Practical Implementation of a Deep Random Generator

Practical Implementation of a Deep Random Generator

Thibault de Valroger (*)

Abstract We have introduced in former work [5] the concept of Deep Randomness and its interest to design Unconditionally Secure communication protocols. We have in particular given an example of such protocol and introduced how to design a Deep Random Generator associated to that protocol. Deep Randomness is a form of randomness in which, at each draw of random variable, not only the result is unpredictable bu also the distribution is unknown to any observer. In this article, we remind formal definition of Deep Randomness, and we expose two practical algorithmic methods to implement a Deep Random Generator within a classical computing resource. We also discuss their performances and their parameters.

Key words. Deep Random, Random Generator, Perfect Secrecy, Unconditional Security, Prior Probabilities, information theoretic security I. Introduction Prior probabilities theory Before presenting the Deep Random assumption, it is needed to introduce Prior probability theory. The art of prior probabilities consists in assigning probabilities to a random event in a context of partial or complete uncertainty regarding the probability distribution governing that random event. The first author who has rigorously considered this question is Laplace [10], proposing the famous principle of insufficient reason by which, if an observer does not know the prior probability of occurrence of 2 events, he should consider them as equally likely. In other words, if a random variable can take several values , and if no information regarding the prior probabilities
is available for the observer, he should assign them
⁄ in any attempt to produce inference from an experiment of . Several authors observed that this principle can lead to conclusion contradicting the common sense in certain cases where some knowledge is available to the observer but not reflected in the assignment principle. If we denote the set of all prior information available to observer regarding the probability distribution of a certain random variable (‘prior’ meaning before having observed any experiment of that variable), and any public information available regarding an experiment of , it is then

possible to define the set of possible distributions that are compatible with the information
regarding an experiment of ; we denote this set of possible distributions as:

The goal of Prior probability theory is to provide tools enabling to make rigorous inference reasoning in a context of partial knowledge of probability distributions. A key idea for that purpose is to consider groups of transformation, applicable to the sample space of a random variable , that do not change the global perception of the observer. In other words, for any transformation of such group, the observer has no information enabling him to privilege | rather than
| as the actual conditional distribution. This idea has been developed by Jaynes [7], in order to avoid the incoherence brought in some cases by the imprudent application of Laplace principle. We will consider only finite groups of transformation, because one manipulates only discrete and bounded objects in digital communications. We define the acceptable groups as the ones fulfilling the 2 conditions below: Stability - For any distribution , and for any transformation , then
Convexity - Any distribution that is invariant by action of does belong to
It can be noted that the set of distributions that are invariant by action of is exactly: {
| | ∑

| } The condition is justified by the fact that in the absence of information enabling the observer to privilege from , he should choose equivalently one or the other distribution, but then of course the average distribution

| | ∑

should still belong to the set of possible distributions knowing . The set of acceptable groups as defined above is denoted:

Let’s consider some examples. Example 1: we consider a 6-sides dice. We are informed that the distribution governing the probability to draw a given side is altered, but we have no information of what that distribution actually is, and we have no available information regarding an experiment. We want nevertheless to assign an a priori probability distribution for the draw of dice. In this very simple case, it seems quite reasonable to assign an a priori probability of
⁄ to each side. A more rigorous argument to justify this decision, based on the above, is the following: let’s consider the finite group of transformation that let the dice unchanged, this group is well known, it is generated by the 3 axis 90° rotations, and has 24 elements. It is clear he

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