Improving randomness characterization through Bayesian model selection

Abstract

Random number generation plays an essential role in technology with important applications in areas ranging from cryptography to Monte Carlo methods, and other probabilistic algorithms. All such applications require high-quality sources of random numbers, yet effective methods for assessing whether a source produce truly random sequences are still missing. Current methods either do not rely on a formal description of randomness (NIST test suite) on the one hand, or are inapplicable in principle (the characterization derived from the Algorithmic Theory of Information), on the other, for they require testing all the possible computer programs that could produce the sequence to be analysed. Here we present a rigorous method that overcomes these problems based on Bayesian model selection. We derive analytic expressions for a model’s likelihood which is then used to compute its posterior distribution. Our method proves to be more rigorous than NIST’s suite and Borel-Normality criterion and its implementation is straightforward. We applied our method to an experimental device based on the process of spontaneous parametric downconversion to confirm it behaves as a genuine quantum random number generator. As our approach relies on Bayesian inference our scheme transcends individual sequence analysis, leading to a characterization of the source itself.

Advanced Statistical Testing of Quantum Random Number Generators

“…Randomness characterization through Bayesian model selection has some clear and natural advantages, as already pointed out in [16], but, unfortunately, it has an important drawback: the number of all possible models for a given length i, given by B 2 i , grows supra-exponentially with i: indeed, for i = 1, we have two possible models, for i = 2, we have 15 possible models, for i = 3, we have instead 4140 possible models, while, for i = 4, we have 10,480,142,147 models. Thus, even if we are able to acquire data for the evaluation of these many models, it becomes computationally impractical to estimate the posterior for all of them using Equation (2).…”

“…Recently, a Bayesian criterion has been introduced [16,17] by some of the authors of the present article to test, from a purely probabilistic point of view, whether a sequence is maximally random as understood within information theory [18]. The method works by exploiting the Borel-normality compression scheme and then recasting the problem of finding possible biases in the sequence as an inferential one in which Bayesian model selection can be applied.…”

“…Each such model determines a unique probability assignation to the elements of Ω (i) , which depends on a set of prior parameters θ. For these parameters, the Jeffreys' prior, P Jeff (θ), turns out to be a convenient choice of prior parameter distribution, as it entails the "Occam Razor principle" in which more complex models are penalized, as well as being mathematically convenient for the case at hand; some other advantages are pointed out in [16,17].…”

“…α , θ P Jeff (θ). One of the most important results of [16] is that this marginalization can be accomplished exactly for all the models and any value of i. Therefore, we can obtain the posterior distribution by a simple application of Bayes' rule:…”

Generating randomness: making the most out of disordering a false order into a real one

“…Current methods either do not rely on a formal description of randomness (e.g., the NIST test suite) or are inapplicable in principle, requiring testing of all possible computer programs that could produce the sequence. A method that behaves like a genuine QRNG and overcomes these difficulties based on Bayesian model selection was proposed [3]. Moreover, hardware TRNGs are used to create encryption keys, and offer advantages over software PRNGs.…”