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Abstract
In this paper, we have fabricated two quantum random number generators (QRNGs) based on different mechanisms. The first one is based on the photon time of arrival and produces high-quality random numbers without the need for post-processing but using expensive equipment. The second one is based on the tunneling effect in a Zener diode and produces random strings with comparable quality but using low-cost equipment. We then evaluated the random sequences from these QRNGs using a set of statistical tests and showed that they are suitable for special applications such as quantum technologies.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Random Number Generators (RNGs) are an essential ingredient for many applications such as telecommunications [1–3], cryptography [4,5], simulations [6,7], games [8], and quantum technologies such as Quantum Key Distribution [9–11], quantum imaging [12,13], and quantum radar [14,15], etc. There are two general types of RNGs based on their mechanism: pseudo-random number generators (PRNGs) which use mathematical algorithms to generate random numbers from an initial seed [16,17], and true random number generators (TRNGs) which extracts random numbers from a physical process [18,19]. There exists a subset of TRNGs that employs quantum phenomena to generate random sequences known as Quantum random number generators (QRNGs) [20]. Although PRNGs produce high-quality random numbers, they are inherently predictable [21]. However, QRNGs produce true random sequences that are unpredictable and incomputable, thus suitable for applications that need unpredictable random numbers such as Quantum Key Distribution, quantum imaging, and quantum radar [22]. There are different types of entropy sources for QRNGs including radioactive decay [23], the quantum mechanical noise in electronic circuits known as shot noise [24], photon arrival times [25,26], quantum vacuum fluctuations [27–29], laser phase fluctuations [30], optical parametric oscillators [31], and amplified spontaneous emission [32], etc.
The Iranian Center for Quantum Technologies (ICQTs) is a newly established institute which conducts researches in quantum technologies [33–35]. Considering ongoing projects (including QKD and quantum imaging), high-quality and low-cost QRNGs were needed. Thus using available equipment, we built two QRNGs using different approaches. We also gathered a comprehensive set of statistical tests [22] and applied them to these QRNGs. In this work, we realized a QRNG based on photon time of arrival which produces high-quality random numbers and does not need post-processing. Like most QRNGs this one is complex and expensive, thus to satisfy our needs we also realized a simple and inexpensive QRNG based on Zener diode which also produces high-quality random numbers. We compared these two QRNGs and showed that the output from the Zener diode QRNG is very close to that of the single-photon QRNG considering its cost.
This paper is organized as follows. In Section 2. we describe the physics behind the randomness and the experimental setup of two different QRNGs realized in this work. In Section 3. we bring the results and compare the QRNGs. Finally, we give a conclusion in Section 4.
2. Materials and methods
2.1 Single photon QRNG
The wave function of a highly attenuated laser state (coherent state) can be represented as a product of all coherent states in time within the coherence time
where $n_b$ is the number of time bins within the coherence time of the laser. We can rewrite the eq 1 in terms of photons generated at different time instances asThe experimental setup (see Fig. 1(a) for schematic and Fig. 1(b) for actual photo) consists of a DFB laser with 1310 nm wavelength and 5 mW output power operating at continuous mode, two variable attenuators, a silicon SPAD, and a time tagger with a resolution of 81 ps. One attenuator is kept fixed while the other one is changed to achieve different attenuation levels.
For extracting the raw data from photon time of arrival, we divided the time segment $T$ into different bin numbers of 4, 8, 16, 32, 64, 128, and 256. We also used different attenuation levels to achieve photon flux of 10, 50, 100, 200, 500, and 1000 kcps and compared the randomness quality of their outputs to find the optimal photon flux. The raw bit rate depends on the number of time bins, photon flux, and extraction method. In our case, the maximum achievable raw bit rate is 8 Mbps.
In the end, we analyzed the output random sequences of length 50 Mb and 1 Gb, using 5 statistical tests including Dieharder [36], NIST SP-800-22 [37], NIST SP-800-90B [38], ENT [39], and Borel normality [40,41].
2.2 Zener diode QRNG
A Zener diode is a semiconductor device consisting of a highly doped p-n junction that allows current to flow if a certain reverse voltage -known as Zener voltage- is applied. While at higher voltages the avalanche breakdown occurs, at lower voltages the reverse conduction occurs due to electron quantum tunneling known as the Zener effect. Thus, if one applies the Zener voltage to a Zener diode, the current fluctuates randomly as the quantum tunneling is a random process [20].
In this paper, we realized a simple, compact, and low-cost QRNG based on Zener diode as an alternative for expensive single photon QRNG. The experimental setup (Fig. 2) consists of a Zener diode, a common emitter amplifier to enhance the quantum noise and an 8-bit Analog-to-Digital Converter (ADC) with a sampling rate of 25 MSa/s. The raw data is sampled by ADC and sent to the computer for post-processing. The SHA-256 algorithm is used to extract the final random data. The raw bit rate is determined by the ADC speed and raw bit extraction method. In our case, the raw bit rate is about 0.5 Mbps. This can be improved to a few Mbps by changing the extraction parameters. For investigating the quality of the generated output, the same 5 statistical tests were applied to the random sequences of length 50 Mb and 1 Gb.
3. Results and discussion
3.1 Single photon QRNG
First, we investigated the effect of changing the number of time bins on the quality of output sequences at a constant photon flux of 50 kcps. As one can see in Fig. 3, the min-entropy is almost constant and independent of the number of bins. Thus, we should look for other quantities to determine the optimal bin number.
We used two quantities from the ENT test as criteria for choosing the optimal bin number, namely error in Monte Carlo value for Pi, and Serial correlation coefficient. The Monte Carlo value for Pi is a measure of performance and Serial correlation coefficient shows any correlations [22,42]. Thus the ideal value for Pi value error and correlation coefficient is zero. As it can be seen in Fig. 4, the minimum error for Pi value (Fig. 4(a)) occurs at 32 and 64 bins, but the correlation coefficient (Fig. 4(b)) of 32 bins is far from the zero line. Thus, we choose 64 as the optimal number of time bins. We repeated this for the lowest and highest photon flux of 10 kcps and 1000kcps respectively, and again we found the 64 as the optimal number of bins.
Fig. 4. Monte Carlo value for Pi error (a) and the Serial correlation coefficient (b) of the Single Photon QRNG at the different number of time bins.
Next, we investigated the photon flux on the quality of output sequences. As we can see in Fig. 5 the minimum Pi value error (Fig. 5(a)) occurs at photon flux of 50 kcps and its correlation coefficient (Fig. 5(b)) is relatively low, Thus we choose 50 kcps as the optimal photon flux.
Fig. 5. Monte Carlo value for Pi error (a) and the Serial correlation coefficient (b) of the Single Photon QRNG at the different intensities (photon counts).
3.2 Zener diode QRNG
As one can see in Fig. 6 when the Zener diode is off, there exists a classical noise, and when the Zener diode is on, the quantum noise is dominant with a Signal to Noise Ratio (SNR) of about 9 dB.
Fig. 6. Quantum and classical noised in the Zener diode QRNG. The Signal to Noise Ratio (SNR) is about 9 dB.
The signal voltage distribution histogram is plotted in Fig. 7 and it can be seen that it follows a symmetric normal distribution.
Fig. 7. The distribution histogram of the quantum signal obtained from the Zener diode QRNG after amplification, where a normal distribution curve is fitted.
3.3 Statistical tests
Now that we have two different QRNGs, we evaluate their output random numbers using various statistical tests. The first test is Dieharder, and its results are presented in Table 1.
Table 1. Results of Dieharder tests for random sequences with 1Gb length.
Dieharder is a hard test to pass and needs large amounts of random data, nonetheless both QRNGs passed it. The next famous test suite is NIST SP800-22 whose results are listed in Table 2. Here we used the significance level of $\alpha =0.01$ and for tests with several P-values, we choose the worst result. Each test is considered passed if the P-value falls in the range $0.01-0.99$.
Table 2. Results of NIST SP800-22 test suite for random sequences with 50Mb length and significance level of $\alpha =0.01$.
As one can see both QRNGs passed the NIST SP800-22 tests. The outputs from both QRNGs also passed the NIST SP800-90B test. Next, we perform the ENT test for optimal configurations. Its results are presented in Table 3.
Table 3. The ENT test results.
As one can see both QRNGs are comparable (or even better) than commercial QRNGs [22], and the results are very close to ideal values. At last, we perform the Borel normality test on the output of our QRNGs to show that they have the necessary condition for algorithmic randomness. The metric value from the Borel normality test and results are presented in Table 4.
Table 4. The Borel normality test results.
As it can be seen from the above results the single-photon QRNG shows a great performance which is very promising for use in QKD setups. The Zener diode QRNG as an inexpensive and simple alternative also shows good performance with close results to the results of the single-photon QRNG.
4. Conclusion
In conclusion, considering our needs for random number sources, we realized two QRNGs; a high quality but complicated and expensive one based on single-photon time of arrival, and a simple and inexpensive one based on quantum tunneling in a Zener diode. By performing five different statistical tests on these two QRNGs we showed that these two QRNGs are comparable to commercial QRNGs and can be used for special applications such as quantum technologies. As the Zener diode QRNG showed a great performance (considering its cost) we decided to develop it using an FPGA to further increase the number generation rate and automate the post-processing.
Acknowledgments
We thank the Iranian Center for Quantum Technologies (ICQTs) and all colleagues who helped us in this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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