).

Figures 5(a) and 5(b) show the band structure and spin density of the
ferromagnetic Zr2S(001) surface, respectively, calculated without including
SOC. We find that the $SS_{1}$ and $SS_{2}$ surface states are spin-polarized
to exhibit surface ferromagnetism. Consequently, the spin density is mostly
distributed around the Zr-1 layer, while it is significantly reduced at the
Zr-2 layer [see Fig. 5(b)]. The calculated spin magnetic moments integrated
within the muffin-tin spheres around Zr-1, $X_{1}$, and $X_{2}$ at the topmost
surface layer are 0.189 ${\mu}_{B}$, 0.056${\mu}_{B}$, and 0.053${\mu}_{B}$,
respectively (see Table I). It is noted that the magnitude of the spin
magnetic moment of Zr-2 decreases to $-$0.014${\mu}_{B}$ and the Zr-1 and Zr-2
spins are antiferromagnetically coupled to each other. By including SOC, we
calculate the magnetic anisotropy energy (MAE) at the Zr2S(001) surface.
Figure 5(c) displays the angular dependence of MAE on the $xy$-, $yz$-, and
$zx$-planes. We find that the easy axis is out-of-plane with a MAE of 0.024
meV per Zr surface atom. As shown in Fig. 5(c), the MAE on the $xy$-plane is
isotropic, whereas that on the $yz$\- or $zx$-plane strongly depends on the
angle ${\phi}$ relative to the $z$ direction. To explore the topological
nature of surface states associated with the gapped bulk DNLs, we calculate
their spin texture at $E_{F}$ with including SOC. Figure 5(d) shows the
calculated Fermi surface at the ferromagnetic Zr2S(001) surface. We find that
the minority-spin $SS_{1}$ surface state forming a hole pocket around the $M$
point and an electron pocket along the ${\Gamma}-K$ path exhibits a helical
spin texture with spin-momentum locking, indicating nontrivial topological
surface states without backscattering [49, 50].

Table 1: Calculated spin magnetic moments (in unit of ${\mu}_{B}$ per Zr atom) of Zr-1 and Zr-2 at the Zr2S(001) surface. The values of $X_{1}$ and $X_{2}$ at the topmost surface layer are also given. | Zr-1 | Zr-2 | $X_{1}$ | $X_{2}$
---|---|---|---|---
Magnetic moment | 0.189 | -0.014 | 0.056 | 0.053
Figure 5: (a) Calculated band structure of the ferromagnetic phase of the
Zr2S(001) surface. Here, the spin-polarized surface states $SS_{1}$ and
$SS_{2}$ are projected onto the Zr-1 $d$ orbitals where the radii of circles
are proportional to the weights of the corresponding orbitals. The grey shaded
region indicates the projection of bulk states. In panel (b), the spin density
at the Zr2S(001) surface is drawn with a contour spacing of 0.005 $e$/Å3. The
angular dependence of MAE with respect to ${\phi}$ on the $xy$-, $yz$-, and
$zx$-planes is displayed in panel (c), where the minimum energy is set to
zero. The spin texture of surface states on the Fermi surface is drawn in
panel (d), where the arrows represent the $S_{x}$ and $S_{y}$ components along
the horizontal and vertical directions, respectively. In panel (d), the blue
lines represent the Fermi surface of the minority-spin $SS_{1}$ surface state,
while the grey shaded region indicates the Fermi surface projected by bulk
states.

Finally, we examine the existence of an anomalous transport behavior at the
ferromagnetic Zr2S(001) surface. Using the Kubo-formula approach in the linear
response scheme [51], we calculate the intrinsic anomalous Hall conductivity
${\sigma}_{xy}$ originating from the Berry curvature. Figure 6(a) shows
${\sigma}_{xy}$ versus energy plot, obtained by integrating the $z$ component
of Berry curvature (${\Omega}_{z}$) of all the occupied bulk and surface
states over the surface Brillouin zone. We find that ${\sigma}_{xy}$ has a
negative value of ${\approx}-$0.5 $e^{2}/h$ at $E_{\rm F}$, while it exhibits
a large positive peak of ${\approx}$1.5 $e^{2}/h$ at $-$0.084 eV below $E_{\rm
F}$. Thus, as hole doping shifts $E_{\rm F}$ to $-$0.084 eV (see Fig. S6 in
the Supplemental Material [43]), ${\sigma}_{xy}$ can increase significantly.
Figures 6(b) and 6(c) display ${\Omega}_{z}$ distributions integrated in the
two different energy ranges: one is the $R_{\rm I}$ region between $-$0.148
and $-$0.015 eV and the other is the $R_{\rm II}$ region between $-$0.015 eV
and $E_{F}$. For the $R_{\rm I}$ region, there are the “hotspots” of positive
${\Omega}_{z}$ arising from the electron pocket of the surface state along the
${\Gamma}-K$ path and the gapped NL1 DNL around the ${\Gamma}$ point.
Meanwhile, for the $R_{\rm II}$ region, the latter gapped DNL mostly
contributes to more positive and negative ${\Omega}_{z}$ values. Therefore,
the gapped NL1 DNL plays an important role in ${\sigma}_{xy}$ around $E_{F}$,
yielding an anomalous electronic transport behavior.

Figure 6: (a) Energy dependence of ${\sigma}_{xy}$ and (b) (c) ${\Omega}_{z}$
distribution in the two energy regions I and II. In panels (b) and (c), red
and blue areas indicate the “hotspots” of positive and negative
${\Omega}_{z}$, respectively. The dashed lines in panels (b) and (c) represent
the Fermi surface of the minority-spin $SS_{1}$ surface state.

##  IV IV. CONCLUSION

Based on first-principles DFT calculations, we have predicted a ferromagnetic
instability at the (001) surface of the 2D layered electride Zr2S, the bulk of
which is nonmagnetic. In contrast to bulk Zr2S where hybridized Zr-4$d$
cationic and interlayer anionic states are located away below $E_{F}$, we
found that the Zr2S(001) surface has a high DOS at $E_{F}$ arising from such
hybridized states, thereby inducing surface ferromagnetism via a Stoner
instability. As a result, the Zr2S(001) surface not only possesses highly
spin-polarized topological surface states associated with massive DNLs but
also hosts an intrinsic anomalous Hall effect originating from the Berry
curvature generated by SOC. Therefore, our findings provide a novel platform
to investigate the intriguing interplay between electride materials, surface
ferromagnetism, and anomalous Hall transport which will be promising for
future spintronics technologies.

Acknowledgements. This work was supported by the National Research Foundation
of Korea (NRF) grant funded by the Korean Government (Grant No.
2022R1A2C1005456), by BrainLink program funded by the Ministry of Science and
ICT through the National Research Foundation of Korea (Grant No.
2022H1D3A3A01077468), and by the National Natural Science Foundation of China
(Grant No. 12074099). The calculations were performed by the KISTI
Supercomputing Center through the Strategic Support Program (Program No.
KSC-2022-CRE-0073) for the supercomputing application research.

S. L. and Y. L. contributed equally to this work.
∗ Corresponding author<EMAIL_ADDRESS>
## References

  * [1] C. Kittel, Theory of the structure of ferromagnetic domains in films and small particles, Phys. Rev. 70, 965 (1946).
  * [2] U. Gradmann, Ferromagnetism near surfaces and in thin films, Appl. Phys. 3, 161 (1974).
  * [3] A. Sundaresan, C.N.R. Rao, Ferromagnetism as a universal feature of inorganic nanoparticles, Nano. Today 4, 96 (2009).
  * [4] J.-H. Cho and M.-H. Kang, Possibility of a ferromagnetic Rh (001) surface: A plane-wave-basis partial-core pseudopotential approach, Phys. Rev. B 52, 13805 (1995).
  * [5] J.-H. Cho and M. Scheffler, Surface relaxation and ferromagnetism of Rh (001), Phys. Rev. Lett. 78, 1299 (1997).
  * [6] Edmund C Stoner F R S, Ferromagnetism, Rep. Prog. Phys. 11, 43 (1947).
  * [7] Z. Song, Z. Fang, and C. Fang, (d-2)-Dimensional Edge States of Rotation Symmetry Protected Topological States, Phys. Rev. Lett. 119, 246402 (2017).
  * [8] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry-protected topological orders in interacting bosonic systems, Science 338, 1604 (2012).
  * [9] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  * [10] B.-J. Yang and N. Nagaosa, Classification of stable three-dimensional Dirac semimetals with nontrivial topology, Nat. Commun. 5, 4898 (2014).
  * [11] Z. Lin, C. Wang, P. Wang, S. Yi, L. Li, Q. Zhang, Y. Wang, Z.Wang, H. Huang, Y. Sun, Y. Huang, D. Shen, D. Feng, Z. Sun, J.-H. Cho, C. Zeng, and Z. Zhang, Dirac fermions in antiferromagnetic FeSn kagome lattices with combined space inversion and time-reversal symmetry, Phys. Rev. B 102, 155103 (2020).
  * [12] L. Liu, C. Wang, J. Li, X.-Q. Chen, Y. Jia, and J.-H. Cho, Two-dimensional topological semimetal states in monolayer Cu2Ge, Fe2Ge, and Fe2Sn, Phys. Rev. B 101, 165403 (2020).
  * [13] R. Yu, Z. Fang, X. Dai, and H. Weng, Topological nodal line semimetals predicted from first-principles calculations, Front. Phys. 12, 127202 (2017).
  * [14] S. Y. Yang, H. Yang, E. Derunova, S. S. P. Parkin, B. Yan, and M. N. Ali, Symmetry demanded topological nodal-line materials, Adv. Phys.: X 3, 1414631 (2018).
  * [15] R. Li, H. Ma, X. Cheng, S. Wang, D. Li, Z. Zhang, Y. Li, and X.-Q. Chen, Dirac node lines in pure alkali earth metals, Phys. Rev. Lett. 117, 096401 (2016).
  * [16] S. Liu, C. Wang, L. Liu, J.-H. Choi, H.-J. Kim, Y. Jia, C. H. Park, and J.-H. Cho, Ferromagnetic Weyl fermions in two-Dimensional layered electride Gd2C, Phys. Rev. Lett. 125, 187203 (2020).
  * [17] L. Liu, C. Wang, S. Yi, D. K. Kim, C. H. Park, and J.-H. Cho, Theoretical prediction of Weyl fermions in the paramagnetic electride Y2C, Phys. Rev. B 99, 220401(R) (2019).
  * [18] K. Lee, S. W. Kim, Y. Toda, S. Matsuishi, and H. Hosono, Dicalcium nitride as a two-dimensional electride with an anionic electron layer, Nature (London) 494, 336 (2013).
  * [19] S. Liu, W. Li, S. W. Kim, and J.-H. Choi, Decisive Role of Interlayer Ionic Couplings for the Electronic Properties of Two-Dimensional Layered Electrides, J. Phys. Chem. C 124, 1398 (2020).
  * [20] S. Liu, C. Wang, H. Jeon, J. Kim, and J.-H. Cho, Interlayer exchange interaction driven topological phase transition in antiferromagnetic electride Gd2O, Phys. Rev. B 105, L041406 (2022).
  * [21] T. Tada, S. Takemoto, S. Matsuishi, and H. Hosono, High-throughput ab initio screening for two-dimensional electride materials, Inorg. Chem. 53, 10347 (2014).
  * [22] T. Inoshita, S. Jeong, N. Hamada, and H. Hosono, Exploration for two-dimensional electrides via database screening and $ab$ $Initio$ calculation, Phys. Rev. X 4, 031023 (2014).
  * [23] J. Zhou, L. Shen, M. Yang, H. Cheng, W. Kong, and Y. P. Feng, Discovery of hidden classes of layered electrides by extensive high-throughput material screening, Chem. Mater. 31, 1860 (2019).
  * [24] Y. Zhang, H. Wang, Y. Wang, L. Zhang, and Y. Ma, Computer-assisted inverse design of inorganic electrides, Phys. Rev. X 7, 011017 (2017).
  * [25] S. H. Kang, D. Thapa, B. Regmi, S. Ren, Y.-M. Kim, S.-G. Kim, and S. W. Kim, Chemically stable low-dimensional electrides in transition metal-rich monochalcogenides: theoretical and experimental explorations, J. Am. Chem. Soc. 144, 4496 (2022).
  * [26] S. H. Kang, J. Bang, K. Chung, C. N. Nandadasa, G. Han, S. Lee, K. H. Lee, K. Lee, Y. Ma, S. H. Oh, S.-G. Kim, Y.-M. Kim, and S. W. Kim, Water- and acid-stable self-passivated dihafnium sulfide electride and its persistent electrocatalytic reaction, Sci. Adv. 6, eaba7416 (2020).
  * [27] P. Chanhom, K. E. Fritz, L. A. Burton, J. Kloppenburg, Y. Filinchuk, A. Senyshyn, M. Wang, Z. Feng, N. Insin, J. Suntivich _et al_., Sr3CrN3: A new electride with a partially filled d-Shell transition metal, J. Am. Chem. Soc. 141, 10595 (2019).
  * [28] J. Wang, K. Hanzawa, H. Hiramatsu, J. Kim, N. Umezawa, K. Iwanaka, T. Tada, and H. Hosono, Exploration of stable strontium phosphide-based electrides: theoretical structure prediction and experimental validation, J. Am. Chem. Soc. 139, 15668 (2017).
  * [29] X. Zhang, Z. Xiao, H. Lei, Y. Toda, S. Matsuishi, T. Kamiya, S. Ueda, and H. Hosono, Two-Dimensional transition-Metal electride Y2C, Chem. Mater. 26, 6638 (2014).
  * [30] J. Park, K. Lee, S. Y. Lee, C. N. Nandadasa, S. Kim, K. H. Lee, Y. H. Lee, H. Hosono, S.-G. Kim, and S. W. Kim, Strong localization of anionic electrons at interlayer for electrical and magnetic anisotropy in two-dimensional Y2C electride, J. Am. Chem. Soc. 139, 615 (2017).
  * [31] M. Hiraishi, K. M. Kojima, I. Yamauchi, H. Okabe, S. Takeshita, A. Koda, R. Kadono, X. Zhang, S. Matsuishi, H. Hosono, K. Hirata, S. Otani, and N. Ohashi, Electronic correlation in the quasi-two-dimensional electride Y2C, Phys. Rev. B 98, 041104(R) (2018).
  * [32] S. Y. Lee, J.-Y. Hwang, J. Park, C. N. Nandadasa, Y. Kim, J. Bang, K. Lee, K. H. Lee, Y. Zhang, Y. Ma, H. Hosono, Y. H. Lee, S.-G. Kim, and S. W. Kim, Ferromagnetic quasi-atomic electrons in two-dimensional electride, Nat. Commun. 11, 1526 (2020).
  * [33] H. Y. Song, B. I. Yoo, J.-H. Choi, S.-H. Kang, J. Bang, W. Li, C. N. Nandadasa, D. Thapa, D. Yoon, M. J. Han, K. H. Lee, S. G. Kim, K. Lee, and S. W. Kim, Van der Waals electride: toward intrinsic two-dimensional ferromagnetism of spin-polarized anionic electrons, Mater. Today Phys. 20, 100473 (2021).
  * [34] S. Liu, C. Wang, H. Jeon, Y. Jia, and J.-H. Cho, Emerging two-dimensional magnetism in nonmagnetic electrides Hf2X (X=S, Se, Te), Phys. Rev. B 105, L220401 (2022).
  * [35] J. Zhang, D. Xu, X. Qiu. N. Zhao, Z. Lu and K. Liu, Magnetic surface on nonmagnetic bulk of electride Hf2S, arXiv preprint arXiv: 2207, 07042 (2022).
  * [36] G. Kresse and J. Hafner, Ab initio molecular dynamics for open-shell transition metals, Phys. Rev. B 48, 13115 (1993).
  * [37] G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996).
  * [38] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  * [39] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996); 78, 1396(E) (1997).
  * [40] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo _et al_., QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter 21, 395502 (2009).
  * [41] A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater. 108, 1 (2015).
  * [42] K. Parlinski, Z. Q. Li and Y. Kawazoe, First-Principles determination of the soft mode in cubic ZrO2, Phys. Rev. Lett. 78, 4063 (1997).
  * [43] See Supplemental Material at http://link.aps.org/supplemental/xxxx for the PDOS of bulk Zr2S, the comparison of the electronic bands obtained using the DFT and tight-binding Hamiltonian calculations, breaking the inversion symmetry in bulk Zr2S, the Stoner instability and Curie temperature at the Zr2S(001) surface, and the hole doping at the Zr2S(001) surface, which includes Refs. [12, 44].
  * [44] G. S. Rushbrooke and P. J. Wood, On the Curie points and high temperature susceptibilities of Heisenberg model ferromagnetics, Mol. Phys. 1, 257 (1958); On the high temperature staggered susceptibility of Heisenberg model antiferromagnetics, 6, 409 (1963).
  * [45] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N Marzari, wannier90: A tool for obtaining maximally-localised Wannier functions, Comput. Phys. Commun. 178, 685 (2008).
  * [46] Q. S. Wu, S. N. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, WannierTools: An open-source software package for novel topological materials, Comput. Phys. Commun. 224, 405 (2018).
  * [47] K. Kim, J. Seo, E. Lee, K.-T. Ko, B. S. Kim, B. G. Jang, J. M. Ok, J. Lee, Y. J. Jo, W. Kang, J. H. Shim, C. Kim, H. W. Yeom, B. Il Min, B.-J. Yang, and J. S. Kim, Large anomalous Hall current induced by topological nodal lines in a ferromagnetic van der Waals semimetal, Nat. Mater. 17, 794 (2018).
  * [48] C. Fang, H. Weng, X. Dai, and Z. Fang, Topological nodal line semimetals, Chin. Phys. B 25, 117106 (2016).
  * [49] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  * [50] X.-Q. Sun, S.-C. Zhang, and Z. Wang, Helical spin order from topological Dirac and Weyl semimetals, Phys. Rev. Lett. 115, 076802 (2015).
  * [51] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).


# A Secure Key Sharing Algorithm Exploiting Phase Reciprocity in Wireless
Channels

Shayan Mohajer Hamidi, , Amir Keyvan Khandani, , and Ehsan Bateni

###### Abstract

This article presents a secure key exchange algorithm that exploits
reciprocity in wireless channels to share a secret key between two nodes $A$
and $B$. Reciprocity implies that the channel phases in the links
$A\rightarrow B$ and $B\rightarrow A$ are the same. A number of such
reciprocal phase values are measured at nodes $A$ and $B$, called shared phase
values hereafter. Each shared phase value is used to mask points of a Phase
Shift Keying (PSK) constellation. Masking is achieved by rotating each PSK
constellation with a shared phase value. Rotation of constellation is
equivalent to adding phases modulo-$2\pi$, and as the channel phase is
uniformly distributed in $[0,2\pi)$, the result of summation conveys zero
information about summands. To enlarge the key size over a static or slow
fading channel, the Radio Frequency (RF) propagation path is perturbed to
create several independent realizations of multi-path fading, each used to
share a new phase value. To eavesdrop a phase value shared in this manner, the
Eavesdropper (Eve) will always face an under-determined system of linear
equations which will not reveal any useful information about its actual
solution value. This property is used to establish a secure key between two
legitimate users.

###### Index Terms:

Physical layer security, Channel reciprocity, secret key generation,

##  I introduction

The inherent broadcast nature of wireless communication makes it vulnerable to
different types of attacks such as Eavesdropping. Traditionally, the data is
secured by using classic encryption schemes, which entail functions and
algorithms that provide a desired level of secrecy for communicating data.
Most commonly used encryption methods rely on the computational hardness of
some mathematical problems, e.g., discrete logarithm. Nevertheless, due to
continual advances in computer technology, and the discovery of new
computational techniques such as Quantum computing, eavesdroppers are becoming
more equipped and intelligent, and thus long-term effectiveness of such
traditional techniques is questionable. As a result, there is a need for an
extra level of secrecy to strengthen the security of communicating data in
wireless communication.

To overcome these challenges, in addition to traditional cryptographic
techniques—which are applied at the third or higher layers of communication
protocols—, Physical Layer Security (PLS) could also be exploited to improve
the security of wireless channels [1]. Most PLS schemes are based on the idea
of leveraging intrinsic randomness in radio channel parameters [2, 3, 4, 5],
and are divided into key-less and key-based methods [6].

###  I-A Key-less PLS

This type of secrecy does not require a key for encrypting data, yet it uses
the channel properties of legitimate users and eavesdroppers to achieve
secrecy [6]. Indeed, this class of techniques, pioneered by Shannon [7],
secures communication from an information-theoretic perspective. Shannon
introduced a noiseless cipher system which is based on transmission limits
subject to both reliability and secrecy. Later, Wyner considered noisy
communication for the Shannon's cipher system, and introduced the so-called
wire-tap channel [8]. The secrecy provided by Wyner's method relies on random
coding existence results. These random codes should have two features: (i)
error-free detection for the legitimate receiver; and (ii) total randomness in
erroneous detection, i.e., a bit error rate (BER) of 50% for the Eve. In
recent years, there have been codes (constructed from linear structures) with
iterative decoding that come quite close to the Shannon limit in terms of
providing error-free detection. However, these codes do not necessarily
satisfy the second requirement of total randomness in erroneous detection.
Random coding for a reliable and secure transmission to achieve the secure
channel capacity has some fundamental differences with the case of channel
capacity, and the design of channel codes that could mimic the random codes
for secure and reliable transmission is in its infancy.

A common metric used in PLS to assess and compare different practical
transmission schemes is the security gap [9]. This refers to the gap in term
of loss in energy to the optimum benchmark for reliability and secrecy
obtained from the underlying information theoretic existence results (due to
Wyner and relevant subsequent works) on the level of secrecy. Many recent
practical coding schemes directly aim to reduce this security gap. In [9], the
authors proposed to use punctured Low-Density Parity-Check (LDPC) codes for
PLS. [10] used scrambling, concatenation, and hybrid automatic repeat-request
to LDPC and Bose–Chaudhuri–Hocquenghem (BCH) codes to farther lessen the
security gap. Using polar and LDPC codes at the same time is also proposed in
[11]. Co-set codes constructed based on convolutional and turbo codes are
proposed in [12]. Also, the possibility of exploiting Serially Concatenated
Low Density Generator Matrix (SCLDGM) codes in PLS schemes is considered in
[13]. An overview of error-control coding techniques for PLS can be found in
[14]. It is seen that, in the best case, when a BER of $50\%$ is desired for
the Eve, security gaps as small as $1~{}dB$ could be achieved by appropriately
designing codes with long length [14].

In fact, the simple-to-compute property of security gap makes it a useful
criterion to avoid potential complexities associated with computing the
original information-theoretic metrics. Nevertheless, there are no known
techniques to quantify the impact of any such gap. This issue is non-existent
in the technique proposed in this paper.

As another approach in key-less PLS, the idea proposed by Goel and Nagi in
[15, 16, 17] achieves the pefect secrecy, where additional artificially
interfering signals are transmitted to the channel along with the original
signals. In this method, the Eve's channel is degraded by injecting artificial
noise into the null space of the legitimate user's channel. However, this
technique sacrifices some power resources which consequently decreases the
main channel capacity.

###  I-B Key-based PLS

On the contrary, key-based PLS schemes do not require an intricate
computation, and they are practically more feasible [18, 3]. Channel
reciprocity is one of the main principles used for the process of generating
key in wireless communications. This feature implies that the transmitted
signal from both legitimate parties experiences almost the same fading, and
thus the same key is generated at the legitimate nodes. Among different
reciprocal features representing unpredictable radio channel parameters,
Received Signal Strength (RSS) and Channel State Information (CSI) are the two
common source of randomness. RSS has been the most favoured channel parameter
used for key generation [19]. However, one of the main shortcomings of key
generation methods based on RSS is that only one RSS value can be obtained
during the channel coherence time, limiting the Key Generation Rate (KGR). On
the other hand, CSI, which mainly refers to Channel Impulse Response (CIR) in
time domain and Channel Frequency Response (CFR) in frequency domain, is
proven to be suitable for key generation [20]. In these schemes, both
amplitude and phase information of CSI could be exploited. It is worth noting
that the channel phase, in comparison with channel amplitude, could
potentially grant a better level of secrecy for the following reasons:

  * •

The distribution of the phase, specifically in our case, will be uniform in
$[0,2\pi)$ [21]. Therefore, any conditional density built by Eve based on its
local observation of phase and/or magnitude conveys zero information about the
phase values shared between legitimate nodes.

  * •

The magnitude has a known probability density function and Eve can form a
conditional density function by conditioning on its local observations of
signal magnitude, while accounting for factors such as the distance that
affects the signal strength.

  * •

The channel phase is more sensitive to the temporal variation of wireless
channels. For instance, a small movement of transmitter, receiver or any other
objects in the vicinity of legitimate users causes a noticeable change in the
phase of the channel, while the amplitude might remain unchanged.

Henceforth, phase is a promising random characteristic of radio channels for
key generation schemes. The idea of using phase reciprocity for key generation
was suggested in [22]. In this work, both parties send $T$ known tones in both
directions and by calculating the successive differences between the phase
angles of the various tones, they establish $T-1$ shared phases. In [23], each
of the legitimate parties send two unmodulated tones with equal phases over
two different frequencies; then, the parties quantize the phase difference
between the two tones to establish a shared key. In continuation of [23],
authors in [24] find the probability that both ends of a link generate the
same quantization index for a particular phase difference. Afterward, [25]
uses pre-coding for secure transmission schemes proposed in [23]. In [25], the
first node sends M sinusoids with the same phase over M different frequencies;
then, the second node calculates the $M-1$ phase differences between the
received $M$ tones to pre-code Phase Shift Keying (PSK) symbols to be sent
back to the first node; then, the first node calculates $M-1$ phase
differences between the received $M$ tones which enables it to decode the
information content of PSK symbols. Inspired by [25], in [26], a secure
transmission scheme for downlink Sparse Code Multiple Access (SCMA) by
rotating the constellations using reciprocal channel phases is proposed.
Furthermore, in [27], the authors suggested to use reciprocal channel phases
to rotate the transmitted signal, and to change the modulation type among a
supported set. Although the pre-coding operation used in [25, 26, 27] is
similar to the masking operation performed in the current article, these works
do not discuss the relevant information hiding capabilities. In general, the
problem with all the above-mentioned methods (which exploit phase reciprocity
to establish a shared key) is that if one legitimate user transmits a sequence
of data in less than the coherence time of the channel, there will be a strong
correlation between the subsequent transmissions which could be exploited by
Eve. This problem is more severe in static or slow-fading channels (where the
channel temporal variation rate is very slow), reducing the effective KGR.

To tackle the above-mentioned problems, some research works have proposed the
idea of random beam-forming to enhance KGR when exploiting reciprocity over a
static channel [28, 29, 30]. The hardware complexities of realization of these
methods are significantly high; because random beam-forming requires multiple
active antennae, each requiring a dedicated base-band and Radio Frequency (RF)
front-end. However, our approach relies on a single antenna and a single RF
front end, and randomized switching of parasitic elements is performed using
low-cost Positive-Intrinsic-Negative (PIN) diodes.

In this article, a new practical system for achieving unconditional secrecy in
key exchange based on phase reciprocity is presented. The main contribution is
in practical implementation of the ideas first introduced in [31, 32], and
continued in [33]. Indeed, the protocols of key sharing discussed in [33] are
enhanced to make them more suitable for practical implementation. Furthermore,
mathematical proof for completely masking the content of PSK symbols, and for
unconditional secrecy obtained in the proposed method is provided.

Our method exploits the fact that the channel phase is uniformly distributed
in $[0,2\pi)$. This allows completely masking (hiding) PSK symbols by
modulo-$2\pi$ addition of phase values (analogous to modulo-2 addition in
binary XOR). In a first step, a number of reciprocal phase values are measured
at the two legitimate nodes connected through a wireless link. Due to channel
reciprocity, the phase values measured at the two ends will be the same except
for small deviations due to various sources of error, such as independent
noise in legitimate parties' devices and time/frequency mismatch. Data bits to
be communicated securely across the channel, after going through Forward Error
Correction (FEC), are mapped to PSK constellation points, and each such PSK
constellation is masked (rotated) using one of the shared phase values. An RF
structure composed of a central transmit antenna surrounded by switchable
parasitic RF elements is presented. This structure is used to perturb the RF
environment of the transmitter and/or receiver antennas. This enables our
method to offer a high KGR, and further, tackle with the problem of phase
correlation in static and slow fading channels.

As mentioned earlier, it is difficult to translate the existent information-
theoretical results which are based on random coding to practical
implementation that guarantees both reliability and security (50$\%$ bit error
for any erroneous transmissions). One of the major benefits associated with
the technique proposed here is that the mismatch between phases can be
corrected by means of standard FEC techniques.

The rest of the paper is organized as follows. Section II provides the proof
for the unconditional secrecy granted by masking PSK symbols. Section III
defines an RF mirror structure by which the channel in the vicinity of the
legitimate nodes is perturbed, resulting in independent channel realizations.
Section IV elaborates on two different antenna configurations to establish
common shared phase values between two legitimate parties. Section V states a
modulation technique for the proposed key exchange protocol. In Section VI,
some experimental results are presented, and finally, Section VII concludes
the paper.

_Notations:_ In the following, entropy is denoted by $H(.)$ and the mutual
information is denoted by $I(.;.)$. We use $u\perp v$ to show that $u$ and $v$
are independent. Notation $\oplus$ is used to represent modulo-$2\pi$
addition.

##  II Perfect secrecy of the proposed key generation method

A binary key should be ideally composed of Independent and Identically
Distributed (i.i.d.) binary digits with the probability of zero and one being
equal to $\frac{1}{2}$. This maximizes the entropy of the key for a given
length of binary digits. This in turn renders eavesdropping as difficult as
possible. Under such an assumption, a key of length $L$ will contain $L$ bits
of information. On the other hand, to combat the effect of channel noise and
other imperfections typical in wireless transmission, the $L$ key bits should
undergo FEC, adding $r$ bits of channel coding redundancy to the original $L$
bits. It follows that $L+r=mN_{p}$, where $N_{p}$ is the number of $2^{m}$-PSK
symbols carrying the key and the associated redundancy added by the deployed
FEC. The original $L$ bits are selected to have the maximum possible
information, i.e., $L$ bits of information. On the other hand, the $r$ bits of
channel coding redundancy are dependent to the original $L$ bits, carrying no
information. Therefore, the content of the encoded key stream of $L+r$ binary
digits contains only $L$ bits of information. This brings up a
question/concern if Eve can use the $r$ redundant bits to extract any
information about the shared key. In the proposed protocol of key sharing,
each $2^{m}$-PSK symbol is rotated by an independent shared phase value
uniformly distributed in $[0,2\pi)$ obtained by the reciprocal property of the
channel between the legitimate users. On the other hand, the following theorem
proves that, if two phases are added modulo-$2\pi$, and if only one of the two
phases is uniformly distributed in $[0,2\pi)$, the summation will not convey
any information about the summands. Consequently, regardless of the FEC
redundancy in the encoded stream of key bits, the addition of PSK
constellations with (reciprocally) shared channel phases will (i) completely
mask the content of each symbol, and (ii) make the received $N_{p}$ symbols
independent. Therefore, FEC does not reveal any information to Eve, and its
only negative side effect is that it decreases the key rate by a factor of
$\frac{L}{L+r}$.

######  Theorem 1.

Assume that angle $x$ is uniformly distributed in $[0,2\pi)$, and $y$ is
another random angle, independent of $x$, distributed in $[0,2\pi)$ with
certain Probability Distribution Function (PDF). Then modulo-$2\pi$ addition
of $x$ and $y$ is also uniformly distributed in $[0,2\pi)$.

###### Proof.

Denote by $f_{\alpha}(.)$ and $F_{\alpha}(.)$ the PDF and Cumulative
Distribution Function (CDF) of the random variable $\alpha$, respectively.
Also, $p\\{E\\}$ represents the probability of an event $E$. First, we compute
$f_{X+Y}(t)$, and then, accordingly, compute $f_{X\oplus Y}(t)$.

Since $x$ and $y$ are two independent random variables, the PDF of their sum,
i.e., $f_{X+Y}(t)$, is computed by convolving their respective PDF as follows

$\displaystyle
f_{X+Y}(t)=\int_{-\infty}^{\infty}f_{X}(t-\tau)f_{Y}(\tau)d\tau.$ (1)

As $x$ and $y$ are defined in $[0,2\pi)$, $f_{X+Y}(t)$ has non-zero values
only in $[0,4\pi)$. To compute (1), the range of $t$ is partitioned into two
parts, namely, $0\leq t\leq 2\pi$ and $2\pi\leq t<4\pi$, and then (1) is
computed separately over each part. Noting that $f_{X}(x)=\frac{1}{2\pi}$ ($x$
is uniformly distributed), for $0\leq t\leq 2\pi$, equation (1) becomes

$\displaystyle f_{X+Y}(t)$
$\displaystyle=\int_{0}^{t}\frac{1}{2\pi}f_{Y}(\tau)d\tau=\frac{1}{2\pi}F_{Y}(t)$
(2)

and for $2\pi\leq t<4\pi$, equation (1) becomes

$\displaystyle f_{X+Y}(t)$
$\displaystyle=\int_{t-2\pi}^{2\pi}\frac{1}{2\pi}f_{Y}(\tau)d\tau=\frac{1}{2\pi}\big{(}F_{Y}(2\pi)-F_{Y}(t-2\pi)\big{)}=\frac{1}{2\pi}\big{(}1-F_{Y}(t-2\pi)\big{)}$
(3)

and otherwise $f_{X+Y}(t)=0$.

In the following, we will compute $f_{X\oplus Y}(t)$ based on $f_{X+Y}(t)$
obtained above (note that to compute $f_{X\oplus Y}(t)$ we need to consider
the modulo-$2\pi$ operation).

For a fixed value $0\leq t_{0}\leq 2\pi$, if $x\oplus y=t_{0}$, then either
(i) $x+y=t_{0}$, or (ii) $x+y=t_{0}+2\pi$ (note that $0\leq x+y<4\pi$). Case
(i) happens when $0\leq x+y\leq 2\pi$ and case (ii) happens when $2\pi\leq
x+y<4\pi$. Therefore, from the Bayes' theorem, $p\\{x\oplus y=t_{0}\\}$ is
computed as follows

$\displaystyle p\\{x\oplus y=t_{0}\\}$ $\displaystyle=p\\{0\leq x+y\leq
2\pi\\}~{}p\\{x+y=t_{0}|0\leq x+y\leq 2\pi\\}$ $\displaystyle+p\\{2\pi\leq
x+y<4\pi\\}~{}p\\{x+y=t_{0}+2\pi|2\pi\leq x+y<4\pi\\}.$ (4)

Thus, $F_{X\oplus Y}(t)$ can be computed as follows

$\displaystyle F_{X\oplus Y}(t)$ $\displaystyle=p\\{0\leq x+y\leq
2\pi\\}~{}p\\{0\leq x+y\leq t|0\leq x+y\leq 2\pi\\}$
$\displaystyle+p\\{2\pi\leq x+y<4\pi\\}~{}p\\{2\pi\leq x+y\leq t+2\pi|2\pi\leq
x+y<4\pi\\}$
$\displaystyle=\big{(}\int_{0}^{2\pi}\frac{1}{2\pi}F_{Y}(\tau)d\tau\big{)}\big{(}\int_{0}^{t}\frac{1}{2\pi}F_{Y}(\tau)d\tau\big{)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+\big{(}\int_{2\pi}^{4\pi}\frac{1}{2\pi}\big{(}1-F_{Y}(\tau-2\pi)\big{)}d\tau\big{)}\big{(}\int_{2\pi}^{t+2\pi}\frac{1}{2\pi}\big{(}1-F_{Y}(\tau-2\pi)\big{)}d\tau\big{)}$
(5)

where the last equation is obtained from (2) and (3). Let us use $M_{y}(.)$ as
the anti-derivative of $F_{y}(.)$. Noting that $f_{X\oplus
Y}(t)=\frac{d}{dt}\big{(}F_{X\oplus Y}(t)\big{)}$, and that
$\frac{d}{dt}\big{(}\int_{t_{0}}^{t}F_{Y}(\tau)d\tau\big{)}=F_{Y}(t)$ for a
constant value of $t_{0}$ [34], from (II) we have

$\displaystyle f_{X\oplus Y}(t)$
$\displaystyle=\frac{1}{2\pi}\big{(}M_{y}(2\pi)-M_{y}(0)\big{)}\big{(}\frac{1}{2\pi}F_{y}(t)\big{)}+\frac{1}{2\pi}\big{(}\frac{2\pi-
M_{y}(2\pi)+M(0)}{2\pi}\big{)}\big{(}\frac{1}{2\pi}F_{y}(t)\big{)}=\frac{1}{2\pi}.$
(6)

Therefore, the Theorem 1 is proved. ∎

##  III  RF-mirrors Structure

Perturbing the RF environment of Transmit/Receive (TX/RX) antenna is an
essential part of the proposed system. Fig. 1 shows the RF structure used for
this purpose. This structure was first introduced in [35] for Media-based
Modulation. A number of switchable parasitic elements, hereafter referred as
RF mirrors, is surrounding one or two transmitting antenna(s). Each RF wall
(mirror) surrounding the center antenna(s), depending on whether it is in the
OFF or ON state, passes the energy to its external surface to propagate
outside the enclosure, or reflects the energy back (interior of the
enclosure). In this manner, the energy bounces back and forth among different
surrounding walls, and in doing so, creates a pseudo-random propagation
pattern. Consequently, the state of each wall, as a result of its interaction
with other walls, will have a multiplicative effect on the total number of
generated antenna patterns. This feature causes an exponential growth for
reflection/propagation patterns in terms of the number of RF mirrors. This is
similar to the effect of forming the image of a mirror in another mirror in
the construction of a kaleidoscope. The key difference is that in our case,
each surrounding mirror in the kaleidoscope can be selectively turned on (to
reflect light back to the interior of the kaleidoscope enclosure), or turned
off (to let the light rays leave the kaleidoscope enclosure in the
corresponding area). We refer to the different antenna patterns created by on-
off mirrors as transmitter states. The randomization phenomenon caused by
pseudo-random selection of on-off mirrors is enhanced by multi-path
propagation between the transmitting unit and its corresponding distant
receiver. In other words, the outgoing RF signal corresponding to each pseudo-
random antenna pattern will take various independent paths in reaching the
distant receiver, resulting in a different complex gain for multi-path fading.
Overall, the proposed structure creates a rich-scattering environment within
the enclosure, which is further enriched through external multi-path
propagation.

It is noteworthy that, unlike the methods in PLS which rely on the magnitude
of the RF signal, our technique relies only on the phase which is much more
prone to changes while propagating in a rich scattering environment. As a
result, the channel phase corresponding to different transmitter states will
be uniformly distributed in $[0,2\pi)$, and also will be independent of each
other. Fig. 2a shows a normalized received constellation, where the
constellation is subtracted by its empirical (complex) mean in order to
eliminate its bias with respect to origin. The reason of subtracting the bias
is that, due to limitations in laboratory (indoor) physical area, the two
nodes exchanging RF signals for key establishment have been placed within few
meters of each other, resulting in a strong (unwanted) Line-of-Sight (LOS)
component. However, the LOS would be absent in an outdoor environment with
larger separation between the transmitter and receiver units. Each point in
Fig. 2a corresponds to a random combination of RF mirrors switched to ON or
OFF state. The red zone in the center of the constellation reflects the
received signal when all RF mirrors are OFF. Also, Fig. 2b shows samples of
antenna patterns corresponding to four different random selection of ON/OFF
mirrors. Fig. 3 shows simulation results for sample in-door and out-door
environments obtained by exporting several antenna patterns into a wireless EM
propagation software (REMCOM wireless InSite).

Figure 1: Structure of RF-mirrors.

(a)

(b)

Figure 2: Randomness caused by the mirror structure. (a) Normalized
constellation obtained from the over-the-air transmission. (b) Samples of
antenna patterns corresponding to four random states of ON/OFF mirrors. Figure
3: Simulation results for sample in-door and out-door environments obtained by
exporting several antenna patterns into wireless EM propagation software
REMCOM wireless InSite.

##  IV Antenna Configurations

The main condition for perfect secrecy is that each transmit antenna should
transmit only once in each channel state, and then its associated RF mirrors
should be changed to a new configuration for the next transmission (to
establish an independent phase value).

In this section, we present two different configurations satisfying these
requirements. In addition, we show that a potential eavesdropper cannot attain
any information when two legitimate nodes are exchanging keys. Similar to
assumptions in other papers [36], it is assumed that the eavesdropper is aware
of the key exchange protocols.

The first configuration uses a single antenna at each legitimate node (two-
antenna system), and the second configuration uses two antennas at each
legitimate node (four-antenna system). As it will be discussed in this
section, the four-antenna system resolves the issue of synchronization
associated with two-antenna system, yet it has a higher hardware complexity.

###  IV-A Two-antenna System

In the two-antenna configuration, the reciprocal phase is measured as the
phase of the channels from Alice to Bob and from Bob to Alice, represented as
Alice$\to$Bob and Bob$\to$Alice, respectively. This protocol of key sharing
relies on using a pilot which is known to both Alice and Bob. Also, the RF
signal is modulated using Orthogonal Frequency-Division Multiplexing (OFDM)
over $S$ frequency tones, i.e., $\\{f_{1},f_{2},...,f_{S}\\}$.

####  IV-A1 Notations and Assumptions

Assume that only Alice's transmitter is equipped with the RF mirror structure
explained in Section III. Note that although having the mirror structure at
both Alice's and Bob's sides enhances the richness of the state space, it is
adequate to have only one side equipped with the mirror structure. Denote by
$K$, the number of mirrors at Alice's side. Therefore, based on whether a
mirror is in the ON or OFF state, $2^{K}$ different states will be realized.
We use $\alpha^{i}$ to show that the phase $\alpha$ is measured when the
mirrors are in state $i$, $1\leq i\leq 2^{K}$. We assume that the Eve is
equipped with $n$ antennas overhearing the messages transmitted between Alice
and Bob. As indicated in Fig. 4, Eve's antennas are denoted by
$\\{\mathsf{e}_{1},\mathsf{e}_{2},...,\mathsf{e}_{n}\\}$. We denote Alice's
and Bob's antennas by $\mathsf{a}$ and $\mathsf{b}$, respectively. When the
Alice's mirrors are in state $i$, we denote by $\mathsf{ab}^{i}$ the phase
that Bob records when Alice acts as the transmitter; and by $\mathsf{ba}^{i}$
the phase that Alice records when Bob acts as the transmitter. Furthermore, we
indicate by $\mathsf{ae}_{k}^{i}$ the phase from Alice's antenna to the
$k^{th}$ antenna of Eve when the mirrors are in state $i$. Note that as the
mirrors are on the Alice's side, Eve's observations from Bob's antenna are
independent of the mirror state. Hence, we denote by $\mathsf{be}_{k}$ the
phase angle from Bob's antenna to the Eve's $k^{th}$ antenna (see Fig. 4).

Figure 4: The two-antenna system. Eve's observations from both Alice's and
Bob's antennas are indicated when the Alice's mirrors are at the state $i$.
Eve's observations from Bob's antenna are independent of the mirror's state as
the mirror structure is on the Alice's side.

####  IV-A2 The protocol of Key sharing

To attain a shared key, each cycle of key sharing is composed of four
transmissions: two consecutive transmissions from Alice to Bob, followed by
two consecutive transmissions from Bob to Alice. In the following, one cycle
of transmission is elaborated by referring to Fig. 5.

Figure 5: The protocol of key sharing in the two-antenna system. The time slot
corresponding to each state in the time domain is divided into $S$ tones in
the frequency domain.

  * •

First transmission: Alice sends a preamble (pilot) to Bob using OFDM
modulation. Then, Bob measures the phase of the received signal over each
tone, i.e., Bob records $\mathsf{ab}^{i_{0}}(f_{l})$ for $1\leq l\leq S$; note
that $\mathsf{ab}^{i_{0}}$ is a function of frequency.

  * •

Second transmission: In this phase, the mirrors must be switched to a new
state $i$. To avoid intersymbol interference (ISI), a cyclic prefix with
duration of $\Delta t$ is used. Note that changing the state of the mirrors
increases the delay spread of the channel, and therefore switching to a new
state must take place in the early parts of the cyclic prefix to ensure that
$\Delta t>\tau_{max}$, where $\tau_{max}$ is the maximum delay spread of the
channel. After the cyclic prefix duration is over, Alice sends another RF
signal to Bob using OFDM modulation. Bob measures the change in the phase of
the pilot in each frequency band with respect to the corresponding tone in the
preamble, i.e., Bob records
$\big{(}\mathsf{ab}^{i}(f_{l})\big{)}\ominus\big{(}\mathsf{ab}^{i_{0}}(f_{l})\big{)}$
for $1\leq l\leq S$.

  * •

Third transmission: Now, the roles of Alice and Bob are reversed. Bob sends
the same preamble to Alice as the one that Alice sent to Bob in the First
transmission. Therefore, in this protocol, it is required to have a preamble
known to both legitimate nodes. To obtain the same preamble, the Alice's
mirrors are returned back to the reference states $i_{0}$, and after the
duration of cyclic prefix $\Delta t$, an RF signal is sent from Bob to Alice.
Alice measures the phase of the received signal, i.e., Alice records
$\mathsf{ba}^{i_{0}}(f_{l})$ for $1\leq l\leq S$.

  * •

Forth transmission: Finally, Alice's mirrors state is changed to $i$, and
after the duration of cyclic prefix (which is a part of OFDM signal), another
RF signal will be sent from Bob to Alice. Alice measures the change in the
phase of the pilots in each frequency band with respect to the corresponding
tone in the preamble, i.e., Alice records
$\big{(}\mathsf{ba}^{i}(f_{l})\big{)}\ominus\big{(}\mathsf{ba}^{i_{0}}(f_{l})\big{)}$
for $1\leq l\leq S$.

######  Claim 1.

Based on the cycle of transmission elaborated above, Alice and Bob can obtain
a shared common phase value as
$\big{(}\mathsf{ab}^{i}(f_{l})\ominus\mathsf{ab}^{i_{0}}(f_{l})\big{)}=\big{(}\mathsf{ba}^{i}(f_{l})\ominus\mathsf{ba}^{i_{0}}(f_{l})\big{)}$.

###### Proof.

First note that the change in the phase of an RF signal from the transmitter
to receiver is affected by the signal traversing through: (i) the RF channel
between the nodes, and (ii) both transmit and receive chains. For Alice and
Bob to obtain a shared phase value based on their recorded phases in the above
four transmissions, the effect of phases induced by receive/transmit chains
must be taken into account. To shed more light, when Alice is the transmitter,
the transmit chain of Alice and the receive chain of Bob contribute to the
phase change of the RF signal. On the other hand, when Bob is the transmitter,
the transmit chain of Bob and the receive chain of Alice contribute to the
phase change. In the following, we will elaborate on how the phases induced by
receive/transmit chains are mitigated in our protocol.

When Bob calculates
$\big{(}\mathsf{ab}^{i}(f_{l})\big{)}\ominus\big{(}\mathsf{ab}^{i_{0}}(f_{l})\big{)}$
in the Second transmission, the phase changes induced by Alice's transmit
chain and Bob's receive chain are cancelled out. Similarly, when Alice
calculates
$\big{(}\mathsf{ba}^{i}(f_{l})\big{)}\ominus\big{(}\mathsf{ba}^{i_{0}}(f_{l})\big{)}$
in the Forth transmission, the phase changes induced by Bob's transmit chain
and Alice's receive chain are cancelled out. On the other hand, the
reciprocity of the channel necessitates that
$\mathsf{ab}^{i}(f_{l})=\mathsf{ba}^{i}(f_{l})$ and
$\mathsf{ab}^{i_{0}}(f_{l})=\mathsf{ba}^{i_{0}}(f_{l})$; therefore
$\big{(}\mathsf{ab}^{i}(f_{l})\big{)}\ominus\big{(}\mathsf{ab}^{i_{0}}(f_{l})\big{)}=\big{(}\mathsf{ba}^{i}(f_{l})\big{)}\ominus\big{(}\mathsf{ba}^{i_{0}}(f_{l})\big{)}$
which could be used as a common shared phase value (note that as the time
interval between two OFDM symbols is very small, the hardware physical
conditions, such as its temperature, do not notably change. Thus, the hardware
behaviour—in terms of the phase change it exerts over the signal—remains the
same between two consecutive transmissions). ∎

In the following, to suppress the effect of Additive White Gaussian Noise
(AWGN), we propose obtaining only one common phase value from the $S$ phases
shared in the above manner (in lieu of using all of the $S$ common phases as
shared values).

First, at the transmit side, the pilots' amplitudes are selected to be $\pm 1$
(there are $S$ pilots in $S$ frequency bands). Then, at the receive side, by
applying appropriate sign changes, the receiver coherently adds up the pilots
in the $S$ frequency bands, and then computes their average. Then, the phase
of the averaged pilots will be used as the only shared phase value. In other
words, Bob computes $\mathsf{ab}^{i}$ and $\mathsf{ab}^{i_{0}}$ as the phase
of the averaged pilots in the First and Second transmissions, respectively.
Similarly, Alice finds $\mathsf{ba}^{i}$ and $\mathsf{ba}^{i_{0}}$ as the
phase of the averaged pilots in the Third and Forth transmissions,
respectively. Thereafter, Bob and Alice use
$\left(\mathsf{ab}^{i}\ominus\mathsf{ab}^{i_{0}}\right)$ and
$\left(\mathsf{ba}^{i}\ominus\mathsf{ba}^{i_{0}}\right)$ as a shared phase
value to establish a common key.

######  Remark 1.

Instead of simple averaging at the receiver, a more complex coding technique
could be applied over the pilots received in $S$ frequency bands.
Nevertheless, averaging the pilots can be regarded as a repetition code in
frequency domain. Note that spectral efficiency is a secondary concern in this
paper because a key can be used over many transmissions before it is changed.

######  Remark 2.

As discussed earlier, the length of the cyclic prefix $\Delta t$ is determined
by the delay spread of the channel. Denote by $\Delta T$ the time length
during which one node sends an RF signal to the other node, as depicted in
Fig. 4. Assuming a fixed available bandwidth, if the number of frequency bands
$S$ decreases, $\Delta T$ also decreases. This, in turn, decreases the
efficiency of the OFDM, as for a fixed-length cyclic prefix, $\Delta T$ has
decreased. On the other hand, if $S$ increases, $\Delta T$ also increases
(assuming a fixed available bandwidth) yielding a higher efficiency.
Nevertheless, the number of sub-bands could be increased to the extent that
the time interval during which one node completes its two consecutive
transmissions, i.e., $2\left(\Delta t+\Delta T\right)$, falls below the
coherence time of the channel. Therefore, the number of sub-bands $S$ should
be cautiously selected.

####  IV-A3 Analysis

Next, we show that a potential eavesdropper equipped with $n$ receive antennas
cannot attain the secret key shared by these two legitimate nodes.

We assume Eve is aware of the protocol of key sharing used by the legitimate
nodes, and therefore she finds the average of her observations over the
utilized $S$ frequency bands. The observations made by Eve upon one cycle of
key sharing between Alice and Bob are as follows:

  * •

$\mathsf{a}$ $\to$ $\mathsf{b}$, when mirrors are at state $i_{0}$: Eve
measures $\mathsf{ae}^{i_{0}}_{k}$,

  * •

$\mathsf{a}$ $\to$ $\mathsf{b}$, when mirrors are at state $i$: Eve measures
$\mathsf{ae}^{i}_{k}$,

  * •

$\mathsf{b}$ $\to$ $\mathsf{a}$, when mirrors are at state $i_{0}$: Eve
measures $\mathsf{be}_{k}$,

  * •

$\mathsf{b}$ $\to$ $\mathsf{a}$, when mirrors are at state $i$: Eve measures
$\mathsf{be}_{k}$,

for $1\leq k\leq n$. In the following paragraphs, we show that based on these
observations, Eve cannot extract any information about the shared phase, i.e.,
$\mathsf{ab}^{i}\ominus\mathsf{ab}^{i_{0}}$. First, we present a Lemma.

######  Lemma 1.

$I\left(\mathsf{ae}_{k}^{i};\left(\mathsf{ab}^{i}\ominus\mathsf{ab}^{i_{0}}\right)|\mathsf{ae}_{k}^{i_{0}}\right)=0$.

###### Proof.

By the definition of conditional mutual information [37], we have

$\displaystyle
I\left(\mathsf{ae}_{k}^{i};\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|\mathsf{ae}_{k}^{i_{0}}\right)=E_{p\left(\mathsf{ae}_{k}^{i},\left(\mathsf{ab}^{{i_{0}}}\ominus\mathsf{ab}^{j}\right)|\mathsf{ae}_{k}^{{i_{0}}}\right)}\left(\log\frac{p\left(\mathsf{ae}_{k}^{i},\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|\mathsf{ae}_{k}^{i_{0}}\right)}{p\left(\mathsf{ae}_{k}^{i}|\mathsf{ae}_{k}^{i_{0}}\right)p\left(\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|\mathsf{ae}_{k}^{i_{0}}\right)}\right)$
(7)

where $E_{p\left(.\right)}\\{.\\}$ represents the expected value over the
probability space of $p\left(.\right)$. On the other hand, by Bayes' formula,
we have

$\displaystyle
p\left(\mathsf{ae}_{k}^{i},\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|\mathsf{ae}_{k}^{i_{0}}\right)=\frac{p\left(\mathsf{ae}_{k}^{i}|\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right),\mathsf{ae}_{k}^{i_{0}}\right)p\left(\mathsf{ae}_{k}^{i_{0}},\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)\right)}{p\left(\mathsf{ae}_{k}^{i_{0}}\right)}.$
(8)

To simplify the terms in (8), it must be noted that by changing the state of
the mirrors at the Alice's side, the RF environment in the vicinity of the
Alice antenna $\mathsf{a}$ is changed (in effect, it changes the antenna
pattern). As mentioned earlier, such a change in the vicinity of the transmit
antenna changes the propagation path to the receiver antenna, further
enhancing the randomness in the end-to-end channel phase. In other words, the
randomness caused by RF mirrors will be augmented when the signal propagates
to the destination and thereby interacts with randomly located objects in its
propagation path. Interactions can be in the form of partial
absorption/reflection which causes a phase shift. Numerous such signal paths,
each experiencing a random phase shift, will add up at the destination
antenna. In other words, the environment external to the transmit antenna is
typically a rich scattering environment which enhances the randomness
initially caused by the change in the Alice's antenna pattern using RF
mirrors. Therefore, the phases that Eve records from Alice's antenna when the
mirrors are at state $i$, i.e., $\mathsf{ae}_{k}^{i}$, are independent from
the those she records when mirrors are in state ${i_{0}}$, i.e.,
$\mathsf{ae}_{k}^{i_{0}}$, meaning that

$\displaystyle\mathsf{ae}_{k}^{i}\perp\mathsf{ae}_{k}^{i_{0}}.$ (9)

Furthermore, it is obvious that as the location of Eve and Bob is different,
$\mathsf{ae}_{k}^{i}\perp\mathsf{ab}^{i}$, and also
$\mathsf{ae}_{k}^{i}\perp\mathsf{ab}^{i_{0}}$, which yields

$\displaystyle\mathsf{ae}_{k}^{i}\perp\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right).$
(10)

In addition, $\mathsf{ae}_{k}^{i_{0}}\perp\mathsf{ab}^{i_{0}}$ and
$\mathsf{ae}_{k}^{i_{0}}\perp\mathsf{ab}^{i}$, which results

$\displaystyle\mathsf{ae}_{k}^{i_{0}}\perp\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right).$
(11)

From equations (9) and (10)

$\displaystyle
p\left(\mathsf{ae}_{k}^{i}|\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right),\mathsf{ae}_{k}^{i_{0}}\right)=p\left(\mathsf{ae}_{k}^{i}\right)$
(12)

and from (11) it is concluded

$\displaystyle
p\left(\mathsf{ae}_{k}^{i_{0}},(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i})\right)=p\left(\mathsf{ae}_{k}^{i_{0}}\right)p\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right).$
(13)

Substituting equations (12) and (13) in (8), results in

$\displaystyle
p\left(\mathsf{ae}_{k}^{i},\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|\mathsf{ae}_{k}^{i_{0}}\right)=p\left(\mathsf{ae}_{k}^{i}\right)p\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right).$
(14)

Therefore, equation (7) is simplified to

$\displaystyle
I\left(\mathsf{ae}_{k}^{i};\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|\mathsf{ae}_{k}^{i_{0}}\right)$
$\displaystyle=E_{p\left(\mathsf{ae}_{k}^{i},\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|\mathsf{ae}_{k}^{i_{0}}\right)}\left(\log\frac{p\left(\mathsf{ae}_{k}^{i}\right)p\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)}{p\left(\mathsf{ae}_{k}^{i}\right)p\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)}\right)$
$\displaystyle=E_{p(\mathsf{ae}_{k}^{i},(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i})|\mathsf{ae}_{k}^{i_{0}})}\log\left(1\right)=0.$
(15)

∎

Note that, Lemma 1 is focused on a single antenna of Eve, namely its $k^{th}$
antenna. To find the information captured by the other Eve's antennas, if
$\mathsf{ae}_{k}^{i_{0}}$ is replaced by $\mathsf{ae}_{l}^{i_{0}}$ for $l\neq
k$ in Lemma 1, using the same reasoning as that discussed above, it follows
that all the equations (7) to (IV-A3) are still correct. Therefore,

$\displaystyle
I\left(\mathsf{ae}_{k}^{i};\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|\mathsf{ae}_{l}^{i_{0}}\right)=0.$
(16)

Since $k$ and $l$ are arbitrary in (16), we have

$\displaystyle
I\left([\mathbf{ae}^{i}];\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|[\mathbf{ae}^{i_{0}}]\right)=0$
(17)

where
$[\mathbf{ae}^{i}]=\\{\mathsf{ae}_{1}^{i},\mathsf{ae}_{2}^{i},...,\mathsf{ae}_{n}^{i}\\}$,
and similarly
$[\mathbf{ae}^{i_{0}}]=\\{\mathsf{ae}_{1}^{i_{0}},\mathsf{ae}_{2}^{i_{0}},...,\mathsf{ae}_{n}^{i_{0}}\\}$
are the vectors of phases that Eve measures on all of its antennas from the
Alice's antenna when the mirrors are in states $i$ and $i_{0}$, respectively.

Thus far, we have considered the Eve's observations from Alice's antenna, but
not those from Bob's. In the following we inspect the Eve's observations from
Bob's antenna. Using the same method and justification as those used in
proving Lemma 1, it could be proven that
$I\left(\mathsf{be}_{k};\left(\mathsf{ab}^{i}\ominus\mathsf{ab}^{i_{0}}\right)|\mathsf{be}_{k}\right)=0$.
Furthermore, with the same reasoning as that explained in proving (17), it is
straight-forward to prove that

$\displaystyle
I\left([\mathbf{be}];\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|[\mathbf{be}]\right)=0$
(18)

where
$[\mathbf{be}]=\\{\mathsf{be}_{1},\mathsf{be}_{2},...,\mathsf{be}_{n}\\}$ is
the vector of phases that Eve receives on all of its antennas from
$\mathsf{b}$ (Bob's antenna), no matter what the state of the mirrors is.

Furthermore, since $\mathsf{ae}_{k}^{i}\perp\mathsf{be}_{k}$, all the
equations (7) to (IV-A3) are still correct. Thus,

$\displaystyle
I\left([\mathbf{ae}^{i}];\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|[\mathbf{be}]\right)=0$
(19)

and similarly,

$\displaystyle
I\left([\mathbf{be}];\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|[\mathbf{ae}^{i_{0}}]\right)=0.$
(20)

From equations (17) to (20), it is obtained that

$\displaystyle
I\left([\mathbf{ae}^{i},\mathbf{be}];\left(\mathsf{ab}^{i_{0}}\ominus\mathsf{ab}^{i}\right)|[\mathbf{ae}^{i_{0}},\mathbf{be}]\right)=0$
(21)

where $[\mathbf{ae}^{i},\mathbf{be}]$ and $[\mathbf{ae}^{i_{0}},\mathbf{be}]$
are all the phases that Eve receives from both $a$ and $b$, i.e., Alice's and
Bob's antennas, when the mirrors are at state $i$ and $i_{0}$, respectively.
Equation (21) states that even if Eve records all the phases received from the
legitimate nodes during the four transmissions of a full cycle, still she
cannot attain any information about the shared key.

###  IV-B Four-antenna System

While having a simple structure, the downside of two-antenna system is the
issue of synchronization between the legitimate nodes in both time and
frequency domains. Time mismatch causes the receiver to inaccurately sample
the received OFDM symbols, diminishing the effect of cyclic prefix. On the
other hand, unmatched frequencies between the local oscillators of the
transmitter and receiver results in inter-carrier-interference.

To tackle this issue, four-antenna system is introduced in this section in
which each of the two legitimate nodes is equipped with two antennas. As will
be elaborated in the following, in such a system, the change in the phase of
the signal is measured between the two antennas located on the same unit, and
consequently, the problem of time and frequency mismatch is not of any
concern.

####  IV-B1 Notations and Assumptions

We use the same notations and assumptions as those used for two-antenna
system. However, as each node has an additional antenna with respect to the
two-antenna system, we indicate the new antennas on Alice's and Bob's sides by
$\alpha$ and $\beta$, respectively. In other words, we denote by
$\\{\mathsf{a},\mathsf{\alpha}\\}$ and $\\{\mathsf{b},\mathsf{\beta}\\}$ the
antennas for Alice and Bob, respectively. Note that similar to the two-antenna
system, the RF mirror structure is only utilized on only of the two sides, say
on Alice's side. The system structure along with the channels between the
legitimate nodes and Eve are depicted in Fig. 6.

Furthermore, similar to the two-antenna system, the signals are transmitted
using OFDM over $S$ frequency bands, and by means of averaging, the effect of
the noise will be reduced. For the sake of notational simplicity, although the
averaging over $S$ frequency bands takes place, the process of averaging is
not expressed explicitly.

Figure 6: Eve's observations in four-antenna system when the Alice's mirrors
are at the state $i$. Eve's observations from Bob's antennas are independent
of the mirror's state.

####  IV-B2 The protocol of Key sharing

As indicated in Fig. 7, two traversing loops, one initiated by Alice and the
other one initiated by Bob, will be completed to share a common phase value.

It should be pointed out that exchanging the signal between the two antennas
on the same side (i.e., transmissions between $\mathsf{a}$ and $\alpha$, or
between $\mathsf{b}$ and $\beta$) takes place through the RF front-end (wired
connection). The propagation of RF signal within the RF front-end of each unit
results in a phase shift which is virtually constant over time. Note that
variations in such interior-to-hardware phase shifts are caused by effects
such as changes in temperature or aging of components which occurs very slowly
over time. Consequently, the net effect is a constant phase difference between
the phase shifts measured at the two legitimate nodes. Such a constant phase
difference can be compensated through an initial calibration. An alternative
is to estimate the constant phase difference as part of the maximum likelihood
or soft-output decoding of the underlying forward error correcting code.
Therefore, by compensating the constant phase difference, hereafter we assume
that $\alpha\mathsf{a}=\mathsf{b}\beta$.

Figure 7: Traversing loops in four-antenna system. Note that
$\alpha\mathsf{a}=\mathsf{b}\beta$ are independent of the mirror's state.

In the first traversing loop, Alice selects its mirrors at a random state $i$.
Then, a pilot with random phase $\theta^{i}$, uniformly distributed in
$[0,2\pi)$, is sent from Alice to traverse the following loop:
$\alpha\to\mathsf{b}\to\beta\to\mathsf{a}$ (note that Alice records
$\theta^{i}$). This is referred to as the first traversing of the loop. Once
the first loop is completed, Alice measures the change in the phase of the
received signal with respect to that of original preamble, and records
$\alpha\mathsf{b}^{i}\oplus\mathsf{b}\beta\oplus\beta\mathsf{a}^{i}$. Note
that the initial random phase $\theta^{i}$ is canceled, and that
$\mathsf{b}\beta$ is independent of the mirror state (as it is caused via an
internal connection). Thereafter, the same loop initiated by Bob with random
phase $\phi^{i}$, uniformly distributed in $[0,2\pi)$, is traversed in a
different direction, namely, $\mathsf{b}\to\alpha\to\mathsf{a}\to\beta$. This
is referred to as the second traversing of the loop. Similarly, Bob measures
the phase shift with respect to its original pilot to obtain
$\mathsf{b}\alpha^{i}\oplus\alpha\mathsf{a}\oplus\mathsf{a}\beta^{i}$. Also,
as discussed earlier, $\alpha\mathsf{a}=\mathsf{b}\beta$. The reciprocity of
the channel implies that $\alpha\mathsf{b}^{i}=\mathsf{b}\alpha^{i}$ and
$\beta\mathsf{a}^{i}=\mathsf{a}\beta^{i}$. Hence, the two legitimate nodes
have measured the same phase shift which would be used as a shared phase
value. To generate a new key, Alice changes the state of its mirrors to a new
random state $j$ and again two traversing loops will be completed to establish
a new shared phase value.

####  IV-B3 Analysis

Now, we show that an eavesdropper equipped with $n$ antennas is not capable of
attaining the shared key between Alice and Bob.

First, we find out what information Eve can obtain by its $k^{th}$ antenna
when both Alice and Bob complete their traversing loops. It is assumed that
Alice mirrors are in state $i$.
First traverse of the loop:

  * •

$\alpha\to\mathsf{b}$: Eve measures
$\alpha\mathsf{e}_{k}^{i}\oplus\theta^{i}$, (i)

  * •

$\beta\to\mathsf{a}$: Eve measures
$\alpha\mathsf{b}^{i}\oplus\mathsf{b}\beta\oplus\beta\mathsf{e}_{k}\oplus\theta^{i}$,
(ii)

Second traverse of the loop:

  * •

$\mathsf{b}\to\alpha$: Eve measures $\mathsf{be}_{k}\oplus\phi^{i}$, (iii)

  * •

$\mathsf{a}\to\beta$: Eve measures
$\mathsf{b}\alpha^{i}\oplus\alpha\mathsf{a}\oplus\mathsf{ae}_{k}^{i}\oplus\phi^{i}$,
(iv)

for $1\leq k\leq n$. Eve's observations obtained over its different antennas
are independent of each other, and therefore they could be considered
separately. Hence, we remove the Eve's antenna indexes in our analysis, i.e.,
we consider a single antenna at the Eve's side. Referring to the observations
(i)-(iv), we express the summation of initial random phases with the phase of
the channel between Eve and legitimate users as
$\mathsf{m}_{1}^{i}=\alpha\mathsf{e}^{i}\oplus\theta^{i}$,
$\mathsf{m}_{2}^{i}=\beta\mathsf{e}\oplus\theta^{i}$,
$\mathsf{m}_{3}^{i}=\mathsf{be}\oplus\phi^{i}$, and
$\mathsf{m}_{3}^{i}=\mathsf{ae}^{i}\oplus\phi^{i}$. Based on Theorem 1, it is
concluded that since both $\theta^{i}$ and $\phi^{i}$ are uniformly
distributed in $[0,2\pi)$, the phase values
$\\{\mathsf{m}_{1}^{i},\mathsf{m}_{2}^{i},\mathsf{m}_{3}^{i},\mathsf{m}_{4}^{i}\\}$
are also uniformly distributed in $[0,2\pi)$. Furthermore, we assume that Eve
is aware of the antenna structure utilized by both legitimate nodes, and thus
she knows the value of $\alpha\mathsf{a}=\mathsf{b}\beta$. Therefore, Eve
forms the following system of equations to find the value of
$\alpha\mathsf{b}^{i}$.

$\displaystyle\alpha\mathsf{b}^{i}\oplus\mathsf{m}_{2}^{i}=y_{1}^{i}$
$\displaystyle\mathsf{b}\alpha^{i}\oplus\mathsf{m}_{4}^{i}=y_{2}^{i}.$ (22)

The system of equations in (IV-B3) has three unknowns—note that because of
channel reciprocity $\alpha\mathsf{b}^{i}=\mathsf{b}\alpha^{i}$—and two
equations; hence, it is under-determined [38]. It is straight-forward that a
system of equations of the form (IV-B3) (where the variables are in $[0,2\pi)$
and the summations are modulo-$2\pi$) can be solved as follows: first, the
system is solved as an ordinary system of linear equations, and second, the
modulo-$2\pi$ of the unknown variables found in the first step will be the
final solution for the original system. On the other hand, the coefficients of
all the unknown variables in (IV-B3) are one, and in order for Eve to solve
such a system, she needs to replace an unknown variable from one equation into
the other equation. This will give Eve the summation of two unknown variables
both having coefficient one. Thus, based on Theorem 1, Eve will face a phase
ambiguity uniformly distributed in $[0,2\pi)$ when determining any of unknown
variables in (IV-B3). This essentially means that Eve cannot extract any
information about the shared key based on her observations.

##  V Modulation

In this section, a simple modulation technique for a secure data exchange
between legitimate users utilizing the proposed key sharing method is
presented. Assume that $\Phi$ is a secret common phase value between the
legitimate nodes that is obtained by one of the two systems proposed in this
paper. Suppose we have $2^{m}$-PSK symbols as our plain text (which could
represent the message, or a bit stream to be used as key). At the receiver, we
rotate each $2^{m}$-PSK constellation point $X$ with the common phase value
$\Phi$ which is equivalent to modulo-$2\pi$ addition of $X$ and $\Phi$:
$Y=X\oplus\Phi$. Any eavesdropper who observes $Y$, cannot extract any
information about $X$ because $\Phi$ is a random variable with uniform
distribution in $[0,2\pi)$, and based on Theorem 1, $Y$ is also a random
variable uniformly distributed in $[0,2\pi)$ and independent from $X$, leading
to $I(X;Y)=H(Y)-H(Y|X)=H(Y)-H(\Phi)=0$. At the receiver side, the legitimate
receiver knowing the common phase value de-rotates each constellation point
accordingly and recovers the message which is equivalent to modulo-$2\pi$
subtraction of $\Phi$ from $Y$ (not considering noise for simplicity of
notation): $X=Y\ominus\Phi$. This process allows legitimate parties to share
$m$ secure bits per common phase value. In the following subsection, an
example of key exchange when the legitimate users utilize Quadrature Phase
Shift Keying (QPSK) is elaborated, and furthermore, the effect of FEC is
discussed.

###  V-A Example for Key Exchange

Assuming there are $\Theta$ shared phases, and the modulation for the
transmission of key is QPSK, with a rate 1/2 FEC. One of the two nodes, acting
as the master, generates $\Theta$ random bits (representing a messege to be
securely communicated, or to be used as the key), passes these bits through
FEC encoder to generate $2\Theta$ bits, maps the resulting $2\Theta$ bits to
$\Theta$ QPSK constellations, rotates each constellation with one of the
shared phase values, and transmits the rotated constellations to the other
node. The receiving node will first de-rotate subsequent constellations with
its local copies of the corresponding phase values, and then demodulate the
constellation and decode the FEC to extract the $\Theta$ bits of information.
Note that, in this example, decoding the FEC is equivalent to decoding it in
an AWGN channel. In addition, it is seen that the redundant bits added by
using FEC are also transmitted by QPSK constellations. Since these
constellations are rotated by one of the shared random phase values, they form
an independent constellation stream at the receiver, each having a phase
ambiguity uniformly distributed in $[0,2\pi)$. As discussed in Section II,
this in turn concludes that the redundant bits—those added by FEC—do not
disclose any information to Eve.

##  VI Empirical Results

(a)

(b)

Figure 8: (a) Setup for implementation on WARP platform. (b) Over-the-air
system performance.

The structure is implemented on two WARP1 software radio platforms operating
at a carrier of 2.4Ghz. Fig. 8a depicts the implementation setup block
diagram. Each RF mirror is a set of 6 conductor patches connected to each
other with high frequency PIN diodes acting as RF switches. If these PIN
diodes are ON, the conductor patches are connected and act as a perfect
conductor wall reflecting the incident wave, whereas if they are OFF, the
disjoint conductor patches act as a set of parasitic elements, resonating with
the antenna, and in turn radiate externally. As mentioned earlier, due to
space limitations, the two nodes were placed within the same room at a
distance of a few meters. As a result, there is a LOS component which
contributes to all channel states including the reference state, and makes the
constellation points spreading around a center point. LOS is a constant and
predictable component, adding a bias to each constellation preventing the
random phase values to be uniformly distributed over $[0,2\pi)$. To overcome
this shortcoming, the bias due to the LOS component is subtracted from the
measurements. In our experiments, transmit power is about 10dBm, which is
partially wasted (absorbed) within the mirror enclosure prior to propagating
externally. The variance of phase error could be reduced by using a higher
transmit power, and/or by averaging over multiple transmissions. Phase error
can be also reduced by discarding points with low signal magnitude, i.e., when
the magnitude of the multi-path fading is too small, and/or a higher amount of
energy is absorbed within the enclosure. Fig. 8b shows an example for over-
the-air phase measurement in which LOS is removed and about $20\%$ of points
of lowest energy are discarded.

##  VII conclusion

This article presents a practical method that exploits phase reciprocity in
wireless channels for perfectly secure key exchange. The proof of perfect
secrecy granted by the proposed method is provided. Also, two different
antenna configurations with their respective protocols of key sharing are
proposed. Unlike known techniques for PLS, which have unsolved challenges in
terms of removing possible errors between the shared keys (without disclosing
information), the proposed structure relies on well established techniques in
wireless transmission such as ``synchronization", ``channel phase measurement"
and ``Forward Error Correction''. The hardware components are simple, low cost
and the rate of key generation is up to the limits governed by Nyquist
sampling theorem.

## References

  * [1] Y.-S. Shiu, S. Y. Chang, H.-C. Wu, S. C.-H. Huang, and H.-H. Chen, ``Physical layer security in wireless networks: A tutorial,'' _IEEE wireless Communications_ , vol. 18, no. 2, pp. 66–74, 2011.
  * [2] U. M. Maurer and S. Wolf, ``Unconditionally secure key agreement and the intrinsic conditional information,'' _IEEE Transactions on Information Theory_ , vol. 45, no. 2, pp. 499–514, 1999.
  * [3] U. M. Maurer, ``Secret key agreement by public discussion from common information,'' _IEEE transactions on information theory_ , vol. 39, no. 3, pp. 733–742, 1993.
  * [4] R. Wilson, D. Tse, and R. A. Scholtz, ``Channel identification: Secret sharing using reciprocity in ultrawideband channels,'' _IEEE Transactions on Information Forensics and Security_ , vol. 2, no. 3, pp. 364–375, 2007.
  * [5] C. Ye, A. Reznik, G. Sternburg, and Y. Shah, ``On the secrecy capabilities of itu channels,'' in _2007 IEEE 66th Vehicular Technology Conference_. IEEE, 2007, pp. 2030–2034.
  * [6] A. Mukherjee, S. A. A. Fakoorian, J. Huang, and A. L. Swindlehurst, ``Principles of physical layer security in multiuser wireless networks: A survey,'' _IEEE Communications Surveys & Tutorials_, vol. 16, no. 3, pp. 1550–1573, 2014.
  * [7] C. E. Shannon, ``Communication theory of secrecy systems,'' _The Bell system technical journal_ , vol. 28, no. 4, pp. 656–715, 1949.
  * [8] A. D. Wyner, ``The wire-tap channel,'' _Bell system technical journal_ , vol. 54, no. 8, pp. 1355–1387, 1975.
  * [9] D. Klinc, J. Ha, S. W. McLaughlin, J. Barros, and B.-J. Kwak, ``Ldpc codes for the gaussian wiretap channel,'' _IEEE Transactions on Information Forensics and Security_ , vol. 6, no. 3, pp. 532–540, 2011.
  * [10] M. Baldi, M. Bianchi, and F. Chiaraluce, ``Coding with scrambling, concatenation, and harq for the awgn wire-tap channel: A security gap analysis,'' _IEEE Transactions on Information Forensics and Security_ , vol. 7, no. 3, pp. 883–894, 2012.
  * [11] Y. Zhang, A. Liu, C. Gong, G. Yang, and S. Yang, ``Polar-ldpc concatenated coding for the awgn wiretap channel,'' _IEEE Communications Letters_ , vol. 18, no. 10, pp. 1683–1686, 2014.
  * [12] A. Nooraiepour and T. M. Duman, ``Randomized turbo codes for the wiretap channel,'' in _GLOBECOM 2017-2017 IEEE Global Communications Conference_. IEEE, 2017, pp. 1–6.
  * [13] ——, ``Randomized serially concatenated ldgm codes for the gaussian wiretap channel,'' _IEEE Communications Letters_ , vol. 22, no. 4, pp. 680–683, 2018\.
  * [14] S. R. Aghdam, A. Nooraiepour, and T. M. Duman, ``An overview of physical layer security with finite-alphabet signaling,'' _IEEE Communications Surveys & Tutorials_, vol. 21, no. 2, pp. 1829–1850, 2018.
  * [15] R. Negi and S. Goel, ``Secret communication using artificial noise,'' in _IEEE vehicular technology conference_ , vol. 62, no. 3. Citeseer, 2005, p. 1906.
  * [16] S. Goel and R. Negi, ``Secret communication in presence of colluding eavesdroppers,'' in _MILCOM 2005-2005 IEEE Military Communications Conference_. IEEE, 2005, pp. 1501–1506.
  * [17] ——, ``Guaranteeing secrecy using artificial noise,'' _IEEE transactions on wireless communications_ , vol. 7, no. 6, pp. 2180–2189, 2008\.
  * [18] R. Ahlswede and I. Csiszár, ``Common randomness in information theory and cryptography. i. secret sharing,'' _IEEE Transactions on Information Theory_ , vol. 39, no. 4, pp. 1121–1132, 1993.
  * [19] J. Zhang, T. Q. Duong, A. Marshall, and R. Woods, ``Key generation from wireless channels: A review,'' _Ieee access_ , vol. 4, pp. 614–626, 2016\.
  * [20] S. Mathur, R. Miller, A. Varshavsky, W. Trappe, and N. Mandayam, ``Proximate: proximity-based secure pairing using ambient wireless signals,'' in _Proceedings of the 9th international conference on Mobile systems, applications, and services_ , 2011, pp. 211–224.
  * [21] A. Goldsmith, _Wireless communications_. Cambridge university press, 2005.
  * [22] J. E. Hershey, A. A. Hassan, and R. Yarlagadda, ``Unconventional cryptographic keying variable management,'' _IEEE Transactions on Communications_ , vol. 43, no. 1, pp. 3–6, 1995.
  * [23] A. A. Hassan, W. E. Stark, J. E. Hershey, and S. Chennakeshu, ``Cryptographic key agreement for mobile radio,'' _Digital Signal Processing_ , vol. 6, no. 4, pp. 207–212, 1996.
  * [24] A. Sayeed and A. Perrig, ``Secure wireless communications: Secret keys through multipath,'' in _2008 IEEE International Conference on Acoustics, Speech and Signal Processing_. IEEE, 2008, pp. 3013–3016.
  * [25] H. Koorapaty, A. A. Hassan, and S. Chennakeshu, ``Secure information transmission for mobile radio,'' _IEEE Communications Letters_ , vol. 4, no. 2, pp. 52–55, 2000.
  * [26] K. Lai, J. Lei, L. Wen, G. Chen, W. Li, and P. Xiao, ``Secure transmission with randomized constellation rotation for downlink sparse code multiple access system,'' _IEEE Access_ , vol. 6, pp. 5049–5063, 2017.
  * [27] S. Althunibat, V. Sucasas, and J. Rodriguez, ``A physical-layer security scheme by phase-based adaptive modulation,'' _IEEE Transactions on Vehicular Technology_ , vol. 66, no. 11, pp. 9931–9942, 2017.
  * [28] M. G. Madiseh, S. W. Neville, and M. L. McGuire, ``Applying beamforming to address temporal correlation in wireless channel characterization-based secret key generation,'' _IEEE Transactions on Information Forensics and Security_ , vol. 7, no. 4, pp. 1278–1287, 2012.
  * [29] L. Cheng, W. Li, and D. Ma, ``Secret key generation via random beamforming in stationary environment,'' in _2015 International Conference on Wireless Communications & Signal Processing (WCSP)_. IEEE, 2015, pp. 1–5.
  * [30] G. Li, A. Hu, J. Zhang, and B. Xiao, ``Security analysis of a novel artificial randomness approach for fast key generation,'' in _GLOBECOM 2017-2017 IEEE Global Communications Conference_. IEEE, 2017, pp. 1–6.
  * [31] A. K. Khandani, ``Two-way (true full-duplex) wireless,'' in _2013 13th Canadian Workshop on Information Theory_. IEEE, 2013, pp. 33–38.
  * [32] ——, ``Full duplex wireless transmission with channel phase-based encryption,'' Feb. 14 2017, uS Patent 9,572,038.
  * [33] A. K. Khandani and E. Bateni, ``A practical, provably unbreakable approach to physical layer security,'' in _2019 16th Canadian Workshop on Information Theory (CWIT)_. IEEE, 2019, pp. 1–6.
  * [34] G. Strang, ``Calculus. 1991, p. 215.''
  * [35] A. K. Khandani, ``Media-based modulation: A new approach to wireless transmission,'' in _2013 IEEE International Symposium on Information Theory_. IEEE, 2013, pp. 3050–3054.
  * [36] S. Mathur, W. Trappe, N. Mandayam, C. Ye, and A. Reznik, ``Radio-telepathy: extracting a secret key from an unauthenticated wireless channel,'' in _Proceedings of the 14th ACM international conference on Mobile computing and networking_ , 2008, pp. 128–139.
  * [37] T. M. Cover, _Elements of information theory_. John Wiley & Sons, 1999.
  * [38] B. N. Datta, _Numerical linear algebra and applications_. Siam, 2010, vol. 116.