⇦Chinese Physics Letters, 2017, Vol. 34, No. 1, Article code 018401 True Random Number Generator Realized by Extracting Entropy from a Negative-Inductance Superconducting Quantum Interference Device * Hao Li(李浩), Jian-She Liu(刘建设)**, Han Cai(蔡涵), Ying-Shan Zhang(张颖珊), Qi-Chun Liu(刘其春), Gang Li(李刚), Wei Chen(陈炜) Affiliations Tsinghua National Laboratory for Information Science and Technology, Department of Microelectronics and Nanoelectronics, Institute of Microelectronics, Tsinghua University, Beijing 100084 Received 18 October 2016 *Supported by the State Key Program for Basic Research of China under Grant No 2011CBA00304, the National Natural Science Foundation of China under Grant No 60836001, and the Tsinghua University Initiative Scientific Research Program under Grant No 20131089314.
**Corresponding author. Email: jsliu@tsinghua.edu.cn Citation Text: Li H, Liu J S, Cai H, Zhang Y S and Liu Q C et al 2017 Chin. Phys. Lett. 34 018401 Abstract A new type of superconductive true random number generator (TRNG) based on a negative-inductance superconducting quantum interference device (nSQUID) is proposed. The entropy harnessed to generate random numbers comes from the phenomenon of symmetry breaking in the nSQUID. The experimental circuit is fabricated by the Nb-based lift-off process. Low-temperature tests of the circuit verify the basic function of the proposed TRNG. The frequency characteristics of the TRNG have been analyzed by simulation. The generation rate of random numbers is expected to achieve hundreds of megahertz to tens of gigahertz. DOI:10.1088/0256-307X/34/1/018401 PACS:84.30.Ng, 85.25.Dq, 85.25.Am © 2017 Chinese Physics Society Article Text Random number generators (RNGs) are of vital importance to information security which require that the encryption cannot be broken by an attacker better than a random chance. Generally, RNGs are classified into two categories, pseudo random number generator (PRNG) based on deterministic computational algorithms and true random number generator (TRNG) by measuring non-deterministic physical phenomena, such as thermal noise,[1] radioactive decay,[2] chaos in laser,[3] and other electromagnetic and quantum effects.[4,5] Since the patterns of PRNGs are discernible when the generated sequences are long enough, in applications that require stringent unpredictability, such as cryptography and certain numerical algorithms, TRNGs are preferred. Several TRNGs using semiconductor circuits,[6,7] single-photon detectors[8,9] and superconducting single flux quantum circuits[10] have been developed. Because of the sensitivity to thermal noise and high-speed operation of Josephson junction, superconductive TRNGs are expected to achieve high generation rate of tens of gigahertz,[11] which is one order higher than the fastest semiconductor circuit counterpart reported,[12] and high generation efficiency of 100%, while the improved TRNGs based on superconducting nanowire single-photon detectors typically generate random numbers at rates of 10 Mbps and efficiency of about 75%.[13] Furthermore, to reduce the risk of side-channel attacks during information communication, it is more suitable to use superconductive TRNGs than other types of TRNG in the superconducting digital circuits and superconducting quantum computing, because they can be fabricated and integrated in a single chip. In this Letter, we propose a new type of superconductive TRNG, which consists of a negative-inductance super-conducting quantum interference device (nSQUID)[14] and a direct-current superconducting quantum interference device (dc-SQUID). The entropy harnessed to generate random numbers comes from the symmetry breaking of the nSQUID. The prototype circuit of this TRNG is fabricated by the Nb-based lift-off process. Low temperature tests have been carried out to verify the basic function of the TRNG at different frequencies. The frequency characteristics of the proposed TRNG have been analyzed and simulated. Block diagram of the proposed superconductive TRNG based on nSQUID is shown in Fig. 1, when proper clock and bias signals are applied, small random fluctuations decide which final state, $-1$ or 1, the initial 'idle' state will fall into. The schematic circuit of the TRNG is shown in Fig. 2(a), which consists of an nSQUID as the entropy source and a readout dc-SQUID. The potential energy of the nSQUID is[15] $$\begin{align} U(\varphi _+,\varphi _-)=\,&E_{\rm J} \Big[\frac{(\varphi _+ -2\pi \phi _{\rm clk})^2}{\beta _{\rm L} (1-m)+2\beta _{\rm q}}+\frac{(\varphi _- -2\pi \phi _{\rm bias})^2}{\beta _{\rm L} (1+m)}\\ &-2\cos \varphi _+ \cos \varphi _-\Big],~~ \tag {1} \end{align} $$ where $E_{J}=I_{\rm c}{\it \Phi} _{0}/2\pi$ is the Josephson energy of $J_{1}$ and $J_{2}$, $I_{\rm c}$ is the critical current, ${\it \Phi}_{0}=2.07\times 10^{-15}$ Wb is the single flux quantum, $\varphi _{\pm}=(\varphi _{1}\pm \varphi _{2})/2$ are the common and differential part of the Josephson phase differences $\varphi _{1}$ and $\varphi _{2}$, $\phi _{\rm clk}=M_{\rm c}I_{\rm clk}/{\it \Phi} _{0}$ and $\phi _{\rm bias}=(M_{\rm b}I_{\rm b}+{\it \Phi}_{\rm bc})/{\it \Phi}_{0}$ are clock and bias flux normalized to ${\it \Phi}_{0}$, $m=M_{12}/L$ is the coupling coefficient of mutual inductance between $L_{1}$ and $L_{2}$, and $\beta _{L}=2\pi LI _{\rm c}/{\it \Phi}_{0}$ and $\beta _{\rm q}=2\pi L _{\rm q}I_{\rm c}/{\it \Phi}_{0}$ are reduced inductance parameters. In our design, we choose $m=0.8$, $\beta _{L}(1+m)=1.73$ and $\beta _{L}(1-m)+2\beta _{\rm q}+0.36$, therefore the nSQUID forms a tunable double-well potential in the $\varphi _{-}$-direction. The barrier height of the double-well potential is tuned by $\phi _{\rm clk}$, and the tilt is decided by $\phi _{\rm bias}$. The process to generate a random number by the nSQUID is shown in Fig. 2(b). When the barrier height is zero, the nSQUID is in the idle state; when the barrier height is raised slightly to form a symmetric double-well potential, a small perturbation will cause the nSQUID to fall from the idle state into either 1 or $-1$ state, which is called the symmetry breaking, while the nSQUID will jump between 1 and $-1$ if the barrier height is smaller compared with the thermal noise. When the barrier height becomes large enough, the nSQUID is in a well-defined state expected to have equal probability to be in 1 or $-1$. The statistical probability of the final state to be $-1$ or 1 can be tuned by $\phi _{\rm bias}$, which is controlled by $I_{\rm b}$.
cpl-34-1-018401-fig1.png
Fig. 1. Block diagram of the TRNG based on an nSQUID. Random sequences would be generated when proper clock and bias signals are applied.
cpl-34-1-018401-fig2.png
Fig. 2. (a) Schematic circuit diagram of the TRNG where the nSQUID serves as an entropy source and the dc-SQUID works as a readout measurement. (b) The process to generate a random number by changing barrier height of the nSQUID.
The chip is fabricated by the Nb-based lift-off process in our lab.[15] The Nb ground layer (100 nm), the Nb/AlO$_{x}$/Nb trilayer (150/10/150 nm), and the Nb wiring layer (250 nm) are used in our fabrication to form grounding, tunneling Josephson junction and wiring of the circuit, respectively. The optical micrograph of the chip is shown in Fig. 3. It is tested in the Oxford dilution refrigerator at 1.6 K with the low-temperature, low-pass $LC$ filter and two-stage, room-temperature voltage amplifiers. As shown in Fig. 4, the probability of the final state to be 1 can be tuned from zero to unit by $I_{\rm b}$. In the test of the TRNG, $I_{\rm b}$ is fixed at $-$414.36$\pm$0.07 μA to make sure the probability of state 1 to be 0.5, the sinusoidal waveform of $\phi _{\rm clk}$ is provided by an arbitrary waveform generator Agilent 33250A and the traces of $\phi _{\rm clk}$, $\phi _{\rm bias}$ and $V_{\rm out}$ are recorded by a 4-channel oscilloscope Tektronix DPO3034. No noises are deliberately induced from room temperature to $I_{\rm b}$ and no high-gain high-bandwidth amplifiers at low temperature are used, because the perturbation which causes symmetry breaking is so small that the thermal noise coming from low-temperature environment is enough.
cpl-34-1-018401-fig3.png
Fig. 3. (a) Optical micrograph of the testing chip. (b) The detailed optical micrograph of the TRNG based on the nSQUID. Here $I_{\rm c1}=I_{\rm c2}=15.7$ μA, $L_{1}=L_{2}=20$ pH, $L_{\rm q}=1.7$ pH, $M_{12}=16$ pH, $M_{\rm c}=0.56$ pH, $M_{\rm b}=1$ pH, $I_{\rm c3}=I_{\rm c4}=14$ μA, $L_{\rm sq}=16$ pH, $R_{1}=R_{2}=1.2$ $\Omega$.
cpl-34-1-018401-fig4.png
Fig. 4. The probability of state 1 among the output data is tuned by the bias current $I_{\rm b}$. When the TRNG works, $I_{\rm b}$ is fixed at $-$414.36$\pm$0.07 μA to make sure the probability of state 1 is 0.5.
The sequences of output generated by the TRNG at low frequency are demonstrated in Fig. 5(a). The autocorrelation function of the generated data, $$\begin{align} R(1)=\frac{1}{N}\sum\limits_{i=1}^{N-1} {x_i \cdot x_{i+1}},~~ \tag {2} \end{align} $$ is used to evaluate the correlation of adjacent bits, where $x$ is either 1 or $-$1, and $N$ is the number of bits. The autocorrelation function $R(1)$ of a 40-bit data generated by the prototype circuit at three different frequencies is calculated in Fig. 5(b), where the dashed lines are the 95% confidence bounds of the $R(1)$ of arbitrary 40-bit random number. When increasing the frequency of $\phi _{\rm clk}$ to several hundreds of hertz, the ratio of output signal $V_{\rm out}$ to noise becomes too small because of the corresponding increase of bandwidth of the filter. Moreover, the feeding line of $\phi _{\rm clk}$ then is not particularly designed for frequency test up to megahertz. Therefore, the frequency characteristics of the proposed TRNG is mainly analyzed in principle at low frequencies and simulated by WRspice[16] at higher frequencies.
cpl-34-1-018401-fig5.png
Fig. 5. (a) The sequence of outputs generated by the TRNG at 10 Hz. (b) The autocorrelation function $R(1)$ of the generated outputs at 1 Hz, 10 Hz and 20 Hz, and the red dashed lines are the 95% confidence bounds of the $R(1)$ of arbitrary 40-bit random number.
The transient response of the proposed TRNG can be modeled by a parallel $RCL$ circuit. Since underdamped Josephson junctions ($\beta _{\rm c}=2\pi I_{\rm c}R^{2}C/{\it \Phi} _{0}>1$) are used, when the $\phi _{\rm clk}$ is turned off at $t_{0}$, the TRNG returns to the idle state with a damped oscillation $$\begin{align} I_{\rm loop} ={(I_1 -I_2)}/2=I_0 e^{-(t-t_0)/\tau}\sin (\omega(t-t_0)+\varphi _0),~~ \tag {3} \end{align} $$ where $I_{\rm loop}$ is the circulating current in the nSQUID, $I_{0}$ is the initial amplitude of the oscillation, $\tau$ is the decay time constant, $\omega$ is the oscillation frequency, and $\varphi _{0}$ is the initial phase. According to the typical response of an under-damped parallel $RLC$ circuit,[17] the decay time constant $\tau$ is 2$RC$, where $R$ is the combined resistance of sub-gap resistance $R_{\rm sg}$ of the Josephson junction and shunting resistance $R_{\rm sh}$, and $C$ is the capacitance of the Josephson junction. The oscillation frequency $$\begin{align} \omega =\frac{1}{\sqrt {LC}}=\Big(\frac{1/L_{J0}+1/(L(1+m))}{C}\Big)^{1/2},~~ \tag {4} \end{align} $$ where $L$ is the combined inductance of the inductance of the Josephson junction at zero bias, $L_{J0}={\it \Phi} _{0}/(2\pi I_{\rm c})$, and half of the loop inductance $L(1+m)$. The response of the TRNG in time domain is simulated by WRspice, as shown in Fig. 6. The parameters of the damped oscillation extracted from the simulation data are in good agreement with the calculated ones of 2$RC$=0.1 ns and $(LC)^{-1/2}$=367.4 rad/ns according to Eqs. (3) and (4).
cpl-34-1-018401-fig6.png
Fig. 6. The response characteristics of the TRNG with $R=87.74$ $\Omega$, $C=569.9$ fF, $I_{\rm c}=15.7$ μA, $L=20$ pH and $m=0.8$. (a) When the clock signal $\phi _{\rm clk}$ is off, the TRNG returns to the idle state with a damped oscillation. (b) The simulation data and fitting curve of the damped oscillation.
If the TRNG switches too fast that the damped oscillation is not sufficiently decayed, the next state will correlate with the last state because of the oscillation. For critical damped ($\beta _{\rm c}=1$) or overdamped ($\beta _{\rm c} < 1$) Josephson junction, it also takes time to decay to the idle state but no oscillation will occur. For a given density of critical current, the voltage cross the critical damped Josephson junction returns to zero faster than underdamped and overdamped Josephson junctions, its decay time constant is also 2$RC$. The frequency characteristics of the proposed TRNG is simulated by WRspice. The 10 Kbit outputs are generated at different clock frequencies, where the thermal noise is modeled by a Gaussian type noise added to $\phi _{\rm bias}$ with $\mu=0$ and $\sigma=0.2m{\it \Phi}_{0}$. The dependences of the autocorrelation function $R(1)$ and output probability on frequency are shown in Fig. 7. Here $R(1)$ increases dramatically when the frequency is larger than a certain value, while the probability of state 1 only fluctuates at 0.5 in a very small range. This certain value of frequency can be defined as the cutoff frequency $f_{\rm c}$ of the proposed TRNGs. The adjacent bits will be correlated when the TRNG switches before sufficient decay of the oscillation of the Josephson junction in the idle state which keeps the phase information of the last state. Therefore, $f_{\rm c}$ is supposed to be limited by the response speed of the Josephson junction. As shown in Fig. 7, $f_{\rm c}$ is about 10 times smaller than $1/2RC$. Considering the speed records of the Josephson junction circuits,[18] $f_{\rm c}$ is expected to achieve tens of gigahertz if the Josephson junctions with high critical current density and critical damping condition ($\beta _{\rm c}=1$) are used.
cpl-34-1-018401-fig7.png
Fig. 7. Dependence of the autocorrelation function $R(1)$ on frequency. The inset shows the corresponding dependence of probability of state 1 on frequency.
In summary, a new type of TRNG based on an nSQUID has been proposed. We have fabricated the prototype circuit by the Nb-based lift-off process. Low-temperature experiments of the circuit have been conducted to verify the basic characteristics of the TRNG. The frequency characteristics of the TRNG have been analyzed and simulated by WRspice. The generation rate of the proposed TRNG is limited by the response speed of the Josephson junction and expected to be hundreds of megahertz to tens of gigahertz. We would like to thank Professor Siyuan Han for helpful discussion and experiment instruction. Hao Li, Han Cai, Yingshan Zhang, Qichun Liu and Gang Li would like to thank Zhejiang Tianjingsheng Foundation, China for student assistantship. References A noise-based IC random number generator for applications in cryptographyA compact and accurate generator for truly random binary digitsAn optical ultrafast random bit generatorThe generation of 68 Gbps quantum random number by measuring laser phase fluctuationsSource-Independent Quantum Random Number GenerationPhysical random number generator based on MOS structure after soft breakdownA 3 $\mu$W CMOS True Random Number Generator With Adaptive Floating-Gate Offset CancellationA high speed, postprocessing free, quantum random number generatorBias-free true random-number generatorSuperconductive Random Number Generator Using Thermal Noises in SFQ CircuitsDemonstration of 30 Gbit/s Generation of Superconductive True Random Number Generator2.4 Gbps, 7 mW All-Digital PVT-Variation Tolerant True Random Number Generator for 45 nm CMOS High-Performance MicroprocessorsBias-free true random number generation using superconducting nanowire single-photon detectorsNegative-inductance SQUID as the basic element of reversible Josephson-junction circuitsJosephson junctions in SPICE3Rapid single flux quantum T-flip flop operating up to 770 GHz
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