_applied sciences_  
_Article_  
Statistical Validation of a Physical Prime Random Number Generator Based on Quantum Noise \({}^{\dagger}\)  
**[PERSON] 1,2\({}^{\copyright}\)**, [PERSON] 1,*\({}^{\copyright}\)**, [PERSON] 1,2\({}^{\copyright}\) and [PERSON] 1\({}^{\copyright}\)  
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seems increasingly unsound [7; 8]. Despite the best algorithmic implementations being able to output uniformly distributed numbers that pass multiple statistical tests, their output is inherently periodic and thus can be predicted by adversaries with enough computational resources [9]. In fact, the applicability of cryptanalytic attacks against PRNGs has progressively been made easier by the increasing computational power and novel techniques such as Machine Learning (ML), which are capable of independently recognizing the correlations in the RNG output [10; 11]. More worryingly, they are particularly susceptible to the insertion of backdoors [12], and weak cryptographic keys have already been found to still be widely in use due to poorly implemented PRNGs [13]. Although there are alternative physical implementations that extract randomness from measuring noisy phenomena, most of these still rely on classical processes such as atmospheric noise [14; 15; 16]. Consequently, they still simply ground their unpredictability on an incomplete description of the physical system and can potentially present highly correlated outputs [2]. Indeed, recently, a ML model was found to be successful in increasing the probability of guessing the outcome of such a RNG whenever the electronic noise source was prominent [11]. These problems are slowly being addressed by the emergence of Quantum Random Number Generators (QRNGs), which extract randomness from the probabilistic properties of quantum measurements [2; 17]. Unlike the approaches based on classical noise, their randomness source is fundamentally unpredictable to any adversary, thus allowing them to yield information-theoretically provable randomness [18]. In fact, several different schemes have already been thoroughly explored, such as measuring amplified spontaneous emission [19; 20], phase laser noise [21; 22], photon arrival times [23; 24], or the quadrature fluctuations of the optical vacuum state [25; 26; 27]. The latter is particularly promising since it can deliver the high throughput rates of potentially several hundred Gbps seen for more complex implementations while requiring a simpler homodyne measurement scheme that can be implemented with off-the-shelf optical devices [28]. Moreover, obtaining the optical vacuum state can be done, within a reasonable approximation, by simply blocking the impinging optical signal, which makes it a source more resilient against environmental perturbations [29]. Due to the lower number of optical components required, these schemes are also more easily integrated. Despite often sacrificing the achievable throughput due to challenges such as the presence of dark currents, these compact schemes are essential to achieve commercially viable implementations [28; 30]. This thus lowers their adoption threshold and allows them to compete against the traditional pseudorandom solutions. Nonetheless, the output statistical quality of these schemes should not be assumed by default but corroborated by extensive characterization. In fact, challenges such as guaranteeing a properly balanced detection or fluctuations in the classical noise floor can lower the available entropy and introduce security loopholes [31]. Although fully device-independent solutions exist, these rely on observing loophole-free Bell inequality violations and thus are tendentiously extremely slow and difficult to implement [32]. This makes them unfeasible for any practical realization, and thus, implementations at least partially based on trusted devices are widely used.