Influence of Post-Processing Techniques on Random Number Generation Using Chaotic Nanolasers

Abstract

In this paper, we propose using a chaotic system composed of nanolasers (NLs) as a physical entropy source. Combined with post-processing technologies, this system can produce high-quality physical random number sequences. We investigated the parameter range for achieving time-delay signature (TDS) concealment in the chaotic system. This study demonstrates that NLs exhibit noticeable TDS only under optical feedback. As mutual injection strength between the master NLs (MNLs) increases, the TDS of the MNLs is gradually suppressed until they are completely concealed. Compared to MNLs, the slave NL (SNL) exhibits better TDS suppression performance. Additionally, we investigated the chaotic and highly unpredictable regions of the SNL, demonstrating that high-quality chaotic signals can be produced over a wide range of parameters. Using TDS hidden and highly unpredictable chaotic signals as the source of random entropy, the effects of different post-processing techniques on random number extraction were compared. The results indicate that effective post-processing can enhance the unpredictability of the random sequence. This study successfully utilized NLs for random number generation, showcasing the potential and application prospects of NLs in the field of random numbers.

Full text

Citation: Zhao, J.; Liu, G.; Li, R.; Mu,
P. Influence of Post-Processing
Techniques on Random Number
Generation Using Chaotic Nanolasers.
Electronics 2024,13, 2712. https://
doi.org/10.3390/electronics13142712
Academic Editors: Abdelali El Aroudi,
Costas Psychalinos, Esteban Tlelo-
Cuautle and Ahmed S. Elwakil
Received: 31 May 2024
Revised: 26 June 2024
Accepted: 9 July 2024
Published: 11 July 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
electronics
Article
Influence of Post-Processing Techniques on Random Number
Generation Using Chaotic Nanolasers
Jing Zhao 1, * , Guopeng Liu 2, Rongkang Li 2and Penghua Mu 2, *
1School of Network & Communication Engineering, Chengdu Technological University,
Chengdu 611730, China
2School of Physics and Electronic Information, Yantai University, Yantai 264005, China;
liuguopeng@s.ytu.edu.cn (G.L.); lirongkang1228@s.ytu.edu.cn (R.L.)
*Correspondence: zjing4@cdtu.edu.cn (J.Z.); ph_mu@ytu.edu.cn (P.M.)
Abstract: In this paper, we propose using a chaotic system composed of nanolasers (NLs) as a
physical entropy source. Combined with post-processing technologies, this system can produce
high-quality physical random number sequences. We investigated the parameter range for achieving
time-delay signature (TDS) concealment in the chaotic system. This study demonstrates that NLs
exhibit noticeable TDS only under optical feedback. As mutual injection strength between the master
NLs (MNLs) increases, the TDS of the MNLs is gradually suppressed until they are completely
concealed. Compared to MNLs, the slave NL (SNL) exhibits better TDS suppression performance.
Additionally, we investigated the chaotic and highly unpredictable regions of the SNL, demonstrating
that high-quality chaotic signals can be produced over a wide range of parameters. Using TDS hidden
and highly unpredictable chaotic signals as the source of random entropy, the effects of different
post-processing techniques on random number extraction were compared. The results indicate
that effective post-processing can enhance the unpredictability of the random sequence. This study
successfully utilized NLs for random number generation, showcasing the potential and application
prospects of NLs in the field of random numbers.
Keywords: nanolaser; chaotic signal; time delay signature; random number
1. Introduction
Random numbers play a crucial role in fields such as cryptography [
1
], data simula-
tion [
2
], and secure communications [
3
–
9
]. They can be categorized into two types: true
random numbers [
10
,
11
] and pseudo random numbers [
12
,
13
]. True random numbers,
also known as physical random numbers, are generated through physical phenomena.
Their generation process is unpredictable and relies entirely on the randomness of nature.
Therefore, true random numbers possess characteristics of being unpredictable, being
non-repetitive, and having high entropy, making them superior in applications requiring
high security. However, their generation rate is relatively slow, which may not meet the
demands of certain high-frequency applications. Additionally, changes in the physical
environment could affect the reliability of random number generators (RNGs). Pseudo
random numbers are generated by deterministic algorithms. Their sequence exhibits sta-
tistical properties resembling randomness, and they are widely used in computer science
and statistics. Common pseudo random number generation algorithms include linear
congruential generators, the Mersenne Twister algorithm, and the Blum Blum Shub gen-
erator. Pseudo random number generation is fast and suitable for real-time applications.
However, pseudo RNGs have finite periods, which can lead to repetition. Additionally,
their deterministic algorithms are vulnerable to being cracked, making them unsuitable for
secure applications. In practical applications, there is a significant demand for RNGs that
are fast, convenient, have high randomness, and have high security. This poses a major
challenge for RNG development.
Electronics 2024,13, 2712. https://doi.org/10.3390/electronics13142712 https://www.mdpi.com/journal/electronics

Electronics 2024,13, 2712 2 of 13
In recent years, chaos has been widely applied in high-speed physical random number
generation [
14
–
17
]. Chaos systems have characteristics such as high sensitivity to initial
conditions, unpredictability, and ergodicity. These features make them a potential tool for
generating high-quality random numbers. Semiconductor lasers (SLs) are the most practical
and important type of lasers, and typically have stable output. However, when subjected to
external perturbations they exhibit rich nonlinear dynamic characteristics, including steady
state, periodicity, period doubling, and chaos. As research on chaotic lasers progresses,
many methods have been proposed by researchers for generating random numbers using
chaotic lasers. Uchida et al. utilized the chaotic characteristics of SLs to achieve high-speed
random number generation through optical feedback mechanisms [18]. They detailed the
implementation principle of this method, the generation rate, and experimental results
regarding the quality of the generated random numbers. This research has attracted
attention from research groups worldwide, sparking a surge of interest in generating high-
speed random numbers using chaotic lasers. Reidler et al. achieved ultrafast random
number generation by optimizing the feedback mechanism and sampling techniques of
the SL [
19
]. They also conducted detailed studies on the relationship between the random
number generation rate and SL parameters. Li et al. investigated a SL with distributed
feedback from a fiber Bragg grating for random bit generation, achieving output rates
ranging from 0.3 to 100 Gbit/s [
20
]. Xiang et al. constructed a ring network composed of
three SLs and used it for random number generation. By linearly combining the outputs of
three SLs, seven channels of chaotic entropy sources can be obtained. With minimal post-
processing techniques introduced, seven-channel random bit outputs can be achieved [
21
].
The currently confirmed techniques for generating chaotic light include optical feed-
back [
22
,
23
], optical injection [
24
,
25
], and optoelectronic feedback [
26
]. Compared to optical
injection and optoelectronic feedback, optical feedback structure has the advantages of
being simple in design, being low cost, and having rich dynamics. However, chaotic
signals outputted by SLs under optical feedback exhibit notable time-delay signature (TDS).
By analyzing TDS, it may be possible to infer the operational state of chaotic SLs or the
generation patterns of random numbers, thereby compromising randomness. Therefore,
researchers have proposed various schemes to suppress TDS. Shore et al. studied the TDS
of chaotic signals in three cascaded vertical-cavity surface-emitting lasers, analyzing the
parameter range of frequency detuning for achieving TDS hiding in this system [
27
]. Wu
et al. proposed a mutually coupled SL system where modulating the coupling strength and
frequency detuning simultaneously generates two chaotic signals with hidden TDS [
28
].
Nguimdo et al. investigated the possibility of hiding TDS in semiconductor ring lasers
(SRLs), achieving promising results [
29
]. Our research group investigated the TDS and
bandwidth of a chaotic system composed of three cascaded SRLs. The study showed that
the cascaded coupling scheme can enhance the dynamical characteristics of the lasers [
30
].
In recent years, novel semiconductor nanolasers (NLs) have become a focal point
of research in the field of optoelectronics. Their unique structures and characteristics
have shown extensive potential across multiple application domains. NLs provide new
possibilities for various application scenarios due to their small size, high performance,
and high-density integration. These attributes make NLs crucial components in many
modern optoelectronic and nanotechnology applications. Currently, there are numerous
reports on the fabrication of various NL structures, as well as the analysis of their nonlinear
dynamics and characteristics. When analyzing the nonlinear dynamics of conventional SLs,
spontaneous emission is often ignored. Considering the microstructure of NLs, Erwin et al.
introduced the Purcell factor
F
and the spontaneous emission coupling factor
β
to describe
the dynamics of NLs influenced by spontaneous emission [
31
]. Satter et al. conducted
studies on the nonlinear dynamic behaviors of NLs under optical feedback, phase-conjugate
feedback, and optical injection, evaluating the impact of relevant parameters on NLs’
dynamic behavior [
32
–
35
]. Han et al. constructed a mutually coupled NL system and
analyzed its dynamic characteristics. They pointed out that the system can maintain
steady-state output at higher mutual injection intensities. Additionally, they conducted an

Electronics 2024,13, 2712 3 of 13
analysis of the dynamic behavior under the combined effects of optical feedback and optical
injection [
36
–
38
]. Xiang et al. investigated the dynamic characteristics of NLs with double
chaotic optical injection, which can output signals with TDS hidden over a wide range
of parameters [
39
]. Our research group investigated the dynamic behavior of mutually
coupled NL systems with open-loop, semi-open-loop, and closed-loop structures [
40
].
Furthermore, we conducted a detailed study on the TDS hiding capability of the closed-loop
mutually coupled NL system, specifically analyzing the impact of parameter mismatches
on chaos synchronization [
41
]. Our research group has also proposed RNGs based on NL
systems in terms of applications [
42
,
43
]. This includes studying the nonlinear dynamic
characteristics of chaotic entropy sources and generating random sequences that have
passed randomness verification. The studies explored the dynamic characteristics of
different systems, laying the foundation for chaos-based applications. However, there is
still a significant amount of exploration space remaining in the study of NLs.
This article proposes a scheme for generating random numbers utilizing a chaotic
system composed of three NLs as a random entropy source. The rest of the article’s structure
is as follows. Section 2presents the theoretical model of the chaotic system and two different
post-processing schemes. In Section 3, we first investigated the parameter ranges for
achieving TDS concealment in the master NLs (MNLs) and slave NL (SNL). Moreover, we
utilized the 0–1 chaos test and permutation entropy (PE) to study the chaotic regions and
highly unpredictable regions of the SNL output. Next, we fixed the parameters to ensure
the output of chaotic signals with TDS concealment and high complexity and studied the
impact of different post-processing methods on random number extraction. In Section 4,
the research results are summarized. The chaotic system effectively achieves concealed TDS
and demonstrates high unpredictability. Combining post-processing techniques enables
the generation of random numbers passing NIST tests. Moreover, effective post-processing
can enhance the randomness of the random sequence.
2. Theoretical Model
Figure 1shows the structure of the chaotic entropy source based on NLs. Two MNLs
achieve chaotic output through optical feedback and mutual injection, and their outputs
are simultaneously injected into the SNL.
Electronics 2024, 13, x FOR PEER REVIEW 3 of 14
conjugate feedback, and optical injection, evaluating the impact of relevant parameters on
NLs’ dynamic behavior [32–35]. Han et al. constructed a mutually coupled NL system and
analyzed its dynamic characteristics. They pointed out that the system can maintain
steady-state output at higher mutual injection intensities. Additionally, they conducted
an analysis of the dynamic behavior under the combined effects of optical feedback and
optical injection [36–38]. Xiang et al. investigated the dynamic characteristics of NLs with
double chaotic optical injection, which can output signals with TDS hidden over a wide
range of parameters [39]. Our research group investigated the dynamic behavior of mu-
tually coupled NL systems with open-loop, semi-open-loop, and closed-loop structures
[40]. Furthermore, we conducted a detailed study on the TDS hiding capability of the
closed-loop mutually coupled NL system, specifically analyzing the impact of parameter
mismatches on chaos synchronization [41]. Our research group has also proposed RNGs
based on NL systems in terms of applications [42,43]. This includes studying the nonlinear
dynamic characteristics of chaotic entropy sources and generating random sequences that
have passed randomness verification. The studies explored the dynamic characteristics of
different systems, laying the foundation for chaos-based applications. However, there is
still a significant amount of exploration space remaining in the study of NLs.
This article proposes a scheme for generating random numbers utilizing a chaotic
system composed of three NLs as a random entropy source. The rest of the article’s struc-
ture is as follows. Section 2 presents the theoretical model of the chaotic system and two
different post-processing schemes. In Section 3, we first investigated the parameter ranges
for achieving TDS concealment in the master NLs (MNLs) and slave NL (SNL). Moreover,
we utilized the 0–1 chaos test and permutation entropy (PE) to study the chaotic regions
and highly unpredictable regions of the SNL output. Next, we fixed the parameters to
ensure the output of chaotic signals with TDS concealment and high complexity and stud-
ied the impact of different post-processing methods on random number extraction. In Sec-
tion 4, the research results are summarized. The chaotic system effectively achieves con-
cealed TDS and demonstrates high unpredictability. Combining post-processing tech-
niques enables the generation of random numbers passing NIST tests. Moreover, effective
post-processing can enhance the randomness of the random sequence.
2. Theoretical Model
Figure 1 shows the structure of the chaotic entropy source based on NLs. Two MNLs
achieve chaotic output through optical feedback and mutual injection, and their outputs
are simultaneously injected into the SNL.
Figure 1. Structure of the chaotic entropy source based on NLs.
The dynamical behaviors of MNLs and SNL can be derived from the NL rate equa-
tions based on the optical feedback and optical injection structures, as follows [33,35]:
Figure 1. Structure of the chaotic entropy source based on NLs.
The dynamical behaviors of MNLs and SNL can be derived from the NL rate equations
based on the optical feedback and optical injection structures, as follows [33,35]:
dIM1,2(t)
dt =ΓhFβNM1,2 (t)
τn+gn(NM1,2(t)−N0)
1+εIM1,2(t)IM1,2 (t)i−1
τpIM1,2(t)
+2kd1,2 pIM1,2(t)IM2,1 (t−τd2,1)cos(θ1,2 (t)) + 2kr1,2 pIM1,2 (t)IM2,1(t−τr2,1)cos(θ3,4 (t)) (1)
dϕM1,2(t)
dt =α
2Γgn(NM1,2(t)−Nth )−kd1,2 pIM2,1(t−τd2,1)
pIM1,2(t)sin(θ1,2 (t)) −kr1,2 pIM2,1 (t−τr2,1)
pIM1,2(t)sin(θ3,4 (t)) (2)

Electronics 2024,13, 2712 4 of 13
dNM1,2 (t)
dt =Idc
eVa
−NM1,2(t)
τn
(Fβ+1−β)−gn(NM1,2(t)−N0)
1+εIM1,2(t)IM1,2 (t)(3)
dIS(t)
dt =ΓhFβNS(t)
τn+gn(NS(t)−N0)
1+εIS(t)IM1,2(t)i−1
τpIS(t)
+2kr3,4pIS(t)IS(t−τr3,4)cos(θ5,6(t)) (4)
dϕS(t)
dt =α
2Γgn(NS(t)−Nth)−kr3 pIS(t−τr3)
pIS(t)sin(θ5(t)) −kr4pIS(t−τr4)
pIS(t)sin(θ6(t)) (5)
θ1,2(t) = 2πfM1,2 τd1,2 +ϕM1,2(t)−ϕM1,2 (t−τd1,2)(6)
θ3,4(t) = 2πfM2,1 τr2,1 +ϕM1,2(t)−ϕM2,1(t−τr2,1)±2π∆f1,2 t(7)
θ5,6(t) = 2πfM1,2 τr3,4 +ϕS(t)−ϕM1,2(t−τr3,4)−2π∆f3,4t(8)
Here, the subscripts
′M′
1
,
′M′
2
,
and ′S′
represent MNL1, MNL2, and SNL, respec-
tively.
I(t)
,
N(t)
, and
ϕ(t)
represent the photon density, carrier density, and phase, respec-
tively.
∆f1,2 =fM1,2 −fM2,1
represents the frequency detuning between the MNLs, and
∆f3,4 =fM1,2 −fSrepresents the frequency detuning between the MNLs and the SNL.
In Equations (1) and (2), the penultimate term represents the optical feedback compo-
nent.
kd1,2
represent the feedback rate, and
τd1,2
denote feedback delay of the two feedback
paths. kd1,2 can be expressed as follows [33]:
kd1,2 =f(1−R)rRext
R
c
2nL (9)
In Equations (1) and (2), the last term represents the mutual injection between MNLs,
while the last two terms in Equations (4) and (5) denote the injection of the two MNLs into
the SNL, respectively. kr1,2,3,4 can be written as follows [35]:
kr1,2,3,4 = (1−R)rRinj
R
c
2nL (10)
The introduction of the autocorrelation function (ACF) quantifies the TDS. The time-
varying photon density ACF can be expressed as described in [
44
–
46
], where
I(t)
represents
the time series, and ∆tis the time shift, as follows:
CM1,2|S(∆t) = DhIM1,2|S(t+∆t)−DIM1,2|S(t+∆t)EihIM1,2|S(t)−DIM1,2|S(t)EiE
shIM1,2|S(t+∆t)−DIM1,2|S(t+∆t)Ei2hIM1,2|S(t)−DIM1,2|S(t)Ei2(11)
Table 1lists the parameter settings used for simulation [
33
,
35
], while the remaining
parameters will be provided in the following sections.
RNGs based on chaotic lasers mainly consists of three parts: a chaotic laser entropy
source, acquisition devices, and post-processing. Extracting random numbers from the
chaotic laser entropy source requires necessary acquisition devices. Typically, photodetec-
tors (PDs) are used to convert chaotic light signals into electrical signals. Subsequently,
these electrical signals are quantized using a 1-bit or multi-bit analog-to-digital converter
(ADC) to generate random binary numbers. The raw signals generated by the chaotic
entropy source may contain some biases and correlations, necessitating post-processing
techniques to improve the quality and randomness of the generated random numbers.
We proposed two different post-processing schemes, as shown in Figure 2, and explored
the influence of different post-processing approaches on extracting random sequences.
In Scheme 1, the random sequence is obtained by directly extracting the least significant
bits (LSBs) from the binary sequence quantized by an 8-bit ADC. Scheme 2 introduces
exclusive-OR (XOR) operation, combining the unshifted sequence with the shifted sequence

Electronics 2024,13, 2712 5 of 13
using XOR to form a new sequence. The final random sequence is obtained by retaining
the LSBs.
Table 1. The parameter values set for NLs in the numerical simulation.
Parameter Description Value
λWavelength of MNL 1591 nm
LCavity Length 1.39 um
VaVolume of Active Region 3.96 ×10−13 cm3
ΓMode Confinement 0.645
gnDifferential Gain 1.65 ×10−6cm3/s
τpPhoton Lifetime 0.36 ps
τd1,2 Feedback Delay 0.2 ns
τnCarrier Lifetime 1 ns
N0Transparency Carrier Density 1.1 ×1018 cm−3
εGain Saturation Factor 2.3 ×10−17 cm3
nRefractive Index 3.4
αLinewidth Enhancement Factor 5
Rext External Facet Power Reflectivity 0.95
RLaser Facet Reflectivity 0.85
cSpeed of Light in Free Space 3×108m/s
Rinj Injection Parameter 0–0.1
fFeedback Coupling Fraction 0–0.9
Electronics 2024, 13, x FOR PEER REVIEW 6 of 14
Figure 2. Block diagram of RNG based on chaotic signals, Scheme 1 (a) and Scheme 2 (b); PD: pho-
todetector; ADC: analog-to-digital convertor; m-LSBs: m least significant bits; XOR: exclusive-OR.
3. Results and Discussion
The rate equations of the chaotic entropy source are numerically solved using the
Fourth-Order Runge–Kua algorithm. Due to the presence of TDS, there may be certain
periodicity or correlation in the random number sequence, thus affecting the randomness
of the random numbers. Hiding the TDS of the chaotic entropy source can improve the
quality and security of the generated random number sequences, making it more suitable
for various applications with high requirements for randomness. Therefore, achieving
TDS hiding in the chaotic system is the focus of this study. Next, we will explore the TDS
of the MNLs and the SNL separately.
To analyze the concealment performance of TDS, we use the ACF for analysis. The
smaller the ACF value, the beer the TDS concealment performance. When it is less than
0.2, it is generally considered successful in suppressing TDS. Based on previous research,
it was found that NLs only under optical feedback cannot entirely conceal the TDS. There-
fore, we explore the influence of increasing mutual injection strength on TDS concealment
of MNLs under fixed feedback parameters. Under the fixed feedback parameter 𝑓=0.02
we compared the time series and ACF of the output from MNLs when the mutual injection
strengths were 0 ns and 100 ns. Figure 3a depicts that, when the mutual injection
strength is 0 ns, corresponding to the case where MNLs are only under optical feed-
back, there is a clear TDS. Figure 3b illustrates that, when the injection strength increases
to 100 ns, the oscillation range of the chaotic signal expands, and the TDS is signifi-
cantly suppressed but still noticeable. This indicates that larger injection strength may
help hide the TDS of MNLs.
Figure 2. Block diagram of RNG based on chaotic signals, Scheme 1 (a) and Scheme 2 (b); PD:
photodetector; ADC: analog-to-digital convertor; m-LSBs: m least significant bits; XOR: exclusive-OR.
3. Results and Discussion
The rate equations of the chaotic entropy source are numerically solved using the
Fourth-Order Runge–Kutta algorithm. Due to the presence of TDS, there may be certain
periodicity or correlation in the random number sequence, thus affecting the randomness
of the random numbers. Hiding the TDS of the chaotic entropy source can improve the
quality and security of the generated random number sequences, making it more suitable
for various applications with high requirements for randomness. Therefore, achieving TDS
hiding in the chaotic system is the focus of this study. Next, we will explore the TDS of the
MNLs and the SNL separately.
To analyze the concealment performance of TDS, we use the ACF for analysis. The
smaller the ACF value, the better the TDS concealment performance. When it is less than
0.2, it is generally considered successful in suppressing TDS. Based on previous research, it
was found that NLs only under optical feedback cannot entirely conceal the TDS. Therefore,
we explore the influence of increasing mutual injection strength on TDS concealment of
MNLs under fixed feedback parameters. Under the fixed feedback parameter
f=
0.02 we
compared the time series and ACF of the output from MNLs when the mutual injection
strengths were 0
ns−1
and 100
ns−1
. Figure 3a depicts that, when the mutual injection
strength is 0
ns−1
, corresponding to the case where MNLs are only under optical feedback,
there is a clear TDS. Figure 3b illustrates that, when the injection strength increases to
100
ns−1
, the oscillation range of the chaotic signal expands, and the TDS is significantly

Electronics 2024,13, 2712 6 of 13
suppressed but still noticeable. This indicates that larger injection strength may help hide
the TDS of MNLs.
Electronics 2024, 13, x FOR PEER REVIEW 7 of 14
Figure 3. The MNL outputs in time domains (a1,b2) and ACF (a2,b2) under different mutual injec-
tion strengths. (a) 𝑘 =𝑘
 =0 ns
 and (b)
𝑘 =𝑘
 = 100 ns.
Next, we analyzed in detail the variation trend of the ACF peak values of the MNLs
and the SNL as the mutual injection intensity increases. We fixed the feedback parameter
at 𝑓 = 0.02 and set the injection intensity from the MNLs to the SNL at 100 ns. From
Figure 4, it can be observed that, when the mutual injection strength is 0 ns, the MNLs
exhibit significant TDS. By increasing the mutual injection strength, the ACF peak values
of the MNLs gradually decrease until they are less than 0.2. However, the ACF peak values
of the SNL were much lower than those of the MNLs. When the mutual injection strength
between the MNLs exceeds 40 ns, the ACF peak value of the SNL is already less than
0.2. In summary, although the TDS of the MNLs is gradually suppressed as the mutual
injection intensity increases, the SNL achieves TDS concealment over a wide range of mu-
tual injection intensities. Therefore, compared to the MNLs, the SNL can beer conceal
the TDS.
Figure 4. The trend of the ACF peak values of the MNLs and the SNL, with varying mutual injection
strength between the MNLs.
To further analyze the TDS concealment performance of the SNL, we studied the
trend of the peak values of the ACF of the SNL with varying injection strengths from the
Figure 3. The MNL outputs in time domains (a1,b1) and ACF (a2,b2) under different mutual injection
strengths. (a)kr1=kr2=0 ns−1and (b)kr1=kr2=100 ns−1.
Next, we analyzed in detail the variation trend of the ACF peak values of the MNLs
and the SNL as the mutual injection intensity increases. We fixed the feedback parameter
at
f=
0.02 and set the injection intensity from the MNLs to the SNL at 100
ns−1
. From
Figure 4, it can be observed that, when the mutual injection strength is 0
ns−1
, the MNLs
exhibit significant TDS. By increasing the mutual injection strength, the ACF peak values
of the MNLs gradually decrease until they are less than 0.2. However, the ACF peak values
of the SNL were much lower than those of the MNLs. When the mutual injection strength
between the MNLs exceeds 40
ns−1
, the ACF peak value of the SNL is already less than
0.2. In summary, although the TDS of the MNLs is gradually suppressed as the mutual
injection intensity increases, the SNL achieves TDS concealment over a wide range of
mutual injection intensities. Therefore, compared to the MNLs, the SNL can better conceal
the TDS.
Electronics 2024, 13, x FOR PEER REVIEW 7 of 14
Figure 3. The MNL outputs in time domains (a1,b2) and ACF (a2,b2) under different mutual injec-
tion strengths. (a) 𝑘 =𝑘
 =0 ns
 and (b)
𝑘 =𝑘
 = 100 ns.
Next, we analyzed in detail the variation trend of the ACF peak values of the MNLs
and the SNL as the mutual injection intensity increases. We fixed the feedback parameter
at 𝑓 = 0.02 and set the injection intensity from the MNLs to the SNL at 100 ns. From
Figure 4, it can be observed that, when the mutual injection strength is 0 ns, the MNLs
exhibit significant TDS. By increasing the mutual injection strength, the ACF peak values
of the MNLs gradually decrease until they are less than 0.2. However, the ACF peak values
of the SNL were much lower than those of the MNLs. When the mutual injection strength
between the MNLs exceeds 40 ns, the ACF peak value of the SNL is already less than
0.2. In summary, although the TDS of the MNLs is gradually suppressed as the mutual
injection intensity increases, the SNL achieves TDS concealment over a wide range of mu-
tual injection intensities. Therefore, compared to the MNLs, the SNL can beer conceal
the TDS.
Figure 4. The trend of the ACF peak values of the MNLs and the SNL, with varying mutual injection
strength between the MNLs.
To further analyze the TDS concealment performance of the SNL, we studied the
trend of the peak values of the ACF of the SNL with varying injection strengths from the
Figure 4. The trend of the ACF peak values of the MNLs and the SNL, with varying mutual injection
strength between the MNLs.

Electronics 2024,13, 2712 7 of 13
To further analyze the TDS concealment performance of the SNL, we studied the trend
of the peak values of the ACF of the SNL with varying injection strengths from the MNLs.
This includes scenarios where the MNL output signals exhibit significant TDS and where
TDS is completely suppressed. From Figure 4, it is evident that there is a noticeable TDS
when the mutual injection strength between the MNLs is 80
ns−1
, while, when the mutual
injection strength is 150
ns−1
, the TDS is completely suppressed. From Figure 5, it can be
observed that, when the MNLs exhibit significant TDS, the value of the ACF drops below
0.2 when the injection strength from the MNLs to the SNL exceeds 20
ns−1
. When the TDS
of the MNLs is significantly suppressed, the ACF remains below 0.2 across the entire range
of injection strengths. This indicates that the TDS of the SNL can be effectively suppressed,
with a more pronounced effect when the TDS of the MNLs is completely suppressed and
injected into the SNL.
Electronics 2024, 13, x FOR PEER REVIEW 8 of 14
MNLs. This includes scenarios where the MNL output signals exhibit significant TDS and
where TDS is completely suppressed. From Figure 4, it is evident that there is a noticeable
TDS when the mutual injection strength between the MNLs is 80 ns, while, when the
mutual injection strength is 150 ns, the TDS is completely suppressed. From Figure 5,
it can be observed that, when the MNLs exhibit significant TDS, the value of the ACF
drops below 0.2 when the injection strength from the MNLs to the SNL exceeds 20 ns.
When the TDS of the MNLs is significantly suppressed, the ACF remains below 0.2 across
the entire range of injection strengths. This indicates that the TDS of the SNL can be effec-
tively suppressed, with a more pronounced effect when the TDS of the MNLs is com-
pletely suppressed and injected into the SNL.
Figure 5. The trend of the peak values of the ACF of the SNL with varying injection strengths from
the MNLs.
The advantage of physical random numbers lies in their genuine unpredictability,
and the unpredictability of random numbers lies in the inherent characteristics of their
generation method. Therefore, the complexity of the output signal from a chaotic entropy
source determines the randomness of the random sequence. The 0–1 chaos test is a nu-
merical method used to determine chaotic behavior [47]. The 0–1 chaos test analyzes time
series data by calculating a statistic called the 0–1 measure. If the value of the 0–1 measure
is close to 1, it indicates that the system exhibits chaotic behavior; if the value is close to 0,
it suggests that the system is quasi-periodic or stable. PE [48] is a tool used for time series
analysis that measures the complexity of a time series to determine the dynamical charac-
teristics of a system. Because it is robust to noise and computationally efficient, PE is par-
ticularly well-suited for handling data from nonlinear and chaotic systems. Low PE indi-
cates that the time series has high predictability, suggesting that the system may be quasi-
periodic or deterministic. High PE indicates that the time series has high complexity, sug-
gesting that the system may be random or chaotic. Figure 6a depicts the chaotic region of
the SNL output, showing that the SNL operates in a chaotic state across almost the entire
parameter range. This is because the MNLs operate in a chaotic state solely under the
influence of optical feedback. The chaotic signals are further enhanced through mutual
injection between the MNLs and additional injection into the SNL. Figure 6b shows the
complexity region of the chaotic signals of the SNL output, which is consistent with the
chaotic region. Highly unpredictable chaotic signals can be output across nearly the entire
parameter range. This lays the foundation for extracting high-quality random numbers in
subsequent steps.
Figure 5. The trend of the peak values of the ACF of the SNL with varying injection strengths from
the MNLs.
The advantage of physical random numbers lies in their genuine unpredictability,
and the unpredictability of random numbers lies in the inherent characteristics of their
generation method. Therefore, the complexity of the output signal from a chaotic entropy
source determines the randomness of the random sequence. The 0–1 chaos test is a nu-
merical method used to determine chaotic behavior [
47
]. The 0–1 chaos test analyzes time
series data by calculating a statistic called the 0–1 measure. If the value of the 0–1 measure
is close to 1, it indicates that the system exhibits chaotic behavior; if the value is close to
0, it suggests that the system is quasi-periodic or stable. PE [
48
] is a tool used for time
series analysis that measures the complexity of a time series to determine the dynamical
characteristics of a system. Because it is robust to noise and computationally efficient, PE
is particularly well-suited for handling data from nonlinear and chaotic systems. Low PE
indicates that the time series has high predictability, suggesting that the system may be
quasi-periodic or deterministic. High PE indicates that the time series has high complexity,
suggesting that the system may be random or chaotic. Figure 6a depicts the chaotic region
of the SNL output, showing that the SNL operates in a chaotic state across almost the entire
parameter range. This is because the MNLs operate in a chaotic state solely under the
influence of optical feedback. The chaotic signals are further enhanced through mutual
injection between the MNLs and additional injection into the SNL. Figure 6b shows the
complexity region of the chaotic signals of the SNL output, which is consistent with the
chaotic region. Highly unpredictable chaotic signals can be output across nearly the entire
parameter range. This lays the foundation for extracting high-quality random numbers in
subsequent steps.

Electronics 2024,13, 2712 8 of 13
Electronics 2024, 13, x FOR PEER REVIEW 9 of 14
Figure 6. The two-dimensional map of the 0–1 chaos test (a) and PE (b) of the SNL.
TDS and unpredictability are important factors affecting the quality of random num-
bers. Similarly, the bandwidth and statistical properties of the chaotic laser entropy source
also have significant effects on random number extraction. The bandwidth of the chaotic
entropy source determines the generation rate of random numbers, while the statistical
properties determine the statistical characteristics of the random sequence. While ensur-
ing complete hiding of the TDS, adjusting relevant parameters to maximize bandwidth
and ensure the chaotic entropy source possesses favorable statistical properties. The spe-
cific parameter seings are as follows: 𝑓 = 0.02, Δ𝑓
=5 GHz, Δ𝑓
=−5 GHz, Δ𝑓
=
0 GHz, Δ𝑓
=−5 GHz, 𝑘
 =𝑘
 =80 ns
,and 𝑘
 =𝑘
 = 50 ns . Figure 7 shows the
time series, ACF curve, power spectrum, and amplitude distribution of the chaotic en-
tropy source under the aforementioned parameter seings. The chaotic system outputs
chaotic signals with large amplitude fluctuations and completely suppression of the TDS.
The effective bandwidth is approximately 100 GHz, and the statistical properties closely
resemble an ideal Gaussian distribution.
Figure 7. Time series (a), ACF (b), power spectrum (c), and amplitude distribution (d) of the chaotic
signals.
Figure 6. The two-dimensional map of the 0–1 chaos test (a) and PE (b) of the SNL.
TDS and unpredictability are important factors affecting the quality of random num-
bers. Similarly, the bandwidth and statistical properties of the chaotic laser entropy source
also have significant effects on random number extraction. The bandwidth of the chaotic
entropy source determines the generation rate of random numbers, while the statisti-
cal properties determine the statistical characteristics of the random sequence. While
ensuring complete hiding of the TDS, adjusting relevant parameters to maximize band-
width and ensure the chaotic entropy source possesses favorable statistical properties.
The specific parameter settings are as follows:
f=
0.02,
∆f1=
5
GHz
,
∆f2=−
5
GHz
,
∆f3=
0
GHz
,
∆f4=−
5
GHz
,
kr1=kr2=
80
ns−1
,
and kr3=kr4=
50
ns−1
. Figure 7
shows the time series, ACF curve, power spectrum, and amplitude distribution of the
chaotic entropy source under the aforementioned parameter settings. The chaotic system
outputs chaotic signals with large amplitude fluctuations and completely suppression of
the TDS. The effective bandwidth is approximately 100 GHz, and the statistical properties
closely resemble an ideal Gaussian distribution.
Electronics 2024, 13, x FOR PEER REVIEW 9 of 14
Figure 6. The two-dimensional map of the 0–1 chaos test (a) and PE (b) of the SNL.
TDS and unpredictability are important factors affecting the quality of random num-
bers. Similarly, the bandwidth and statistical properties of the chaotic laser entropy source
also have significant effects on random number extraction. The bandwidth of the chaotic
entropy source determines the generation rate of random numbers, while the statistical
properties determine the statistical characteristics of the random sequence. While ensur-
ing complete hiding of the TDS, adjusting relevant parameters to maximize bandwidth
and ensure the chaotic entropy source possesses favorable statistical properties. The spe-
cific parameter seings are as follows: 𝑓 = 0.02, Δ𝑓
=5 GHz, Δ𝑓
=−5 GHz, Δ𝑓
=
0 GHz, Δ𝑓
=−5 GHz, 𝑘
 =𝑘
 =80 ns
,and 𝑘
 =𝑘
 = 50 ns . Figure 7 shows the
time series, ACF curve, power spectrum, and amplitude distribution of the chaotic en-
tropy source under the aforementioned parameter seings. The chaotic system outputs
chaotic signals with large amplitude fluctuations and completely suppression of the TDS.
The effective bandwidth is approximately 100 GHz, and the statistical properties closely
resemble an ideal Gaussian distribution.
Figure 7. Time series (a), ACF (b), power spectrum (c), and amplitude distribution (d) of the chaotic
signals.
Figure 7. Time series (a), ACF (b), power spectrum (c), and amplitude distribution (d) of the
chaotic signals.

Electronics 2024,13, 2712 9 of 13
The characteristics of the chaotic entropy source and the post-processing technologies
together determine the quality of the random number sequence. Next, we compared the
performance of obtaining random numbers under two different post-processing schemes.
In the first scheme, we converted the optical signals into electrical signals using a PD.
Subsequently, we employed an 8-bit ADC to quantize the signals into a binary sequence
and extracted the LSBs to obtain the final random sequence. In the second scheme, after
sampling and quantizing the signals into an 8-bit binary sequence, we first shift the binary
sequence and then performed XOR processing on the binary sequences before and after the
shift. Finally, we extract the LSBs to obtain the random number sequence.
Quantizing chaotic signals using an 8-bit ADC is a process of converting continuous
chaotic signals into discrete digital signals. An 8-bit ADC has 2
8=
256 digitization levels,
which means that the entire input range of the analog signal is divided into 256 discrete
voltage intervals. Each quantization level corresponds to a unique 8-bit binary number,
ranging from 00000000 (0) to 11111111 (255). However, this quantization process can be
affected by issues such as uneven binary distribution and weak randomness. To address
these issues, it is common practice to extract the LSBs. The uniformity of the amplitude
distribution determines the balance of random bits 0 and 1. Figure 8shows the results of
extracting the LSBs in Scheme 1. When retaining eight LSBs, the amplitude distribution
remains consistent with the original chaotic entropy source. When the highest significant
bit is discarded, the amplitude distribution still exhibits non-uniformity. However, as the
number of discarded LSBs increases, the uniformity of the random sequence improves.
Similarly, in Scheme 2, when retaining eight and seven LSBs after the XOR operation, the
amplitude distribution is also non-uniform (Figure 9). Similarly, by discarding the LSBs, the
random sequence also becomes increasingly balanced. Below, we validate the randomness
of the obtained random sequence.
Electronics 2024, 13, x FOR PEER REVIEW 10 of 14
The characteristics of the chaotic entropy source and the post-processing technolo-
gies together determine the quality of the random number sequence. Next, we compared
the performance of obtaining random numbers under two different post-processing
schemes. In the first scheme, we converted the optical signals into electrical signals using
a PD. Subsequently, we employed an 8-bit ADC to quantize the signals into a binary se-
quence and extracted the LSBs to obtain the final random sequence. In the second scheme,
after sampling and quantizing the signals into an 8-bit binary sequence, we first shift the
binary sequence and then performed XOR processing on the binary sequences before and
after the shift. Finally, we extract the LSBs to obtain the random number sequence.
Quantizing chaotic signals using an 8-bit ADC is a process of converting continuous
chaotic signals into discrete digital signals. An 8-bit ADC has 2= 256 digitization lev-
els, which means that the entire input range of the analog signal is divided into 256 dis-
crete voltage intervals. Each quantization level corresponds to a unique 8-bit binary num-
ber, ranging from 00000000 (0) to 11111111 (255). However, this quantization process can
be affected by issues such as uneven binary distribution and weak randomness. To ad-
dress these issues, it is common practice to extract the LSBs. The uniformity of the ampli-
tude distribution determines the balance of random bits 0 and 1. Figure 8 shows the results
of extracting the LSBs in Scheme 1. When retaining eight LSBs, the amplitude distribution
remains consistent with the original chaotic entropy source. When the highest significant
bit is discarded, the amplitude distribution still exhibits non-uniformity. However, as the
number of discarded LSBs increases, the uniformity of the random sequence improves.
Similarly, in Scheme 2, when retaining eight and seven LSBs after the XOR operation, the
amplitude distribution is also non-uniform (Figure 9). Similarly, by discarding the LSBs,
the random sequence also becomes increasingly balanced. Below, we validate the random-
ness of the obtained random sequence.
Figure 8. The amplitude distribution when retaining m-LSBs in Scheme 1: (a) m = 8; (b) m = 7; (c)
m = 6; (d) m = 5; (e) m = 4; (f) m = 3; (g) m = 2; (h) m = 1.
Figure 8. The amplitude distribution when retaining m-LSBs in Scheme 1: (a) m = 8; (b) m = 7;
(c)m=6;(d)m=5;(e)m=4;(f)m=3;(g)m=2;(h)m=1.
The NIST randomness tests are standard tests used to assess and validate the quality
of RNGs. They comprise a total of 15 tests, including various classical statistical methods
such as frequency tests, sequence tests, block tests, independence tests, and so on. We select
1000 sets of 1 Mbit random number samples for testing, with the significance level
∝
set at
0.01. Passing the NIST test requires that the uniformity p-value of each subtest item is larger
than 0.0001 and the sample pass proportion falls within the range of 0.99 ±0.0094392.

Electronics 2024,13, 2712 10 of 13
Electronics 2024, 13, x FOR PEER REVIEW 11 of 14
Figure 9. The amplitude distribution when retaining m-LSBs in Scheme 2: (a) m = 8; (b) m = 7; (c)
m = 6; (d) m = 5; (e) m = 4; (f) m = 3; (g) m = 2; (h) m = 1.
The NIST randomness tests are standard tests used to assess and validate the quality
of RNGs. They comprise a total of 15 tests, including various classical statistical methods
such as frequency tests, sequence tests, block tests, independence tests, and so on. We se-
lect 1000 sets of 1 Mbit random number samples for testing, with the significance level ∝
set at 0.01. Passing the NIST test requires that the uniformity p-value of each subtest item
is larger than 0.0001 and the sample pass proportion falls within the range of 0.99 ±
0.0094392.
The original output of the chaotic laser may not be a perfectly uniformly distributed
random sequence, but rather exhibit some bias. Therefore, post-processing techniques are
typically applied to transform and adjust the generated random numbers. Effective post-
processing can enhance the uniformity of the random sequence, increase randomness, and
thereby allow for more LSBs to be retained in generating random numbers. In Figure 10,
we compared the number of NIST subtest items passed when retaining eight to one LSBs
under two different post-processing schemes. In Scheme 1, retaining one to four LSBs al-
lowed passing all tests in the NIST test suite. However, in Scheme 2, retaining one to five
LSBs allowed passing all tests in the NIST test suite, which means an additional LSB can
be retained. From this, it can be inferred that both post-processing schemes can generate
physical random numbers based on this chaotic system. However, the second scheme can
more effectively increase the random number generation rate. Table 2 lists the detailed
results of preserving the most LSBs that can be subjected to NIST testing in different
schemes.
Figure 9. The amplitude distribution when retaining m-LSBs in Scheme 2: (a) m = 8; (b) m = 7;
(c)m=6;(d)m=5;(e)m=4;(f)m=3;(g)m=2;(h)m=1.
The original output of the chaotic laser may not be a perfectly uniformly distributed
random sequence, but rather exhibit some bias. Therefore, post-processing techniques are
typically applied to transform and adjust the generated random numbers. Effective post-
processing can enhance the uniformity of the random sequence, increase randomness, and
thereby allow for more LSBs to be retained in generating random numbers. In Figure 10,
we compared the number of NIST subtest items passed when retaining eight to one LSBs
under two different post-processing schemes. In Scheme 1, retaining one to four LSBs
allowed passing all tests in the NIST test suite. However, in Scheme 2, retaining one
to five LSBs allowed passing all tests in the NIST test suite, which means an additional
LSB can be retained. From this, it can be inferred that both post-processing schemes can
generate physical random numbers based on this chaotic system. However, the second
scheme can more effectively increase the random number generation rate. Table 2lists
the detailed results of preserving the most LSBs that can be subjected to NIST testing in
different schemes.
Electronics 2024, 13, x FOR PEER REVIEW 12 of 14
Figure 10. The number of NIST subtests passed when retaining eight to one LSBs in Scheme 1 and
Scheme 2.
Table 2. The NIST test results for Scheme 1 and Scheme 2.
Statistical Test In Scheme 1, Four LSBs Are Retained. In Scheme 2, Five LSBs Are Retained.
p-Value Proportion Result p-Value Proportion Result
Frequency 0.352107 0.994 Success 0.777265 0.987 Success
Block frequency 0.570792 0.990 Success 0.308561 0.990 Success
Cumulative sums 0.635037 0.992 Success 0.695200 0.987 Success
Runs 0.591409 0.986 Success 0.635037 0.994 Success
Longest runs 0.428095 0.992 Success 0.915317 0.989 Success
Rank 0.705466 0.993 Success 0.781106 0.994 Success
Fast Fourier transform 0.011875 0.993 Success 0.492436 0.990 Success
Non-overlapping template 0.664168 0.982 Success 0.007975 0.982 Success
Overlapping template 0.558502 0.988 Success 0.326749 0.994 Success
Universal 0.317565 0.989 Success 0.307077 0.986 Success
Approximate entropy 0.542228 0.991 Success 0.363593 0.985 Success
Random excursions 0.191505 0.984 Success 0.012181 0.982 Success
Random excursions variant 0.278122 0.986 Success 0.025588 0.982 Success
Serial 0.595549 0.990 Success 0.184549 0.984 Success
Linear complexity 0.469232 0.989 Success 0.373625 0.991 Success
4. Conclusions
This paper proposes a scheme to generate random numbers using a chaotic system
composed of three NLs as the random entropy source. Two MNLs achieve chaotic output
through optical feedback and mutual injection, and their outputs are simultaneously in-
jected into the SNL. We investigated the TDS concealment performance of both the MNLs
and the SNL separately. Under only simple optical feedback, the MNLs exhibit a signifi-
cant TDS. With the increase in mutual injection strength, the TDS of the MNLs are gradu-
ally suppressed. Compared to the MNLs, the TDS concealment performance of the SNL
is beer, achieving almost complete TDS concealment across the entire parameter range.
Additionally, we explored the chaotic regions and complexity of the chaotic signals out-
put by the SNL. The results show that SNL can output highly complex chaotic signals over
a wide range of parameters. Next, we proposed two post-processing schemes for extract-
ing random numbers. The research indicates that both post-processing schemes can gen-
erate random numbers that pass NIST tests. However, in Scheme 2, one more LSB can be
retained, compared to Scheme 1, when subjected to NIST testing. Therefore, effective post-
processing can enhance the performance of RNGs. This study confirms the feasibility of
NLs as random entropy sources in random number generation, providing new solutions
Figure 10. The number of NIST subtests passed when retaining eight to one LSBs in Scheme 1 and
Scheme 2.

Electronics 2024,13, 2712 11 of 13
Table 2. The NIST test results for Scheme 1 and Scheme 2.
Statistical Test In Scheme 1, Four LSBs Are Retained. In Scheme 2, Five LSBs Are Retained.
p-Value Proportion Result p-Value Proportion Result
Frequency 0.352107 0.994 Success 0.777265 0.987 Success
Block frequency 0.570792 0.990 Success 0.308561 0.990 Success
Cumulative sums 0.635037 0.992 Success 0.695200 0.987 Success
Runs 0.591409 0.986 Success 0.635037 0.994 Success
Longest runs 0.428095 0.992 Success 0.915317 0.989 Success
Rank 0.705466 0.993 Success 0.781106 0.994 Success
Fast Fourier transform 0.011875 0.993 Success 0.492436 0.990 Success
Non-overlapping template 0.664168 0.982 Success 0.007975 0.982 Success
Overlapping template 0.558502 0.988 Success 0.326749 0.994 Success
Universal 0.317565 0.989 Success 0.307077 0.986 Success
Approximate entropy 0.542228 0.991 Success 0.363593 0.985 Success
Random excursions 0.191505 0.984 Success 0.012181 0.982 Success
Random excursions variant 0.278122 0.986 Success 0.025588 0.982 Success
Serial 0.595549 0.990 Success 0.184549 0.984 Success
Linear complexity 0.469232 0.989 Success 0.373625 0.991 Success
4. Conclusions
This paper proposes a scheme to generate random numbers using a chaotic system
composed of three NLs as the random entropy source. Two MNLs achieve chaotic output
through optical feedback and mutual injection, and their outputs are simultaneously
injected into the SNL. We investigated the TDS concealment performance of both the
MNLs and the SNL separately. Under only simple optical feedback, the MNLs exhibit a
significant TDS. With the increase in mutual injection strength, the TDS of the MNLs are
gradually suppressed. Compared to the MNLs, the TDS concealment performance of the
SNL is better, achieving almost complete TDS concealment across the entire parameter
range. Additionally, we explored the chaotic regions and complexity of the chaotic signals
output by the SNL. The results show that SNL can output highly complex chaotic signals
over a wide range of parameters. Next, we proposed two post-processing schemes for
extracting random numbers. The research indicates that both post-processing schemes can
generate random numbers that pass NIST tests. However, in Scheme 2, one more LSB can
be retained, compared to Scheme 1, when subjected to NIST testing. Therefore, effective
post-processing can enhance the performance of RNGs. This study confirms the feasibility
of NLs as random entropy sources in random number generation, providing new solutions
for applications in information security, cryptography, and random simulation. Moving
forward, we will intensify our research on the security of RNGsbased on NLs to ensure
their safety and reliability.
Author Contributions: Methodology, J.Z., G.L. and P.M.; validation, R.L. and P.M.; investigation,
G.L., J.Z. and P.M.; writing—original draft preparation, G.L. and R.L.; writing—review and editing,
J.Z. and P.M. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the Project: The Talent Project of Chengdu Technological
University (2023RC013).
Data Availability Statement: Data are contained within the article.
Conflicts of Interest: The authors declare no conflicts of interest.

Electronics 2024,13, 2712 12 of 13
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