The generation of pseudo-random numbers is crucial in many fields of science like cryptography, where cryptographically secure pseudo-random numbers are needed e.g. [1] or scientific computations, where often numbers from another than uniform distribution are crucial e.g. [2, 3].

There are many published algorithms that enable the derivation of values from the given probability distribution. One of the most popular is the method of inverse cumulative distribution function determined by the equation [4]:
$$\begin{array}{}X={F}^{-1}(U),\end{array}$$(1)

where *U* is a random variable from the uniform distribution on interval (0, 1), *F*^{−1} is a quantile function and *X* is a random variable with distribution corresponding to *F*.

Another very popular method for pseudo-random numbers generation is the rejection (also called acceptance and rejection) method, which is the implication of the following observation [5]:

if a random point (*X*, *Y*) is uniformly distributed in the region *G*_{f} between the graph of the density function *f* and the *x*-axis, then random variable *X* has density *f*.

Additionally, in professional literature the methods which allow the generation of pseudo random numbers from a concrete distribution can be found, for example, from the normal one [6, 7, 8, 9]. One of such algorithms is the Box-Muller transformation given by the equations [6]:
$$\begin{array}{}{N}_{1}=\sqrt{-2\mathrm{ln}{U}_{1}}\mathrm{cos}\left(2\pi {U}_{2}\right)\text{\hspace{0.17em}and\hspace{0.17em}}{N}_{2}=\sqrt{-2\mathrm{ln}{U}_{1}}\mathrm{sin}\left(2\pi {U}_{2}\right),\end{array}$$(2)

where *N*_{1} and *N*_{2} are standard normal random variables, whereas *U*_{1} and *U*_{2} are random variables from uniform distribution.

Apart from these classical methods, there are also ways of constructing chaotic maps solving the so-called inverse Frobenious-Perron problem [10, 11, 12], which enables the construction of recurrences with predefined invariant densities. Iterating such dynamical systems is an easy way of generating pseudo-random numbers. One of such recurrences is in the following form [13]:
$$\begin{array}{}{x}_{k+1}={F}^{-1}\left(U\left(F({x}_{k})\right)\right),\end{array}$$(3)

where *F* is a given cumulative distribution function, *F*^{−1} is the inverse function to *F* and *U* is the skew tent map. The skew tent map (also called as the asymmetric tent map) is given by the relation:
$$\begin{array}{}{x}_{k+1}=f({x}_{k})=\left\{\begin{array}{cc}\frac{{x}_{k}}{p}& 0<{x}_{k}\le p\\ \frac{1-{x}_{k}}{1-p}& p<{x}_{k}<1\end{array}\right..\end{array}$$(4)

For each value of parameter *p* ∈ (0, 1), the recurrence (4) is chaotic and has a uniform distribution of the iterated variable. Due to these properties, reccurence (4) is very popular as a component of pseudorandom number generators in cryptographic applications [14, 15, 16].

Transformations in the form of (3) were analyzed in [17]. The derived results indicate that for values of parameter *p* close to 0 or 1, the desired probability distribution of the iterative variable cannot be derived. The reason is a small -– close to zero – value of the Lyapunov exponent, which measures the rates of convergence or divergence of nearby trajectories. The Lyapunov exponent of the dynamical system *x*_{k+1} = *f*(*x*_{k}) is given by the formula:
$$\begin{array}{}{\displaystyle \lambda =\underset{m\to \mathrm{\infty}}{lim}\frac{1}{m}\sum _{i=0}^{m-1}\mathrm{ln}|{f}^{\mathrm{\prime}}({x}_{i})|.}\end{array}$$(5)

Furthermore, methods for generating pseudo-random numbers with the use of chaotic maps related only to a specific distribution can be shown, for example the normal distribution with the use of the Weierstrass recurrence, which was firstly shown in [18] and futher analized in [19]. The Weierstrass recurrence can be expressed by the formula:
$$\begin{array}{}{\displaystyle {x}_{k+1}=\sum _{i=0}^{N}{a}^{i}\mathrm{cos}({b}^{i}\pi {x}_{k}),}\end{array}$$(6)

where 0 < *a* < 1, *b* is a odd number and
$\begin{array}{}ab>1+\frac{3}{2}\pi .\end{array}$ As shown in [19], iterating (6) with parameter value *a* close to 1, generates values from the normal distribution.

Another method which applies chaotic maps in pseudo-random numbers generation was shown in [20], where values from uniform distribution are generated. This method may be described by the following procedure:

#### Method 1

*Let* *U*^{n} (*x*) *denote the n*-*th iteration of the chaotic map with a uniform distribution starting from initial condition x*. *Furthermore*, *let*:
$$\begin{array}{}X=\{{x}_{0},{x}_{1},\dots ,{x}_{N-1}\}\end{array}$$(7)

*be a certain pseudo-random set of numbers from continuous distribution with finite support*. *In such case*, *the set*
$$\begin{array}{}U=\{{u}_{0}={U}^{n}\left(|a{x}_{0}|\right),{u}_{1}={U}^{n}\left(|a{x}_{1}|\right),\dots ,{u}_{N-1}={U}^{n}\left(|a{x}_{N-1}|\right)\}\end{array}$$(8)

*where a is a normative coefficient*, *has the distribution similar to uniform*.

The above procedure enables the ”flattening” of continous distribution, i.e. reducing it to the uniform distribution. Furthermore, the accuracy of this process depends on the number of iterations n - if it is too small, then the obtained distribution only ”flattens” the oryginal density functions of (7). The transformation *f* may be chosen as the skew tent map (4). Other examples of chaotic maps with uniform distribution may be found in [21, 22]. Likewise, as recurrence (4), they consist of several independent functions, which may be called as *branches*.

While analyzing the above-described method a natural question arises: Is the process of reduction of any distribution to the uniform distribution reversible? If yes, then in consequence, a new method enabling the generation of pseudo-random numbers from any distribution could be derived. The fact that the transformation described in (8) is a 1D chaotic map means that it is irreversible. However, by additional assumptions the process may become reversible, which is discussed in the next section of this paper.

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