1. Introduction
The ability to authenticate electronic information-processing devices based on their unique characteristics attracts the attention of researchers who work to ensure the cybersecurity of information systems. The uniqueness of electronic devices at the physical level makes it possible to carry out authentication using various methods, namely physically non-cloneable functions for the Internet of Things (IoT) [
1,
2], error vector trajectories of the Global System for Mobile Communications (GSM) mobile phone signals [
3], the spectrum of noise in the signal of radioactivity sensors [
4], spontaneous electromagnetic radiation from operating mobile phones, light-emitting diode (LED) screens, laptops [
5], wireless fidelity transmitters [
6], etc. Authentication accuracy depends on the chosen method and conditions of experiments, but usually does not reach 100 percent.
The implementation of the idea of authentication by individual characteristics is based on a preliminary measurement of a physical quantity; for example, electromagnetic interference from operating units of the device, the so-called internal electrical noise. The method of authentication for electronic information-processing devices (personal computers) by the individual forms of correlograms of their internal electrical noise is known [
7]. The device authenticator—noise bit template—is calculated from the normalized autocorrelation function of noise. In the future, bit templates could be compared with each other using the selected metric; this is the Hamming distance in the simplest case. An important property of the bit template set, obtained from sequential measurements of the internal noise of a single device, is their closeness in the sense that the Hamming distances between possible pairs of bit templates are small, while the distances between pairs of templates for different devices are much larger. This makes it possible to reliably distinguish these devices, i.e., to authenticate them. The reference templates from each electronic device are pre-recorded on the server. During authentication, the device presents a real-time template, which is compared with a reference template. Authentication is confirmed if the distance between templates is less than a threshold value for the claimed device. Bit template variations provide dynamic authentication because the templates do not repeat exactly, and thus the authenticator reuse attack is eliminated. There are known experiments, a result of which made it possible to authenticate stationary personal computers of the same series with an accuracy of 98.6% [
8].
An integrated sound card can be used to measure the internal noise of computers. Usually, noise bit templates of desktop computers are stable over time. For laptops the situation is slightly different. If the laptop gets into a location with a strong external electromagnetic field that significantly affects the internal noise of the laptop, then authentication errors occur [
8]. In addition, not all electronic information-processing devices have integrated ADCs, for example, many microprocessors do not have an integrated ADC. For them, the use of internal noise as a sign of authentication is not possible. Therefore, in this study, the problem of modeling authentication features based on Poisson pulse sequence generators was formulated.
Generators of random or pseudo-random pulse sequences have been used for a long time to solve a wide range of problems in science and technology. Almost all standard program libraries have the embedded generators of pseudo-random sequences, which users could utilize. One of the most important generators is the Poisson pulse sequence generator (PPSG). These generators are widely used in different branches of techniques for simulating different processes that have a random temporal and spatial nature [
9], for sociological and scientific research [
10,
11]. Such generators are effectively used to solve cybersecurity problems [
12,
13], to simulate the output signals of dosimetric detectors when designing them, and testing devices for measuring the parameters of ionizing radiation [
14,
15,
16,
17,
18,
19], because the number of radioactive decay particles detected by the detector over a period of time is subject to the Poisson distribution law.
In recently published works quite effective principles of realization of software and hardware PPSG are presented. Their structures, based on the use of pseudo-random number generators (PRNGs), were proposed [
20,
21,
22,
23,
24,
25,
26,
27] and methods for assessing the quality of their output signals were developed [
28,
29,
30,
31]. In this case the effectiveness of the possible application of the PPSG significantly depends on the quality of the designed generator and on the main characteristics of its output sequence.
The aim of this study was to model bit templates for the authentication of electronic information-processing devices based on a Poisson pulse sequence generator. The following tasks were solved to achieve this aim.
Optimization of the parameters of the PPSG’s structural elements in order to obtain the specified parameters of the output pulse sequence. Definition of the limits of control code values, specification of the range of values of the average frequency of the output pulse sequence, which corresponds to Poisson’s law of distribution.
The bit template simulator was proposed based on the Poisson pulse sequence generator. Bit sequences with given characteristics were the result of the simulator.
The simulation experiment was carried out to test the required properties of bit templates.
In this research, based on previously obtained results concerning Poisson pulse sequence generators and the development of the control code theory, the needed sequences were programmatically generated. Sample device bit templates were simulated based on these sequences and authentication was also performed programmatically. The examination consisted of calculating pairs of possible Hamming distances between the templates of the same device (intradistances) and for different devices (interdistances) and comparing them one with another.
Comparing the set of intradistances with the set of interdistances confirmed the main idea, that the generated templates could be unmistakably classified as being related to different devices. The threshold of distances was determined, according to which classification was made for specific parameters of the PPSG.
In this research the Poisson Pulse Sequence Generator was used for the first time to create device authentication templates based on the principle of biometrics. Compared with the best practices, which were using the measured values—electromagnetic radiation, internal electrical noise—the proposed method had several advantages. Benefits included 100% authentication, significantly more devices, and no time delay for measurements.
2. Materials and Methods
2.1. Structural Scheme PPSG and the Principle of Its Operation
The generator [
16,
17,
18], whose structural scheme is illustrated in
Figure 1, consisted of a modified additive Fibonacci generator (MAFG), which contained registers Rg1–Rg5, adders Ad1–Ad3, logical scheme LS, as well as a comparing scheme CS and logical element &. All the structural MAFG elements, except LS, worked in binary-decimal code.
On MAFG output, e.g., on Rg5 output, a sequence of pseudo-random numbers was formed in accordance with the following expression:
where
,
,
,
are the numbers in registers Rg4, Rg3, Rg2, Rg1, correspondingly,
, and
is the number of decades of the scheme’s structural elements. The value of the variable
a is determined by the logical equation
where
is the value of bits of the binary-decimal number in Rg5. The number of members of Equation (2) can be selected from the range
.
The theoretical average value of the pulse frequency at the PPSG output is determining from the following Equation [
16]:
where
G is the control code,
is the clock pulse frequency.
2.2. Output Signal Parameters and Internal Parameters of the Generator and Their Relationship
The main parameters of the output pulse sequence were as follows:
average value of frequency ;
value range of ;
step of frequency changing −;
the repetition period of the pulse sequence;
compliance of the pulse sequence with the Poisson distribution law.
The parameters of the output pulse sequence were determined by the following internal parameters of the generator (
Figure 1):
the number of decades of the MAFG structural elements;
initial settings of the registers Rg1–Rg5;
number of members of Equation (2) involved in the implementation of the logic scheme LS.
The three internal parameters were clearly defined:
Based on the principle of PPSG construction, it could be argued that the repetition period of the output pulse sequence was equal to the repetition period of numbers in the MAFG output.
The repetition period and statistical characteristics of the sequence of numbers in the MAFG output determined the compliance of the output PPSG pulse sequence with the Poisson distribution law. However, that compliance significantly depended on the average frequency of an output sequence , whose theoretical value was determined by Equation (3) and, therefore, depended on the correlation between control code value G and value . In fact, when value G was approaching the value , then was approaching the clock frequency and, under such conditions, the output sequence started losing its pseudo-random properties. From the other side, the lower the frequency of the output sequence , the greater the time interval needed to be to determine its statistical characteristics. In this case such an interval should not exceed the repetition period of this sequence. Thus, in principle (theoretically), the original PPSG pulse sequence might conform to the Poisson distribution law for arbitrarily small average values , however, the sequence repetition period should be of a sufficiently large value. As a limit, if the average value went to zero, the repetition period should go to infinity.
These statements were practical in nature, satisfied most PPSG applications, and are confirmed below by specific calculations and simulation results. Theoretically, a more general approach to determine the correspondence of the output sequence to the Poisson distribution law could be considered, taking into account the value of the average repetition frequency, repetition period, observation time, and the chosen method of estimating statistical characteristics. However, such an approach needs to be refined to be applied in practice.
Taking into account the above, the average frequency , the range of its values, and the step change could be calculated theoretically using Equation (3). The real values of these quantities were determined as a result of simulation and/or experimentally.
2.3. Estimation Method for Statistical Characteristics of the Output Signal
This research was carried out using a generalized method of studying the parameters of the output PPSG pulse sequence for compliance with the Poisson distribution law using Pearson’s test [
32].
In accordance with the proposed method, the flow of input pulses of the PPSG was divided into
n equal groups, each of which consisted of
pulses. The maximum number of groups was
. The groups of input pulses corresponded to the groups of output pulses with the number of pulses
. The proposed method was based on the classical testing method of the hypothesis of the distribution of the general totality according to Poisson’s law using Pearson’s criterion (
criterion) [
32,
33,
34]. In this case, taking into account the specifics of the PPSG construction, the following additions were proposed:
we fixed nominal (theoretical) average value of numbers , regardless of the control code value G;
the value was variable, depended on the value G, and was determined by the equation
As a result of the application of this method we obtained the value
. According to the tables of critical distribution points of
[
33,
34], according to the selected level of significance
(usually
is assigned one of the three following values: 0.1; 0.05; 0.01), the number of degrees of freedom
k could be obtained using the critical value
. If
there was no reason not to accept the hypothesis that the pulse flux corresponded to the Poisson distribution law.
When determining the statistical characteristics of the PPSG output signal in the range of values of the control code G, it was useful to average the last (current) h values of . Obtained by such a way, variable was comparable with . The averaging of the values was necessary for a certain “smoothing” of the results. Based on the simulation experience, one could select value , which could be changed if needed for a clearer (more integrated) determination of the control code range G, in which the output pulse sequence corresponded to the Poisson distribution law.
When designing a PPSG, it is also useful to pre-determine the statistical characteristics of the number sequence, in this case at the MAFG output. This could be achieved using standard statistical tests, such as NIST statistical tests [
22,
23,
24,
25,
26,
27,
32,
35].
2.4. Defining the Limits of the Range of the Control Code Values
Lower and upper limits of the control code values G, in which the statistical characteristics of the output pulse sequence corresponded to the Poisson distribution law, could be determined based on the following.
The sequence evaluation time should not be longer than its repetition period
. That is, based on the above methodology, the following inequality must be satisfied:
From Equation (4) and inequality (5) we obtain
This meant that the value
was the smallest integer number satisfying Inequality (6). As a result of PPSG simulation, it was found that the value
satisfied the following condition:
In this case the value of the coefficient s was determined separately for a concrete number of MAFG decades q, and depended on the initial settings of the registers Rg1–Rg5, the number of involved members of Equation (2) and, under certain conditions, was close to 0.1.
3. Results
3.1. Investigation of the PPSG Based on MAFG When
3.1.1. Determining the Repetition Period of the MAFG
At a fixed number of decades, the MAFG repetition period of a pseudo-random sequence of numbers in its output and, thus, the repetition period of the pulse sequence in the output of the PPSG, also depended on the number of involved members of Equation (2) and from the initial settings of the registers Rg1–Rg5.
The performed investigations showed that the initial settings of the registers affected the statistical characteristics of the output sequence. The values of these settings obtained as a simulation result, when the statistical characteristics were satisfactory, is shown below.
Dependence of the repetition period
on the used number of members from Equation (2) was significant. Some confirmed results are presented in
Table 1, which were obtained for such initial states of registers Rg1–Rg5, correspondingly 1, 0, 0, 0, 0.
Optimization of equation choosing for the output signal LS was a separate partial task requiring additional research. Its solution would also affect the speed of the generator.
3.1.2. Determination of Statistical Characteristics and the Range of Values of the Control Code
Figure 2 illustrates the investigation results of PPSG statistical characteristics based on the MAFG for
.
Here the following notations were used:
The results were obtained at the following values of the method parameters for evaluating the quality of the pulse sequence: , , .
The output signal of the logic circuit of the LS was formed by the following expression:
as a result of the search for initial states of various variants of registers Rg1–Rg5, it was found that the value of these settings was satisfactory—
G, 0, 0, 0, 0, correspondingly. That is, the option in which the initial settings depended on the control code. This was for such initial settings for which results are presented in
Figure 2.
Thus, the range of the control code values G −
, in which the original pulse sequence corresponded to the Poisson distribution law, in this case (when
), was determined by the equation
In this case, the value
, determined as a result of simulation, coincided with the value
, defined theoretically by the expression (6):
3.2. Dependence of the Average Value of the Output Signal Frequency on the Control Code
This section is divided by subheadings. It should provide a concise and precise description of the experimental results and their interpretation, as well as the experimental conclusions that can be drawn.
Figure 3a illustrates the dependence of the average frequency of the PPSG output pulse sequence on the control code
G, while
Figure 3b illustrates a fragment of that dependence.
Here solid lines show the dependences obtained by simulation, and dotted lines show theoretical values, calculated on the basis of Equation (3). Solid and dotted lines in
Figure 3a almost coincide. To specify the calculations, it was accepted that
. All real dependences were obtained for the condition of formation of the LS output signal correspondingly with logical Equation (8) and explained initial states Rg1–Rg5:
G, 0, 0, 0, 0, correspondingly.
Thus, the dependences of the values of the average frequency of the output pulse sequence of the generator from the control code, obtained as a result of simulation, were close to theoretical. That practically allowed the use of Equation (3) while determining the average frequencies of the PPSG output signal.
3.3. Investigation of the PPSG Based on MAFG When
3.3.1. Determining the MAFG Repetition Period
Dependence of the repetition period
on the number of involved members of Equation (2) is presented in
Table 2. The following initial states of the registers Rg1–Rg5, correspondingly 1, 0, 0, 0, 0, were obtained.
3.3.2. Determination of Statistical Characteristics and the Range of Values of the Control Code
Investigation results of the PPSG statistical characteristics based on MAFG for
are presented in
Figure 4.
The results were obtained for the same values of the parameters of the quality assessing method of the pulse sequence, as in the previous case (for ): , , .
Output signal of the logic scheme LS was formed according to the following expression:
where the initial settings of registers Rg1–Rg5 were correspondingly the following:
G, 0, 0, 0, 0.
Control code range values
G −
, in which the output pulse sequence corresponded to the Poisson distribution law, in this case (when
), was determined by the following equation.
In this case the value
, determined as a result of simulation, coincided with the value
, determined theoretically by Expression (6):
3.4. Dependence of the OUTPUT Signal Frequency Average Value on the Control Code
Figure 5a illustrates the dependence of the output of the PPSG’s pulse sequence average frequency on the control code
G, while
Figure 5b shows a fragment of this dependence.
Here, similarly to
Figure 3, solid lines illustrate the dependences obtained by simulation, and dotted lines illustrate theoretical values, calculated on the basis of Equation (3). Solid and dotted lines in
Figure 5a almost coincide. It was accepted that.
. All the real dependences were obtained under the condition of LS output signal formation according to the logic of Equation (11) and the above-justified initial states Rg1–Rg5:
G, 0, 0, 0, 0, correspondingly.
The fundamental difference between the dependencies presented in
Figure 3 and
Figure 5 is that they were obtained using different values for the control code step changing
G −
: in
Figure 3 (for
) when
; while in
Figure 5 (for
) when
.
A decrease in value
for
, led to some ambiguity in establishing the average frequency value of the PPSG output sequence. This is illustrated in
Figure 6 where the dependences were similar to the dependences in
Figure 5, for
.
3.5. Comparing PPSG Characteristics Based on MAFG for and
Increasing the number of decades of the generator could significantly increase the repetition period of the sequence of numbers in the MAFG output and, thus, also the pulse sequence period of the PPSG output. However, this did not lead automatically to an increase in the generator’s “distinguishing ability” concerning the established value of the output sequence average frequency , which was actually setting the ability to specify the changing step .
The performed research showed that “distinguishing ability”, at a fixed value for the number of decades q, depended on the initial settings of the registers Rg1–Rg5 and on the involved members of Equation (2), which determined the logic of signal generation on the output of the LS scheme. In this case, the statistical characteristics of the original sequence depended on these parameters. Taking into account the above considerations, improving a generator’s “distinguishing ability” could be the subject of a separate study.
As far as increasing the number of decades from to , during the above-mentioned conditions, in fact did not lead to an increase in the PPSG’s “distinguishing ability” (decreasing ) and expanded the range . For future work, it would be worth considering the possibility of the practical use of this generator when .
3.6. Using the PPSG Based on the MAFG When
Figure 7 shows the structural scheme of the device, in which to expand the range of average values of the output frequency, an additional frequency divider FD was introduced.
At the FD output the clock pulse sequence was formed for the PPSG, the frequency of which was determined by the equation
where
is the division factor FD and
is the reference generator frequency.
Some generator parameters are presented in the
Table 3, of which one is presented in
Figure 7, when
= 1 MHz. The PPSG was implemented based on the MAFG for
, while its internal parameters corresponded to the above: the output signal of the logic scheme LS was formed by the expression (8), and the initial states of the registers Rg1–Rg5 were
G, 0, 0, 0, 0, correspondingly. This allowed us to tell whether the statistical characteristics of the output pulse sequence in the given ranges of values
, corresponded to Poisson’s law of distribution.
The construction of PPSGs based on MAFGs, all elements of which, except those of the LS, work in binary-decimal code, improves significantly the quality of the output sequence. This was confirmed by a generalized technique for studying parameters of the PPSG output pulse sequence for compliance with the Poisson distribution law using the Pearson test. Investigations of the proposed solutions illustrated that the dependences of the average frequency values of the generator’s output pulse sequence from the control code, obtained as a result of simulation, were close to the theoretical ones. It was shown that the number of decades was enough to choose , because greater numbers of decades did not actually lead to an increased “distinguishing ability” for the PPSG; while scheme realization would be more complicated in that case. In order to expand the output frequency average values the introduction of a division factor into the PPSG structural scheme was proposed, which would be divided by the frequency of the reference generator. The question of the selecting number of the equation members to calculate the logical variable a, the value of which was obtained at the LS output, was rather significant. The number of data members of the equation and approaches to their choice significantly affected the size of the repetition period . Further research is needed in this direction in order to improve the initial characteristics of the PPSG and increase its performance.
5. Conclusions
As a result of this research we executed the modeling of bit templates for information-processing electronic device authentication on the basis of the pulse Poisson sequences generator. For the purposes of the study, the Poisson pulse sequence generator was developed based on a modified additive Fibonacci generator. The developed generator had improved statistical characteristics for the output pulse sequence and expanded capabilities for solving specific practical problems.
The proposed simulator scheme contained two generators. The generator for the value of the control code G = 10,000 formed a bit sequence A, fragments of which had properties of the real-time templates of each device. The generator for the value of the control code G = 100,000 formed a bit sequence B, fragments of which reflected the difference between the series of real-time templates of different devices. In the bit template of the device, these properties were preserved by applying the action of the direct sum of fragments of sequences A and B.
An imitation experiment to generate templates for two devices confirmed the effectiveness of the proposed simulator. The properties of the generated bit templates allowed them to be used for the purpose of unambiguous authentication of information-processing electronic devices.
Further research will focus on protecting such bit templates from a variety of attacks. From the authors’ point of view, the direction of detecting acoustic traps in speech recognition systems is also promising for the application of Poisson pulse sequence generators [
36,
37].