Information warfare in general and electronic warfare in particular have assumed the paramount role in modern war settings. One of the main techniques to secure electronic communication systems is the spread-spectrum method. The robustness of the pseudorandom codes used in spread-spectrum communications is considered insufficient. The paper examines the creation of an entropy source for use in random bit generators for creating true random sequences for spread-spectrum and cryptographic applications. It is shown that the real source of entropy can only be based on some fundamental physical processes; therefore, the random bit generator is designed on the base of a thermal noise generated by the active impedance in an electrical circuit. It is shown that the characteristics of the entropy source can be brought so close to those of the ideal random sources that it would take a very long time for the observer to detect the difference between the proposed entropy source and the ideal one. The designed bit generator can be used in modern communication links for spread-spectrum and cryptographic applications.
modern battlefields [
Electronic communication systems and networks play a crucial part in
modern
military
applications [
Secondly, spoofing can be highly effective. This approach means that valid messages in the communication links are replaced by misleading or deceptive ones.
Thirdly, an alternative approach to the jamming of communication links is to intercept
transmitted traffic of the enemy and use it for the aims of intelligence instead of simply disrupting
(jamming) those links [
Wide use of fiber optic links to prevent useful signal suppression, as well as to preclude the interception and injection of confusing or disrupting radiations into the communication links.
Use of highly directional antennas (adaptive antenna arrays) with low-level side lobes
creates substantial problems for the enemy for useful signal detection and jamming [
Use of encryption to encode sensitive messages, so that they cannot be compromised.
One can observe that at present, in addition to the listed above, the majority of electronic
communication systems widely deploy different information technologies that prevent spoofing
[
Although fiber optic links are more and more used on the battlefield (to control drones as well
as to get visual information back to the operator), these links cannot be employed in many military
settings yet. Highly directional antennas usually have big dimensions and, therefore, are visible to
the enemy forces and can be easily destroyed [
It is well-known that the widespread utilization of modern encryption methods has mostly
rendered the interception of the opposing party messages very challenging, if not altogether
impossible. As a result, due to the inability to use the enemy’s electronic communications to
intercept the messages and try to deploy spoofing, the only viable approach is to use intensive
jamming techniques [
As a result of the analysis made above, the vast majority of electronic communication systems
commonly use an approach known as spread-spectrum communications [
The nature of spread-spectrum communications is such that the signal is spread over a wide
radiofrequency band by deploying a pseudorandom code [
The opposite procedure involves the process of correlating the received pseudorandom code
with the beforehand stored copy of this code at the receiver’s side [
This correlation process, as the vital part of this spread-spectrum technique, possesses the
advantage that it suppresses interference (non-correlated) by the spreading factor, which can be in
the range of 20 to 60 dB [
As a result, the spread-spectrum technique provides the following benefits for electronic
communication channels [
Strong resistance to interference and jamming [
Multiple spread-spectrum signals transmission through communication channels using the code division multiple access (CDMA) method.
Low probability of intercept characteristics.
Highly accurate range measurements.
The two main types of spread-spectrum methods are known as direct sequence and frequency
hopping [
Direct sequence spreading is more widely used [
Therefore, to secure the integrity of information over electronic communication links for both
spread-spectrum techniques and encryption methods as well, one needs to obtain a code that
would be random and not pseudorandom, as is the case at present [
An efficient source of entropy can only be based on some fundamental physical process, that is, an immutable natural phenomenon, and not on some technological achievement. For example, the thermal noise voltage at the ends of a conductor (resistor) is generated by a fundamental natural phenomenon, while the noise voltage determined by the effect of avalanche breakdown of a reverse-biased p-n junction is the result of a technological achievement.
Let’s consider a noise generator based on thermal noise.
One should not forget that the thermal noise voltage is created by the active component of the complex resistance of any circuit, regardless of how many resistors there are and whether there are any at all. The thermal noise electromotive force (EMF) et is equal to (in modern notation) et=√ 4 kTRB , where k is the Boltzmann constant (1.38e–23), T is the temperature in degrees Kelvin, R is the active component of the circuit resistance in Ohms, and B is the noise bandwidth in Hz.
Typically, a thermal noise generator contains a resistor and an amplifier. The equivalent circuit of such a generator is shown in Fig. 1. In this figure, R is the active component of the circuit resistance, generating the EMF et; ea is the source of the amplifier’s noise voltage; i a is the source of the amplifier’s noise current; KU is the amplifier’s voltage gain (we assume the input resistance of the amplifier to be infinite); Uo is the amplifier’s output voltage. For the noise at the output to be as random as possible, one needs to minimize the noise figure (NF), which determines the contribution of the amplifier’s noise to the output signal:
NF =1+
Pa , Pt where Pa is the amplifier’s noise power, Pt is the thermal noise power generated by the active component of the input circuit resistance. Considering the bandwidths of all the noises and the gain for all the noises to be the same, we can express NF through the parameters of the noise sources as follows:
It is important to consider several possible options and, as a result, to choose the best practical realization among them for the equivalent circuit shown in Fig. 1. Let us remind ourselves here that et in this figure is formed by the active component of the complex resistance of the amplifier input circuit, regardless of its configuration.
One should consider several circuit diagrams of a generator based on thermal noise. In all of them hereinafter, R is the noisy resistor, Rf is the resistor in the op-amp feedback circuit, et is the source of thermal noise EMF, and Uo is the op-amp output voltage. The first circuit diagram is shown in Fig. 2. For this circuit, we consider the resistance Rf to be significantly higher than the resistance of the noisy resistors and, since the EMFs of the resistor’s noise et are independent, then .
In this case, Uo remains unchanged.
Let us consider the second circuit, shown in Fig. 3. . that is, Uo will decrease by the root of 2 times.
In the third circuit, shown in Fig. 4, the output voltage of the op-amp can be expressed as follows: √2 et Rf ,
Ri
In this case, by doubling the value of the resistor R, we obtain the following: that is, Uo will increase by the root of 2 times.
Let us consider the fourth option, in the form of the circuit diagram shown in Fig. 5. The output voltage of the op-amp in Fig. 5 can be expressed in the following way: , that is, Uo will increase by 2 times.
Let us consider once again the equivalent circuits of the noise generators taken with account of the amplifier noise. There are two possible options. The first one is shown in Fig. 6. In this circuit, R is the active component of the circuit resistance, generating the EMF et; ea is the source of the amplifier’s noise voltage; ia is the source of the amplifier’s noise current; KU is the amplifier’s voltage gain (it is assumed that the amplifier’s input resistance to be infinite), and Uo is the amplifier’s output voltage.
For the noise at the output to be as random as possible, that is, so that the amplifier’s noise is represented in the output voltage of the op-amp to a minimum degree, one needs to minimize the noise factor NF, which determines the contribution of the amplifier’s noise to the output signal in the following way:
NF =1+
Pa P ,
t where Pa is the amplifier’s noise power, Pt is the thermal noise power generated by the active component of the input circuit resistance.
Considering the bandwidths of all noises and the gain for all noises to be the same, we can express NF through the parameters of the noise sources as follows: e2 a +i2a R
R NF =1+ .
4 kT
And if, given ea and ia, we have chosen the value of R that minimizes NF, then knowing the required value of Uo, we can calculate the required gain coefficient KU.
Let us consider the second variant of the equivalent circuit that is shown in Fig. 7. In Fig. 7, R is the resistance of the noisy resistor generating the EMF et, Ri is the input resistance of the amplifier generating the EMF ei.
According to Ohm’s law for the complete circuit, the maximum power from the source R can be received under the condition Ri = R.
Thus, the circuit of the noise generator shown in Fig. 2 should be considered the best option, in which (with sufficiently large values of the resistance of the resistors Rf) the equality of the active impedances of the source and the receiver of the noise signal is observed. This is true because the resistors R in the circuit of each input simultaneously represent the output resistances of the thermal noise sources and the input resistances of the amplifier.
At a sampling frequency of Fs, the sampling theorem is valid not only for signals whose spectrum is limited by frequencies from 0 to Fs/2, but also for signals in bands from Fs/2, to Fs, from Fs to 3Fs/2, and so on. These bands are called Nyquist zones: 1st zone, 2nd zone, 3rd and so on.
Modern analog-to-digital converters (ADCs) are technologically catching up with this capability and allow working not only in the 1st zone but also in zones with much higher numbers. For example, assume that we need to process a signal with a bandwidth of 30 MHz, in the frequency range from 450 to 480 MHz. To achieve this, we need to install a bandpass filter at the ADC input that selects the required frequency range of 450–480 MHz, and this filter has to operate at a sampling frequency of 60 MHz in the 16th Nyquist zone as if this range lies in the region from 0 to 30 MHz. It is only necessary to remember that for odd Nyquist zones, the signal spectrum remains unchanged, and for even ones it is mirrored, that is, when the frequency of the original signal changes from the lower boundary to the upper boundary of the zone, the frequency of the resulting signal presented in the first zone will change from the upper boundary to the lower. In this case, a one-to-one correspondence is maintained between the analog signal as a continuous function and the signal presented (restored from) by digital samples (with an accuracy of up to the Nyquist zone).
Now we will violate the sampling theorem. Let us take a signal with a band from 0 to 480 MHz and digitize it without any filters with a frequency of Fs = 60 MHz. We will obtain the sum of the signals in 16 Nyquist zones, and the components of the sum from the even zones will be mirrorreflected. This phenomenon is called the aliasing effect and is considered harmful.
The power of the resulting signal will be equal to the sum of the power of the components of all Nyquist zones, and the one-to-one correspondence between the analog signal and the sample values will be irreversibly lost.
Fig. 8 and Fig. 9 show the graphs of a continuous noise signal with a bandwidth of 30 MHz and a signal with a bandwidth of 480 MHz, respectively (the bold dots show the samples with a sampling frequency of 60 MHz).
If we take a broadband signal from a physical noise source and subject it to analog-to-digital conversion with a sampling frequency significantly lower than the upper frequency of the noise signal, we get the aliasing effect.
What it gives one is as follows: 1. Everything is normalized, that is, summing up many independent random variables, even not quite normal ones, leads to a normal distribution of the probability density of the resulting signal. 2. Not quite white noise is whitened to the degree of averaging.
The noise power increases by n times compared to using a narrow-band analog filter in the 1st Nyquist zone.
The correlation between the original signal samples decreases to a power of times.
The main inference is that the signal, possibly not ideal, is idealized to a power corresponding to the number of steps of averaging. In practice, 16 of them is more than enough.
Let us there be two independent binary sequences a and b with probabilities of zero and one, respectively, p0, p1 and bias d = |p0–p1|. Then, when summing modulo 2 sequences a and b, the bias of the probability of zero and one for the summed sequence will be equal to 2d2. In the general case, the expectation of the bias of the sum of m independent sequences with the same biases will have the order of the m-th power of the expectation of the bias of one sequence.
For the confidence probability β and the number of elements in the sequence N, we define the admissible interval ε, into which the bias d must fall when p0 = p1, as, ε =
To secure that with probability β the shift dm of the sum of m sequences of length N does not go beyond the permissible interval ε, i.e. to satisfy the condition dm<ε, we can define the minimum required number m of summed sequences as ln d m=⌈ ⌉
ln ε .
Below are the values of m for a range of values of β, d, and N. Individual ADC digits can be selected as independent sequences for summation (to eliminate possible correlations between digits, we will delay the values of different digits by a different number of clock cycles). If an m-digit ADC is used, then the estimate of the value of the permissible interval ε for a given value of d will be:
For example, for an 8-bit ADC with m = 8 and d<1%, we obtain an interval estimate of ε = 1e–16.
According to the law of the iterated logarithm (the limit law of probability theory), the following condition may be violated for some sufficiently large n if the sequence differs from the ideal Bernoulli sequence ε =exp ( m⋅ln d ) . xn< √( 2 lnln n )/ n
. ,
Here xn is the estimate of the deviation of the probability of the sequence values from the expectation value of 0.5 in n experiments, equal to n−1 ∑ ( bk−0 . 5 ) x = k=0
n where bk is the k-th bit of the sequence.
Our degree of difference is the value xn, comparable with the boundaries of the interval ε = 1e– 16, by the value of which we can easily calculate n = 1.0e33.
If our generator produces random bits at a rate of 60 Mbit/s, then the time to detect the difference between our sequence and the ideal Bernoulli sequence will be:
1.0e33/60000000/3600/24/366=5e18 years.
It looks like the result is more than sufficient.
Even for m = 8 and d<5%, we get an interval estimate of ε = 4e–11, for which we calculate n = 1e22 and 1.0e22/60000000/3600/24/366 = 5.2 million years.
So, our proposed entropy source consists of a noise generator in the form of a resistor with an amplifier and an ADC, part of bits of which are delayed for a different number of cycles and modulo 2 added, forming an output sequence of random bits, as is shown in Fig. 10. This is enough, no additional or other elements are required for the entropy source, they will be superfluous.
We have built a block diagram of an entropy source with characteristics close to the ideal ones. Such a source can become the basis of a truly random bit generator, close to an ideal one.
To complete the generator, it is necessary to add power nodes (the power supply of the noise generator must be especially stable (filtered), or without fluctuations; power node control circuits; a noise source quality control node; nodes implementing tests for switching on, initialization, periodic and on-demand operator tests, continuous tests of primary sequences, tests of the output random sequence; nodes for blocking the generator operation in the case of violation of the conditions for its correct functioning; a controller for exchange with the computer (technician). It is also necessary to implement electrical and mechanical requirements for preventing external electromagnetic interference and other possible adverse effects on the generator.
In addition, the generator must be equipped with drivers, a library of application programming interface functions for creating applications, and test software.
Then it will be a fully-fledged non-deterministic generator of truly random bit sequences. These sequences can be used for spread-spectrum techniques in modern robust electronic communication links and cryptographic applications as well. 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.