**4. Discussion**

*4.1. Structural Scheme of the Simulator for the Authentication Bit Templates and the Principle of Its Operation*

Real bit templates of the internal electrical noise of desktop computers (PC), which are calculated according to the normalized autocorrelation function, have a length of 1000 bits and contain approximately the same number of zero bits "0" and single bits "1". When comparing a pair of real-time templates of one PC, it turns out that they match for most positions. Only a few positions will have inverted bits. The positions of the inverted bits do not match for different pairs of templates. A comparison of the real-time bit noise templates of two different PCs showed much less similarity. The Hamming distances between the noise templates of different PCs were 5–10 times larger than the distances between the real-time noise templates of each PC. The developed Poisson pulse sequence generator made it possible to reproduce these properties.

The generator for *q* = 6 and *G* = 10,000 formed a bit sequence *A*, which contained mainly zero bits "0", and the number of single bits "1" for every thousand bits was an average of 10. The positions of the single bits in each fragment of 1000 bits did not match. Therefore, to simulate the real-time templates of the same device, it was advisable to choose the control code of the generator *G* = 10,000. On average, the Hamming distance between a

pair of such fragments will be 20. If at *q* = 6 the value of the control code *G* = 100,000, for the generated sequence *B*, the number of single bits "1" per thousand bits was on average equal to 100. The Hamming distance between two 1000-bit fragments of sequence *B* will be on average 200. The formation of fragments of bit sequences *A* and *B* is shown in Figure 8.

**Figure 8.** Derivation of groups *A* and *B* of bit sequences with a length of 1000 bits for the subsequent formation of bit templates.

From the beginning, the generation process was set, so the first 1000 bits were discarded for both sequence *A* (*A*0) and sequence *B* (*B*0).

A combination (direct sum) of fragments of sequences *A* and *B* was used to form bit templates. For each electronic device, a reference template was first created, and the real-time templates were compared with it. To form a reference bit template for electronic device *N*, there was a need to combine one 1000-bit fragment of sequence *B*, for example *BN* with one 1000-bit fragment of sequence *A*, for example *A*1, Figure 9.

**Figure 9.** Formation of reference and real-time templates by two simulators, *ti* is the time of template formation.

The fragment AM was used instead of *A*1 to form the real-time template *M* of electronic device *N*. The bit templates of the electronic device *N* were calculated by the expression

$$BT\_N^M = B\_N \oplus A\_M.\tag{15}$$

The structural scheme of the bit template simulator is shown in Figure 10.

**Figure 10.** The structural scheme of the bit template simulator based on the PPSG.

The simulator functions were as follows. First, sequences *A* and *B* were generated and stored in the memory. To generate them, one could use one PPSG, which was started first with a control code *G* = 10,000, and then with a control code *G* = 100,000. Sequences were written to memory. Two PPSGs and two memory blocks were used to illustrate the process of forming and storing the necessary sequences in the scheme of Figure 10. The request Q for the template arrived at the control block, which sent a request to the memory for the required 1000-bit fragments of the *A M* of sequence *A* and *BN* of sequence *B*. Fragments *A M* and *BN* arrived at the adder, where the template *BT M N* was formed.

#### *4.2. Results of the Simulation Experiment*

Comparative analysis of the standard template set *BT*<sup>1</sup> *J* of individual devices J was performed for ten devices. The Hamming distances *H BT*<sup>1</sup> *J* , *BT*<sup>1</sup> *I* between pairs of standard template devices J and I were calculated. The calculated distances led to the following results in Table 4.

**Table 4.** The characteristics of the distance distribution between standard templates.


The results in Table 4 were obtained for the 90 distances, I = 1..10, J = 1..10,I = J. Each template was characterized by the template group—standard templates and real-time templates. For successful device authentication distances within each group needed to be significantly lower than the distances between standard templates. To check the adequacy of the proposed simulator model, two types of comparisons should be performed: the first type compares distances inside of the each group (for the each device), while the second one compares distances between different groups (between the different devices).

Simulation experiments were performed to generate templates for two devices, 10 templates for each. The distances between different templates *M* = *K* of one device *N* (group *H*1, intradistances) and distances between different templates *M* = *K* of two devices *N* = *L* (group *H*2, interdistances) were found, only 90 distances for each group. The distances between bit templates were calculated by the following expressions.

$$\begin{aligned} H1(M, N) &= H(BT\_N^M, BT\_N^K), \\ H2(M, N) &= H(BT\_N^M, BT\_L^K). \end{aligned} \tag{16}$$

The distances for *M* = *K* corresponded to the comparison of the reference template with itself for the same device and the comparison of the reference templates of two devices, and were not taken into account.

The results of calculations using Expression (16) are presented in Figure 11. The left side of the figure shows the distances between pairs of templates for the same device. (intradistances), whose numbers are indicated by columns and rows. The right side of the figure illustrates the distances between pairs of templates for different devices (interdistances).

The threshold value must be set in such a way to provide reliable authentication. For our calculations, as could be seen from Figure 12, the threshold value needed to exceed the maximum distance value of the intradistance group and be less than the minimum distance value of the interdistance group. In that case FRR and FAR were equal to zero.

The results of calculations of the distance distribution of group *H*1 for *N* = 1 and group *H*2 for *N* = 1, *L* = 2 are presented in Figure 12 as histograms.



**Figure 11.** Distances between pairs of templates of the same device (**top**) and two different devices (**bottom**).

**Figure 12.** Histograms of the distance distribution between bit templates of one device (intradistances) and two devices (interdistances).

The characteristics of the distance distribution are shown in Table 5.

**Table 5.** The characteristics of the distance distribution.


The distance between the histograms was 150 bits, providing unambiguous authentication between the two devices. The average values were in good agreemen<sup>t</sup> with the theoretical estimates.

The repetition period for the developed generator under certain conditions was not less than 10<sup>9</sup> (Table 2). This allowed the estimation of the possible number of authenticated devices, which was determined by the repetition period and the length of the fragments of the sequence *B*, which was 106. The same estimate was valid for the number of authentication requests for each of the devices. These estimates determined the class of tasks for which the proposed model could be applied. For example, it could be a large enough network with up to a million devices. If you accept the service life of each device as 10 years, then such a device could be authenticated up to 250 times a day.

Let us compare the obtained simulation results with the existing practice, which uses authentication by internal electrical noise [8]. For comparison, the following parameters were selected: authentication reliability, the number of devices in the corporate network that could be simultaneously authenticated, the bit template calculation time. The results are presented in the Table 6.

**Table 6.** Comparison results by efficiency parameters.


As can be seen, the proposed method in the article provided better performance compared to the practice of authentication by internal electrical noise.
