Editing Logistic map (section)

===Pseudorandom number generator===
In the fields of computer simulation and information security, the creation of pseudorandom numbers using a computer is an important technique, and one of the methods for generating pseudorandom numbers is the use of chaos. <!--[ 324 ]--> Although a pseudorandom number generator based on chaos with sufficient performance has not yet been realized, several methods have been proposed. <!--[ 324 ]--> Several researchers have also investigated the possibility of creating a pseudorandom number generator based on chaos for the logistic map. <!--[ 234 ]--><!--[ 325 ]--><!--[ 326 ]-->

Parameter r = 4 is often used for pseudorandom number generation using the logistic map.<!--[ 327 ]--><!--[ 328 ]--><!--[ 329 ]--> Historically, as described below, in 1947, shortly after the birth of electronic computers, Stanisław Ulam and John von Neumann also pointed out the possibility of a pseudorandom number generator using the logistic map with r = 4.<!--[ 330 ]--> However, the distribution of points for the logistic map <math>f_{r = 4}</math> is as shown in equation ( 3-17 ), and the numbers that are generated are biased toward 0 and 1.<!--[ 234 ]--> Therefore, some processing is required to obtain unbiased uniform random numbers.<!--[ 234 ]--> Methods for doing so include:

A method for converting the obtained values to a uniform distribution using the tent map ( 4-8 ). <!--[ 327 ]-->
The resulting number is converted to either 0 or 1 using a threshold, as in the coin tossing analogy above, and this process is repeated to obtain a uniformly random bit stringm <!--[ 329 ]-->
In addition, the sequences <math>x_n</math> and <math>x_n +1</math> obtained by the logistic map are strongly correlated, which makes it problematic for pseudorandom sequences. <!--[ 234 ]--> One way to solve this is to generate the sequence <math>x_0, x_1, x_2, ... </math> for each iteration of the map, rather than generating the sequence <math>x_0, x_{\tau}, x_{2 \tau} ,...</math> for some number of iterations τ > 1  <!--[ 234 ]--> For example, it is said that good pseudorandom numbers can be obtained for method 1 with τ > 10 or τ > 13, <!--[ 234 ]--> and for method 2 with τ > 16. <!--[ 329 ]-->

A common problem with digitally calculating chaos using a computer is that, because a computer has a finite calculation precision, it is in principle impossible to obtain a truly aperiodic sequence, which is the nature of chaos, and instead outputs a finite periodic sequence. <!--[ 325 ]--> Even if aperiodic sequences cannot be obtained in principle, sequences with as long a period as possible are desirable for generating pseudorandom numbers. <!--[ 325 ]--> However, when the periodicity of the sequence actually output by the logistic map <math>f_{r=4}</math> in single-precision floating-point calculations was investigated, it was reported that the period of the sequence actually output is much smaller than the maximum period possible from the number of bits allocated, and from this point of view, it has been pointed out that pseudorandom number generation by the logistic map is inferior to existing pseudorandom number generators such as the Mersenne Twister. <!--[ 325 ]--> In addition, with the logistic map, <math>f_{r=4}</math> there is a risk that the value will fall to the fixed point 0 during the calculation and remain constant. <!--[ 331 ]--> On the other hand, the logistic map always takes values in the open interval (0, 1), so it can be calculated without problems not only with floating point but also with fixed point, and can enjoy the advantages of fixed point arithmetic. <!--[ 331 ]--> It has been pointed out that fixed point has a longer period than floating point for the same number of bits, and that unintended convergence to 0 can be eliminated. <!--[ 331 ]-->