Quantum Entropy Model of an Integrated Quantum-Random-Number-Generator Chip

We present the physical model for the entropy source of a quantum-random-number-generator chip based on the quantum fluctuations of the photon number emitted by light-emitting diodes. This model, combined with a characterization of the chip, estimates a quantum min-entropy of over 0.98 per bit without postprocessing. Finally, we show with our model that the performances in terms of security are robust against fluctuations over time.

Figure 1

A schematic representation of the QRNG. All the components are embedded on a single chip.

Figure 2

(a) The simulated ADC output distribution in the case in which there is no noise, with $K=0.8192$ (obtained from the factory-given parameters of the chip). (b) The 2-bits probability distribution simulated from (a), giving a min-entropy per bit $H_{\min }=0.982$.

Figure 3

A schematic representation of the noise sources in the chip. Dark electrons are added to the electrons generated by the LED. The total number of electrons is converted into a voltage with a factor $K$. After conversion, noise from the readout circuit is added before the signal is digitized with the ADC.

Figure 4

(a) The ADC output distribution $Z$ given by one pixel of the array for various values of the light intensity. (b) The variance of $Z$ versus its mean value for the distributions of (a).

Figure 5

The noise distribution of one of the pixels.

Figure 6

(a) The probability distribution of the Pearson correlation factors measured between all pairs of pixels (in this case, $12800\times (12800-1)/2$ pairs). The standard deviation $\sigma$ on the correlation factor is $3.16\times {10}^{-3}$, which corresponds to the uncertainty expected for the size of our data. (b) The autocorrelation of four pixels from the array. The solid and dashed gray lines represent, respectively, the confidence intervals of $\sigma$ and $2.57\sigma$.

Figure 7

The quantum entropy as a function of the mean photon number simulated based on the classical noise characterization of pixel 1.

Figure 8

The probability of failure and the quantum entropy per bit of an array of 64 pixels uniformly illuminated as a function of the mean photoelectron number.

Figure 9

Typical results for the different entropy estimators on the NIST non-IID tests. The tests are carried out on 10-Mbyte samples.