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### Generating random numbers

A random number generator is a device that produces sequences of numbers complying with the definitions proposed above. There exist two main classes of generators: software and physical generators. From a general point of view, software generators produce so-called pseudo random numbers. Although they may be useful in some applications, they should not be used in most applications where randomness is required.

### Physical sources of randomness

In applications where pseudo-random numbers are not appropriate, one must resort to using a physical random number generator. When using such a generator, it is essential to consider the physical process used as the randomness source. This source can be either based on a process described by classical physics or by quantum physics. Classical physics is the set of theories developed by physicists before the beginning of the XXth century and which describes macroscopic systems like falling coins. Quantum physics is a set of theories elaborated by physicists during the first half of the XXth century and which describes microscopic systems like atoms or elementary particles. Some examples of generators based on each of these theories, along with their advantages, are presented below, after a brief discussion of biased random number sequences.

A problem encountered with physical random number generators is their bias. A binary generator is said to be biased when the probability of one outcome is not equal to the probability of the other outcome. Bias arises because of the difficulty to devise precisely balanced physical processes. It is however less of a problem than one might expect at first sight. There exists some post-processing algorithm that can be used to remove bias from a sequence or random numbers.

The simplest of these unbiasing procedures was first proposed by Von Neumann [1]. The random bits of a sequence are grouped in subsequences of two bits. Whenever the two bits of a subsequence are equal, it is discarded. When the two bits are different and the subsequence starts with a 1, the subsequence is replaced by a 1. When it starts with a 0, it is replaced by a 0. After this procedure, the bias is removed from the sequence.

The cost of applying an unbiasing procedure to a sequence is that it is shortened. In the case of the Von Neumann procedure, the length of the unbiased sequence will be at most 25% of the length of the raw sequence. It was mentioned above that randomness tests basically all amount to verifying whether the sequence can be compressed. An unbiasing procedure can be seen as a compression procedure. After its application, the bias is removed and no further compression is possible, guaranteeing that the sequence will pass the tests. Other unbiasing procedures exist. The one proposed by Peres [2] for example is significantly more efficient than the Von Neumann procedure.

Macroscopic processes described by classical physics can be used to generate random numbers. The most famous random number generator – coin tossing – indeed belongs to this class. However, it is very important to realize that classical physics is fundamentally deterministic.

### Processes described by quantum physics randomness revealed by simplicity

Contrary to classical physics, quantum physics is fundamentally random. It is the only theory within the fabric of modern physics that integrates randomness. This fact was very disturbing to physicists like Einstein who invented quantum physics. However, its intrinsic randomness has been confirmed over and over again by theoretical and experimental research conducted since the first decades of the XXth century.

When designing a random number generator, it is thus a natural choice to take advantage of this intrinsic randomness and to resort to the use of a quantum process as source of randomness. Formally, quantum random number generators are the only true random number generators. Although this observation may be important in certain cases, quantum random number generators have other advantages. This intrinsic randomness of quantum physics allows selecting a very simple process as source of randomness. This implies that such a generator is easy to model and its functioning can be monitored in order to confirm that it operating properly and is actually producing random numbers. Contrary to the case where classical physics is used as the source of randomness and where determinism is hidden behind complexity, one can say that with quantum physics randomness is revealed by simplicity.

Until recently the only quantum random number generator that existed were based on the observation of the radioactive decay of some element. Although they produce numbers of excellent quality, these generators are quite bulky and the use of radioactive materials may cause health concerns. The fact that a simple and low cost quantum random number generators did not exist prevented quantum physics to become the dominant source of randomness.

### Optical quantum random number generator

Optics is the science of light. From a quantum physics point of view, light consists of elementary "particles" called photons. Photons exhibit in certain situations a random behavior. One such situation, which is very well suited to the generation of binary random numbers, is the transmission upon a semi-transparent mirror. The fact that a photon incident on such a component be reflected or transmitted is intrinsically random and cannot be influenced by any external parameters. The figure below schematically shows this optical system.

Figure 1: Optical system used to generate random numbers.

[1] Von Neumann, J., "Various techniques used in connection with random digits", Applied Mathematics Serires, no. 12, 36-38 (1951).
[2] Peres, Y., Ann. Stat., 20, 590 (1992).