True Random Binary Sequence Generator for Secure Communications

Slavko Šajić1, Branislav M. Todorović2, Nebojša Maletić3

Abstract – The true random binary sequence generator provides a secure, non-reproducible source of true random binary sequences for applications with strong security requirements, such as for generating encryption keys and for secure communications. In this paper we present a realization of true random binary sequence generator. Thermal noise found in chaotic physical system is used as a source of true randomness. NIST tests for verification of randomness are applied. Some experimental results are presented.

Keywords – Secure communications, cryptography, true random sequence, generator

I. INTRODUCTION

Random binary sequences are widely used in the field of secure communications [1], [2]. There are two different types of random generators used: pseudorandom and true random binary sequence generators.

A pseudorandom binary sequence generator (PRBSG) generates sequences of bits with properties that are approximately close to random. [3], [4]. Some of them are realized by using shift registers [3], while the others are algorithmic based [5], [6]. Pseudorandom binary sequence is completely determined by a relatively small set of initial states and feedback network or recursive relation. Since pseudorandom binary sequences exhibit a behavior that relies on a finite number of states and transitions between those states, they cannot produce true random outputs as they are finite state mechanisms. Pseudorandom sequence is deterministic and after N elements it starts to repeat itself, where N denotes the period of the pseudorandom sequence. Pseudorandom binary sequences are important in practice for simulations. Also, they are widely used in spread spectrum communications [2] as well as in cryptographic applications [7]. An important class of pseudorandom sequences is, so called, class of chaotic sequences [8]-[12]. Although there is no universally accepted mathematical definition of chaos, a commonly used one for chaotic sequence says that it is random-like deterministic sequence which is generated

1Slavko Šajić is with the Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina, E-mail: sajic@etfbl.net

2Branislav M. Todorović is with RT-RK, Institute for Computer Based Systems, Fruškogorska 11, 21000 Novi Sad, Serbia, E-mail: branislav.todorovic@rt-rk.com

3Nebojša Maletić is with the Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina, E-mail: nebojsa.maletic@etfbl.net

sequentially by using a mapping function Xn+1 = f(Xn) and an initial value X0, but whose distribution looks like white noise [13].

Chaotic sequence has merit that knowing only two information, namely: a mapping function and an initial value, the same sequence can be regenerated. These sequences are also widely used in spread spectrum communications and cryptography [13]-[16].

Although some PRBSG produces sequences which pass all statistical pattern tests for randomness, they cannot be claimed as true random binary sequence generator (TRBSG) [17].

In the field of cryptographic applications, the need for true random binary numbers arises as modern communication systems increasingly employ electronic transactions and digital signature application for authenticity. It is of high importance to secure privacy during these operations. That was the reason for developing true random binary sequence generator, which should indicate high unpredictability for usage in encryption for digital communications. A true random binary sequence generator is a hot topic in the last years [18]-[23].

It is a postulate that true random numbers cannot be generated mathematically. Hence, computer algorithms cannot be used for that. The generation of true random binary sequences based on non-deterministic physical mechanisms is of paramount importance for cryptography and secure communications.

We refer to true random binary sequence generator as random engine built on microscopic phenomena such as thermal noise or other quantum phenomena. In other words, TRBSGs can be realized on the basis of physical processes which are based on the laws of quantum mechanics, such as radioactive decay, photon emission, thermal noise, radio noise, etc. These physical processes are theoretically unpredictable in practice. The unpredictability is justified by the chaos theory [8]. This theory suggests that even though microscopic phenomena are deterministic, real-world macroscopic systems evolve in ways that cannot be predicted in practice because one would need to know the initial conditions at microscopic level to an accuracy that grows exponentially over time.

In last few years, different approaches to true random binary sequences generation have been explored, as well as for finding test to check their randomness.

In this paper we present one realization of true random binary sequence generator. As a source of true randomness we use thermal noise. In order to examine randomness of binary sequence generated by the proposed generator, NIST set of tests [24] is used. Some numerical results are presented.

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II. A METHOD FOR TRUE RANDOM BINARY

SEQUENCE GENERATION

Generating encryption keys from a pseudorandom source constitutes a security risk that can be removed with true random binary sequence generator. TRBSGs are very important for cryptography and secure communication. To be used in cryptology, random binary sequence generators must meet strict requirements so potential attacker and those who know design of RBSG can not be able to predict the output of RBSG. TRBSG generates an infinite sequence of mutually independent bits. When generator is restarted, it never reproduces earlier generated sequence. There is no mathematic algorithm which can be used to describe random sequence of bits.

A method for true random binary sequence generation using a thermal noise source is given on Fig.1.

Fig.1. True random binary sequence generator

By its nature, thermal noise is random process with zero mean and Gaussian distribution of amplitudes. Generated noise is amplified with wideband amplifier. Since the thermal noise has a relatively uniform spectral power density over wide frequency range (up to 1011 Hz), it is desirable that amplifier’s bandwidth be as large as possible. Amplified noise excites a ultrafast comparator with zero decision threshold. At the output of comparator binary sequence of random bit duration is obtained.

The following assumptions are made: (1) number of passes through zero in an interval is independent on the number of passes through zero in some other non-overlapping interval, (2) probability of more then one pass through zero in an interval is infinitely small relative to single pass in the same interval, and (3) if in some interval τ pass through zero appears, then it is equal to the probability that the same pass occurs in a subinterval dτ of τ. Number of passes through zero in time interval τ is described by Poisson’s distribution and autocorrelation function of binary signal at the output of ultrafast comparator is as follows [25]:

ceAR 22 , (1)

where A is signal amplitude and c average number of passes through zero per second. In proposed configuration, the average number of intersection with time axis c depends on bandwidth of wideband amplifier and the speed of comparator. Furthermore, autocorrelation function is decreasing by exponential law tends to zero with τ increasing. This means that the random process with autocorrelation function given by (1) has no periodic spectral components.

If binary signal at the output of comparator is sampled every To sec, where 0R To , true random binary

sequence with binary symbol duration of To is generated. For example, if the number of intersections with time axis is c 107 and sampling period is 1 sTo , the value of

autocorrelation function at τ = To is 10-9. So, assumption of mutually uncorrelated samples is justified. Thus, generated binary signal can serve as entropic source.

The proposed scheme can be used to generate true random binary sequences of high bit rates, depending on the speed of electronic circuits that process thermal noise. In applications where the speed is not crucial, proposed generator scheme can use frequency limited noise sources instead of thermal noise source. Autocorrelation of band-limited white noise is a sin /x x function, where the first null is equal to inverse value of doubled maximal frequency in the spectrum of band-limited noise. If the sampling is done at Nyquist rate, samples are mutually uncorrelated. In this case, wideband amplifier and ultrafast comparator are not required. Band-limited noise sources occur in many electronic devices; hence, such configuration of TRBSG is very suitable for practical implementation. Especially convenient are narrowband radio channels where output signal of demodulator, when there is not present RF signal at input, provides spectrally limited noise sources which can be used for true random binary sequence generation. Convenience of this generator is that it does not require additional mapping (digital post-processing) to improve certain statistical properties.

III. MEASUREMENTS RESULTS

There are many different statistical tests, but no one can declare the generated sequence as “random”. It is only possible to conclude weather the sequence is showing some characteristics that the binary sequence generated from the random generator would show. NIST (National Institute of Standards and Technology) set of tests [24] is the most widely used to examine randomness of binary sequence generated by the generator. It consists of 15 different statistical tests, whose names are shown in Table I.

In the proposed configuration, as a noise source output of radio receiver with 125 kHz channel wide is used. Based on the measured value of c (the average number of passes through zero in unit time) normalized autocorrelation function at the output of ultrafast comparator is calculated and shown in Fig.2 (curve 1). Also, autocorrelation function of noise at the output of receiver is shown (curve 2).

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TABLE I RESULTS OF SEQUENCES TESTING BY NIST TESTS

To = 5 µs To = 10 µs To = 20 µs To = 200 µs

Test No.

Test Name P value

Random P value

Random P value

Random P value

Random

1 Frequency 0.888 Yes 0.262 Yes 0.829 Yes 0.928 Yes 2 Frequency Block 1.000 Yes 0.945 Yes 0.684 Yes 0.822 Yes 3 Runs 0.000 No 0.000 No 0.634 Yes 0.593 Yes 4 Longest Runs of Ones 0.000 No 0.029 Yes 0.445 Yes 0.900 Yes 5 Rank 0.470 Yes 0.164 Yes 0.274 Yes 0.867 Yes 6 DFT 0.000 No 0.486 Yes 0.927 Yes 0.340 Yes 7 NonOverlappingTemplateMatching - No - No - Yes - Yes 8 Overlapping Template Matching 0.000 No 0.000 No 0.616 Yes 0.924 Yes 9 Universal 0.000 No 0.639 Yes 0.249 Yes 0.730 Yes 10 Linear Complexity 0.986 Yes 0.831 Yes 0.963 Yes 0.690 Yes

0.000 0.000 0.341 0.963 11 Serial

0.000 No

0.061 No

0.353 Yes

0.974 Yes

12 Approximate Entropy 0.000 No 0.000 No 0.391 Yes 0.724 Yes 13 Cumulative Sums 0.993 Yes 0.505 Yes 0.866 Yes 0.929 Yes 14 Random Excursions - No - No - Yes - Yes 15 Random Excursions Variant ‐  Yes - No ‐  Yes  ‐  Yes 

0 10 20 30 40 50 60 70 80 90 100-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

[s]

Rn()

curve 1

curve 2

Fig.2. Autocorrelation functions of binary signal at the output of comparator (curve 1) and at the output of receiver (curve 2)

The results of NIST tests applied to generated binary signals are shown in Table I. Four different sampling periods

5,10,20,200To s are considered. For each test P value is

calculated and comment about randomness is given. For tests 7, 14 and 15 only comments are shown. P values are not given (-) because these tests have more than one P value (in this case 148, 8 and 18 respectively). Results show better fulfilments of NIST statistical tests with increase of sampling period. If <20 s , value of autocorrelation function does not

provide statistical independence between samples. Hence, random sequences generated with sampling periods To 5 s

and 10To s did not pass some of NIST tests. Sequences

generated with sampling periods To 20 s

s

have passed all

NIST tests. Also, it is noticeable that with decreasing value of

autocorrelation function, P (probability) values as indicators of randomness are generally increasing.

Furthermore, autocorrelation function of noise at the output of receiver (as shown on Fig.2, curve 2) shows that statistical independence between samples can be made if sampling is done in zeros of autocorrelation function which can be the subject of further research.

IV. CONCLUSION

In this paper a method for generating true random binary sequences with thermal noise source is presented. In addition to thermal noise source, this configuration allows the use of other sources of noise. Binary sequence generation rate depends on the bandwidth of the noise and the speed of electronic circuits used for noise processing. Wider noise bandwidth provides faster decreasing of autocorrelation function of signal at the input of sampling device, which allows higher random sequence generation rate. Proposed scheme with 125 kHz band radio channel as noise source without presence of RF signal at receiver’s input is realized. An average number of passes through zero, c, based on which the autocorrelation function of signal at the output of comparator is drawn, is measured. For sequence generation, different sampling periods To are considered. Sequences are tested according to NIST procedure. Sequences generated with sampling periods To 20 have passed all NIST tests.

This value of To corresponds to ( )R 10-3. Finally, the proposed solution is suitable for practical

implementation, especially in case when spectrally limited noise source is available and there is no strict demand on sequence generation rate.

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