This paper discusses creating a pseudorandom sequence generator using the natural logarithm of the number 2 (ln 2) calculator. Pseudorandom sequence generators are key elements in cryptography, modeling, and numerical methods, where high-quality randomness is required. Traditionally, various mathematical algorithms are used for this purpose, but we propose a new approach based on the numerical properties of ln 2. The paper describes in detail the method of computing ln 2 using the Taylor series and demonstrates how these calculations can be integrated into a pseudorandom sequence generator. The main idea is to use the ln 2 approximation to initialize the generator, allowing for the creation of number sequences with a high degree of randomness. The use of ln 2, known for its mathematical stability and accuracy, opens new horizons for generating numbers that are important for many scientific and engineering applications. The presented test results show that the proposed method provides uniform distribution and passes the standard NIST statistical tests. This demonstrates the potential of using mathematical constants and their numerical computations to improve the characteristics of pseudorandom sequence generators. Our approach offers the possibility of creating generators with improved characteristics without significantly increasing computational complexity. Additionally, we discuss potential directions for improving the generator, including optimizing the algorithm and expanding to other mathematical constants. This approach not only enhances the quality of pseudorandom sequences but also provides new tools for research in number theory and computational mathematics. An important aspect is that the proposed method provides high generation speed, making it attractive for use in real-world applications where computation time is a critical parameter. Thus, our generator may find wide application in various fields, including cryptographic protocols, simulation algorithms, and other numerical methods that require high-quality randomness and computational efficiency.
Pseudorandom Number Generators (PRNGs) and
Pseudorandom Sequence Generators (PSGs) are key
elements in many scientific and technical fields. They play
a crucial role in modern technologies, providing the basis
for numerous applications in computer science,
cryptography, statistical sampling, modeling, and
simulations [1–7]. These generators enable the creation of
number sequences that, while deterministic, appear random,
which is critically important for ensuring data security,
model accuracy, and algorithm reliability. In a world where
information is becoming increasingly valuable,
understanding and using PRNGs and PSGs is essential for
developing effective solutions in various fields, from finance
to gaming. Thus, the importance of pseudorandom number
generators is hard to overestimate, as they provide the
foundation for innovation and development in many sectors
[8]. Particularly noteworthy is their importance in
cybersecurity, where they are also a key element, and are
used in solving various tasks, namely for data encryption
[
This series converges relatively slowly, so more efficient
methods are typically used for practical calculations [
For numerical computation of this integral, methods
such as the rectangle (midpoint), trapezoidal, or Simpson’s
rule are applied, which allows for obtaining more precise
values [
One of the most powerful methods of computing
Ln 2 is the Newton-Raphson method, which is used for
finding the roots of equations and can be adapted for
computing logarithms [
A clear downside is that the use of Euler’s constant for computation is required, which itself is transcendental and cannot be precisely calculated. Therefore, it is necessary to calculate the constant itself simultaneously, which increases the number of operations. On the other hand, in such a case, one iteration can generate a much larger sequence of numbers than other algorithms.
With the advent of computers, an obvious application
became the computation of mathematical constants. In one
of the first works on this topic [
The research problem involves seeking a new approach to
generating pseudorandom numbers that would ensure
highquality randomness and computational efficiency. One such
approach is the use of the numerical properties of
mathematical constants [
This paper aims to develop and test a pseudorandom number generator based on the computation of ln 2 using the Taylor series. The research tasks include describing the methodology for computing ln 2, integrating these computations into a pseudorandom number generator, conducting testing, assessing the quality of the generated sequences, and analyzing the results.
Lastly, this solution offers robust change and feature
management capabilities. This means that it can easily adapt
to evolving business needs, with the ability to incorporate
new features and make necessary changes in a timely and
efficient manner. This flexibility ensures the solution
remains relevant and continues to deliver value over time
[
Historically, the concept of logarithms was introduced in
the 17th century by the Scottish mathematician John Napier,
who first developed logarithmic tables [
One of the classic methods for computing is using the Taylor series. For example, one can use the expansion of the logarithmic function into a Taylor series around 1: For x = 1, we get: (3) (5) (6) (7)
This represents a series of transformations over the
classical Taylor series expansion [
Later, other variations of this formula were found for
other constants [
Also, a representative of this type of formula is the
aforementioned formula (5). These formulas have allowed
for the effective calculation of constants starting from any
hexadecimal number [
Over time, the basic principle of calculation has not
changed, as the use of such formulas allows obtaining
precise values, although it requires a large amount of
computation. For example, the record for the number of
digits after the decimal was set in 2021 by William Ekols [
Also, do not forget the software that was used to
calculate such several digits. This software is called
ycruncher and was created in 2009 [
The most recent record as of now for computing the
number of digits after the decimal is 3·1012 digits after the
comma, set by Jordan Ranous on February 12, 2024 [
In conclusion, it can be noted that in calculating ln(2), we face two extremes: formulas that allow efficiently, in terms of resource use, to calculate this constant but for accuracy lose their speed or those that use a large number of resources. Ideally, finding a compromise would be optimal, but when we face the task of precise calculation of ln(2), this option does not exist. Accordingly, by shifting the focus from precise calculation of ln(2) to using knowledge about this constant for generating pseudorandom sequences, we can achieve a compromise and obtain the desired result.
In this paper, we will take a detailed look at using the
Taylor series for generating pseudorandom sequences and
demonstrate an algorithm that allows efficiently obtaining
binary random sequences of great length that pass NIST
statistical tests through which the quality of the generated
pseudorandom sequence can be best assessed [
4.1. Analysis of the Taylor series for ln 2 As mentioned earlier, the constant ln(2) is decomposed into the following Taylor series:
Input: sequence_length n: = 1 numerator_buffer = 0 denominator_buffer = 1 sequence[sequence_length] Repeat sequence_length times sequence_item = n * (n+1) numerator_buffer = numerator_buffer sequence_item+denominator_buffer denominator_buffer * = sequence_item If (numerator_buffer “1” >= denominator_buffer: sequence[i-th] = 1 numerator_buffer = (numerator_buffer “1” denominator_buffer If numerator_buffer == 0: denominator_buffer = 1
After performing a series of operations, we obtain such a definition of the series:
Let’s move on to the binary representation of ln 2. In essence, it is: where xi is the value of the ith bits.
However, it should be noted here that we are not interested in calculating the exact value in its mathematical essence, we are interested in the non-periodicity of the bit sequence, what it contains, respectively, the operations of multiplication/division by 2, offsets, adjustments at a random place in the sequence, etc. affect the non-periodicity of the sequence and can be used.
Accordingly, the problem can be reduced to finding a way to obtain a sequence of bits from (13).
Let’s consider one way to solve this problem:
The binary number system includes only two values: 0 and 1, respectively, they can be used for logical “Yes” or “No”.
The proposed method determines whether a specific iteration of formula (13) is in the interval [xi,xi + 1] and localizes this interval. In it is the principle, it resembles a mixture of Newton’s method and arithmetic coding. 4.2. Development and improvement of the algorithm for the generation of pseudorandom sequences based on the calculation of the Taylor series
The operation of the algorithm can be represented as follows: Otherwise: sequence[i-th] = 0 numerator_buffer “ = 1 n+ = 2 Output: sequence
Let’s consider this algorithm in more detail. Let’s assume that a segment of length is given. We iterate the series. We check whether the iteration value belongs to the interval [0.5, 1]. If it does, one is entered into the sequence, and the segment is localized by subtracting the doubled numerator from the denominator. If it is not, then zero is entered into the sequence, and the segment is localized by doubling the numerator.
Overall, this method does not ensure the exact calculation of any type of Taylor series, but due to its intricate structure, it can be used as a basis for generating pseudorandom numbers. This is because it involves a series of manipulations with non-periodic constants, which empirically should enable the generation of pseudorandom sequences based on them. Additionally, the technical aspect of the algorithm’s operation, which includes the overflow of some variables, should not be overlooked. Although this deprives us of calculation precision, it introduces a certain randomness. To verify the quality of the algorithm, let’s analyze the results of testing the obtained sequences using series (3) with the NIST statistical test suite:
As can be seen, in some key aspects, the obtained sequence does not meet the statistical standards of randomness. This is caused by technical limitations because, at high n values, n(n+1) overflows the variable and starts to acquire a certain pattern, losing the aspect of randomness.
This issue can be prevented by using libraries for
working with large numbers (for example, the GNU
Multiple Precision Arithmetic Library [
Let’s analyze the series iteration in the given algorithm. For each iteration, the value of n(n+1) is calculated, which transforms into n2+n. As can be seen, the value in the denominator of the series increases quadratically, which is the reason for the low speed and loss of randomness characteristics in the later iterations of the algorithm. One can try to simplify this series by removing one of the multipliers and checking the statistical characteristics of the sequence obtained in this case the n(n+1) contains two multipliers: even and odd. The best option would be to leave the odd multiplier because, in the case of an even one, the first bit of the iteration value of the series will always be 0, and when calculating the numerator, the obtained value will be even, which represents a certain pattern of the pseudorandom sequence.
As a result, we get the following improved algorithm:
n: = 1 numerator_buffer = 0 denominator_buffer = 1 sequence[sequence_length] Repeat sequence_length times sequence_iteration = n numerator_buffer = numerator_buffer sequence_iteration+denominator_buffer denominator_buffer *= sequence_iteration If (numerator_buffer “1) >= denominator_buffer: sequence[i-th] = 1 numerator_buffer = (numerator_buffer “1” denominator_buffer If numerator_buffer == 0: denominator_buffer = 1 Otherwise: sequence[i-th] = 0 numerator_buffer “=1 n+ = 2 Output: sequence[] * —
After testing the improved algorithm using the NIST statistical test suite, we obtained the following results:
Analyzing the test results, we can conclude that the algorithm generates sequences that meet the statistical standards for pseudorandom sequences. Let’s move on to comparing the performance between the basic version and the improved one.
Comparison of the Improved Algorithm with the Basic Version:
RAM: 16 GB Sequence Generation Performance: 10,000 bits generation: Basic: 0m0.003 seconds Improved: 0m0.002 seconds 1,000,000 bits generation: Basic: 0m0.105 seconds Improved: 0m0.081 seconds
From these results, we see that the improved algorithm consistently performs faster than the basic version at all tested sequence lengths, demonstrating its efficiency. This shows significant advantages, especially when the algorithm scales to larger data sizes, suggesting that the modifications made to simplify the series calculation contribute to a reduction in computational time while maintaining or enhancing the randomness quality of the sequences.
As can be seen from the figure, at any sequence length, the improved algorithm shows better performance, and considering that it also passes the NIST statistical tests, this indicates its significant advantage over the basic algorithm. This improvement not only enhances efficiency but also ensures that the algorithm maintains robust statistical properties, making it highly effective for applications requiring high-quality pseudorandom sequences.
Algorithm Development: A new algorithm based on the Taylor series has been proposed that provides the generation of pseudorandom sequences. This approach is based on the numerical properties of the natural logarithm of number 2 (ln 2), which is mathematically stable and accurate. Using ln 2 to initialize the generator allows achieving a high degree of randomness in the created sequences.
Algorithm Analysis: A detailed analysis of the developed algorithm was conducted, which includes checking its statistical characteristics and testing for compliance with NIST requirements. Testing showed that the algorithm could not initially provide a uniform distribution of pseudorandom numbers, leading to its improvement.
Algorithm Improvement: The basic algorithm has been improved, which provides better performance and improved statistical characteristics of the generated sequences. Optimization of the algorithm allows for significantly reducing the computational complexity, making it effective for use in real-world applications where computation time is a critical parameter.
The results of this research are an important step towards improving the reliability and quality of pseudorandom number generators. The proposed approach may find wide application in various fields such as cryptography, numerical modeling, simulations, and other numerical methods that require high-quality randomness and computational efficiency. Furthermore, the improved algorithm proposed in this paper can be used to create new generators or to enhance existing solutions, for example through optimization of calculations or application of new generation methods. Future research may focus on expanding the algorithm to other mathematical constants, which may further improve the quality of pseudorandom numbers. It is also possible to create an algorithm based on formula (5) using intervals (for example, as in Hamming matrices) or using other Taylor series for generating new pseudorandom sequences. Using such methods opens new horizons for the development of number theory and computational mathematics, providing powerful tools for solving a wide range of tasks in various fields of science and technology, especially for information protection.