High-Speed Noise-Based Random Bit Generator by Removing 1/f Noise with Differential Comparison

Jianzhong Zhang, Xiaoyu Yu, Mingjiang Zhang, Yi Liu and Zhuping Li

(Key Laboratory of Advanced Transducers and Intelligent Control System, Ministry of Education and Shanxi Province, Institute of Optoelectronic Engineering, College of Physics & Optoelectronics,Taiyuan University of Technology, Taiyuan 030024, China)

Abstract: A design scheme of high-speed physical random bit generator is proposed by utilizing a wideband white noise as an entropy source. The difference operation between the wideband noise signal and its delayed signal is done to produce a series of binary code by a differential comparator.The D flip-flop,which is triggered by a clock, samples the output of the comparator. A random bit sequence at rates of up to 720 Mbit/s is obtained after the exclusive-OR operation. The differential comparison on noise signals can effectively eliminate 1/f characteristics of the amplified noise, and correct the probability density distribution deviation of noise signal amplitude. The quality of the resulting random sequence is verified using common tests of statistical randomness.

Key words: differential comparison; noise source; 1/f noise; random bit generator

Random bits or numbers have important roles in cryptographic applications. Good cryptography requires high-quality random numbers. At present, most cryptographic protocols can be implemented by a large number of pseudorandom numbers, which are generated by expanding short seeds into long bit sequences using deterministic algorithms[1]. However, this may pose serious problems in security application because a potential attacker can make useful predictions about output pseudorandom numbers through partial knowledge of the initial state. Thus, to ensure the ultimate security of encrypted data, one must turn to the one-time pad that is theoretically unbreakable. Such a cipher requires a truly random sequence, which is generated from a physical process.

To date, design schemes of random number generator based on all kinds of physical entropy sources have been proposed and demonstrated. These usually could be grouped into two categories. One is the measurement of macroscopic effects of an underlying noise ruled by statistical mechanics.For example, thermal noise[2-3], timing jitter in electrical oscillators[4-5], radioactive decay[6], atmospheric noise[7], quantum noise manifested as shot noise[8], quantum effects in optics[9-10]or laser phase noise[11-12]. The other is the sampling of a strictly nonlinear process such as chaotic electronic circuits[13-15]or chaotic laser[16-20]inherently exhibiting chaotic behavior and sensitivity to initial conditions which are generally unknown or unmeasurable. Among these systems, the relatively simple, reliable and low-cost random number generators are implemented by using noise-based electric devices. Ref.[2] demonstrated that a 10 kbit/s random bit stream was extracted from the amplified noise of large resistors. Ref. [3] reported that random sequences at bit rates up to 1.4 Mbit/s were generated from unpredictable digital signals that were derived from a fundamental noise mechanism. Currently, most commercial random number generators which use resistor or diode noise as the source of randomness can achieve random bit streams at rates of 4 Mbit/s[21-22]. The generation rates of such systems are mainly limited by the bandwidth of the amplifier. The enhancement of the amplifier bandwidth can increase the generationrate,but a large number of 1/f noises are introduced,which will degrade the randomness of the generated random sequence.

In this paper, the amplified white noise from an amplifier is employed as a physical entropy source.The differential comparison on noise signals can effectively suppress 1/f characteristics of the amplified noise. A random bit sequence at rates of up to 720 Mbit/s is finally generated with verified randomness.

1 Experimental Setup

The experimental system used to generate random bit sequences is shown in Fig.1. As the noise source, this may be any source which provides an analog random variable. Here, two cascade amplifiers (TLA-000040G38) are utilized, one of which is used to generate noise signal, and the other is employed to amplify the noise. At the front end of the first amplifier, a 50 Ω terminal is connected to prevent the interference of the external deterministic electromagnetic signal on internal noise components of the amplifier. The output noise signal from two cascade amplifiers is divided into two beams through a T-type connector. These two beams of noise signals are differentially coupled to a 1-bit ADC consisting of a comparator (ADCMP567) and a D flip-flop (MC10EP52), respectively. The binary signal is obtained by comparing the differential logic inputs of the comparator. At the same time, an extra coaxial cable is inserted into an input terminal of the comparator to ensure that two differential input noise signals are uncorrelated. Finally, to improve the randomness of the random sequence, two binary digital signals obtained from the two independent noise signals are combined using Boolean exclusive-OR (XOR) operation. The bit rate of the random sequence is determined by the input clock frequency of the D flip-flop. The clock module (AD9516-1) can generate multi-frequency clock signals, including 120 MHz, 360 MHz, and 720 MHz, with the maximum clock frequency reaching 2.87 GHz. Thus, our random bit generator obtains multi-rate random bit sequence output. In our experiment, the temporal waveforms of the amplified noise and random sequence are observed and recorded by an oscilloscope with a 20 GHz bandwidth and a 40-GS/s sampling rate (LeCroy SDA 820Zi-A). The corresponding radio-frequency (RF) spectra of the amplified noise and random sequence are measured by a spectrum analyzer (Hewlett Packard 8563E, 26.5 GHz bandwidth).

2 Noise Characteristics

The noise signal with peak-to-peak value of about 20 mV is generated when the first amplifier is driven by a DC voltage of 12 V. The output noise is amplified by the second amplifier with the typical parameters, such as 38 dB gain,±2.5 dB gain flatness, and a -3 dB bandwidth of approximately 4 GHz. The final noise signal with peak-to-peak value of approximately 2 V is obtained. The corresponding RF spectra of the noise signals are shown in Fig.2a. The circle and crosslines indicate the intrinsic noise of the amplifier and the background noise of the spectrum analyzer, respectively. The triangle line denotes the amplified noise. Over the frequency range from 0 to 4 GHz, the amplified noise is more than 38 dB larger than the background noise. This means that the amplified noise dominates the noise entropy source in the process of random bit extraction. It can go far beyond the lowest input level of the subsequent comparator to ensure that random bits are generated by differential comparison. In amplifier circuits, there are three common noise sources: thermal noise, shot noise and Flicker noise[23]. Thermal noise and shot noise are spectrally flat noises and referred to as white noises, which exist in high frequency region. Flicker noise is also called 1/f noise and plays an important role in low frequency range. Here, we focus on the RF spectral density of 1/f noise, which can be written as

S(f)=A/fr

(1)

whereAandrare the amplitude and frequency index of 1/f noise, respectively. The optimized parameter,Aandr, can be obtained through the fitting of experimentally measured noise RF spectral density data (fi,Si)i=1,n, which leads to the minimum value of relative errorχ2.

(2)

whereNis the length of the measured data. By utilizing Eq.(2) and the experimentally measured noise RF spectral data in Fig.2a,A=0.000 092 andr=0.51 are achieved. The corresponding fitting line of 1/f noise in the amplified noise is denoted by the circle line in Fig.2b. In order to highlight the 1/f noise, the low frequency part of the amplified noise in Fig.2a is enlarged and shown as the square line in Fig.2b. Over the frequency range lower than 1.1 GHz, the occurrence of 1/f noise can lead to the deviation of 0 and 1 ratio in the generated random sequence. Here, a differential comparison method is utilized to suppress the 1/f characteristic of the noise, and the detailed hardware operation is mentioned in the previous section. The triangle line in Fig.2b denotes the noise RF spectrum of the amplified noise after the differential comparison. We can see that the low components of 1/f noise are effectively eliminated, ensuring the equality of the ratio of 0 and 1.

Fig.2 RF spectra of the different noise signals

In most cases, the amplifier noise has a Gaussian probability density functionp(x)，which is expressed by

(3)

whereμandσrepresent the mean and standard deviation of the noise signal, respectively. For a given constant mean valueμ, the parameterσdetermines the shape of the Gaussian distribution.The larger the value ofσ, the smoother the curve.

Time traces and statistical histograms of the output amplified noises are illustrated in Fig.3. Fig.3b and Fig.3d show time traces of two independent noise signals connected to the differential inputs of the comparator, respectively, where two noise signals have about 7.5 ns delay time. Fig.3a and Fig.3c illustrate the corresponding statistical histograms.Note that the DC component of two noise signals having the peak-to-peak voltage value of approximately 1 V has been isolated off. From the measured statistical histograms, we can see that the two signals have nearly identical amplitude distribution. The solid curve superposed on the measured voltage histogram shows the best-fit Gaussian distribution. When carrying out the fitting, the Gaussian distribution is shifted to a mean of zero, to account for the fact the amplifier is AC-coupled. The best-fit Gaussian distribution is obtained withσ=0.17. Fig.3f and Fig.3e show the calculated difference between the two noise signals and the corresponding statistical distribution of voltages, respectively. Unlike the single noise signals shown in Fig.3b and Fig.3d, the differential voltage has a more symmetric distribution. The shape parameterσof the best-fit Gaussian distribution reaches 0.42. This means that the smoother the distribution curve obtained, the more conveniently analogue noise signals are converted to random numbers.

Fig.3 Time traces and statistical histograms of the amplified noises

3 Generation of Random Sequences

An example of random bit generation at the maximum rate achievable with the amplified noise in this setup is given. The generation rate is 720 Mbit/s, corresponding to a clock with a frequency of 720 MHz. To improve the randomness of single random sequence, the post-processing of XOR operation is required. Fig.4a shows an eye diagram of generated random bit sequence. The measured eye diagram is obviously opening. Fig.4b illustrates random bit patterns with 500×500 bits in a two-dimensional plane. Bits “1” and “0” are converted to white and black dots, respectively, and placed from left to right (and from top to bottom). It can be seen that there are no obvious patterns as we would expect that the ratio of 1 and 0 is roughly equal. To further evaluate the statistical properties of random bit sequences, we used the standard statistical tests for random number generators provided by the National Institute of Standard Technology (NIST)[24]. The tests were carried out using 1 000 samples of 1 Mbit data for NIST tests. The typical results of NIST tests are shown in Fig.5. The test results show that random bit sequences generated from the experiment can pass all of the NIST tests. Besides, we recorded a bit file of 11 Mbit for the ENT tests[25]. The ENT results are entropy=1.000 000 bits per bit (optimum compression would reduce the bit file by 0 percent).χ2distribution is 0.11 (randomly would exceed this value 73.95 percent of the time). Arithmetic mean value of data bits is 0.500 0 (0.5=random). Monte Carlo value for Pi is 3.141 218 037 (error 0.00 percent). The serial correlation coefficient is 0.000 012 (the value is 0.0 if totally uncorrelated).

Fig.4 Eye diagram and random dot diagram of random bit sequence at 720 Mbit/s

Fig.5 Results of NIST special publication 800-22 statistical tests

4 Discussions

In the experiment, the amplified noise is dividedinto two beams:the noise signalV1(t) and its delayed signalV2(t). The two noise signals are coupled to the two differential input terminals of the comparator, respectively. The random bit, 0 or 1, is determined according tothe sign of the differential signalV1(t)-V2(t). To have a low correlation of two adjacent bits of the generated random sequence, two differential input noise signals put into the comparator should be mutually independent. Actually, the correlation between the noise signal and its time-delay signal can be assessed by the correlation coefficient of autocorrelation trace of noise signal. Fig.6 shows the autocorrelation trace of the amplified noise signal. Thex-coordinate, i.e. the delay time, can determine the delay line length between the noise signal and its time-delay signal. From Fig.6, it can be found that to ensure the independence of two noise signals into the comparator, the delay time is at least 1.05 ns and the corresponding delay line length is about 21 cm. The theoretical interpretation of delay time selection between the chaotic signal and its time-delay signal has been reported in previously published work[19].

Fig.6 Autocorrelation function of the amplified noise

Sequences generated at rates of 720 Mbit/s or lower passed the NIST statistical tests, but sequences generated at higher rates did not. The limitation of the ultimate rate is the bandwidth of the amplified noise signal, that is, the gain bandwidth of the amplifier. So, faster rates can be achieved by utilizing the broader gain bandwidth of the amplifier. At present, optical transmission links operating at 10 Gbit/s are commercially available, and systems operating at higher bit rates of 40 Gbit/s or more are under investigation for telecommunications applications. The information encryption of such systems requires high-speed random numbers capable of matching these bit rates. It was recently reported that extremely fast random bits at equivalent rates of several or even a few 100 Gbit/s are generated using chaotic lasers and high-speed ADCs[17,20]. Instead of applying a simple differential comparison, these systems use the output of an ADC in order to produce multiple bits per sample. Here, we also investigated random number generation using a high-speed 8-bit ADC based on the amplified noise. The time traces shown in Fig.3a and Fig.3b were collected on a 20 GHz, 40 GS/s, 8-bit oscilloscope. According to the extraction theory of random bits in Ref.[17], the noise signal from cascade amplifiers and its delayed signal are converted to two digital 8-bit signals, respectively. Corresponding pairs of bits in the two 8-bit digital signals, one of which has the reversed 8-bit order, are combined by bitwise XOR operation. All of 8 bits from each sample are then selected and interleaved to generate a single bit sequence. The resulting sequence at an equivalent rate of 320 Gbit/s (= 8 bit×40 GS/s) is confirmed to pass all of the standard NIST and ENT tests for randomness. Compared with chaotic light sources, the amplifiers utilized will be easily integrated with the subsequent ADC devices. The overall integration of random number generator will be our next research work.

5 Conclusion

In this paper, a fast physical random number generator is implemented by utilizing the wideband white noise as a physical entropy source. The noise signal is converted to a series of binary code by a 1-bit ADC consisting of a differential comparator and a D flip-flop. A random bit sequence, at rates of up to 720 Mbit/s,is finally obtained after the XOR operation. The differential comparison on noise signal is employed to eliminate 1/f characteristics of the amplified noise. In addition, the method can correct the probability density distribution deviation of noise signal amplitude, and prevent common mode electromagnetic interference and the average power drift. The generated random sequence can pass commonly used standard tests for randomness. We further investigate that the amplified noise signal is converted to a binary sequence by the oscilloscope’s 8-bit ADC. All of 8 bits from each sample are interleaved to generate random bit sequence with a higher rate of 320 Gbit/s. High-speed noise-based physical random bit generation will help the development of high-speed secure communications.