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  <front>
    <journal-meta>
      <journal-title-group>
        <journal-title>I. Rozlomii);</journal-title>
      </journal-title-group>
    </journal-meta>
    <article-meta>
      <title-group>
        <article-title>The method for verifying firmware integrity in IoT devices for secure boot using lightweight hash functions⋆</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Inna Rozlomii</string-name>
          <email>inna-roz@ukr.net</email>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Emil Faure</string-name>
          <email>e.faure@chdtu.edu.ua</email>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Andrii Yarmilko</string-name>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Serhii Naumenko</string-name>
          <email>sergey.naumenko94@vu.cdu.edu.ua</email>
        </contrib>
      </contrib-group>
      <pub-date>
        <year>2046</year>
      </pub-date>
      <volume>000</volume>
      <fpage>0</fpage>
      <lpage>0002</lpage>
      <abstract>
        <p>The paper is devoted to the development of a method for verifying the integrity of firmware for IoT devices, focused on the conditions of limited computing resources and minimal power consumption. The method is based on the use of lightweight hash functions, such as SPONGENT, PHOTON, QUARK and LESAMNTA-LW, which provide computational efficiency on low-end microcontrollers. A multi-level approach is proposed, in which firmware segments are evaluated by weight coefficients depending on their criticality for security, and the aggregated control value is formed taking into account the importance of each segment. To increase protection against replay attacks, session markers are integrated into hashing, which add context dependency. The adaptive nature of the check allows you to dynamically change the depth of analysis depending on the state of the device-for example, the battery charge level or processor load. The experimental part of the work covers testing the method on popular microcontrollers STM32F072, ESP8266 and ATmega328P. The study included an assessment of verification time, memory consumption, power consumption, and resistance to firmware modification attacks and reuse of previous values attacks. Special attention is paid to partial verification scenarios, which are relevant for devices with limited resources. The proposed method is considered suitable for a wide range of IoT applications, including autonomous sensors, energy modules, medical devices, and transportation systems. The results demonstrate the possibility of effective secure boot even on platforms without hardware cryptography support.</p>
      </abstract>
      <kwd-group>
        <kwd>eol&gt;limited resources</kwd>
        <kwd>IoT</kwd>
        <kwd>lightweight hashing</kwd>
        <kwd>session tokens</kwd>
        <kwd>weighted aggregation</kwd>
        <kwd>STM32F072</kwd>
        <kwd>ESP8266</kwd>
        <kwd>ATmega328P</kwd>
      </kwd-group>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>1. Introduction</title>
      <p>
        The Internet of Things (IoT) today encompasses a vast array of devices, from consumer electronics
to medical implants and industrial sensors [
        <xref ref-type="bibr" rid="ref1 ref2 ref3 ref4 ref5 ref6">1–6</xref>
        ]. As the number of IoT devices increases, so does
the level of threats to them, especially when it comes to firmware modification, which is one of the
key targets of attacks [
        <xref ref-type="bibr" rid="ref7 ref8">7, 8</xref>
        ]. Firmware modification or replacement can lead to loss of functionality,
leakage of confidential data, or complete device takeover [
        <xref ref-type="bibr" rid="ref9">9</xref>
        ]. Ensuring firmware integrity is
becoming a prerequisite for Secure Boot, but in practice its implementation is complicated by the
limited resources of IoT devices, in particular limitations in terms of memory, processing power,
and power consumption [
        <xref ref-type="bibr" rid="ref10 ref11">10, 11</xref>
        ].
      </p>
      <p>
        Traditional cryptographic hash functions, such as SHA-2 or SHA-3, demonstrate high resistance
to attacks, but their use in resource-constrained environments is of little use [
        <xref ref-type="bibr" rid="ref12">12</xref>
        ]. They consume a
significant amount of RAM, require a powerful processor, and increase the overall boot time [
        <xref ref-type="bibr" rid="ref13 ref14 ref15">13–
15</xref>
        ]. These limitations have encouraged research groups to develop lightweight hash functions
aimed at IoT needs, including SPONGENT, PHOTON, QUARK, and LESAMNTA-LW [
        <xref ref-type="bibr" rid="ref16 ref17 ref18">16–18</xref>
        ]. They
allow for significant reductions in computational costs and power consumption while maintaining
an acceptable level of resistance to cryptanalytic attacks.
      </p>
      <p>Against the background of these limitations, there is considerable scientific interest in analyzing
the possibility of using lightweight hash functions for integrity verification, as they are specifically
designed for resource-constrained environments. Their application opens up new prospects for
embedded systems, allowing for secure booting even where the use of classical algorithms is
impractical or technically impossible.</p>
      <p>Figure 1 presents a general diagram of a secure boot of an IoT device using a lightweight hash
function, showing the main stages—storing a check hash, calculating the hash of the loaded
firmware, and comparing the values before launching the main program.</p>
      <p>The scheme clearly shows the sequence of actions: first, a reference hash value is read from
non-volatile memory, then a hash of the current firmware version is calculated, after which both
values are compared. In case of a match, the system proceeds to the next stage of loading,
otherwise it blocks the loading or activates recovery procedures.</p>
      <p>The specified scheme illustrates the key stages of implementing Secure Boot in an IoT device,
emphasizing the role of the hash function as the central element responsible for verifying the
authenticity of the loaded code. Importantly, this approach allows integrating the verification
mechanism at the software level without the need for hardware crypto modules, while maintaining
the limited amount of memory and low power consumption that are critical for autonomous sensor
nodes, medical implants and other similar systems.</p>
      <p>The aim of the research is to develop a method for verifying firmware integrity for IoT devices,
based on the use of lightweight hash functions and taking into account the specifics of
resourceconstrained environments, in particular, minimizing computational costs, memory, and power
consumption while ensuring an appropriate level of cryptographic stability.</p>
    </sec>
    <sec id="sec-2">
      <title>2. Related works</title>
      <p>
        Providing Secure Boot in IoT devices has attracted considerable attention from researchers in
recent years [
        <xref ref-type="bibr" rid="ref19 ref20 ref21">19–21</xref>
        ]. The main goal of Secure Boot is to ensure that only verified and authentic
code is executed by the device during startup. According to [
        <xref ref-type="bibr" rid="ref22">22</xref>
        ], the principle of operation of
Secure Boot is based on cryptographic verification of digital signatures or hash values of program
code stored in the trusted memory of the device. Works [
        <xref ref-type="bibr" rid="ref23 ref24">23, 24</xref>
        ] demonstrate the implementation of
such approaches based on microcontrollers with hardware support for cryptography, for example,
Trusted Platform Module (TPM) or ARM TrustZone, however, these solutions are not suitable for
low-end microcontrollers due to their high cost and power consumption.
      </p>
      <p>
        Current methods for ensuring integrity in microcontrollers are mostly focused on a compromise
between the level of security, computational costs and resource consumption. For example, [
        <xref ref-type="bibr" rid="ref25">25</xref>
        ]
describes a software implementation of Secure Boot for STM32 microcontrollers, where the
SHA-256 hash function is used to verify the integrity of the firmware. The results of the study
showed that the use of SHA-256 allows for high cryptographic stability, but leads to an increase in
the device boot time by 30–50% depending on the firmware size, and also requires a significant
amount of RAM (several tens of kilobytes), which is critical for microcontrollers with limited RAM.
The article [
        <xref ref-type="bibr" rid="ref26">26</xref>
        ] considers approaches to optimizing such solutions, including partial hashing, when
only the most critical code segments are checked, or a phased integrity check per segment, which
allows distributing the load on computing resources. However, the authors emphasize that such
simplifications potentially create new attack vectors, for example, selective substitution of
uncontrolled segments or attacking actions during intermediate checks.
      </p>
      <p>
        The development of lightweight hash functions has become a separate research area, since they
are specifically designed for use in embedded and sensor systems [
        <xref ref-type="bibr" rid="ref27">27</xref>
        ]. In [
        <xref ref-type="bibr" rid="ref28">28</xref>
        ], a classification of
such functions by design approaches was carried out: sponge-constructions, Davies–Meyer block
schemes and double-block Hirose. SPONGENT [
        <xref ref-type="bibr" rid="ref29">29</xref>
        ], which belongs to sponge-constructions,
demonstrates a noticeable reduction in computational costs compared to SHA-2 due to the use of a
compact state block and simple bitwise operations XOR, AND, ROT, which allows minimizing
memory consumption to the level of 1–2 kB. PHOTON [
        <xref ref-type="bibr" rid="ref30">30</xref>
        ], built on the principle of Substitution–
Permutation Network (SPN), combines a small hardware implementation area and resistance to the
main types of attacks, such as differential and linear cryptanalysis, which is confirmed by
numerous studies on ARM Cortex-M and AVR microcontrollers. QUARK [
        <xref ref-type="bibr" rid="ref31">31</xref>
        ], designed as a
serialized stream cipher-like design, is particularly effective in scenarios with tight energy
constraints, as it requires a minimum number of clock cycles per byte of data. LESAMNTA-LW
[
        <xref ref-type="bibr" rid="ref32">32</xref>
        ] is positioned as a solution for ARM Cortex-M platforms, providing a balance between speed
(due to extensive modular addition and permutation operations) and resistance to collisions and
preimage attacks, while maintaining a small code size and low memory consumption.
      </p>
      <p>
        Despite the presence of a significant amount of research, there are open problems that need to
be solved. These include choosing the optimal hash function for specific IoT scenarios,
mathematical modeling of the security-resources ratio, adapting Secure Boot for microcontrollers
without hardware cryptography support, and providing protection against attacks at the physical
level (e.g., fault injection). Recent advances in access control mechanisms emphasize the
importance of integrating policy-as-code frameworks to enforce role-based and attribute-based
access control, thereby enhancing the security of IoT environments [
        <xref ref-type="bibr" rid="ref33">33</xref>
        ]. Additionally,
improvements in device identification and authentication accuracy through electromagnetic (EM)
measurements provide promising avenues for strengthening hardware-level trustworthiness in
constrained IoT devices [
        <xref ref-type="bibr" rid="ref34">34</xref>
        ]. Furthermore, the design of combined pseudo-random sequence
generators, as well as generators based on mathematical constants such as ln 2, contribute to the
development of lightweight and secure cryptographic primitives suitable for IoT firmware integrity
verification [
        <xref ref-type="bibr" rid="ref35 ref36">35, 36</xref>
        ]. These approaches collectively support the enhancement of secure boot
processes and firmware integrity validation under resource constraints. Further research should
focus on finding methods that consider the balance between cryptographic robustness, hardware
platform limitations, and practical time and power consumption requirements.
      </p>
    </sec>
    <sec id="sec-3">
      <title>3. Method for verifying firmware integrity in IoT devices</title>
      <p>The proposed method for verifying the integrity of firmware in IoT devices is based on the concept
of multi-level, weighted and context-dependent verification, which allows taking into account the
heterogeneity of the importance of individual sections of the code, the hardware limitations of the
device and the current mode of its operation. A feature of the method is that for each code
segment, weight coefficients are determined that reflect its criticality for system security. The
aggregated control value is formed on the basis of calculated local hashes taking into account these
weights, which allows increasing the depth of verification for the most vulnerable components
without a significant increase in the load on the system.</p>
      <p>Unlike classic Secure Boot schemes, where a single hash value of the entire firmware is
compared, the proposed method uses lightweight hash functions, for example, SPONGENT,
PHOTON, QUARK or LESAMNTA-LW, which provide a balance between cryptographic stability
and resource efficiency. This opens up the possibility of implementing integrity checking even on
low-power microcontrollers, such as STM32F0, ESP8266, AVR, which have a limited amount of
RAM (up to several kilobytes) and computing resources. The method takes into account not only
memory limitations, but also minimizing power consumption, which is especially relevant for
autonomous sensors, medical implants, IoT modules in power grids and transport systems.</p>
      <p>A key element of the approach is the adaptive selection of the depth of the check depending on
the state of the device, for example, the battery charge level, temperature regime or processor load.
In scenarios with limited energy resources, a partial check mode is activated, where only the most
critical segments are checked, while under normal conditions a full check is performed. This
approach allows the IoT device to dynamically balance between protection and autonomy,
maintaining the appropriate level of security in conditions of limited resources.</p>
      <p>Additionally, the method involves the implementation of context markers (e.g., time tokens or
session identifiers) that are integrated into the hashing process. This increases the system's
resistance to replay attacks, since the verification is carried out not only on the basis of the code,
but also taking into account the session context, which significantly complicates the preparation of
fake firmware. The proposed approach combines the concepts of cryptographic robustness,
resource efficiency, and adaptability, which allows it to be scaled for a wide range of IoT
applications.</p>
      <p>The algorithm of the proposed method for verifying the integrity of the firmware includes
multi-level hashing with weighted aggregation, adaptive selection of the verification depth, and the
use of context markers to protect against replay attacks. Let the firmware F be divided into
segments F = {f 1 , f 2 , .. , f n}, each of which is assigned a weight coefficient wi, reflecting its
n
criticality, with ∑ w i =1.</p>
      <p>i=1</p>
      <p>The choice of an aggregation model based on weight coefficients is associated with the need to
take into account the criticality of functional modules. The coefficients wi are formed based on a
preliminary risk analysis, which takes into account the role of the segment in system security, the
frequency of access to resources and the probability of attacks. This approach allows you to
achieve a flexible balance between security and resource costs.
where S ⊆ { 1 , … , n } is a subset of the most important segments selected by the criterion wi
exceeds the set threshold value. The aggregation model is selected at the system configuration
stage in accordance with the target usage scenario.</p>
      <p>At the reference profile formation stage, the following is performed:
1.</p>
      <p>Calculation of local hashes H ( f i ) for each segment using a lightweight hash function
H ( x ), such as SPONGENT or PHOTON.</p>
      <p>n
2. Formation of an aggregated hash value—H agg= ∑ wi ∙ H ( f i ).
i= 1
3. Writing H agg to trusted non-volatile memory together with a time or session token T used
as a salt value.</p>
      <p>During device boot, the following is performed:</p>
      <p>Calculation—H cagg= ∑</p>
      <p>j ∈ C ( F )
binds hashing to the session token.</p>
      <p>c e
Compare H agg and H agg and make a decision (3).</p>
      <p>Reading H cagg and T e from trusted memory.</p>
      <p>Depending on the state of the device (e.g., battery level), determine the set C ( F ) ⊆ F to be
tested.</p>
      <p>e
w j ∙ H ( f j ∨ T ), where ∨ is a concatenation operation that
To form the aggregated control value, the weighted average aggregation model (1) is used.</p>
      <p>where H i is the local hash value of the i-th segment, wi is its weighting factor, n is the number
of segments. This scheme allows to increase the contribution of critical components to the final
result, while reducing the influence of secondary modules. Alternatively, for devices with very
tight constraints, it is possible to use a simple additive model (2).</p>
      <p>,
(1)
(2)
(3)
D ( H cagg , H eagg)= {1 , if H cagg= H eagg .</p>
      <p>e
0 , if H cagg ≠ H agg</p>
      <p>The computational complexity of the algorithm for full verification is estimated as O ( n ∙ C H ),
where n is the number of segments, C H is the average complexity of calculating the hash for one
segment. For partial verification, which is activated when resources are reduced, the complexity
decreases in proportion to the number of verified segments C ( F ) ∨ . This allows reducing the
verification time by 30–60% depending on the selected configuration.</p>
      <p>Particular attention is paid to the selection of optimal wi—for example, larger values are
assigned to areas responsible for communications, hardware control, or secure data storage, while
auxiliary modules (e.g., user interfaces) are given minimal weight. The method is focused on
checking only static segments, i.e. parts of the code that do not change during execution. Dynamic
modules that are loaded during operation are not covered by the current model and require
separate solutions, such as digital signature verification or runtime integrity monitoring.</p>
      <p>Figure 2 presents a flowchart of the algorithm, which demonstrates the process of multi-level
validation, adaptive segment selection, and the use of session tokens. This algorithm allows for
high scalability—it is suitable for both simple sensors and more powerful edge devices, where
advanced cryptographic operations are available. The introduction of weighting factors and context
markers increases resistance to attacks even in scenarios with limited energy and computing
power.</p>
      <p>If a mismatch in control values is detected, the device enters a protected mode, which involves
blocking the launch of the main code, recording the event in the system log and, if possible,
transmitting a message to an external monitoring system. For some categories of devices, for
example, in medical applications, automatic activation of the firmware recovery mode with
restoration of a previously saved valid firmware version is provided.</p>
    </sec>
    <sec id="sec-4">
      <title>4. Experimental evaluation of the effectiveness of the method</title>
      <sec id="sec-4-1">
        <title>4.1. Experimental conditions</title>
        <p>The experimental evaluation of the method’s effectiveness was carried out on hardware platforms
representing typical classes of resource-constrained IoT devices. Three microcontrollers were used
for the study:</p>
        <p>STM32F072 (32-bit, Cortex-M0, 48 MHz frequency, 64 KB Flash, 16 KB SRAM, 12-bit ADC,
USB, SPI, I2C);
ESP8266 (32-bit, Tensilica L106, 80 MHz frequency, 64 KB instruction memory, 96 KB data
RAM, Wi-Fi 2.4 GHz, UART, SPI);
ATmega328P (8-bit, AVR, 20 MHz frequency, 32 KB Flash, 2 KB SRAM, 10-bit ADC, UART,
SPI, I2C).</p>
        <p>The choice of these microcontrollers is justified by their representativeness for a wide range of
IoT applications: STM32F072 represents a class of energy-efficient 32-bit MCUs, ESP8266—devices
with wireless data transmission, and ATmega328P—popular 8-bit MCUs, widely used in low-end
sensor nodes. Such a set provides comprehensive coverage of various architectures and classes of
resource constraints.</p>
        <p>The software environment included the IDE STM32CubeIDE (version 1.14.1), Arduino IDE
(version 2.3.2) and ESP-IDF (version 5.1.2), compilers gcc-arm-none-eabi, avr-gcc and
xtensa-lx106elf-gcc, respectively. To measure the indicators, the internal timers of the microcontrollers were
used, as well as an external current consumption meter Nordic Power Profiler Kit II, connected to a
3.3 V power supply.</p>
        <p>The collection of energy and time indicators was carried out with an average error of ±2%,
confirmed by the calibration of the Nordic Power Profiler Kit II. To ensure the repeatability of the
experiment, automated launch and measurement scripts were used, which allowed minimizing the
influence of the human factor.</p>
        <p>The following hash functions were used in the experiment: SPONGENT-160, PHOTON-128/16,
QUARK-D, LESAMNTA-LW and for comparison SHA-256. The firmware for testing included
segmented areas: system drivers, network stacks, program kernel, data processing modules. The
total size of the firmware varied: STM32F072—32 KB, ESP8266—48 KB, ATmega328P—28 KB.
Experimental options with full hashing and with partial (critical segments only) were used for
evaluation.</p>
        <p>The weighting factors for the segments were determined based on a preliminary risk analysis:
the highest values were assigned to the communication and control modules, the middle ones to
the program core, and the lowest values to the auxiliary interface components. This allowed
emulating the real operating conditions of IoT devices.</p>
        <p>The following indicators were measured: execution time of the full integrity check and partial
(with weighting factors), the amount of RAM used, the average power consumption per session,
the additional load on the CPU, the number of detected modification attempts. Additionally, the
impact of the partial hashing mode on the overall system performance was assessed, in particular,
delays in the start sequence (boot delay) and the frequency of false positive or false negative
activations of the integrity detector.</p>
        <p>Figure 3 shows a fragment of the experimental environment—STM32CubeIDE with a running
project, where the multi-level integrity check function is implemented.</p>
        <p>Special attention was paid to modeling scenarios with real threats, in particular, attempts to
modify the firmware at the critical segment level and replay attacks using previously stored hash
values without taking into account the session context. This allowed not only to estimate the
overall resources required for the algorithm to work, but also to test its stability in typical
application conditions of IoT devices</p>
      </sec>
      <sec id="sec-4-2">
        <title>4.2. Experiment results</title>
        <p>The results of the experimental verification are presented in the table, which shows the average
time for a full firmware integrity check using different algorithms, as well as the time for partial
verification of critical segments. For SHA-256, the time on the STM32F072 was 184 ms, on the
ESP8266—242 ms, on the ATmega328P—315 ms. For SPONGENT-160, the time was 96 ms, for
PHOTON-128/16—103 ms, for QUARK-D—112 ms, for LESAMNTA-LW—98 ms. In the partial
verification mode (approximately 30% of the segments), the time decreased to 58 ms, 69 ms, and 84
ms, respectively.</p>
        <p>The distribution of verification execution time for full and partial modes using different hash
functions is shown in Figure 5. It shows a performance comparison on three hardware platforms,
demonstrating the advantages of lightweight algorithms over the classic SHA-256, as well as the
effect of using partial hashing of critical segments.</p>
        <p>In addition to time, the amount of RAM was measured, which ranged from 2.1–2.8 KB for
lightweight algorithms, while SHA-256 required 6.4–7.1 KB, exceeding the resource capabilities of
the ATmega328P. Power consumption during partial verification was reduced by 35–50% compared
to full SHA-256 verification. Resistance to attacks was high: the algorithm detected 100% of
modifications of critical segments and 96% of replay attacks without a valid session token.</p>
      </sec>
    </sec>
    <sec id="sec-5">
      <title>5. Discussion</title>
      <p>The results show that the proposed method of verifying the integrity of the firmware based on
lightweight hash functions provides a significant reduction in verification time, memory usage, and
power consumption compared to classical algorithms such as SHA-256. The effect of using partial
hashing of critical segments is particularly noticeable, which allows reducing resource
consumption by 30–60% depending on the configuration, while maintaining a high level of
cryptographic stability.</p>
      <p>The analysis showed that the use of a weighted average hash value aggregation model increases
the sensitivity of verification to modifications of the most important components, while simple
additive aggregation may be appropriate for scenarios with minimal resources. The introduction of
session tokens significantly increases resistance to replay attacks, which is confirmed by
experimental simulation.</p>
      <p>
        Comparison with existing approaches described in the literature demonstrates the advantages of
the proposed solution: in [
        <xref ref-type="bibr" rid="ref25 ref26">25, 26</xref>
        ], classic Secure Boot schemes based on SHA-256 provide a high
level of protection, but are not adapted to microcontrollers with limited resources due to significant
computational costs and memory size. In turn, partial hashing without weight analysis, as in [
        <xref ref-type="bibr" rid="ref26">26</xref>
        ],
potentially opens up new attack vectors. The proposed method combines the advantages of
lightweight hashing with adaptive selection of the check depth, which allows for a flexible balance
between security and efficiency. At the same time, the method has certain limitations. The current
implementation is focused exclusively on checking static firmware segments, while dynamically
loaded modules remain outside its scope. For such components, it is advisable to use digital
signatures or runtime integrity monitoring, which is planned to be investigated in future works.
Another aspect is the integration of the proposed method into typical IoT device bootloaders,
which may require additional optimization for specific hardware architectures. Overall, the results
of the experimental evaluation confirm the high practical significance of the developed method for
use in autonomous sensor systems, medical implants, smart energy modules, and transport IoT
solutions, where not only the level of security, but also the efficiency of use of computing and
energy resources is critical.
      </p>
    </sec>
    <sec id="sec-6">
      <title>Conclusions</title>
      <p>The paper develops and investigates a method for verifying firmware integrity for IoT devices that
takes into account the specifics of environments with limited computing resources. The proposed
approach combines the use of lightweight hash functions (SPONGENT, PHOTON, QUARK,
LESAMNTA-LW), weighted average aggregation of hash values, and adaptive selection of
verification segments depending on the current state of the device, for example, the battery level or
processor load. The introduction of session tokens additionally enhances resistance to replay
attacks, which is an important advantage compared to other existing solutions.</p>
      <p>The results of experimental evaluation on different hardware platforms (STM32F072, ESP8266,
ATmega328P) demonstrated a significant reduction in verification time, power consumption, and
occupied RAM compared to classical algorithms such as SHA-256. The efficiency of partial hashing
of critical segments allows for flexible adaptation of the verification to the limitations of a specific
device, while maintaining high accuracy of modification detection. It is important to emphasize
that the method can be scaled for different classes of IoT devices—from low-end sensors to edge
platforms with advanced capabilities.</p>
      <p>Prospects for further research include the development of mechanisms for checking
dynamically loaded modules, integration with digital signatures, and research into the system's
resistance to attacks at the physical level, in particular fault injection. A separate direction is the
optimization of the software implementation of the method to minimize overhead and ensure
compatibility with typical microcontroller loaders.</p>
    </sec>
    <sec id="sec-7">
      <title>Acknowledgements</title>
      <p>This research was funded by the Ministry of Education and Science of Ukraine under grant
0125U000637.</p>
    </sec>
    <sec id="sec-8">
      <title>Declaration on Generative AI</title>
      <p>While preparing this work, the authors used the AI programs Grammarly Pro to correct text
grammar and Strike Plagiarism to search for possible plagiarism. After using this tool, the authors
reviewed and edited the content as needed and took full responsibility for the publication’s content.
Also generative artificial intelligence tools were used exclusively for creating the diagram
presented in Figure 5.</p>
    </sec>
  </body>
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