The logistic map is the most basic recurrence formula exhibiting various levels of chaos depending on its parameter. It has been used in population demographics to model chaotic behavior. Here we explore this model in the context of randomness simulationÂ and revisit a bizarre non-periodic random number generator discovered 70 years ago, based on the logistic map equation. We then discuss flaws and strengths in widely used random number generators, as well as how to reverse-engineer such algorithms. Finally, we discuss quantum algorithms, as they are appropriate in our context.

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Logistic map

The logistic map is defined by the following recursion

X(k) = r X(k-1) (1 - X(k-1))

with one positive parameter r less or equal to 4. The starting value X(0) is called the seed, and must be in [0, 1]. The higher r, the more chaotic the behavior.Â AtÂ r = 3.56995...Â is the onset of chaos. At this point, from almost all seeds, we no longer see oscillations of finite period. In other words, slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.

When r=4, an exact solution is known, see here. In that case, the explicit formula is

The case r=4 was used ...

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We are interested here in factoring numbers that are a product of two very large primes. Such numbers are used by encryption algorithms such as RSA, and the prime factors represent the keys (public and private) of the encryption code. Here you will also learn how data science techniques are applied to big data, including visualization, to derive insights. This article is good reading for the data scientist in training, who might not necessarily have easy access to interesting data: here the dataset is the set of all real numbers -- not just the integers -- and it is readily available to anyone. Much of the analysis performed here is statistical in nature, and thus, of particular interest to data scientists.Â

Factoring numbers that are a product of two large primes allows you to testÂ the strength (or weakness) of these encryption keys. It is believed that if the prime numbers in question are a few hundred binary digits long, factoring is nearly impossible: it would require years of computing power on distributed systems, to factor just one of these numbers.

While the vast majority of big numbers have some small factorsÂ and are thus easier to break, the integers that we are dealing ...

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