Mutually testing source-device-independent quantum random number generator

Jialin Cheng; Jiliang Qin; Shaocong Liang; Jiatong Li; Zhihui Yan; Xiaojun Jia; Kunchi Peng

doi:10.1364/PRJ.444853

Abstract

Quantum random numbers have an incomparable advantage over pseudo-random numbers since randomness originates from intrinsic property of quantum mechanics. The generation rate and the security of quantum random numbers are two significant indicators of a quantum random number generator (QRNG) for practical applications. Here we propose a mutually testing source-device-independent QRNG by simultaneously measuring a pair of conjugate quadratures from two separate parts of an untrusted continuous-variable quantum state. The amounts of randomness of the quadratures can be mutually estimated by each other via entropic uncertainty principle. Instead of randomly toggling between the conjugate quadratures of one state for collecting different types of data, two quadratures can generate check data and raw bits simultaneously and continuously in this mutually testing manner, which enhances the equivalent generation rate of private random bits to around 6 Gbit/s with a 7.5 mW laser beam. Moreover, the overall security is also improved by adjusting the conditional min-entropy in real time according to the continually monitored fluctuations of the local oscillator and the randomly measured electronic noise of homodyne detectors.

1. INTRODUCTION

Simulation, computation, and number theory may have a demanding requirement for the generation rate of the exploited random numbers following the right statistics, while other applications, such as the classical communications based on the RSA cryptosystem, the current quantum key distribution [1,2], signature schemes [3,4], and quantum secret sharing [5], place greater demand on the security or privacy of random numbers. It has been reported that thousands of servers are attacked by inferior hackers for the lousy randomness of pseudo-random numbers every year [6]. It can be argued that the private randomness is a potential necessary condition of any secure and secret communications. Even in daily life, the security and the non-repeatability of random numbers can also be crucially significant. Defective products can pass the quality inspection exploiting random sampling just because the illegitimate manufacturer gets the random numbers for sampling and changes the order of products beforehand. An illegal gambler may get a bigger bang for the buck by receiving partial random numbers and adopting the best guessing strategy. A lottery draw may be rigged by an unlawful manager, who obtains the side information of the random number generator during routine maintenance.

Leaning toward the mainstream viewpoint, random numbers generated by classical systems are all pseudo-random numbers, while only quantum theory can offer true randomness [7,8]. The earliest quantum random number generator (QRNG) is based on radioactive decay [9], which generates true random numbers with a low bit rate taking demanding safety measures. Moreover, the source generating quantum random numbers (i.e., entropy source) can be quantum tunneling [10,11], Majorana fermions [12], quantum fluctuations of the collective spin of an alkali-metal vapor [13], phase noise of a single-mode laser [14], etc. However, there have been demonstrations of random numbers generated on a mobile phone [15], in which the random numbers are generated from resolving photon-number distribution of a camera. Most QRNGs are currently based on quantum optics since different quantum states with inherent randomness offer a rich choice of implementations and complete analytical methods [7,8,16 –25]. The polarization of single photons [26], temporal [27, 28] or spatial mode [29,30] of photons can generate random bits; photon counting [31], amplified spontaneous emission [32,33], and stimulated Raman scattering [34] can also be the entropy source. In recent years, there have been reports of a quantum random number generator with a photonic integrated chip [21] and even a quantum random number cloud platform [35]. Among all quantum resources, measurement of the vacuum noise via homodyne detection is one of the most efficient and economical methods with high-speed and secure bits generation [36 –38]. For a general entropy source, it is far from trivial to estimate the min-entropy. Consequently, there have been many QRNG protocols [7,8]. In practical terms the measurement device can be characterized easily by the user at close range, while the entropy source and local oscillator (LO) can be distributed remotely for the convenience of users. Therefore, the entropy source and LO are more vulnerable to attack. Based on this consideration, the QRNG protocol with an untrusted source and indeterminate LO is needed, i.e., the source-device-independent (SDI) QRNG. In the SDI QRNG, the secure randomness needs to be estimated and quantified in the presence of the eavesdropper attacking the source. Finally, the private random bits can be extracted by the Toeplitz-matrix hashing algorithm from the raw random numbers.

The entropic uncertainty principle (EUP) [39,40] gives a lower bound on the conditional min-entropy that estimates the amount of secure randomness in the presence of the eavesdropper and is used to calibrate the randomness extractor in the SDI QRNG where the entropy source is completely untrusted [41 –43]. A pair of non-commutative operators $\hat{P}$ and $\hat{Q}$ are normally seen as the data quadrature and check quadrature, respectively. Estimating the conditional min-entropy of data quadrature is achieved by randomly switching conjugate quadratures of one state for measuring. Up to now, QRNGs based on the EUP have been realized in discrete-variable system [44,45] and continuous-variable system [41]. Then an SDI QRNG based on generic positive operator valued measurements with heterodyne measurements has been proposed [42]. Very recently, the squeezed state has been applied in the SDI QRNG [43].

In fact, the check quadrature $\hat{Q}$ that estimates the randomness of the data quadrature $\hat{P}$ also contains the secure randomness estimated by the data quadrature $\hat{P}$ . In contrast, the mutually testing QRNG does not require random switching and can further increase the generation rate of private bits. A comparison between mutual testing and random switching can be found in Appendix A.

The implementation of our protocol is shown in Fig. 1. An untrusted coherent state (CS) is divided into two identical parts (CS1 and CS2 with equal power and fluctuation) in which quadrature $\hat{Q}$ and quadrature $\hat{P}$ are severally measured. The entropy source has not been characterized while the equipartition process must be completely trusted. The eavesdropper may replace the untrusted coherent state by other states (for instance, a submode of a two-mode squeezed state) for side information, which results in an inevitable increase of insecure quantum noise on the probably impure CS1 and CS2. The attack on the source can be eliminated via the EUP with a trusted division process. The quadratures are measured using balanced homodyne detectors (BHDs) and all data are collected by an oscilloscope (OSC). The details about the measurement of quadratures can be found in Appendix B. The data of quadrature $\hat{Q}$ measured on CS1 are used as the check data to estimate the data of quadrature $\hat{P}$ measured on CS2, and the quadrature $\hat{P}$ measured on CS2 in turn is used to estimate quadrature $\hat{Q}$ measured on CS1. The data of both quadrature $\hat{Q}$ measured on CS1 and $\hat{P}$ measured on CS2 are used to generate the raw random bits. Eventually the secure random bits are obtained by applying randomness extractors to the raw data, where the extractors are constructed with two Toeplitz matrices calibrated by the quantum conditional min-entropy. It should be noted that the quadrature $\hat{Q}$ of one quantum state generally cannot be used to estimate the quadrature $\hat{P}$ of the others. It is not in accordance with the EUP. However, for a pair of continuous-variable quantum states with equal power and fluctuation, the distributions of corresponding quadratures should be the same, which can be achieved by setting a trusted equipartition process to the unknown source. Therefore, the quadrature of one state can be used to test the conjugate quadrature of the other state and the relation is mutual. All the data of the pair of quadratures can generate random numbers with mutual testing and estimating in real time.

Figure 1.Flow diagram of the experimental structure. An untrusted coherent state (CS) is divided into two identical and probably impure parts, CS1 for measuring quadrature $\hat{Q}$ and CS2 for quadrature $\hat{P}$ . The collected data of one quadrature are chosen as the check quadrature to estimate the conditional min-entropy of the conjugate quadrature of the other state since the dividing process is completely trusted and the distributions of two parts are identical. After two randomness extractors, the secure random bits are obtained. CS, coherent state; OSC, oscilloscope.

When an untrusted state is used as the entropy source, the amount of private random bits based on the EUP is given by [40,41,46 –51] $H_{\min} (P_{δ p} | E) \geq - \log_{2} c (δ q, δ p) - H_{\max} (Q_{δ q}) = H_{low} (P_{δ p} | E),$ (1) $c (δ q, δ p) = \frac{1}{2 π} δ q δ p S_{0}^{(1)} {(1, \frac{δ q δ p}{4})}^{2},$ (2)where $H_{\min} (P_{δ p} | E)$ is the quantum conditional min-entropy of quadrature $\hat{P}$ . The term $c (δ q, δ p)$ denotes “incompatibility” of quadratures $\hat{P}$ and $\hat{Q}$ , and $H_{\max} (Q_{δ q})$ is the max-entropy expressing the user’s lack of knowledge about quadrature $\hat{Q}$ . $H_{low} (P_{δ p} | E)$ is the lower bound on the conditional min-entropy, which bounds the amount of secure and private random numbers. Both the $δ p$ and $δ q$ mean the measurement accuracy of quadratures $\hat{P}$ and $\hat{Q}$ , and $S_{0}^{(1)} {(1, \frac{δ q δ p}{4})}^{2}$ is the 0th radial prolate spheroidal wave function of the first kind [40,41,52]. Similarly, quadrature $\hat{Q}$ can be estimated by quadrature $\hat{P}$ .

In fact, the coherent state before division is an uncharacterized state in our mutually testing QRNG protocol. Therefore, the total amount of extractable randomness of two separated impure states (CS1 and CS2, in Fig. 1) can be represented as $H_{\min}^{(T)} (Q_{δ q}^{(1)} \cup P_{δ p}^{(2)} | E) = H_{\min}^{(1)} (Q_{δ q}^{(1)} | E) + H_{\min}^{(2)} (P_{δ p}^{(2)} | E) \geq - 2 \log_{2} c (δ q, δ p) - H_{\max}^{(1)} (P_{δ p}^{(1)}) - H_{\max}^{(2)} (Q_{δ q}^{(2)}),$ (3)where $H_{\min}^{(T)} (Q_{δ q}^{(1)} \cup P_{δ p}^{(2)} | E)$ is the total amount of extractable randomness, and $H_{\min}^{(i)} (K_{δ k}^{(i)} | E)$ and $H_{\max}^{(i)} (K_{δ k}^{(i)})$ are the conditional min-entropy and max-entropy of quadrature $\hat{K}$ ( $\hat{K} = \hat{P}, \hat{Q}$ ) of the $i$ th ( $i = 1, 2$ ) coherent state, respectively.

Taking into account the finite-size effects and the security of the extractor during the randomness extraction, the bound on the conditional min-entropy should be further lowered as $H_{\min}^{ϵ} (P_{δ p} | E) \geq H_{low} (P_{δ p} | E) - \frac{4}{\sqrt{n_{p}}} \sqrt{\log_{2} (\frac{2}{ϵ^{2}})} \log_{2} [2^{1 + \frac{H_{\max} (Q_{δ q})}{2}} + 1],$ (4)where $H_{\min}^{ϵ} (P_{δ p} | E)$ is the smooth quantum conditional min-entropy, $n_{p}$ is the number of measurements for the quadrature $\hat{P}$ , and $ϵ$ is the security parameter.

It is a remarkable fact that the electronic noise may be controlled by the eavesdropper, so it needs to be measured at random times. Moreover, the fluctuations of the LO need to be monitored since the fluctuations can add insecure noise to the measured noise of the quadratures via an imperfect homodyne detector from the eavesdropper [53]. The collected data consist of the vacuum noise, extra noise introduced by LO fluctuations, and electronic noise of the detection system: $σ_{T}^{2} = σ_{V}^{2} + σ_{LO}^{2} + σ_{E}^{2},$ (5)where $σ_{T}^{2}$ is the total noise contained in the measured data. The electronic noise $σ_{E}^{2}$ can be measured by randomly blocking all the light paths of the detectors according to the random seed. The extra noise $σ_{LO}^{2}$ can be estimated by analyzing the LO fluctuations in real time. The extra noise and electronic noise are eventually regarded as impurity of the vacuum noise.

2. PRINCIPLE AND EXPERIMENTAL SETUP

The schematic of the experimental setup of the mutually testing SDI QRNG is illustrated in Fig. 2. A laser beam at the wavelength of 1342 nm from the $Nd : {YVO}_{4}$ laser (Yuguang Company) passes through a mode-cleaner with a finesse of 300 for spatiotemporal filtering and intensity stability. The output beam is divided into the signal beam and LO via a beam splitter (BS) with reflectivity of about 98%. The intensity of the signal beam is decreased further, and about 2% of the LO is used to monitor the power and fluctuations to resist the attacks from the eavesdropper. The signal beam and LO then are split into two identical parts respectively for data collection via a pair of identical BHDs. The BHD1 (BHD2) was used to measure quadrature $\hat{Q}$ ( $\hat{P}$ ) with the relative phase 0 ( $π / 2$ ) between the coherent state and the LO. The interference signals are detected by the broadband BHDs (PDB480C-AC, Thorlabs) with two photodiodes FD150 (Fermionics Opto-Technology). The LO power and measurement bandwidth for each of the two BHDs are set as 7.5 mW and 1 GHz (from 3 MHz to about 1 GHz), respectively, since the signal-to-noise ratio of the homodyne detector should be maximized and the response is close to uniform in a bandwidth of at least 1 GHz. Then the signals are filtered, amplified, and collected by two 1.9 MHz low-pass filters, two broadband amplifiers, and an OSC. After the min-entropy of the quadrature $\hat{Q}$ ( $\hat{P}$ ) of one coherent state is mutually estimated by the quadrature $\hat{P}$ ( $\hat{Q}$ ) of the other coherent state, the two sets of data are put into the extractor for terminal uniform-distribution random bits. Finally, few random bits are injected into the chopper to measure the electronic noise of the measurement device randomly.

Figure 2.Experimental schematic configuration for mutually testing SDI QRNG. The pink area is a private space that no eavesdropper has access to. The black and blue curves represent the electric and data cables, respectively. The coherent state is generated via a laser and MC. The laser beam is divided into the signal beam and LO via a 98:2 BS. Both the signal beam and the LO are split in half via two 50:50 BSs. Two BHDs are used to measure the quadrature $\hat{P}$ and $\hat{Q}$ of the two coherent states with the phase differences (0 and $π / 2$ ) between the signal beam and LO, respectively. All data are recorded by an OSC, and the post-processing is achieved via a PC. Laser, $Nd : {YVO}_{4}$ ; MC, mode-cleaner; 98:2, 98:2 beam splitter; 50:50, 50:50 beam splitter; LO, local oscillator; BHD, balanced homodyne detector; HR, mirror with high reflectivity; OSC, oscilloscope; PC, personal computer.

3. RESULTS

In our experiment, the electronic noise is basically maintained because there is no interference in the surroundings of the detectors from the eavesdropper. The measured shot noises of two BHDs severally contain ( $5.8 \pm 0.1$ )% and ( $5.9 \pm 0.1$ )% of electronic noise and less than $10^{- 4} %$ of the extra noise caused by the power fluctuations of the LO. The measuring time of electronic noise needs to be randomly chosen; the average duration we set is equal to 1/20 total measuring time. The nominal resolution of the OSC is 10 bits. Taking into account the high-speed dual-channel sampling, we use a conservative bit depth of 6 bits. In a round of our experimental process, we measure the data of $\hat{P}$ and $\hat{Q}$ quadratures, each with 1 million samples obtained. The smooth conditional min-entropies of quadratures are both estimated as $3.13 \pm 0.05$ bits; more details can be found in Appendix C. The randomness extractor is applied to extract terminal random bits from the raw data. Eventually, the equivalent generation rate of private random bits is around 6 Gbit/s.

Figure 3.Red, blue, and black curves show the autocorrelations calculated from the raw bits, the downsampled bits, and the extracted bits, respectively. The three data streams have the same length of $5 \times 10^{7}$ .

4. CONCLUSION

In summary, we experimentally demonstrate a mutually testing SDI QRNG based on an untrusted source and a trusted equipartition process. The lower bounds on the amount of secure randomness of a pair of conjugate quadratures of two identical states are mutually estimated with the simultaneously measured data of the two quadratures; thus, it is convenient for users to avoid switching the measurement types of quadrature components. An SDI QRNG protocol based on the EUP is exploited to make the terminal random bits secure from the eavesdropper’s attacks on the entropy source. Moreover, the electronic noise of the homodyne detection system and the fluctuations of the LO are monitored to eliminate the interference around the detectors and resist the attacks on the LO to improve the total security. A coherent state is divided into two identical parts in which the quadrature $\hat{P}$ on one part and the quadrature $\hat{Q}$ on the other part can be measured simultaneously. It is noteworthy that the EUP cannot be applied to two quadratures from two different states in principle [40,41]. However, taking into account a trusted equipartition process operated on an untrusted state, for instance, by exploiting a completely trusted beam splitter, two identical impure coherent states are split from the untrusted state, then the distributions of corresponding quadratures of different states should be the same. Consequently, the max-entropy of a quadrature of one state is equal to that of the same quadrature of the other one. Accordingly, the conjugate quadratures from two states can be mutual check quadratures for each other. Eventually, measuring a pair of quadratures of a coherent state (two separate coherent states, actually) simultaneously can double the generation rate of the SDI QRNG based on the vacuum. It allows us to construct a certified SDI QRNG with a generation rate greater than 7.5 Tbit/s via a 20 W laser beam and 1250 integrated boxes containing all stuff in the private space in Fig. 2 with an external data-collecting and data-processing system.

APPENDIX A: BETWEEN MUTUAL TESTING AND RANDOM TOGGLING

\hat{P}

Figure 4.Comparison of the data acquisitions and appropriate time sequences of mutually testing and randomly toggling manners. The red and blue points represent the measured data of quadratures $\hat{P}$ and $\hat{Q}$ , respectively. (a), (b) The data acquisitions on the conjugate quadratures $\hat{P}$ and $\hat{Q}$ in mutually testing manner. (c) The data acquisitions for raw random numbers on the quadratures $\hat{Q}$ in randomly toggling manner. The data of quadratures $\hat{P}$ are used to estimate the randomness of quadratures $\hat{Q}$ and never generate random numbers. (d) Time sequences. Black and red curves represent the time sequences for randomly measuring electronic noise and check quadrature, respectively.

APPENDIX B: MEASUREMENT OF QUADRATURES WITH HOMODYNE DETECTION

s

Figure 5.Schematic of the balanced homodyne detection. The difference current is converted into an amplified voltage signal by a transimpedance amplifier.

i_{a} = {\hat{a}}^{†} \hat{a} = \frac{1}{2} ({\hat{s}}^{†} \hat{s} + e^{i ϕ} {\hat{s}}^{†} \hat{L} + e^{- i ϕ} {\hat{L}}^{†} \hat{s} + {\hat{L}}^{†} \hat{L}),

i = i_{a} - i_{b} = e^{i ϕ} {\hat{s}}^{†} \hat{L} + e^{- i ϕ} {\hat{L}}^{†} \hat{s} .

L

APPENDIX C: Supplement to entropic uncertainty principle and estimation of generation rate

\hat{Q}

H_{low} (P_{δ p} | E) = - \log_{2} c (δ q, δ p) - 2 \log_{2} \sqrt{\frac{δ q}{σ \sqrt{π}}} ϑ_{3} (0, e^{- \frac{{(δ p)}^{2}}{2 σ^{2}}}),

δ p

10^{- 12}

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