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Artificially Time-Varying Differential MIMO for Achieving Practical Physical Layer Security

September 2021
IEEE Open Journal of the Communications Society PP(99):1-1

DOI:10.1109/OJCOMS.2021.3112486

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Authors:

Naoki Ishikawa

Yokohama National University

Jehad M. Hamamreh

Antalya Bilim University

Eiji Okamoto

Nagoya Institute of Technology

Chao Xu

University of Southampton

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In this paper, we propose a differential multiple-input multiple-output (MIMO) scheme based on the novel concept of chaos-based time-varying unitary matrices to demonstrate—for the first time in the literature—the ability of differential encoding in achieving practical physical layer security even without the need for using channel estimation. In the proposed scheme, an erroneous secret key, which is extracted from the wireless nature, is used to initialize a chaos sequence that is responsible for generating artificially time-varying unitary matrices capable of obfuscating the transmitted data symbols from illegitimate eavesdroppers. Contrary to conventional studies, the key agreement ratio in this study is assumed to be imperfect, which is often true and very realistic in high-mobility scenarios. Following this, we conceive a new calibration algorithm for reconciling the chaotic sequence generated at the legitimate parties, thus making this calibration algorithm a unique, novel solution to the key sharing problem of conventional chaos-based communication techniques, which has been overlooked over the past few decades. It is found out that differential encoding obviates additional complexity and insecurity in dealing with channel estimation, whereas an eavesdropper must tackle the complicated differentially encoded patterns, which have an exponentially increasing complexity order. In addition, the obtained simulation results demonstrate that the proposed scheme can outperform conventional chaos-based MIMO schemes that assume perfect channel knowledge.

Chaos transition x i over time, for which the initial values of x 0 = 0.24, x 0 + 10 −6 , and x 0 + 10 −3 were considered.

…

Eve's channel reliability upon increasing the channel correlation ρ and the security gap. The simulation parameters were the same as those used in Fig. 6(a).

…

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Journal of the Communications Society

Artiﬁcially Time-Varying Differential MIMO

for Achieving Practical Physical Layer Security

Naoki Ishikawa (Member, IEEE), Jehad M. Hamamreh (Member, IEEE), Eiji Okamoto (Member, IEEE),

Chao Xu (Senior Member, IEEE), and Lixia Xiao (Member, IEEE).

In this paper, we propose a differential multiple-input multiple-output (MIMO) scheme based on the novel concept of chaos-based

time-varying unitary matrices to demonstrate—for the ﬁrst time in the literature—the ability of differential encoding in achieving

practical physical layer security even without the need for using channel estimation. In the proposed scheme, an erroneous secret

key, which is extracted from the wireless nature, is used to initialize a chaos sequence that is responsible for generating artiﬁcially

time-varying unitary matrices capable of obfuscating the transmitted data symbols from illegitimate eavesdroppers. Contrary to

conventional studies, the key agreement ratio in this study is assumed to be imperfect, which is often true and very realistic in

high-mobility scenarios. Following this, we conceive a new calibration algorithm for reconciling the chaotic sequence generated at

the legitimate parties, thus making this calibration algorithm a unique, novel solution to the key sharing problem of conventional

chaos-based communication techniques, which has been overlooked over the past few decades. It is found out that differential

encoding obviates additional complexity and insecurity in dealing with channel estimation, whereas an eavesdropper must tackle

the complicated differentially encoded patterns, which have an exponentially increasing complexity order. In addition, the obtained

simulation results demonstrate that the proposed scheme can outperform conventional chaos-based MIMO schemes that assume

perfect channel knowledge.

Index Terms—MIMO, differential modulation, differential space-time block codes, physical layer security, physical layer encryption,

chaos theory, phase ambiguity, constrained capacity, secrecy rate, security gap.

I. INT ROD UC TI ON

RADIO waves can propagate over long distances. Even

a low-power signal that is transmitted by a household

Wi-Fi device can reach as far as 100 meters if a line of

sight path is present [1]. In public Wi-Fi networks, it is easy

for eavesdroppers to obtain standardized 802.11 frames and

retrieve private information streams [2–4]. Because the private

information is encrypted in the transport layer, we feel safe

using wireless network, but this security is not guaranteed

forever. For example, the widely used public-key encryption

method, RSA [5], has been threatened by the invention of

Shor’s algorithm [6], which performs integer factoring in

polynomial time using a quantum computer [7]. Therefore, it is

necessary to invent practical wireless communication method

in the physical layer that reinforces security.

Operational wireless systems generally rely on encryption-

based methods to secure communications, where a secret

key is exchanged in advance. Physical layer communication

schemes that require a perfect secret key are classiﬁable

into the physical layer encryption (PLE) category [8]. As a

N. Ishikawa is with the Faculty of Engineering, Yokohama National

University, 240-8501 Kanagawa, Japan (e-mail: ishikawa-naoki-fr@ynu.ac.jp).

J. M. Hamamreh is with the Department of Electrical and Electronics

Engineering, Antalya Bilim University, 07468 Antalya, Turkey (e-mail: je-

had.hamamreh@gmail.com). E. Okamoto is with the Department of Electrical

and Mechanical Engineering, Nagoya Institute of Technology, 466-8555

Nagoya, Japan. (e-mail: okamoto@nitech.ac.jp). C. Xu is with the School of

Electronics and Computer Science, University of Southampton, Southampton

SO17 1BJ, U.K. (e-mail: cx1g08@ecs.soton.ac.uk). L. Xiao is with the Wuhan

National Laboratory for Optoelectronics, Huazhong University of Science

and Technology, Wuhan 430074, China. (e-mail: lixiaxiao@hust.edu.cn). The

work of N. Ishikawa was supported in part by the Japan Society for the

Promotion of Science (JSPS) KAKENHI under Grant No. 19K14987. The

work of J. M. Hamamreh was supported in part by the Scientiﬁc and

Technological Research Council of Turkey (TUBITAK) under Grant No.

119E392. The work of L. Xiao was supported in part by the National Science

Foundation of China under Grant No. 62001179.

pioneering PLE study, Dedieu et al. proposed a seminal chaos-

based communication system [9], designated as chaos shift

keying (CSK). The original CSK system of [9] was proposed

for wired communications using Chua’s analog circuit, which

generates a chaotic carrier. Based on this CSK philosophy [9],

Okamoto et al. proposed the chaos MIMO technique [10–13].

This technique generates a Gaussian-distributed constellation

that is difﬁcult to perceive and eavesdrop because the Gaussian

symbols are naturally hidden by additive Gaussian noise.

Furthermore, the chaos MIMO arrangement obtains channel

coding gain by exploiting a unique chaos modulation structure.

In parallel, Kaddoum et al. proposed the MIMO-CSK concept

[14] for a 2×2setup, where chaos is used to spread data

symbols. Note that all the above chaos-based schemes [10–

14] require precise estimates of channel state information

(CSI). The estimation of CSI imposes potential risk as the

eavesdropper may use pilot symbols to synchronize received

signals and obtain accurate CSI between the transmitter and

the eavesdropper [15, 16].

Key-based encryption techniques that rely on computational

security may be cracked by future supercomputers [8, 17, 18].

In order to overcome this limitation, physical layer security

(PLS) methods that frequently update private keys have been

conceived [19], which rely on the true randomness of wireless

channels. The information-theoretic foundation of secret-key

agreement in public channels was ﬁrst established by Maurer

and Wolf [20–22]. Most of secret-key agreement or generation

methods require the assumptions of time division duplex

(TDD) channel reciprocity and near-perfect CSI [23–25].

The strong assumptions on CSI have hindered the industrial

applications of PLS [26]. When considering realistic CSI

errors, the key agreement ratio between legitimate parties is

far from perfect in practice [27–29]. Hence, it is a challenging

task to achieve the key agreement ratio of 100% in high-

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mobility scenarios. A long-overlooked problem here is that

this erroneous key cannot be used for all the conventional

encryption methods. To improve the key agreement ratio,

the legitimate parties have to exchange thousands of probe

symbols [27]. We have to eliminate this channel estimation

process because it increases both the communication overhead

and risk.

The classic differential MIMO can eliminate the channel

estimation process [30, 31]. The pioneering scheme [30]

established in the early 2000s relies on square unitary ma-

trices. By contrast, in 2017, a new approach of nonsquare

differential MIMO [32–36] was proposed. This new approach

maps the classic unitary matrix to a nonsquare matrix, which

beneﬁcally improves the transmission rate linearly. Because

of the differential structure, the resultant constellation might

reach an inﬁnite cardinality [37–39]. This structure is naturally

useful for improving the wireless communication security.

Nevertheless, no report of the relevant literature has described

a study of the differential MIMO in the context of both PLE

and PLS.1

Against this background, we propose a chaos-based differ-

ential MIMO system that is free from the additional complex-

ity and insecurity of dealing with channel estimation. More

explicitly, the data-carrying matrices are obfuscated using

a specially designed artiﬁcially time-varying unitary matrix

concept, which is generated by a chaos sequence. Unlike

the conventional chaos-based PLE family [10–14, 40–43],

our proposed system extracts a noisy key from the wireless

channel, which is used as the initial condition of a chaos

sequence. Following this invention, a low-complexity chaos

calibration is conceived to continue estimation of the original

chaos sequence. We also conceive a simple real-valued key

generation method and analyze the minimum security level

achieved by the proposed system.

The major contributions of this paper are summarized in

twofold.

1) Our proposed scheme is the ﬁrst chaos-based scheme

that is free from key establishment in advance. All of the

schemes of the conventional chaos-based family must

rely on the perfect pre-shared key because the chaos

sequence is sensitive to error. For example, the small

error of 2−1022 added to the initial condition causes

mismatches between the legitimate parties.2We resolve

this issue by conceiving a chaos calibration algorithm

that updates a chaos sequence at the receiver. This

calibration process can be interpreted as information

reconciliation of a chaotic sequence. The advantage of

this process is that there is no additional overhead be-

cause the reconciliation is performed using data matrices

instead of transmitting probe signals.

2) We prove and reveal for the ﬁrst time that the classic

differential encoding is suitable for achieving practi-

1The differential counterpart of MIMO-CSK [40] has also been proposed

for the synchronizing a chaos sequence both at a transmitter and receiver.

However, it is noteworthy that the term differential differs from the classic

modulation deﬁnition, as described in Section III-B.

22−1022 is the absolute minimum of a 64-bit ﬂoating point number, which

is speciﬁed by IEEE 754.

cal PLS. The security level depends on the length of

differentially encoded matrices. This special encoding

expands the search space exponentially. Most existing

PLS methods assume perfect knowledge of CSI both at

transmitter and receiver, which is impractical in high-

mobility scenarios. We resolve this CSI issue by the

nonsquare differential structure, which mitigates both the

communication overhead and risk.

We must report that the proposed arrangement also has the

following shortcomings, as discussed in Section VI.

1) For a case in which the eavesdropper is in the same

position as the legitimate receiver, the proposed scheme

cannot provide security because the eavesdropper would

have a near-perfect estimate of the channel coefﬁcients

between the legitimate transmitter and receiver. In this

case, the legitimate user can physically eliminate the

eavesdropping device or can halt secret communications.

2) Despite the fact that the search space is increased expo-

nentially because of the employment of differential en-

coding, the proposed scheme might become susceptible

to a brute-force attack when the length of differentially

encoded matrices is short. In this unrealistic case, an

eavesdropper having perfect channel state information

(PCSI) can obtain a part of private information.

The remainder of this paper is organized as follows. Sec-

tion II deﬁnes the common system model used for this study.

Section III reviews the classic chaos theory and the conven-

tional chaos-based MIMO schemes. Section IV proposes our

chaos-based differential MIMO that relies on the novel time-

varying basis and the calibration algorithm. Section V presents

an attack algorithm for the proposed scheme. Section VI

demonstrates the performance superiority over conventional

schemes in terms of secrecy rate and reliability. Finally,

Section VII concludes this paper.

We note that italicized symbols represent scalar values. Bold

symbols represent vectors and matrices. Table I presents a list

of mathematical symbols used for this study.

II. SY ST EM MO DE L

This section presents a description of a general system

model common to Sections III and IV. Without loss of

generality, a narrow-band system model is considered in this

paper, but extension to the wide-band scenario in the context

of orthogonal frequency division multiplexing (OFDM) is

straightforward.3

We assume that the legitimate transmitter, Alice, is equipped

with Mantennas, whereas the legitimate receiver, Bob, is

equipped with Nantennas. Additionally, we assume that

eavesdropper Eve has unlimited capabilities of computers,

such as cloud-based computing resources and supercomputers.

The received signal block at Bob is given as [44]

Y(i) = H(i)S(i) + V(i)∈CN×M,(1)

3Note that the nonsquare differential coding described later is particularly

suitable for high-mobility OFDM scenarios [33, 34].

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TABLE I

LIS T OF IM PO RTANT M ATHEM ATIC AL SY MB OLS

BBinary numbers

RReal numbers

CComplex numbers

ZIntegers

M∈ZNumber of transmit antennas

N∈ZNumber of receive antennas

T∈ZNumber of time slots in a codeword

W∈ZFrame length

D∈ZNumber of data-carrying codewords (= W−M)

B∈ZInput bitwidth

R∈RTransmission rate

Reﬀ ∈REffective transmission rate

Y∈ZNon-zero integer value that determines security

Y(i)∈CN×TBob’s received signal block

H(i)∈CN×MBob’s channel matrix

V(i)∈CN×TBob’s additive noise

YE(i)∈CN×TEve’s received signal block

HE(i)∈CN×MEve’s channel matrix

VE(i)∈CN×TEve’s additive noise

S(i)∈CM×TSpace-time codeword

X(i)∈CM×MSpace-time codeword

S(i)∈CM×MUnitary space-time codeword

E1(i)∈CM×1Square-to-nonsquare projection

W1∈CM×1First column of DFT matrix

Y(i)∈CN×MEstimation of H(i)˜

S(i)

b∈BBB-length input bits

σ2

v∈RNoise variance

ϵe∈RAccuracy of shared real-valued keys

ρ∈RChannel correlation coefﬁcient

n∈ZBitwidth of a ﬂoating-point number

i∈ZTransmission index (≤W)

h, h′∈CSingle channel coefﬁcient

α(i)∈RForgetting factor

xi∈RPure chaos solution

ˆxi∈RCalibrated counterpart of xi

i, xr

i∈Rxiat transmitter and receiver

x′

i∈RSecond-order Chebyshev polynomial function

yi∈RChaos sequence having a uniform distribution

ˆyi∈RCalibrated counterpart of yi

where irepresents a transmission index, H(i)∈CN×M

denotes a channel matrix that obeys the i.i.d. Rayleigh fad-

ing CN(0,1), and S(i)∈CM×Tstands for a space-time

codeword. The codeword S(i)is transmitted by Mantennas

over Ttime slots. Speciﬁc construction methods for S(i)

and the corresponding detectors are described in Sections III

and IV. Furthermore, the additive noise V(i)is assumed to

follow the i.i.d. complex Gaussian distribution, CN(0, σ2

v).

The signal-to-noise ratio (SNR) is calculated as 1/σ2

v, i.e.,

10 ·log10(1/σ2

v)[dB] because we have the power constraint

of Ei∥S(i)∥2

F/T = 1. The transmission index starts from

i= 1 and ﬁnishes at a frame length i=W. After extracting

a new private key from the channel, it restarts from i= 1.

Similar to Bob, the received signal block at Eve is given as

YE(i) = HE(i)S(i) + VE(i)∈CN×M,(2)

where Eve’s channel matrix is deﬁned as

HE(i) = ρH(i) + p1−ρ2H′(i),(3)

and H′(i)is an independent channel matrix following

CN(0,1). Here, the channel correlation ρrepresents the sim-

ilarity between the Alice–Bob channel H(i)and the Alice–

Eve channel HE(i). When Eve is near Bob, ρis close to 1.

Eve’s channel matrix HE(i)becomes similar to Bob’s channel

matrix H(i).

In this paper, we model calculation complexity based on

Donald Knuth’s big Omega Ω(·)notation [45], which repre-

sents an asymptotic lower bound. Conventional studies [44]

often only evaluate the total number of real-valued mul-

tiplications. They ignore other operations such as division

and elementary functions, which are as costly as multi-

plication. This asymptotic analysis is useful for estimating

the implementation complexity and power consumption of

circuits [44, 46]. According to [47], the addition and sub-

traction cost Ω(n), where nis the bitwidth of a ﬂoating-

point number. The multiplication costs Ω(nlog n), while the

division costs Ω(nlog nlog n). The square root operation √·

costs Ω(nlog n). The elementary functions such as exp(·),

arcsin(·), and arctan(·)cost Ω(nlog nlog n). For example,

the complexity of (a+bj)(a′+b′j)for a, b, a′, b′∈R

is calculated as 4nlog n+ 4n≥Ω(nlog n). The calcula-

tion of H(i)S(i)costs N T (4Mn log n+ 2(M−1)n)≥

Ω(N T M n log n).

III. CON VE NT IO NAL CH AOS TH EO RY AN D ITS

APP LI CATI ON S TO MIMO

Since 1993, chaos theory has been applied to wireless com-

munications for enhancing security [9–15, 41–43]. Although

a chaotic mathematical model is deterministic, it is sensitive

to initial conditions. Moreover, it is almost impossible to

predict future trajectory. This sensitivity can be quantiﬁed

by the Lyapunov exponent [48]. Additionally, the chaotic

sequence is bounded within a region and is non-periodic.

Therefore, the initial condition works as a secret key for secure

communications.

In von Neumann’s seminal study [49], the logistic map

xi+1 = 4xi(1 −xi)(4)

is used to generate a random digit. The initial condition x0

must be 0< x0<1. The chaotic sequence of (4) is called

a pure chaos solution. Its Lyapunov exponent is calculated as

loge(2) ≈0.6931, which is higher than the value of 0.0714

[48] of the simpliﬁed Rossler equation [50]. The probabilistic

distribution of xiis a non-uniform function of [49]

p(x) = 1

πpx(1 −x),(5)

which differs from a uniform distribution. Because the exact

solution of (4) is given as [51]4

xi= sin22iarcsin(√x0):= f(i, x0),(6)

the chaotic sequence of xican be transformed into a uniform

distribution as [51]

yi=2

πarcsin (√xi),(7)

where we have the uniform distribution of p(y)=1for 0<

y < 1.

4This equation might overﬂow because of the calculation of 2i. For our

simulations, we calculate f(i, x)in a recursive manner, i.e., f(0, x) = xand

f(i, x)=4f(i−1, x)(1 −f(i−1, x)).

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0 5 10 15 20 25

Index i

0.0

0.2

0.4

0.6

0.8

1.0

Chaos transition xi

Fig. 1. Chaos transition xiover time, for which the initial values of x0=

0.24,x0+ 10−6, and x0+ 10−3were considered.

As a simple example, Fig. 1 portrays the transition of xi,

deﬁned in (4), where the index is increased from i= 0 to 25.

Three initial values were considered as shown in Fig. 1: x0=

0.24,0.24 + 10−3and 0.24 + 10−6. As shown in Fig. 1, both

the initial values x0= 0.24 and 0.24 + 10−6caused almost

identical xifor 0≤i≤13, whereas both sequences exhibited

signiﬁcantly different transitions for i > 13. In the x0+ 10−3

case, it exhibited different transitions for i > 4. Therefore, the

accuracy of the initial value determines the agreement interval

of the chaos sequence. Based on this fact, we propose a new

practical calibration algorithm in Section IV-F.

A. MIMO-CSK [14]

The MIMO-CSK scheme of [14] uses the second-order

Chebyshev polynomial function of

x′

i= 1 −2x′2

i−1,(8)

where the initial condition x′

0must be within [−1,1]. It is

noteworthy that the mean of x′

iis zero. The variance is 0.5.

The original contribution of [14] considers the direct sequence

spread spectrum instead of OFDM. The chaotic sequence of

(8) is used as a spread sequence over the time domain. The

spreading factor is varied from β= 2 to 50 in [14]. As

described in this paper, we limit the spreading factor to β= 1

for simplicity.5In this case, the 2×2space-time matrix is

generated by S(i) = P(i)X(i)[14], where we have

P(i) = √2·diag(x′

2i, x′

2i+1)(9)

and the orthogonal space-time block code (OSTBC) of [52]

X(i) = 1

√2s1(i)−s∗

2(i)

s2(i)s∗

1(i).(10)

Here, s1(i)and s2(i)denote complex-valued symbols with

L-ary phase-shift keying (PSK) or quadrature amplitude mod-

ulation (QAM). The transmission rate of (10) is R= log2(L).

5As one might expect, this limitation worsens performance. Although

MIMO-CSK is the product of an important pioneering study, it is not a

performance baseline.

Note that the mean transmission power is calculated as

Eih∥S(i)∥2

Fi/T = 2/2=1, which is the same as in other

MIMO techniques. The maximum-likelihood detector is given

X(i) = arg min

X∥Y(i)−H(i)P(i)X∥2

F,(11)

where the chaotic sequence diag(x′

2i, x′

2i+1)is known per-

fectly at the receiver with no noise.

Although the original contributions of [14] considered only

the M= 2 case, it can be extended to the M > 2cases as

S(i) = P(i)X(i),(12)

where we have

P(i) = √2·diag(x′

i·M, x′

i·M+1,··· , x′

i·M+M−1)(13)

and an M×Mdata-carrying matrix X(i). We must multiply

√2in (13) for any Mbecause the variance of x′

iis 0.5. For

example, the Bell Laboratories layered space-time scheme [53]

is deﬁned as X(i) = [s1(i)s2(i)·· · sM(i)]T/√M∈CM×1.

In the OSTBC case having M= 4, the data-carrying matrix

is given as [54]

X(i) = 1

√2





s1(i)−s∗

2(i) 0 0

s2(i)s∗

1(i) 0 0

0 0 s1(i)−s∗

2(i)

0 0 s2(i)s∗

1(i)





.(14)

The transmission rate of (14) is R= log2(L)/2. Following

the complexity model described in Section II, the detection

complexity of (11) is lower bounded by Ω(2RNM n log n).

B. MIMO-DCSK [41]

The MIMO differential CSK (DCSK) scheme [41] has been

proposed for spread spectrum communications, which require

no CSI either at the transmitter or receiver. The conventional

MIMO-CSK [14] requires the receiver to reproduce the orig-

inal chaos sequence generated by the transmitter. To address

this synchronization issue, Kaddoum et al. proposed DCSK for

a MIMO setup. It generates a space-time codeword of [41]

S(i) = x′

4is1(i)x′

4i+1 x′

4i+2 −s∗

2(i)x′

4i+3

x′

4is2(i)x′

4i+1 x′

4i+2 s∗

1(i)x′

4i+3 (15)

when the spreading factor is 1. By transmitting a chaos se-

quence directly, this scheme helps the receiver to reproduce the

chaos sequence. Such DCSK-related studies [41, 43, 55, 56]

differ from the differential STBC concept established in the

early 2000s [30, 31]. Therefore, we do not consider the DCSK

family in our performance comparisons.

C. C-MIMO [10–13]

The C-MIMO scheme has been proposed for improving the

security of MIMO communications, where PCSI is necessary

at the receiver. An initial condition is generated using a pre-

shared key. The key is processed many times by the Dirac

transformation [10, 13]. The resultant constellation follows the

complex-valued Gaussian distribution.

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The C-MIMO scheme requires a pre-shared key c0∈Cthat

obeys 0<Re[c0]<1and 0<Im[c0]<1. The B=M T -

length input bits b= [b1, b2,·· · , bB]∈BBis mapped to a

set of complex-valued symbols s= [s1, s2,··· , sB]∈CB.

This set is then mapped to an M×Tspace-time codeword.

Each symbol skfor k= 1,2,·· ·. Also, B is deﬁned by

two independent chaos sequences Re[zl]and Im[zl]. Both

sequences are initialized by Re[z0] = Γ(Re[ck−1], bk−1)and

Im[z0] = Γ(Im[ck−1], bkmod B), where we have [13]

Γ(a, b) = 









a(b= 0)

1−a(b= 1 and a > 1/2)

a+ 1/2 (b= 1 and a≤1/2)

.(16)

Then, both sequences are generated as6

Re[zl] = 2 ·Re[zl−1] mod (1 −10−16 ) and

Im[zl] = 2 ·Im[zl−1] mod (1 −10−16 )(17)

for l= 1,2,·· · , Ns, Ns+ 1, and Ns= 100 [13]. The C-

MIMO symbol skfor k= 1,2,·· · , B is given by [13]

sk=r−log c(x)

kcos 2πc(y)

k+jsin 2πc(y)

k,

(18)

where we have [13]











ck= Re[zNs+b(k+B/2) mod B] + jIm[zNs+b(k+B/2+1) mo d B]

c(x)

k= arccos(cos(37π(Re[ck] + Im[ck])))/π

c(y)

k= arcsin(sin(43π(Re[ck]−Im[ck])))/π +1

(19)

By virtue of the Box–Muller transform in (18), skfollows

the complex Gaussian distribution CN(0,1). Finally, the code-

word associated with B=M T -length bits bis given as

S(i) = 1

√M







s1sM+1 ·· · sM T −M+1

s2sM+2 ·· · sM T −M+2

.....

sMs2M·· · sM T





∈CM×T.

(20)

The normalized transmission rate is calculated as R=M

[bit/symbol]. The detection complexity is lower bounded by

Ω(2RT N Mn log n), where the complexity of generating (20)

is ignored for simplicity.

IV. PROPOSED CHAOS-BAS ED DI FFE RE NT IA L MIMO

The proposed scheme has a common structure with the

conventional nonsquare differential scheme of [32–34, 36].

It invokes a chaos-based time-varying basis and a chaos

calibration algorithm. Fig. 2 shows (a) the transmitter and (b)

the receiver of our proposed system. As shown in Fig. 2, our

system extracts a secret initial key from the wireless channel,

which is denoted by xt

0at the transmitter and xr

0at the receiver.

At the transmitter, an input bit sequence b(i)is mapped to a

differentially-encoded square matrix ˜

S(i). In parallel, a chaos

6The modulo operation is extended to real numbers as xmod y:= x−

y· ⌊x/y⌋.

Input

bits Delay

Proposed

time-varying

mapping

 E è

E

EFs

 E Unitary matrix

mapper (21)

Conventional differential encoding

ó5E

Initial

key

Fs s

TÜ

Delay

Basis

mapper (35)

ELr

EPr

 E

(a) Transmitter

0Output

bits

 E

ó5E



E

Detector (28)

Delay EFs

Basis mapper (35)

Chaos calibrator (46)

Initial

key

Fs s

TÜ

Delay

ELr

EPr

ÜTÜ

(b) Receiver

Fig. 2. Schematic of the proposed system.

sequence xt

iis used to generate a time-varying basis E1(i)∈

CM×1. Then, the square matrix ˜

S(i)∈CM×Mis mapped into

a nonsquare matrix ˜

S(i)E1(i)∈CM×1. Conventional studies

[32–34] adopted a static basis E1∈CM×1instead of this

time-varying counterpart. At the receiver, the chaos sequence

iis initialized by xr

0. It is used to estimate the private bit

sequence ˆ

b(i). Because the chaos sequence xr

imight include

errors, it is calibrated by the proposed algorithm and is used

for the next time slot.

A. Encoding at the transmitter

We ﬁrst introduce the encoding process which supports the

time-varying basis. The B-length input bit sequence b∈BB

is associated with an M×Msquare data-carrying matrix of

[30]

X(i) = diag exp j2πb

2Bu1,·· · ,exp j2πb

2BuM (21)

:= X(b),(22)

which is known as diagonal unitary code (DUC). Here, the

code index is given as b= (b)10, where (·)10 denotes

the binary to decimal conversion. Additionally, Mdiversity-

maximizing factors 0< u1≤ ·· · ≤ uM≤2B/2∈Zare

designed to maximize the diversity product of [30]

min

b∈{1,··· ,2B−1}

m=1

sin πbum

2B

.(23)

Although this optimization is a time-consuming task, the

designed factors are available from the open-source library

used in [57].7For example, in the (M, B) = (4,4) case, the

designed factors are [u1, u2, u3, u4] = [1,3,5,7]. Note that

7https://github.com/ishikawalab/wiphy/blob/master/wiphy/code/duc.py

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(21) can be replaced with all of the sophisticated differential

family which relies on sparse unitary matrices [37–39, 58, 59].

However, to simplify our analysis, we limit X(i)of (21) to

the classic DUC.

The time-varying basis E1(i)∈CM×1varies as the

transmission index increases from i= 1 to i=W, where

Wis the frame length. Details of the construction method

of E1(i)are presented in Section IV-C. This basis E1(i)is

the ﬁrst column of E(i)∈CM×M. Later, other columns are

represented by E(i)=[E1(i)E2(i)·· · EM(i)] ∈CM×M.

For 1≤i≤Mblocks, the baseband symbol of (1) is deﬁned

S(i) = Ei(M)∈CM×1.(24)

This equation implies that the unitary matrix E(M)∈CM×M

is transmitted in the ﬁrst Mtime slots. Although this matrix

E(M)∈CM×Mis equivalent to the conventional reference

symbol, the overall performance will not change when increas-

ing W[34], which is similar to the classic differential MIMO

family. For M+ 1 ≤i≤Wblocks, the baseband symbol is

deﬁned as

S(i) = ˜

S(i)E1(i)∈CM×1,(25)

where we have the M×Mmatrix of

S(i) = (IM(i≤M)

S(i−1)X(i) (i > M).(26)

The effective transmission rate is calculated as Reﬀ = (W−

M)/W ·R= (1 −M /W )·R, whereas the ideal transmission

rate is R=B[bit/symbol]. In this paper, we use W= 20 ·M

to keep the rate loss at 5%. The frame lengths of W= 100·M

and 1000 ·Mare also possible. However, these simulations

might become time-consuming.

B. Decoding at the receiver

For 1≤i≤Mblocks, the estimate of H(i)˜

S(i)∈CN×M

is updated as

Y(i) = ˆ

Y(i−1) + Y(i)EH

i(M)∈CN×M,(27)

where the initial value is a zero matrix, i.e., ˆ

Y(0) = 0N×M.

Then, for i>Mblocks, the data-carrying matrix X(i)of

(21), which is associated with the input bits b, is estimated

by the maximum-likelihood detector of8

X(i) = arg min

X



Y(i)−ˆ

Y(i−1)XE1(i)



F,(28)

where we have

Y(i) = ˆ

Y(i−1) ˆ

X(i) + (1 −α(i)) D(i)EH

1(i)∈CN×M,

(29)

α(i) = min N·σ2

v/∥D(i)∥2

F,0.99∈R,(30)

8If the coherent time is extremely short, the noncoherent detection will

cause an error ﬂoor, which is similar to the coherent detection. Refer to [33]

for a study in high-mobility scenarios and [34] for a study in millimeter-wave

scenarios.

and

D(i) = Y(i)−ˆ

Y(i−1) ˆ

X(i)E1(i)∈CN×1.(31)

The adaptive forgetting factor α(i)must be within the range

of (0,1), which minimizes the error of 



ˆ

Y(i)−H(i)˜

S(i)



As given, the estimate of CSI, H(i), is not included in

(28). Instead of H(i), this detector is used to calculate

Y(i)≈H(i)˜

S(i), which yields a low-complexity detection.

Speciﬁcally, the detection complexity of (28) is lower bounded

by Ω(2RNM2nlog n). This complexity is higher than those

of the conventional chaos-based schemes having T= 1.

However, these schemes must carry out complex channel

estimation that is not considered for this study.

C. Proposed chaos basis

The proposed chaos basis inherits a basic property from

the conventional basis proposed in [33]. Similarly to the

conventional method, the following M×Mstatic discrete

Fourier transform (DFT) matrix is generated [33]:

W=1

√M







1 1 ·· · 1

1ω·· · ωM−1

1ω2·· · ω2(M−1)

.....

1ωM−1·· · ω(M−1)(M−1)







,(32)

where ω= exp(−2πj/M). Later, the ﬁrst column of Wis

denoted as

W1= [1 1 ·· · 1]T

| {z }

Mrows

/√M(33)

for simple notation. The conventional static DFT basis is

generated by [33]

E(i) = W,(34)

whereas the proposed time-varying unitary matrix is generated

E(i) = exp(j2πyiY)W(35)

= exp(j4 arcsin (√xi)Y)W(36)

:= exp(jθ(xi))W(37)

where yiis a chaos sequence deﬁned in (7) and Yis a

non-zero arbitrary integer. If |Y| ≥ 2, (35) becomes a non-

invertible function of yi, which improves security. The tradeoff

between security and reliability is discussed in Section VI.

Finally, the time-varying DFT basis is generated as E1(i) =

exp(2πyiY j)W1∈CM×1.

Fig. 3 shows the transitions of the chaos DFT basis (35)

having (M, T ) = (4,1) and Y= 8, where the time index was

increased from i=M/T + 1 = 5 to W= 80. As shown in

Fig. 3, the ﬁrst row of E1(i)was distributed uniformly on a

circle.

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−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

Real part

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

Imaginary part

Fig. 3. Transition of the chaos DFT basis E1(i)∈C4×1(35), where the

ﬁrst row of E1(i)was presented in the I/Q domain.

D. Generation of a real-valued secret key from the true

randomness of wireless nature

In the literature, several key generation methods have been

proposed. They rely on the true randomness of wireless nature

[19], with characteristics such as the random received signal

strength (RSS) [60], the random channel coefﬁcients [27],

the diversity of MIMO [28], the MIMO channel ﬂuctuations

[29], and distributed antennas [61]. Most of these methods are

highly reliable. However, achieving the key agreement ratio

of 100% in high-mobility scenarios, where the beneﬁts of

differential coding can be exploited, is a challenging task. As

described in Section III, any chaos-based system requires at

least one initial real-valued key, which is represented as a 64-

bit ﬂoating-point number, for example. Here, optimal mapping

between a shared 64-bit key and a real-valued key remains

unknown. One bit error might result in a large difference in the

real-valued counterpart. Therefore, we must consider a real-

valued key generation method that differs from conventional

binary key generation methods.

Because the proposed scheme is free from the channel

estimation process, we opt to use the randomness of RSS

to generate a real-valued key, which is inspired by the key

generation method presented earlier in the literature [60].

The following RSS-based key generation method is used to

model the error of shared keys, the effects of SNR and the

channel correlation between Alice-Bob and Alice-Eve channel

matrices.

RSS-Based Example: We speciﬁcally examine a single

receive antenna. Its RSS value is mapped to a [0,1] real-valued

key. To be more speciﬁc, the initial condition of (4) at the

transmitter is modeled as

0= exp − |h+ϵeσvv|2,(38)

where h∼ CN(0,1) represents a single channel coefﬁcient,

v∼ CN (0,1) stands for an additive noise, and ϵe∈Rdenotes

the accuracy of shared keys. Because the Rayleigh fading

channel is assumed for this study, |h|follows the Rayleigh

distribution; also, xt

0follows a uniform distribution [0,1] at

high SNRs. Obviously, the phase information of hcan be

useful in the same manner as [62], which is not considered

in this paper for simplicity. By contrast, at the receiver, the

corresponding real-valued key is generated as

0= exp −ρh +p1−ρ2h′+ϵeσvv′

2,(39)

where h′∼ CN(0,1) and v′∼ CN(0,1) respectively denote

single channel and noise components. The channel correlation

ρ∈[0,1] also determines the accuracy. Bob invariably has

ρ= 1, whereas Eve might have ρ∈[0,1). The error between

0and xr

0improves as SNR = 1/σ2

vincreases. In the MIMO

context, the methods described in several papers rely on the

assumption of PCSI at both the transmitter and the receiver,

i.e., ϵe= 0. This assumption is optimistic because the effects

of the additive noise [63] and the mismatch of the TDD

channel reciprocity cannot be ignored. As shown in Fig. 1,

even a small error induces a mismatch in the chaos sequences

at the transmitter and the receiver.

As might be inferred from (38) and (39), when Eve is near

Bob, i.e., ρis close to 1, Eve can estimate the secret key

generated at Bob. Assuming this simple but vulnerable RSS-

based method, one can discuss the minimum security level that

can be guaranteed under the proposed system. Actually, the

security level can be improved using artiﬁcial noise [27] and

beamforming [64] with the sacriﬁce of additional complexity.

The key generation example above can be replaced with such

a sophisticated method.

E. Secrecy rate of the proposed scheme

In [65], Wang et al. deﬁned the secrecy rate as

Cs= max(0, IB−IE).(40)

Here, IBdenotes the average mutual information (AMI)

between Alice and Bob, while IEdenotes the AMI between

Alice and Eve. Since the AMI for nonsquare differential

coding is still unknown, we assume coherent detection at the

receiver and calculate the AMI for the proposed time-varying

codewords. The AMI IBcan be derived by extending the

deﬁnition of [66] as follows:

IB=B−1

W−M

i=M+1

f=1

EH,V

log2

g=1

exp ηB[i, f, g ]

σ2

v

,(41)

where we have

ηB[i, f, g ] = −



HX(f)−X(g)exp(jθ(xt

i))W1+V



+∥V∥2

F.(42)

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Similarly, the AMI IEcan be derived by

IE=B−1

W−M

i=M+1

f=1

EH,V

log2

g=1

exp ηE[i, f, g ]

σ2

v

,(43)

where we have

ηE[i, f, g ] =

− ∥HX(f)exp(jθ(xt

i)) −X(g)exp(jθ(xr

i))W1+V∥2

+∥HX(f)exp(jθ(xt

i)) −X(f)exp(jθ(xr

i))W1+V∥2

(44)

F. Low-complexity calibration of the chaos sequence

The difference between xt

0and xr

0induces severe communi-

cation errors. For that reason, all the conventional chaos-based

systems used a pre-shared key for initializing chaos sequences

at the transmitter and receiver. To address this issue, we

propose a novel calibration algorithm for the chaos sequence at

the receiver. For each candidate Xin (28), the corresponding

E1(i) = exp(2πyr

iY j)W1in (28) is calibrated. Speciﬁcally,

the detector tries to calibrate xr

iand yr

i= 2 arcsin pxr

i/π,

which might contain errors. The latter yr

ican be calibrated by

solving

ˆyr

i= arg min

y



Y(i)−exp(j2πyY )ˆ

Y(i−1)XW1



F(45)

for a given X. This optimization problem has multiple solu-

tions because of the phase ambiguity induced by Y. Here, (45)

is solvable by a low-complexity closed-form equation of

ˆyr

i=θy+ ˆnπ

2πY ,(46)

where we have θy= arctan(−Im[a]/Re[a]),a=

tr[Y(i)Hˆ

Y(i−1)XW1], and ˆn=⌊2yr

iY−θy/π + 0.5⌋.

Finally, the receiver obtains the calibrated chaos sequences

ˆyr

iand ˆxr

i= sin2(ˆyr

iπ/2). After obtaining ˆyr

i, the receiver

updates E1(i) = exp(2πˆyr

iY j)W1of (28).

The complexity of (45) is lower bounded by

Ω(NgN n log n), where Ngis the search space size of

y; it is set to a large value such as 103or 104. By contrast,

the complexity of (46) is negligible by virtue of its closed-

form calculations. Therefore, the lower-bound for overall

ML complexity with the calibration algorithm is the same as

that of the conventional nonsquare differential decoding, as

analyzed in Section IV-B.

In summary, the proposed detection process with the chaos

calibration algorithm is outlined in Algorithm 1.

V. ATTACK AL GO RI TH M AN D SEC UR IT Y ANALYSIS

We conceive an attack algorithm for the proposed scheme,

for which we assume that Eve has PCSI [67] and inﬁnite SNR,

although Bob has no CSI and realistic SNR. In practice, it

is a challenging task for Eve to obtain a precise estimate of

CSI because the proposed scheme transmits no ﬁxed reference

Algorithm 1 Proposed ML detector with the chaos calibration

algorithm of Section IV-F.

Input: Y(i),ˆ

Y(i−1),ˆxr

i−1,W1, B, Y , N, σ2

Output: ˆ

b(i),ˆ

Y(i),ˆxr

Initialization:

1: τmin = +∞ {τmin is the minimum of (26)}

2: xr

i= 4ˆxr

i−11−ˆxr

i−1{update xr

iusing (2)}

3: yr

i= 2 arcsin(pxr

i)/π {update yr

iusing (5)}

ML detection for ˆ

X(i) = X(bmin):

4: for b= 0 to 2B−1do

5: a= tr hY(i)Hˆ

Y(i−1)X(b)W1i{calibrate yr

6: θy= arctan(−Im[a]/Re[a])

7: ˆn=⌊2yr

iY−θy/π + 0.5⌋

8: ˆyr

i= (θy+ ˆnπ)/(2πY ){obtain the calibrated ˆyr

9: E1(i) = exp(2πˆyr

iY j)W1{update E1(i)}

10: D=Y(i)−ˆ

Y(i−1)X(b)E1(i)

11: τ=∥D∥2

F{calculate ML detection norm using (26)}

12: if (τ < τmin )then

13: τmin =τ,bmin =b, and Dmin =D

14: ˆxr

i= sin2(ˆyr

iπ/2)

15: ˆ

E1(i) = E1(i)

16: end if

17: end for

Finalization:

18: ˆ

b(i) = (bmin)2{obtain the estimated bits}

19: α(i) = min N·σ2

v/τmin,0.99{update ˆ

Y(i)}

20: ˆ

Y(i) = ˆ

Y(i−1)X(bmin)+ (1 −α(i)) Dmin ˆ

1(i)

21: return ˆ

b(i),ˆ

Y(i),ˆxr

symbol. If reference symbols are not available, then the

receiver can exploit a sophisticated blind channel estimation

method [68, 69]. However, this blind estimation is possible

only if all the transmitted space-time matrices are semi-unitary

[68]. As described in an earlier report [68], results showed

that 50 unknown unitary matrices were necessary to obtain

a precise CSI for a 4×4MIMO scenario. Another blind

estimation method [69] required 2000 OFDM symbols, each

of which had 64 subcarriers. It is unrealistic to apply these

blind estimation approaches to high-mobility scenarios, where

differential schemes work efﬁciently.

A. Attack algorithm

Eve has PCSI HE∈CN×M, which is unrealistic, as

described above. This fact enables coherent detection at Eve,

although the transmitted matrix ˜

S(i)of (26) is differentially

encoded. Later, the ﬁrst Mreceived symbols are denoted

by ¯

YE= [YE(1) YE(2) ·· · YE(M)], which converges to

HEE(M)when SNR →+∞. The secret key x0can be

estimated by solving

ˆx0= arg min

xg1(x),(47)

where we have

g1(x) = 

¯

YE−exp (jθ(x)) HEW

2

F(48)

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and θ(x)deﬁned in (37). However, this optimization problem

returns multiple solutions because we have |Y|>1. The phase

ambiguity imposed by |Y|>1enhances security.

To address this challenge, Eve must perform high-

complexity joint optimization over i= 1,··· , W blocks.

Letting d= [d1, d2,·· · , dD]be a set of integers indicating

data-carrying matrices and letting D=W−Mbe the number

of these matrices, then as deﬁned in (22), each integer in d

ranges from 0to 2B−1. Accordingly, the number of possible

patterns of dis calculable as 2BD . Eve estimates ˆx0and ˆ

simultaneously by solving

ˆx0,ˆ

d= arg min

(x,d)g1(x) +

i′=1

g2(i′, x, d),(49)

where we have

g2(i′, x, d) = ∥YE(M+i′)−exp (jθ (f(i′, x))) HEF(i′,d)∥2

(50)

and

F(i′,d) = X(d1)X(d2)·· ·X(di′)

| {z }

i′matrices

W1.(51)

Note that f(·,·)is deﬁned in (6), X(·)is deﬁned in (22), and

W1is deﬁned in (33).

B. Security analysis

The complexity of (49) is extremely high because of the

global optimization imposed by ˆx0∈[0,1] and the large

search space of ˆ

d∈ZD.

a) Global optimization for continuous ˆx0:Since the

objective function of (49) is non-convex, Eve has to perform

a global optimization for ˆx0, such as the brute-force search

with a step size of 10−64, the dual annealing method [70]

with a large number of iterations, and the differential evolution

method [71]. The ﬁrst derivative of (49) is given as

dx "g1(x) +

i′=1

g2(i′, x, d)#

=−2·d

dxRe"exp(jθ(x))tr( ¯

EHEW)

i′=1

exp(jθ(f(i′, x)))tr(YH

E(M+i′)HEF(i′,d))#,(52)

which contains a degree D+ 1 polynomial function, and

cannot be solved algebraically. For example, if we consider

the D=W−M= 80 −4 = 76 case, the ﬁrst deriva-

tive d/dx(exp(jθ(f(76, x)))), where f(76, x)is a degree 77

polynomial function, yields a lot of solutions, which makes

this global optimization difﬁcult.

b) Brute-force search for discrete d:To solve the opti-

mization problem of (49) and to resolve the phase ambiguity,

Eve must perform a brute-force combinatorial search for

estimation of d. The search space for dis calculable as

2B(W−M)= 2BD , which increases exponentially with the

transmission rate R=Band the number of transmit antennas

M. For example, in the (M, R, D) = (4,1,2) case, we have

22= 4 patterns: d= [d1, d2] = [0,0],[0,1],[1,0],[0,1].

Each set determines the combination of differentially en-

coded symbols as X(0)X(0) ,X(0)X(1) ,X(1)X(0) ,X(1)X(1) .

To maximize the effective transmission rate, Reﬀ = (1 −

M/W )·R= (1 −M /W )·R, the frame length is set to a large

value such as W= 20M,100M, or 1000M. In each case, the

search space becomes 2BD = 2R(W−M)= 219RM ,299RM ,

and 2999RM . Here, Eve must prepare 2304,21584 , and 215984

candidates. Because the National Security Agency in the USA

recommended the key length of 256 bits as the advanced

encryption standard9, the search space of the proposed scheme

is sufﬁciently large.

In summary, the overall complexity of (49) is lower bounded

by Ω(2RDNgD2M2Nn log n), where Ngdenotes the maxi-

mum iteration limit of the global optimization method. Al-

though we considered the simplest and the worst real-valued

key generation method in Section IV-D, the minimum security

level achieved by the proposed system is sufﬁciently high.

VI. PE RF OR MA NC E COM PARI SO NS

We investigated the performance of the proposed scheme in

terms of the secrecy rate and bit error ratio (BER). Speciﬁcally,

the proposed scheme having the time-varying chaos basis of

(35) was considered, where the novel detector of (28) and the

chaos calibration algorithm of (46) were used. Additionally,

we considered two conventional chaos-based MIMO schemes:

MIMO-CSK [14] described in Section III-A and C-MIMO

[10–13] described in Section III-C. Here, PCSI at the le-

gitimate receiver was assumed to beneﬁt these conventional

schemes. We also considered the classic differential star-

QAM (SQAM) [72] and the conventional nonsquare DUC

(N-DUC) [33], both of which worked efﬁciently without CSI.

As a reference, the massive MIMO (M-MIMO) cryptography

method of [73] was considered, although it required PCSI at

both the transmitter and receiver. It is noteworthy that the M-

MIMO cryptography is designed particularly for large-scale

scenarios, but is also beneﬁcial for small-scale scenarios by

virtue of its linear precoding [73].

For our simulations, we assumed the ideal Rayleigh fading

channel model as described in Section II. The numbers of

transmit and receive antennas were, respectively, M= 4 and

N= 4. The transmission rate was R= 4 [bit/symbol]. The

frame length was W= 20 ·M= 80. The conventional

MIMO-CSK scheme used (12) with two 256-QAM symbols,

whereas the conventional C-MIMO scheme used (20) with

(M, T , Ns) = (4,1,100) and (4,2,100). The M-MIMO

cryptography method used 16-QAM symbols. The proposed

scheme used (35) with Y= 8 to generate a time-varying chaos

basis. The data-carrying unitary matrix of (21) was generated

by [u1, u2, u3, u4] = [1,1,1,1] for R=B= 2,[1,3,5,7]

for R= 4,[1,21,24,25] for R= 6, and [1,35,41,119] for

R= 8.

First, we investigated the performance of the proposed

scheme in channel-coded scenarios. Fig. 4 portrays an AMI

comparison where we considered our proposed scheme and

four other reference curves: the Shannon capacity, 16–SQAM

9https://apps.nsa.gov/iaarchive/programs/iad-initiatives/cnsa-suite.cfm

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−20 −15 −10 −50 5 10 15 20

SNR [dB]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

AMI [bit/symbol]

Shannon capacity

16-SQAM (T= 1) [72]

Conv. C-MIMO (T= 1) [13]

Conv. C-MIMO (T= 2) [13]

Proposed (T= 1)

Fig. 4. AMI comparison for which we considered the frame length of W=

M+ 1 = 5 and the transmission rate of R=B= 4 [bit/symbol].

−20 −15 −10 −50 5 10 15 20

SNR [dB]

Secrecy rate [bit/symbol]

Proposed (R= 2)

Proposed (R= 4)

Proposed (R= 6)

Proposed (R= 8)

Fig. 5. Secrecy rate comparison upon increasing the transmission rate R=B

[bit/symbol], where other parameters are the same as those used in Fig. 4.

[72], and the C-MIMO scheme [10] having T= 1 and 2.

Here, we calculated AMI in the same manner as [44]. Because

the constrained AMI of the nonsquare differential coding is

not yet known, we assumed PCSI at the legitimate receiver

and used the frame length of W=M+ 1 = 5, which

implies that the effects of differential encoding and decoding

were not considered. Additionally, we assumed that Alice’s

and Bob’s initial keys (xt

0, xr

0)were identical. As shown in

Fig. 4, our proposed scheme having T= 1 outperformed

the C-MIMO scheme having T= 1 and achieved the same

AMI as the C-MIMO scheme having T= 2 that required

additional decoding complexity. We observed the same trend

in the (M, R) = (8,8) case. In fact, the C-MIMO has the

advantage of Gaussian-distributed symbols, which are difﬁcult

for Eve to perceive, whereas the proposed scheme generates

constant-envelope symbols.

Following Fig. 4, we investigated the achievable secrecy rate

of the proposed scheme in Fig. 5, where the transmission rate

was increased from R= 2 to 8[bit/symbol] and where other

simulation parameters were identical to those used in Fig. 4.

Here, we calculated the secrecy rate deﬁned by an earlier

study [65], which was introduced in Section IV-E. We assumed

that Bob’s and Eve’s channel matrices were independent, and

assumed that both SNRs were identical. As shown in Fig. 5,

the secrecy rate improved upon increasing the transmission

rate from R= 2 to 8monotonically. The corresponding

ratios of the information leaked to Eve were, respectively,

19.83%, 20.39%, 12.37%, and 9.61% of the transmitted bits.

This observation suggests that Eve is unable to decode all

the private information correctly when we use the powerful

channel coding technique with the coding rates of 1/2,2/3,

3/4, or 5/6, for example.

Second, as shown in the panels of Fig. 6, we investigated

the BER performance of the proposed scheme. Here, we

considered (a) perfect key and (b) erroneous key scenarios.

Additionally, we investigated the performance of Eve’s attack

algorithm as shown in Fig. 6(c).

For Fig. 6(a), an ideal condition was assumed: Alice

and Bob had the same generated key xt

0=xr

0. Only the

conventional schemes of MIMO-CSK [14], C-MIMO [10],

and M-MIMO cryptography [73] had PCSI. Other schemes,

including our proposal, had no CSI. As shown in Fig. 6(a), the

proposed scheme with the time-varying chaos basis achieved

the same performance as that of the conventional N-DUC

scheme of [33]. This ﬁnding implies that the use of chaos basis

induces no performance penalty. The conventional MIMO-

CSK scheme having PCSI exhibited worse performance than

the classic differential SQAM. Actually, this is true because

MIMO-CSK relies on the diagonal matrix (13) composed of

a chaos sequence. Since this diagonal matrix is not unitary,

the minimum Euclidean distance of the resultant space-time

matrix becomes a small value.10 The conventional C-MIMO

scheme having T= 1 and 2achieved competitive perfor-

mances. Speciﬁcally, the C-MIMO scheme having T= 2

achieved the best performance in the sacriﬁce of complexity,

as analyzed in Section III-C, and outperformed M-MIMO

cryptography, which required PCSI at the transmitter. In the

T= 1 case, our proposed noncoherent scheme exhibited a

similar trend to that of the coherent C-MIMO scheme having

PCSI. This is particularly noteworthy because a noncoherent

system generally exhibits the well-known 3 [dB] loss, unlike

its coherent counterpart. Because the C-MIMO symbols follow

a Gaussian distribution, which is a good property for improv-

ing security, the resultant BER might become a little worse

despite having PCSI.

In Fig. 6(b), Alice and Bob obtained a secret key from the

wireless channel, as described in Section IV-D. Both extracted

keys mutually differed. The difference was determined by the

model of (39), where the error metric of ϵe= 10−1,10−2,

and 10−10 were considered. The proposed scheme remains

free from the estimation of a full channel matrix H(i)∈

CN×M, but it requires an RSS value to generate a secret

key xt

0and xr

0, as described in Section IV-D. As shown

in Fig. 6(b), the proposed scheme without the calibration

10Note that the original MIMO-CSK scheme was conceived for spread

spectrum communications [14].

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0 5 10 15 20 25 30 35 40

SNR [dB]

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

BER

T= 1

T= 2

Diﬀ. 16-SQAM (ref.) [72]

Conv. M-MIMO crypt. (ref.) [73]

Conv. CSK (w/ PCSI) [14]

Conv. C-MIMO (w/ PCSI) [13]

Conv. N-DUC (w/o CSI) [33]

Proposed (w/o CSI)

(a) Perfect key scenario (W= 80).

0 5 10 15 20 25 30 35 40

SNR [dB]

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

BER

ǫe= 10−10

ǫe= 10−1

ǫe= 10−2

4.6dB

Conv. CSK (PCSI) [14]

Conv. C-MIMO (PCSI) [13]

Proposed w/o key errors

Proposed w/o calibration

Proposed w/ calibration

(b) Erroneous key scenario (W= 80).

0 5 10 15 20 25 30 35 40

SNR [dB]

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

BER

Bob’s BER w/o CSI

Eve has the same CSI and key as Bob

Eve’s attack algorithm (w/ PCSI and dual annealing)

Eve’s attack algorithm (w/ PCSI and diﬀ. evolution)

Fig. 6. BER comparisons for which we considered three scenarios, where

the transmission rate was R= 4 [bit/symbol].

0 50 100 150 200 250

Index of possible dpatterns

10−2

10−1

100

101

102

Detection norm (49) at Eve’s SNR = 100 [dB]

Bob selected a sub-optimal solution

and obtained the correct bits.

Eve selected the global

optimal solution and

obtained incorrect bits.

Fig. 7. Detection norms of Eve’s attack algorithm (49) for all the possible

dpatterns, where the parameters were the same as those used in Fig. 6(c),

the size of search space for dwas 28= 256, and the dual annealing method

[70] was used.

algorithm exhibited an error ﬂoor where the key contained

the small error of ϵe= 10−10, which were similar to other

conventional schemes. By contrast, the proposed scheme with

the calibration algorithm achieved practical BER performance,

even though we considered the high errors of ϵe= 10−1and

10−2. The SNR gap separating the perfect key scenario was

4.6[dB] at BER = 10−3.

In Fig. 6(c), we investigated the performance of Eve’s attack

algorithm described in Section V. In Figs. 6(a) and (b), we

considered a realistic frame length W= 20 ·M= 80. By

contrast, in Fig. 6(c), we changed the frame length W= 80

to 6to enable the brute-force search at Eve. The corresponding

search space was reduced from 24(80−4) = 2304 to 24(6−4) =

28. In this unrealistic setup, the effective transmission rate was

reduced from 3.80 to 1.33 [bit/symbol]. Since the brute-force

search with a step size of 10−64 is infeasible, we used the dual

annealing method [70] and the differential evolution method

[71].11 As shown in Fig. 6(c), Eve was able retrieve the same

information as Bob when she had the same CSI as him. When

Eve had no knowledge of CSI between Alice and Bob, and

used the attack algorithm with the PCSI between Alice and

Eve, she also succeeded in decoding 92.67% of information.

Here, the performance of the dual annealing outperformed that

of the differential evolution method. In summary, the proposed

scheme offers limited security when Eve has the same CSI as

Bob or has PCSI between her and Alice. The proposed scheme

does not transmit ﬁxed reference symbols and semi-unitary

space-time matrices, as described in Section V. Consequently,

it is difﬁcult for Eve to obtain accurate CSI, especially when

we consider high-mobility scenarios.

In Fig. 6(c), we observed a high error ﬂoor of BER =

7.33·10−2when Eve had PCSI and SNR →+∞. To elucidate

characteristics of this error ﬂoor, in Fig. 7, we calculated Eve’s

detection norm (49) at SNR = 100 [dB], where all the 28=

11Speciﬁcally, we used the corresponding functions

scipy.optimize.dual annealing and scipy.optimize.differential evolution

with the maximum iteration limit of Ng= 100.

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0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

Security gap [dB]

0.0

0.2

0.4

0.6

0.8

1.0

Eve’s channel reliability

ρ= 99.99%

ρ= 0.0%

Proposed (R= 4)

Proposed (R= 6)

Fig. 8. Eve’s channel reliability upon increasing the channel correlation ρ

and the security gap. The simulation parameters were the same as those used

in Fig. 6(a).

256 patterns for dwere considered. Ideally, the detection norm

(49) converges to zero at a high SNR. However, as shown

in Fig. 7, we observed many local optimal solutions, which

revealed that the optimization of (49) was not straightforward.

This ambiguity resulted from the high complexity of (49) and

its numerical errors. Speciﬁcally, the |Y|>1setup yields

multiple solutions and induces the phase ambiguity for Eve. As

a result, Eve selected the global optimal solution and obtained

incorrect bits, whereas Bob selected a sub-optimal solution

and obtained the correct bits. This promising result can be

expected in general.

Finally, in Fig. 8, we investigated the effects of Eve’s

channel correlation and the security gap, which is deﬁned

by SNRBob −SNREve in dB [74, 75]. The simulation pa-

rameters were fundamentally the same as those used for

Fig. 6(a), except for the correlation ρbetween the Alice–

Bob and Alice–Eve channel matrices. Speciﬁcally, Bob’s

channel model is deﬁned in (1), whereas Eve’s channel

model is deﬁned in (2). Here, the correlation coefﬁcient was

ρ= 99.99%,99.9%,99.0%,90.0%, and 0.0%. Because the

BER curves were difﬁcult to differentiate, we showed the

channel reliability instead. The channel reliability is deﬁned

as 1−2·BER. It is useful to calculate the channel capacity as

demonstrated in [76]. In the same manner as [75], we deﬁne

the BER ≥0.4region as safe. The channel reliability must

be less than 0.2. As presented in Fig. 8, the proposed scheme

having R= 4 exhibited high channel reliability at Eve and

required the security gap of about 11.1[dB] to reach the

safe region. By contrast, the proposed scheme having R= 6

required the security gap of about 3.0[dB] in the ρ≤90.0%

case. Its performance is comparable to those of conventional

PLS methods [74, 75].

VII. CON CL US IO NS

In this paper, we proposed the chaos-based differential

MIMO to alleviate the channel estimation overhead that would

help the eavesdropper obtain an exact full channel matrix.

Uniquely, the proposed scheme extracts an initial condition

of the pure chaos sequence from wireless nature, which is

the ﬁrst attempt in the literature. Due to the sensitivity to

initial conditions, conventional chaos-based communication

systems must exchange a common secret key in advance

with no exception. In our work, the extracted noisy key is

used to generate an artiﬁcially time-varying unitary matrix,

which obfuscates private data symbols. The keys extracted

respectively for each transmitter and receiver might become

different. To address this mismatch issue, we then proposed the

low-complexity calibration algorithm for the chaos sequence at

the receiver. Additionally, we conceived the brute-force attack

algorithm for the proposed scheme. Our security analysis

revealed that, because of its differential encoding structure,

this attack algorithm was much more complex than the existing

standard encryption method. Despite the fact that the proposed

scheme requires no channel estimation, it outperformed the

representative chaos-based scheme that had perfect channel

estimates. It was found that the proposed calibration algorithm

worked properly even if the extracted key contained non-

negligible errors. However, our proposed system was unable

to provide security if the eavesdropper had similar channel

coefﬁcients to a legitimate receiver, i.e., ρ≥99.99%, or if

the frame length was extremely short, such as W≤M+ 2.

Based on our analysis, we conclude that differential encod-

ing can achieve practical physical layer security in wireless

communications.

ACK NOW LE DG ME NT

The authors are indebted to the Editor and the anonymous

reviewers for their invaluable suggestions and comments,

which further improved this paper.

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Naoki Ishikawa (Member, IEEE) was born in Kana-

gawa, Japan, in 1991. He received the B.E., M.E.,

and Ph.D. degrees from the Tokyo University of

Agriculture and Technology, Tokyo, Japan, in 2014,

2015, and 2017, respectively. In 2015, he was an

academic visitor with the School of Electronics and

Computer Science, University of Southampton, UK.

From 2016 to 2017, he was a research fellow of

the Japan Society for the Promotion of Science.

From 2017 to 2020, he was an assistant professor

in the Graduate School of Information Sciences,

Hiroshima City University, Japan. Since 2020, he has been an Associate

Professor with the Faculty of Engineering, Yokohama National University,

Japan. His research interests include massive MIMO, physical layer security,

and quantum speedup for wireless communications. He was certiﬁed as an

Exemplary Reviewer of IEE E TR ANS ACT IO NS ON CO MM UNI CATI ONS 2017.

He received the Yasujiro Niwa Outstanding Paper Award from Tokyo Denki

University in 2018, the Telecom System Technology student Award (honorable

mention) from Telecommunications Advancement Foundation of Japan in

2014, and the Outstanding Paper Award for Young C&C Researchers from

NEC C&C Foundation in 2014.

Jehad M. Hamamreh (Member, IEEE) is the

Founder and Director of WISLAB, and A. Profes-

sor with the Electrical and Electronics Engineering

Department, Antalya Bilim University. He received

his Ph.D. degree in telecommunication engineering

and cyber systems from Istanbul Medipol Univer-

sity, Turkey, in 2018. Previously, he worked as

a Researcher at the Department of Electrical and

Computer Engineering at Texas A&M University.

He is the inventor of more than 20+ Patents and

an author of more than 75+ peer-reviewed scientiﬁc

papers along with several book chapters. His innovative patented works won

the gold, silver, and bronze medals by numerous international invention

contests and fairs.

His current research interests include wireless physical and MAC lay-

ers security, orthogonal frequency-division multiplexing and multiple-input

multiple-output systems, advanced waveforms design, multidimensional mod-

ulation techniques, and orthogonal/non-orthogonal multiple access schemes

for future wireless systems. He is a serial referee for various scientiﬁc journals

as well as a TPC member for several international conferences. He is an Editor

at Researcherstore, RS-OJICT journal, and Frontiers in Communications and

Networks.

Eiji Okamoto (Member, IEEE) received the B.E.,

M.S., and Ph.D. degrees in Electrical Engineering

from Kyoto University in 1993, 1995, and 2003,

respectively. In 1995 he joined the Communications

Research Laboratory (CRL), Japan. Currently, he

is an associate professor at Nagoya Institute of

Technology. In 2004 he was a guest researcher at

Simon Fraser University. He received the Young

Researchers’ Award in 1999 from IEICE, and the

FUNAI Information Technology Award for Young

Researchers in 2008. His current research interests

are in the areas of wireless technologies, satellite communication, and mobile

communication systems.

This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/

Journal of the Communications Society

Chao Xu (Senior Member, IEEE) received a MSc

degree with distinction in Radio Frequency Com-

munication Systems and a Ph.D. degree in Wireless

Communications from the University of Southamp-

ton, UK in 2009 and 2015, respectively. He is

currently a research fellow working at University

of Southampton, UK. His research interests include

index modulation, reduced-complexity MIMO de-

sign, noncoherent detection and turbo detection. He

was awarded the Best M.Sc. Student in Broadband

and Mobile Communication Networks by the IEEE

Communications Society (United Kingdom and Republic of Ireland Chapter)

in 2009. He also received 2012 Chinese Government Award for Outstanding

Self-Financed Student Abroad and 2017 Dean’s Award, Faculty of Physical

Sciences and Engineering, the University of Southampton.

Lixia Xiao (Member, IEEE) received the B.E., M.E.,

and Ph.D. degrees from the UESTC in 2010, 2013,

and 2017, respectively. From 2016 to 2017, she was

a visiting student with the School of Electronics

and Computer Science, University of Southampton.

From 2018 to 2019, she has been a Research Fellow

with the Department of Electrical Electronic En-

gineering, University of Surrey. She is currently a

Full Professor with the Wuhan National Laboratory

for Optoelectronics, Huazhong University of Science

and Technology. In particular, she is very interested

in signal detection and performance analysis of wireless communication

systems. Her research interests include wireless communications and com-

munication theory.

Efficient Secure NOMA Schemes Based on Chaotic Physical Layer Security for Wireless Networks

Article

Full-text available

Dec 2022

Non-orthogonal multiple access (NOMA) is considered by the 3GPP as a potential technology for beyond 5G wireless networks to support massive connectivity with robust reliability. However, the massive increase in connected users critically leads to data security problems owing to the high possibility of eavesdropping. In this paper, uplink secure NOMA (sNOMA) schemes based on chaotic physical layer security (PLS) are proposed to achieve twofold secrecy of integrated channel coding and secret key approaches. Different code-domain and power-domain techniques are employed for sNOMA designs over realistic fading channel environments. Equal power allocation strategy is used for the transmitted chaotic signals in code-domain sNOMA (CD-sNOMA), whereas dynamic power control is employed in power-domain sNOMA (PD-sNOMA) and the hybrid code-power-domain sNOMA (CPD-sNOMA) approaches. The former design adopts joint maximum Likelihood (ML) signal detection while the later scenarios utilize integrated receiver designs based on successive interference cancellation (SIC) and ML techniques. For the proposed sNOMA schemes, power control algorithms are presented to optimize the system performance over constrained total received power and target error rate. Numerical results validate the effectiveness of sNOMA designs compared with the benchmark systems under the worst-case secrecy of unauthorized receiver with complete knowledge of the transmission scheme and associated channel. Valuable tradeoffs are demonstrated between the achieved error rate, connectivity, security gap, and complexity. Moreover, the utilized chaotic signals offer robust and cost-effective PLS solutions with a huge key-space to combat the most powerful brute-force eavesdropping attacks.

Noncoherent Massive MIMO with Embedded One-Way Function Physical Layer Security

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We propose a novel physical layer security scheme that exploits an optimization method as a one-way function. The proposed scheme builds on nonsquare differential multiple-input multiple-output (MIMO), which is capable of noncoherent detection even in massive MIMO scenarios and thus resilient against risky pilot insertion and pilot contamination attacks. In contrast to conventional nonsquare differential MIMO schemes, which require space-time projection matrices designed via highly complex, discrete, and combinatorial optimization, the proposed scheme utilizes projection matrices constructed via low-complexity continuous optimization designed to maximize the coding gain of the system. Furthermore, using a secret key generated from the true randomness nature of the wireless channel as an initial value, the proposed continuous optimization-based projection matrix construction method becomes a one-way function, making the proposed scheme a physical layer secure differential MIMO system. An attack algorithm to challenge the proposed scheme is also devised, which demonstrate that the security level achieved improves as the number of transmit antennas increases, even in an environment where the eavesdropper can perfectly estimate channel coefficients and experiences asymptotically large signal-to-noise ratios.

Covert Communications Without Pre-Sharing of Side Information and Channel Estimation Over Quasi-Static Fading Channels

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We propose a new covert communication scheme that operates without pre-sharing side information and channel estimation, utilizing a Gaussian-distributed Grassmann constellation for noncoherent detection. By designing constant-amplitude symbols on the Grassmann manifold and multiplying them by random variables, we generate signals that follow an arbitrary probability distribution, such as Gaussian or skew-normal distributions. The mathematical property of the manifold enables the transmitter's random variables to remain unshared with the receiver, and the elimination of pilot symbols that could compromise covertness. The proposed scheme achieved higher covertness and achievable rates compared to conventional coherent Gaussian signaling schemes, without any penalty in terms of complexity.

Secure authentication in MIMO systems: exploring physical limits

Article

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May 2024

Multiple-input multiple-output (MIMO) technology is employed to improve the reliability and capacity of wireless communication systems. However, the wireless communication environment creates vulnerabilities to spoofing attacks. Furthermore, the authentication challenges posed by the heterogeneous characteristics of wireless applications increase as diverse technologies facilitate the growing number of Internet of Things (IoT) devices. To address these challenges, adaptive physical-layer authentication (PLA) leveraging the inherent antenna diversity in MIMO systems is examined, and an information-theoretic perspective on PLA in MIMO systems is given. The real and imaginary components of the received reference signals are used as attributes with a single-class classification support vector machine (SCC-SVM). It is shown that the authentication performance improves with the number of antennas, and the proposed scheme provides robust authentication.

Design of an M-Ary DLCSK Communication System Using Deep Transfer Learning

Article

Full-text available

Jan 2023

Conventional coherent chaos-based communication systems require synchronization of chaotic signals, which is still practically unattainable in a noisy environment. Moreover, in non-coherent schemes, a part of the bit duration is spent sending non-information-bearing reference samples, which deteriorates the Bit Error Rate performance (BER) of these systems. To tackle these problems, this paper designs an M-ary Deep Learning Chaos Shift Keying (M-ary DLCSK) system. The proposed receiver uses a Convolutional Neural Network (CNN)-based classifier that recovers M-ary modulated data. The trained NN model grasps different chaotic maps, estimates channels, and classifies the received signals effectively. Moreover, we consider a Transfer Learning (TL) framework that enhances the noise performance and classification results. Due to the generalization capabilities of TL, the trained NN can work in different Signal-to-Noise Ratio (SNR) conditions without the need for re-training. We compare the BER performance, complexity, and bandwidth efficiency of the M-ary DLCSK receiver with existing receivers. The results demonstrate that the M-ary DLCSK receiver is the first practical system that achieves the theoretical BER performance of the coherent CSK systems under Rayleigh fading channels. Moreover, the proposed system provides a considerable performance advantage compared to the existing DL-based receivers under Rayleigh fading channels. For example, the BER performance of 8-ary DLCSK shows a gain of 0.1 over the Long Short-Term Memory (LSTM)-aided DNN systems at the target Eb/N0=14dB. These features make M-ary DLCSK an attractive candidate for several applications, such as Massive Multiple-Input Multiple Output (MIMO), Vehicle-to-everything (V2X), Quantum, and optical communication systems.

Design of an M-ary DLCSK Communication System using Deep Transfer Learning

Article

Full-text available

Sep 2023

Conventional coherent chaos-based communication systems require synchronization of chaotic signals, which is still practically unattainable in a noisy environment. Moreover, in non-coherent schemes, a part of the bit duration is spent sending non-information-bearing reference samples, which deteriorates the Bit Error Rate performance (BER) of these systems. To tackle these problems, this paper designs an M-ary Deep Learning Chaos Shift Keying (M-ary DLCSK) system. The proposed receiver uses a Convolutional Neural Network (CNN)-based classifier that recovers M-ary modulated data. The trained NN model grasps different chaotic maps, estimates channels, and classifies the received signals effectively. Moreover, we consider a Transfer Learning (TL) framework that enhances the noise performance and classification results. Due to the generalization capabilities of TL, the trained NN can work in different Signal-to-Noise Ratio (SNR) conditions without the need for retraining. We compare the BER performance, complexity, and bandwidth efficiency of the M-ary DLCSK receiver with existing receivers. The results demonstrate that the M-ary DLCSK receiver is the first practical system that achieves the theoretical BER performance of the coherent CSK systems under Rayleigh fading channels. Moreover, the proposed system provides a considerable performance advantage compared to the existing DL-based receivers under Rayleigh fading channels. For example, the BER performance of 8-ary DLCSK shows a gain of 0.1 over the Long Short-Term Memory (LSTM)-aided DNN systems at the target E b /N 0 = 14dB. These features make M-ary DLCSK an attractive candidate for several applications, such as Massive Multiple-Input Multiple Output (MIMO), Vehicle-to-everything (V2X), Quantum, and optical communication systems.

Covert Communications Without Pre-Sharing of Side Information and Channel Estimation Over Quasi-Static Fading Channels

Article

Jan 2025

We propose a new covert communication scheme that operates without pre-sharing side information and channel estimation, utilizing a Gaussian-distributed Grassmann constellation for noncoherent detection. By designing constant-amplitude symbols on the Grassmann manifold and multiplying them by random variables, we generate signals that follow an arbitrary probability distribution, such as Gaussian or skew-normal distributions. The mathematical property of the manifold enables the transmitter’s random variables to remain unshared with the receiver, and the elimination of pilot symbols that could compromise covertness. The proposed scheme achieved higher covertness and achievable rates compared to conventional coherent Gaussian signaling schemes, without any penalty in terms of complexity.

Improved Chaotic Modulation Encrypting Method for Uplink Multi-User Massive MIMO Systems with Large-Scale Fading

Conference Paper

Aug 2024

A physical layer security scheme for 6G wireless networks using post-quantum cryptography

Article

Feb 2024
COMPUT COMMUN

Walid Abdallah

The sixth generation (6G) of mobile networks is poised to revolutionize communication capabilities with its infinite-like reach. These networks will feature an ultra-dense topology, accommodating a wide range of devices, from macro devices like satellites to nano-devices integrated within the human body. However, the extensive data traffic handled by 6G networks, a significant portion of which is sensitive in nature, presents a security challenge. This paper presents the design and implementation of an encryption scheme that ensures the confidentiality of data transmission in the physical layer of 6G networks. The proposed physical layer security architecture relies on lattice cryptography, wherein each user is associated with a pair of bases (a public basis and a private basis) and a set of orthogonal sub-carriers. The encryption process involves projecting the vector of the transmitted data’s quadrature amplitude modulation (QAM) symbols onto the user’s public basis. A random error vector is then added before applying the inverse Fourier transform of the orthogonal frequency division multiplexing (OFDM) technique. The security of this encryption scheme hinges on the complexity of the closest vector problem in an integer lattice. To evaluate its efficacy, we analyze the security of our lattice-based physical layer encryption concerning the properties of the base pairs and error vectors. Our findings indicate that the proposed scheme offers satisfactory security protection against eavesdropping attacks by significantly enhancing the confidentiality of the transmitted signal. Furthermore, we assess the performance of our design through numerical experiments, demonstrating its resilience against various security attacks.

Error Probability Analysis for Time-Varying Chaos Unitary Matrix-Based Differential MIMO System

Article

Jul 2022

The time-varying (TV) chaos unitary matrix based differential multiple-input multiple-output (D-MIMO) system is capable of achieving beneficial performance gains without channel state information (CSI). To elaborate a little further, the chaos sequence is embedded into the data-carrying matrices by a unique time-varying matrix. By employing differential coding during two adjacent signal blocks, a non-coherent detector can be developed without CSI knowledge. However, its theoretical performance analysis becomes challenging due to its complex construction. To address this issue, the theoretical average bit error probability (ABEP) lower and upper bounds of the TV-D-MIMO system are analyzed using the moment generating function (MGF) over generalized fading channel. The experimental results are shown that the derived ABEP upper and lower bounds are tight with the simulation results, which is helpful for future performance evaluation.

A New Frontier for IoT Security Emerging From Three Decades of Key Generation Relying on Wireless Channels

Article

Full-text available

Jul 2020

The Internet of Things (IoT) is a transformative technology, which is revolutionizing our everyday life by connecting everyone and everything together. The massive number of devices are preferably connected wirelessly because of the easy installment and flexible deployment. However, the broadcast nature of the wireless medium makes the information accessible to everyone including malicious users, which should hence be protected by encryption. Unfortunately the secure and efficient provision of cryptographic keys for low-cost IoT devices is challenging; weak keys have resulted in severe security breaches, as evidenced by numerous notorious cyberattacks. This paper provides a comprehensive survey of lightweight security solutions conceived for IoT, relying on key generation from wireless channels. We first introduce the key generation fundamentals and protocols. We then examine how to apply this emerging technique to secure IoT and demonstrate that key generation relying on the randomness of wireless channels is eminently suitable for IoT. This paper reviews the extensive research efforts in the areas of theoretical modelling, simulation based validation and experimental exploration. We finally discuss the hurdles and challenges that key generation is facing and suggest future work to make key generation a reliable and secure solution to safeguard the IoT.

Sixty Years of Coherent Versus Non-Coherent Tradeoffs and the Road From 5G to Wireless Futures

Article

Full-text available

Dec 2019

Sixty years of coherent versus non-coherent tradeoff as well as the twenty years of coherent versus non-coherent tradeoff in Multiple-Input Multiple-Output (MIMO) systems are surveyed. Furthermore, the advantages of adaptivity are discussed. More explicitly, in order to support the diverse communication requirements of different applications in a unified platform, the 5G New Radio (NR) offers unprecendented adaptivity, abeit at the cost of a substantial amount of signalling overhead that consumes both power and the valuable spectral resources. Striking a beneficial coherent versus non-coherent tradeoff is capable of reducing the pilot overheads of channel estimation, whilst relying on low-complexity detectors, especially in high-mobility scenarios. Furthermore, since energy-efficiency is of salient importance both in the operational and future networks, following the powerful Index Modulation (IM) pholosophy, we conceive a holistic adaptive pholosophy striking the most appropriate coherent/non-coherent, single-/multiple-antenna and diversity/multiplexing tradeoffs, where the number of RF chains, the Peak-to-Average Power Ratio (PAPR) of signal transmission and the maximum amount of interference tolerated by signal detection are all taken into account. We demonstrate that this intelligent tripple-fold adaptivity offers significant benefits in next-generation applications of mmWave and Terahertz solutions, in space-air-ground integrated networks, in full-duplex techniques and in other sophisticated channel coding assisted system designs, where powerful machine learning algorithms are expected to make autonomous decisions concerning the best mode of operation with minimal human intervention.

Light-Weight Physical Layer Enhanced Security Schemes for 5G Wireless Networks

Article

Full-text available

Sep 2019

Due to the broadcast nature of wireless radio propagation channels, 5G wireless networks face serious security threats. Security mechanisms that leverage physical layer characteristics have been considered as potential complements to enhance 5G wireless network security. This article proposes physical layer security enhancements to defend against security attacks from the three most significant aspects: wireless secure communication, physical layer assisted authentication and secret key distribution. By integrating physical layer security with novel techniques and application scenarios in 5G communication, we elaborate several promising strategies, including massive MIMO beamforming with security code communication, self-adaptive mobile and cooperative secure communication with dynamic channel prediction, physical layer assisted authentication, malicious node detection and key generation of massive MIMO millimeter wave system. Research opportunities and future challenges are further discussed.

IMToolkit: An Open-Source Index Modulation Toolkit for Reproducible Research Based on Massively Parallel Algorithms

Article

Full-text available

Jul 2019

Naoki Ishikawa

This paper presents a proposal of an open-source index modulation (IM) toolkit, which facilitates reproducible research and accelerates open innovation in IM studies. The proposed toolkit is implemented based on massively parallel algorithms that are designed for state-of-the-art graphics processing units (GPUs). Since high-performance GPUs are available at low cost, along with the intensive development in deep learning, this toolkit achieves large-scale but significantly fast Monte Carlo simulations at low cost. Two large-tensor-based parallel algorithms are introduced for bit error ratio and average mutual information simulations. Additionally, the design of active indices is newly formulated into an integer linear programming problem that guarantees optimality, which is applicable to the generalized spatial modulation and subcarrier-index modulation schemes. Performance comparisons demonstrated that the proposed GPU-aided algorithms were up to 145 times faster than the conventional CPU-aided efficient counterparts. Furthermore, the designed active indices achieved the theoretical optimum performance in contrast to widely used conventional methods. A comprehensive database of these designed active indices is released online and is available to any researcher.

Differential-Detection Aided Large-Scale Generalized Spatial Modulation Is Capable of Operating in High-Mobility Millimeter-Wave Channels

Article

Full-text available

Apr 2019

A large-scale differential-detection aided generalized spatial modulation (GSM) system is proposed, which relies on a novel Gram-Schmidt basis set and an adaptive low-complexity detector, and is evidently suitable for high-mobility millimeter-wave (mmWave) channels. We consider non-stationary time-varying mmWave channels and assume that the beam-angles remain relatively fixed, while the channel coefficients vary rapidly. In this scenario, it is a challenging task to find the accurate estimates of channel coefficients for digital beamforming, which becomes an even more severe problem, as the numbers of subarrays and subcarriers increase. Our analog-beamforming-aided nonsquare differentially-detected scheme achieves a higher transmission rate than the conventional coherent multiple-input multiple-output schemes because the pilot overhead and the complex-valued feedback are eliminated. Our simulation results following the IEEE 802.11ad specifications show that the performance of our proposed nonsquare differential GSM improved upon increasing the number of subarrays, where the maximum transmission rate of 16 [bps/Hz] was considered.

Wireless Secret Key Generation for Distributed Antenna Systems: A Joint Space-Time-Frequency Perspective

Article

May 2021

Wireless secret key generation has emerged as a promising technique for Internet of Things (IoT) systems to establish shared encryption keys between the server and legitimate mobile user. This paper focuses on the use of multi-domain joint information to achieve a high key generation rate (KGR) and the implementation of a reliable, low-complexity secret key generation mechanism for distributed antenna systems (DAS) with orthogonal-frequency division multiplexing (OFDM). We present a space-time-frequency channel state information (CSI)-based key generation scheme based on a two-step approach of adaptive link selection and stepwise decorrelation algorithms. The performance is evaluated in terms of KGR, key disagreement rate (KDR), randomness, and computational complexity by using both a standardized channel model and real-world measurements. Numerical results show that our proposed low-complexity algorithms effectively utilize the space-time-frequency CSI to multiply the KGR in both indoor and outdoor environments. Through adaptive link selection in DAS, the KDR is maintained within a correctable range, thereby ensuring the validity of generated keys in dynamic environments. Further applying stepwise decorrelation reduces the computational complexity by more than half while satisfying all eight key generation randomness tests in the NIST test suite.

Artificial Noise-Aided MIMO Physical Layer Authentication with Imperfect CSI

Article

Jan 2021

Fingerprint embedding at the physical layer is a highly tunable authentication framework for wireless communication that achieves information-theoretic security by hiding a traditional HMAC tag in noise. In a multiantenna scenario, artificial noise (AN) can be transmitted to obscure the tag even further. The AN strategy, however, relies on perfect knowledge of the channel state information (CSI) between the legitimate users. When the CSI is not perfectly known, the added noise leaks into the receiver’s observations. In this paper, we explore whether AN still improves security in the fingerprint embedding authentication framework with only imperfect CSI available at the transmitter and receiver. Specifically, we discuss and design detectors that account for AN leakage and analyze the adversary’s ability to recover the key from observed transmissions. We compare the detection and security performance of the optimal perfect CSI detector with the imperfect CSI robust matched filter test and a generalized likelihood ratio test (GLRT). We find that utilizing AN can greatly improve security, but suffers from diminishing returns when the quality of CSI knowledge is poor. In fact, we find that in some cases allocating additional power to AN can begin to decrease key security.

Differentially-Encoded Rectangular Spatial Modulation Approaches the Performance of Its Coherent Counterpart

Article

Sep 2020

A simplified rectangular differential spatial modulation (S-RDSM) scheme is conceived for massive multiple-input multiple-output (MIMO) systems dispensing with the channel state information (CSI). In the proposed S-RDSM scheme, the information bits are first mapped to a conventional SM symbol and then rectangular differential encoding is invoked between a pair of SM symbols. Then a non-coherent detector relying on a forgetting factor is developed, which requires no CSI at the receiver. Explicitly, a low-complexity hard limited maximum likelihood (HL-ML) detector is conceived for our generalized SRDSM scheme, which is characterized by our theoretical analysis. Furthermore, we derive the optimal forgetting factor in closed form, which is capable of significantly reducing the complexity of the associated optimization. Finally, the upper bounds of the average bit error probability (ABEP) are derived using the moment generating function (MGF), and are validated by our simulation results. Both the theoretical and simulation results have shown that the proposed S-RDSM system outperforms the existing non-coherent schemes, despite operating at 10% of the benchmarker’s complexity, whilst approaching the performance of its coherent SM counterpart at a comparable complexity.

On the Relationships Between Average Channel Capacity, Average Bit Error Rate, Outage Probability, and Outage Capacity Over Additive White Gaussian Noise Channels

Article

Feb 2020

Ferkan Yilmaz

In the theory of wireless communications, average performance measures (APMs) are widely utilized to quantify the performance gains / impairments in various fading environments under various scenarios, and to comprehend how the factors arising from design/implementation affect system performance. To the best of our knowledge, it has not been yet discovered in the literature how these APMs relate to each other. In this article, having been inspired by the work of Verdu et al. [1], we propose that one APM can be calculated using the other APMs instead of using the end-to-end SNR distribution. Particularly, using the Lamperti’s transformation (LT), we propose a tractable approach, which we call LT-based APM analysis, to identify a relationship between any two given APMs such that it is irrespective of SNR distribution. Thereby, we introduce some novel relationships among average channel capacity (ACC), average bit error rate (ABER) and outage probability / capacity (OP / OC) performances, and accordingly present how to obtain ACC from ABER performance and how to obtain OP / OC from ACC performance in fading environments. We demonstrate that the ACC of any communications system can be evaluated empirically without using end-to-end SNR distribution. We consider some numerical examples and simulations to validate our newly derived relationships.

Space-Time Block Coded Rectangular Differential Spatial Modulation: System Design and Performance Analysis

Article

May 2019

In this paper, a novel scheme dubbed space-time block coded rectangular differential spatial modulation (STBC-RDSM) is proposed for multiple-input and multiple-out (MIMO) systems, which combines space-time block coding (STBC) and rectangular differential spatial modulation (RDSM) to reap their respective benefits, while avoiding the drawbacks of conventional differential spatial modulation (DSM) systems. More specifically, in the proposed STBC-RDSM scheme, information bits are conveyed via the rectangular differentially encoded antenna index matrices, as well as the STBC blocks. Furthermore, a low-complexity detection scheme is proposed. Our simulation results demonstrate that STBC-RDSM outperforms its conventional DSM counterparts in various spectral efficiencies. Finally, a closed-form union bound on the bit error rate (BER) is derived and validated by our simulation results.

Artificially Time-Varying Differential MIMO for Achieving Practical Physical Layer Security

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