# Characterization of a Quantum Random Number Generator Based on Vacuum Fluctuations
## 4 Results and Discussion
### Noise Characterization  
The electronic components within the homodyne detector induce correlation between samples taken at different instants of time in the digitizer. To ensure uncorrelated measurements and avoid randomness overlapping, a sampling rate of 39.06 MSa/s was chosen. Ideally, a sampling rate inferior to twice the detector's bandwidth should be used, as described by the Nyquist sampling theorem. In Figure 6, the correlation coefficients were evaluated over \(15.6\times 10^{6}\) data points for a maximum delay of 100 samples. As can be seen, these rapidly drop to below \(1.9\times 10^{-3}\) at 12 dBm and \(6.8\times 10^{-4}\) for electronic noise samples. Although they never reach zero due to the signal's finite bandwidth, these low values hint at a negligible correlation between samples, which is essential for any reliable random number generator.  
For this implementation, we obtain a QCNR value of approximately 5 dB  
seen in Figure 7. Means of \(0.1004\,\mathrm{mV}\) and \(-0.1360\,\mathrm{mV}\) were calculated for the homodyne and electronic noise samples, respectively. These values could hint at a slightly unbalanced detection but are low enough to be disregarded. Each represented bin corresponds to a unique ADC value, which shows the concentration of measurements on a small number of bins due to the ADC resolution and acquisition range available.  
Here, the ADC resolution and acquisition range also present another relevant consideration. For a given resolution, the acquisition range must be properly chosen as to not risk endangering the security of the implementation. For instance, a wide acquisition range will lead to the concentration of all samples on a small number of quantization levels [43]. Thus we misrepresent the Gaussian distribution and necessarily limit the randomness that can be extracted. The necessity to lower the extraction ratio in this situation is made obvious by the consideration that, with increasing acquisition ranges, the homodyne and electronic noise distributions will be progressively indistinguishable, since measurements are mapped to the same limited number of bins. In an extreme case, guessing the yielded outcome will become trivial to an adversary, as the limited number of unique values greatly increases the probability of finding a measurement in each particular bin. This can partially be compensated by increasing the ADC resolution and, thus, diminishing the outcome probability of each bin. On the other hand, a narrow acquisition range increases the probability of a measurement falling outside the considered interval and, consequently, leads to saturation of the first and last bins. In this case, the probability of these particular results increases, which could be used by an observant adversary to increase its chances of predicting the QRNG outcome. For the chosen ADC, at most, only 256 unique values can be represented. Furthermore, the selected \(\pm 50\,\mathrm{mV}\) acquisition range results in a large portion of the ADC range not being utilized. This further limits the performance of our implementation and, ideally, a more appropriate interval should be considered [43].