## 3 Results and Discussion
### Figures of Merit  
Given the input noise previously characterized, the total number of primes found over the entire 1 GB dataset for each of the scenarios considered is represented in Figure 4. Here, the solid line represents a lower bound for the total number of primes found estimated through Equation (4). Here, it should be noted that the total number of primes found should asymptotically approximate this bound for increasing bit-lengths in agreement with the approximation taken for the prime-counting function in Section 2.2. As theoretically expected, this figure consistently decreases for larger prime lengths but remains relatively constant across the four scenarios for each prime length. A notable exception can be seen for the 32-bit prime output. Here, the normally distributed sources yield significantly fewer numbers than the uniformized scenarios. In fact, for the raw CS and QS cases, a total of 187.9 M and 191.7 M numbers were, respectively, obtained, representing decreases of 3.89% and 1.94% when compared with the average of 195.5 M seen across the uniformized scenarios. This reduction can naturally be explained by the biased input distribution of the raw randomness sources and is consistent with the smaller variance of the CS, which implies a smaller probability of yielding extreme candidate numbers. Despite this, this performance difference is not seen in the time that it takes to find each prime, which stays consistent across the four datasets considered. Instead, the search time exponentially increases with the prime length as described by the decreasing generation rate, expressed in Equation (4), and the algorithmic complexity of the Miller-Rabin primality test. Figure 4 clearly illustrates this. For instance, although finding each prime took an average of 108.8 us across all cases considered, 1019.2 us were necessary for a length of 512 bit, highlighting the challenge of achieving physical RNG schemes able to deliver prime lengths suitable for cryptographic applications. A more relevant figure of merit than the overall number of primes obtained is the diversity of the prime output sequences since it allows the identification of a prevalence of repeated outputs. Here, to allow a comparison between all sources and prime lengths considered, we have taken 10 M prime numbers from each case and compared the number of unique values. In fact, while for larger primes, every output is unique across all cases considered, this is not true for the 32 and 64-bit lengths. As seen in Figure 5, the diversity of primes is particularly low for the 32-bit case, where the number of unique values is only a small fraction of the total. As with what was seen in the overall number of primes, for the unprocessed randomness sources, this observation can be expected since, due to the  
Figure 4: (**a**) Total number of primes found, and (**b**) the respective average search time required, with increasing prime sizes for each of the randomness sources considered. In the former, the solid line represents the estimated lower bound for the total number of prime numbers generated. All the values obtained are additionally represented for each case. normally distributed input noise, certain ADC codes have a higher probability of being observed. For smaller primes, which are constituted from fewer samples, this significantly reduces the diversity of candidate numbers seen, and thus, the probability of observing repeated prime numbers increases. This is obviously more easily observed in the 32-bit case, where every candidate number is only obtained from 2 ADC samples. However, the same phenomena can be seen for the 64-bit prime sequence represented in Figure 5b. The uncorrelated nature of the QS shows here an immediate advantage, yielding approximately 827.5% more unique RNs than the classical case. In fact, this difference cannot be explained by the higher noise variance of the homodyne signal.