Compact quantum random number generator based on superluminescent light-emitting diodes

Random numbers play an extremely important role in daily life and have widespread applications in the fields of science and engineering,¹ such as numerical simulations, lottery games, and cryptography. Based on the computational algorithms, statistically random bits can be obtained from the classical pseudo-random number generators (PRNGs), which have been widely applied in modern digital electronic information systems. However, due to the deterministic algorithms, the random bits produced by the PRNGs are therefore deterministic and not truly random. Thus, the PRNGs are not suitable for certain applications where the true randomness is required,^2,3 for instance, the quantum key distribution (QKD), where true random numbers are essential to guarantee the unconditional security.

Different from the PRNGs, the true random number generators (TRNGs) extract randomness from the unpredictable physical processes.^4–9 Especially, a significant type of the TRNGs is the quantum random number generator (QRNG), which extracts the randomness from the fundamental quantum processes and provides true random numbers based on the measurement of the unpredictable quantum noise.^10–12 Untill now, a variety of quantum entropy sources have been proposed and utilized to develop the QRNG, including measuring the photon paths,^13,14 the photon arrival time,^15,16 the photon numbers,^17,18 the quantum phase fluctuations,^19–24 the vacuum fluctuations,^25,26 and the amplified spontaneous emission (ASE) noise.^27–30 

Thus far, though various QRNG schemes have been demonstrated and some commercial products are available on the shelf, some defects in terms of practical applications cannot be ignored, including the low generation rate, the weak real-time performance, and the large package size. For instance, the commercial QRNG (named “Quantis”) produced by ID Quantique is packed in small size;³¹ however, its real-time generation rate is only 4-16 Mbps, which is not adequate for high-speed applications (e.g., QKD systems). Meanwhile, the generation rates of some QRNG schemes reach up to 10 Gbps, which, however, is not obtained in real-time and is an offline result. Here “offline” means that the raw data are first acquired and stored in a high-performance oscilloscope and then delivered to and processed in a computer to obtain the final random bits, yielding an equivalent random number generation rate. Therefore, the off-line generation rate is only a theoretically equivalent result which can be referred to evaluate the potential of the QRNG and hence cannot be employed in real-time applications. For instance, by measuring the laser phase noise, Zhang et al. proposed a QRNG scheme to get an off-line random number generation rate of 68 Gbps,¹⁹ and a real-time QRNG based on this protocol is proposed to obtain a real-time generation rate of 3.2 Gbps,²⁰ which is a significant work in developing practical QRNGs. However, the real-time QRNG proposed in Ref. 20 contains many different components (e.g., the active feedback control is needed) and is relatively complicated, which increases the total cost and sacrifices the compactness and flexibility for practical applications because of its large package size.

Based on all mentioned above, the generation rate, the real-time performance, and the compactness are three key features that need to be considered primarily when designing and realizing a practical QRNG. In this paper, with the balance for the three features taken into consideration, we design and realize a practical QRNG based on measuring the ASE noise of the superluminescent light emitting diodes (SLEDs). The QRNG is realized and enclosed in a metal box sized 140 mm × 120 mm × 25 mm and a real-time generation rate of 1.2 Gbps is achieved. The final random bit sequences have passed the NIST and DIEHARD statistical tests.

The design block diagram of the QRNG module is depicted in Fig. 1(a). The QRNG module consists of three printed circuit boards (PCBs). The first PCB is designed to obtain quantum noise, which includes a 14-pin butterfly packaged SLED (Inphenix, IPSDD1308) with the home-made current driver and temperature controller. The second PCB is designed to detect and amplify the quantum noise by employing a high-bandwidth photo-detector (PD) (Beijing Light sensing Technology Ltd., LSIPD-A75-B-SMFA) and a low noise microwave amplifier (AMP) (Analog Devices, HMC589AST89E). The last PCB is designed to acquire and post-process the raw random data, which is realized by employing an analog to digital converter (ADC) (Analog Devices, AD9461) and a field programmable gate array (FPGA) (Xilinx, XC6SLX75TFGG484), and then transmit the final random sequences through a USB3.0 peripheral controller chip (Cypress Perform, CYUSB3014-BZXI). All the PCBs are packed in an enclosure sized 140 mm × 120 mm × 25 mm as shown in Fig. 1(b).

The SLED is chosen as the quantum entropy source in our setup, which is a semiconductor device with properties intermediate between a light emitting diode and a laser. According to the luminescence mechanism, the spontaneous emission (SE) of the SLED is coupled to the waveguide cavity and amplified, and its randomness is reflected in the intensity fluctuation of the light field. The output optical intensity of the SLED originates from the SE of massive atoms. According to the quantum mechanics, the SE of different atoms is an essentially independent and random process. Therefore, after the amplification of the SE of massive atoms, the instantaneous output optical intensity of the SLED is also essentially random.

The physical process of the SLED in a 3D framework including SE, waveguide coupling in the Fabry–Pérot cavity, and gain amplification can be modeled by electrical and optical equations.³⁰ For a Fabry–Pérot cavity from 0 to L with the reflectivity r₁ and r₂ on the two faces, respectively, the spontaneous emission power for a Δω interval can be approximated as³⁰ 

where n_sp is the spontaneous emission factor, g_n is the model gain for the lateral mode n, and $ℏ$ is Planck’s constant. From Eq. (1), the emission spectrum is extremely wide; therefore, an approximate Gaussian white noise band can be obtained.

The SLED has very wide response bandwidth (up to 12.5 THz), while the response bandwidth (1.5 GHz) of PD is much smaller than the SLED bandwidth. Therefore, the partial response of the ASE noise by the PD can be considered as white noise.²⁷ By deducing, the power spectrum density of the output current can be approximated as²⁷ 

where B_BP and B_LP represent the 3 dB bandwidths of the PD and AMP, R is the detection efficiency, and S₀ is the spectral density of the input ASE noise, respectively. From Eq. (2), the photocurrent noise power spectrum is Gaussian. The experiments in some published literature show that the intensity noise spectrum of SLED is also Gaussian as in Refs. 21 and 27.

After the detection of the ASE noise from the SLED and the amplification of the photocurrent from the PD, the raw data are acquired by the ADC and the statistical histogram of the raw data is shown in Fig. 2. From this figure, we can see that the statistical distribution of the raw data is approximated to Gaussian distribution, which fits with the results in Refs. 21 and 27.

Due to the presence of classical noise and the imperfections of the device, the randomness of the raw data acquired by the ADC is inevitably impaired. For example, the raw data are probably biased. In order to extract the intrinsic quantum randomness and obtain a true random number sequence, we should further make use of some algorithms to post-process the raw data. In the proposed QRNG, for hardware efficiency, we employ the bitwise exclusive OR (XOR) operation and m-least-significant-bit (m-LSB) method²³ to extract the randomness, which is integrated in the FPGA for real-time processing. In detail, we first perform the XOR operation on the raw data for every 2 samples acquired by the ADC and then based on the min-entropy (H_min) of the data after the XOR operation, we reserve the corresponding m-LSBs (⁠$m=Hmin$⁠) to achieve the final random bit sequences.

The XOR operation can effectively reduce the sequence offset e(n) and the Kth-order autocorrelation coefficient α_Z(k).²¹ We assume that the expectation value for the less relevant or independent binary random variables X and Y is E(X) = u, E(Y) = v, respectively, and the correlation coefficient between X and Y is ρ. We can obtain a new random variable Z by applying the XOR operation on X and Y, i.e., Z = X ⊕ Y, and the expectation value of Z is

Based on the assumption that X and Y are less relevant or independent, ρ should be close to 0 and can be ignored, which means the expectation value of Z can be approximated as E(Z) ≈ 1/2 − 2(u − 1/2) (v − 1/2). Therefore, the bias of Z is

where e_X, e_Y, and e_Z are the bias of X, Y, and Z, respectively. It can be seen that after the XOR operation, the bias of sequence Z is less than either of that of X or Y.

Similarly, the Kth-order autocorrelation coefficient of Z can be obtained as follows:

where Z₀ represents the sequence Z itself and Z_k represents the sequence obtained by shifting the sequence Z to left by k bits, that is, Z_k(i) = Z₀(i + k). Apparently, if the expectation value of the raw sequence approaches 1/2, the autocorrelation coefficient of the new sequence will be significantly reduced after the XOR operation.

Meanwhile, after the ADC sampling, the least significant bits are of better randomness. The explanation is as follows. Supposing X₁, X₂, …, X_n are n random variables, according to the chain rule of entropy,³² we can obtain the Shannon entropy of the sequence,

For the m-LSB method, we can get

Then for m < n, the mean entropy can be obtained as follows:

Compared with the raw random sequence X₁, X₂, …, X_n, the sequence X₁, X₂, …, X_m is of higher mean entropy, which infers that better randomness is obtained after the m-LSB method in terms of the Shannon entropy.

In order to determine how many bits can be reserved for each sample after the XOR operation, the min-entropy, i.e., $Hmin=−log2maxx∈0.1nPrX=x$⁠, is calculated. In our experiment, the typical value of the min-entropy of the sequence after the XOR operation is between 10 and 11, which means that 10 LSBs can be reserved for each sample after the XOR operation. After the XOR operation and 10-LSB method, the final random sequence is obtained and the typical statistical histogram of the final random sequence is depicted in Fig. 3, which shows an approximate uniform distribution.

In the schematic diagram of the proposed QRNG shown in Fig. 1(a), the SLED with a center wavelength of 1310 nm is operated at an optimum driving current of 100 mA and an optimum temperature of 25 °C. After the detection of the PD and the amplification of the AMP, the raw data are acquired by the ADC at a sampling rate of 125 Msps. After the XOR and the 10-LSB post-processing operation, which is integrated in the FPGA, the final random number sequence with a real-time generation rate of 1.2 Gbps is achieved. The autocorrelation coefficient curves of the raw data and the final random sequence are also calculated and shown in Fig. 4, which indicates that the autocorrelation coefficient of the sequence is significantly reduced after the post-processing.

In order to verify the randomness of the final random sequence, two test batteries are employed, the NIST-STS and the DIEHARD. The NIST-STS consists of a set of randomness tests, evaluating the p-value to quantify the randomness of the sequence. It is usually considered to pass a particular test when p-value ≥ 0.01(the default setting of the STS and a common value used in cryptography).³³ The DIEHARD test is considered successful if 0.01 ≤ p-value ≤ 0.99 is satisfied.³⁴ The corresponding test results are shown in Figs. 5 and 6, respectively. The random bits generated by the proposed QRNG have passed all the tests.

To evaluate the stability of the QRNG, the SLED is kept working in the optimum state (temperature = 25 °C, driving current = 100 mA) for a long time. The raw random data and the final random data are both acquired at the working time of 0 h, 24 h, 48 h, and 72 h, respectively. With the raw random data, the statistical distribution and the min-entropy after the XOR operation are obtained, which are shown in Figs. 7 and 8. From the figures, we can see that the performance of the QRNG is significantly stable in terms of the statistical distribution and the min-entropy after the XOR operation. Meanwhile, we employed the NIST and DIEHARD tests to verify the randomness of the final random sequences and the final random sequences acquired at different times have passed all the tests, which infers that the randomness of the QRNG is also validated. Overall, the experimental results indicate that the QRNG can work relatively stable for at least 72 h.

In conclusion, based on measuring the ASE noise of the SLED, we present a compact and high-speed QRNG with a real-time generation rate of 1.2 Gbps. The final random bit sequences have passed all the NIST-STS and the DIEHARD tests. Compared with previous results, the proposed QRNG is advantageous in terms of the real-time performance, high generation rate, and small package size, which indicates significant potential for practical applications. Hence, the proposed QRNG may provide a valuable direction for QRNG engineering research.

This work is supported by the National Natural Science Foundation of China (Grant Nos. 61771439, 61501414, 61702469, and 61602045), the National Cryptography Development Fund (Grant No. MMJJ20170120), the Sichuan Youth Science and Technology Foundation (Grant No. 2017JQ0045), and the Foundation of Science and Technology on Communication Security Laboratory (Grant No. 6142103040105).