Pseudorandom sequence generator based on the computation of ln 2⋆ Ivan Opirskyy1,∗,†, Oleh Harasymchuk1,†, Olha Mykhaylova1,†, Oleksii Hrushkovskyi1,† and Pavlo Kozak1,† 1 Lviv Polytechnic National University, 12 Stepana Bandery str., 79000 Lviv, Ukraine Abstract 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. Keywords pseudorandom sequence generator, pseudorandom number generators, mathematical algorithms, Taylor series, NIST statistical tests, mathematical constants, cryptographic protocols, simulation algorithms, computational efficiency1 1. Introduction generators is hard to overestimate, as they provide the foundation for innovation and development in many sectors Pseudorandom Number Generators (PRNGs) and [8]. Particularly noteworthy is their importance in Pseudorandom Sequence Generators (PSGs) are key cybersecurity, where they are also a key element, and are elements in many scientific and technical fields. They play used in solving various tasks, namely for data encryption a crucial role in modern technologies, providing the basis [9], authentication [10, 11], key generation, digital for numerous applications in computer science, signature creation algorithms, and in testing and evaluating cryptography, statistical sampling, modeling, and security [12–18]. Therefore, developers of such generators simulations [1–7]. These generators enable the creation of face high demands for the quality of the output sequences: number sequences that, while deterministic, appear random, unpredictability, statistical independence, cryptographic which is critically important for ensuring data security, robustness, and maximum generation speed. Ensuring a model accuracy, and algorithm reliability. In a world where high quality of randomness is an important task, as it affects information is becoming increasingly valuable, the reliability and accuracy of many algorithms and systems understanding and using PRNGs and PSGs is essential for where the generated sequences will be applied. developing effective solutions in various fields, from finance Traditionally, various mathematical algorithms and to gaming. Thus, the importance of pseudorandom number methods are used to generate pseudorandom numbers, CQPC-2024: Classic, Quantum, and Post-Quantum Cryptography, August 0000-0002-8461-8996 (I. Opirskyi); 0000-0002-8742-8872 6, 2024, Kyiv, Ukraine (O. Harasymchuk); 0000-0002-3086-3160 (O. Mykhaylova); 0009-0007- ∗ Corresponding author. 3626-9780 (O. Hrushkovskyi); 0009-0005-0432-4541 (P. Kozak) † These authors contributed equally. © 2024 Copyright for this paper by its authors. Use permitted under ivan.r.opirskyi@lpnu.ua (I. Opirskyi); garasymchuk@ukr.net Creative Commons License Attribution 4.0 International (CC BY 4.0). (O. Harasymchuk); mykhaylovaolga1@gmail.com (O. Mykhaylova); oleksii.hrushkovskyi.kb.2022@lpnu.ua (O. Hrushkovskyi); pavlo.kozak.kb.2022@lpnu.ua (P. Kozak) CEUR Workshop ceur-ws.org ISSN 1613-0073 79 Proceedings including the linear congruential method [19–21], shift This series converges relatively slowly, so more efficient register generators [22–24], the Lagrange method [25], methods are typically used for practical calculations [39]. Mersenne Twister algorithms [26], and others [27–30]. Another approach is based on numerical integration However, existing approaches do not always meet the methods. The natural logarithm can be defined as a definite requirements for uniform distribution and passing integral: statistical tests, so researchers continuously search for new methods, ways, and algorithms to generate pseudorandom sequences that meet the growing demands for their quality. (3) . For numerical computation of this integral, methods 2. Problem statement such as the rectangle (midpoint), trapezoidal, or Simpson’s The research problem involves seeking a new approach to rule are applied, which allows for obtaining more precise generating pseudorandom numbers that would ensure high- values [40]. quality randomness and computational efficiency. One such One of the most powerful methods of computing approach is the use of the numerical properties of Ln 2 is the Newton-Raphson method, which is used for mathematical constants [31]. The natural logarithm of finding the roots of equations and can be adapted for number 2 (ln 2) has a range of unique properties that make computing logarithms [41]. Starting with the equation it promising for use in generating pseudorandom numbers. ey = 2, where y = ln 2, we can formulate a function Existing studies demonstrate the effectiveness of using f(x) = ex – 2 and apply the Newton-Raphson method to solve logarithms of mathematical constants in cryptography and for numerical methods [32–34], but the integration of ln 2 into pseudorandom number generators remains insufficiently explored. . (4) This paper aims to develop and test a pseudorandom A clear downside is that the use of Euler’s constant 𝑒 for number generator based on the computation of ln 2 using computation is required, which itself is transcendental and the Taylor series. The research tasks include describing the cannot be precisely calculated. Therefore, it is necessary to methodology for computing ln 2, integrating these calculate the constant itself simultaneously, which increases computations into a pseudorandom number generator, the number of operations. On the other hand, in such a case, conducting testing, assessing the quality of the generated one iteration can generate a much larger sequence of sequences, and analyzing the results. numbers than other algorithms. Lastly, this solution offers robust change and feature With the advent of computers, an obvious application management capabilities. This means that it can easily adapt became the computation of mathematical constants. In one to evolving business needs, with the ability to incorporate of the first works on this topic [42], the following formula new features and make necessary changes in a timely and was used to compute ln 2: efficient manner. This flexibility ensures the solution remains relevant and continues to deliver value over time [35]. (5) . 3. Research analysis This represents a series of transformations over the classical Taylor series expansion [43]. In 1995, there was an Historically, the concept of logarithms was introduced in unprecedented breakthrough in the calculation of constants. the 17th century by the Scottish mathematician John Napier, French mathematician Simon Plouffe discovered a series who first developed logarithmic tables [36]. His work that allowed the computation of the ith hexadecimal digit of significantly simplified the process of multiplying and π [44]. The formula is as follows: dividing large numbers, which was extremely useful for astronomy, navigation, and other sciences. Natural logarithms, based on the number 𝑒 (approximately equal to (6) 2.71828), appeared later and became important in . mathematical analysis thanks to the works of Leibniz and Later, other variations of this formula were found for Euler [37, 38]. other constants [35], among which we are, of course, One of the classic methods for computing 𝑒 is using the interested in ln(2), which is represented by the formula: Taylor series. For example, one can use the expansion of the logarithmic function into a Taylor series around 1: (7) . (1) Also, a representative of this type of formula is the . aforementioned formula (5). These formulas have allowed For x = 1, we get: for the effective calculation of constants starting from any hexadecimal number [45], ensuring low resource consumption, but in practice, the approximation to the exact . (2) result occurs slowly. Some formulas ensure the accuracy of calculations at the expense of using more computer 80 resources. For example, in 1997 a record of 10,079,926 digits was set [46], which compared to the results of 1962 [42], i.e., . (11) 3863 digits, is a significant breakthrough. This was achieved After performing a series of operations, we obtain such using the Mercator series [47], and in this case, by the a definition of the series: formula: . (12) , (8) That has the general form: where . (13) (9) . Let’s move on to the binary representation of ln 2. In Over time, the basic principle of calculation has not essence, it is: 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 (14) , digits after the decimal was set in 2021 by William Ekols [48] and represents 1.5·1012 digits after the comma. To calculate where xi is the value of the ith bits. such several digits, it took 98.9 days [49], and for the However, it should be noted here that we are not correctness check—61.7. And this is on a machine that has interested in calculating the exact value in its mathematical 48 cores and 256 GB of RAM. essence, we are interested in the non-periodicity of the bit Also, do not forget the software that was used to sequence, what it contains, respectively, the operations of calculate such several digits. This software is called y- multiplication/division by 2, offsets, adjustments at a cruncher and was created in 2009 [50]. The main advantage random place in the sequence, etc. affect the non-periodicity of this software is its optimization and maximum efficiency of the sequence and can be used. in resource use thanks to various techniques [51]. For the Accordingly, the problem can be reduced to finding a calculation, this formula is used: 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 (10) and 1, respectively, they can be used for logical “Yes” or . “No”. The most recent record as of now for computing the The proposed method determines whether a specific number of digits after the decimal is 3·1012 digits after the iteration of formula (13) is in the interval [xi,xi + 1] and comma, set by Jordan Ranous on February 12, 2024 [52]. A localizes this interval. In it is the principle, it resembles a machine with 2 × Intel Xeon Platinum 8460H (a total of 80 mixture of Newton’s method and arithmetic coding. cores) and 512 GB of RAM was used. The computation took 42.7 days, while the correctness check took 58.3 days. 4.2. Development and improvement of the In conclusion, it can be noted that in calculating ln(2), algorithm for the generation of we face two extremes: formulas that allow efficiently, in pseudorandom sequences based on terms of resource use, to calculate this constant but for the calculation of the Taylor series accuracy lose their speed or those that use a large number of resources. Ideally, finding a compromise would be The operation of the algorithm can be represented as optimal, but when we face the task of precise calculation of follows: ln(2), this option does not exist. Accordingly, by shifting the focus from precise calculation of ln(2) to using knowledge Input: sequence_length about this constant for generating pseudorandom sequences, n: = 1 we can achieve a compromise and obtain the desired result. numerator_buffer = 0 In this paper, we will take a detailed look at using the denominator_buffer = 1 Taylor series for generating pseudorandom sequences and sequence[sequence_length] demonstrate an algorithm that allows efficiently obtaining Repeat sequence_length times binary random sequences of great length that pass NIST sequence_item = n * (n+1) statistical tests through which the quality of the generated numerator_buffer = numerator_buffer * pseudorandom sequence can be best assessed [53–57]. sequence_item+denominator_buffer denominator_buffer * = sequence_item 4. The main part If (numerator_buffer “1” >= denominator_buffer: 4.1. Analysis of the Taylor series for ln 2 sequence[i-th] = 1 As mentioned earlier, the constant ln(2) is decomposed into numerator_buffer = (numerator_buffer “1” - the following Taylor series: denominator_buffer If numerator_buffer == 0: denominator_buffer = 1 81 Otherwise: transforms into n2+n. As can be seen, the value in the sequence[i-th] = 0 denominator of the series increases quadratically, which is numerator_buffer “ = 1 the reason for the low speed and loss of randomness n+ = 2 characteristics in the later iterations of the algorithm. One Output: sequence can try to simplify this series by removing one of the multipliers and checking the statistical characteristics of the Let’s consider this algorithm in more detail. Let’s sequence obtained in this case the n(n+1) contains two assume that a segment of length is given. We iterate the multipliers: even and odd. The best option would be to leave series. We check whether the iteration value belongs to the the odd multiplier because, in the case of an even one, the interval [0.5, 1]. If it does, one is entered into the sequence, first bit of the iteration value of the series will always be 0, and the segment is localized by subtracting the doubled and when calculating the numerator, the obtained value will numerator from the denominator. If it is not, then zero is be even, which represents a certain pattern of the entered into the sequence, and the segment is localized by pseudorandom sequence. doubling the numerator. As a result, we get the following improved algorithm: Overall, this method does not ensure the exact calculation of any type of Taylor series, but due to its Input: sequence_length intricate structure, it can be used as a basis for generating n: = 1 pseudorandom numbers. This is because it involves a series numerator_buffer = 0 of manipulations with non-periodic constants, which denominator_buffer = 1 empirically should enable the generation of pseudorandom sequence[sequence_length] sequences based on them. Additionally, the technical aspect Repeat sequence_length times of the algorithm’s operation, which includes the overflow of sequence_iteration = n some variables, should not be overlooked. Although this numerator_buffer = numerator_buffer * deprives us of calculation precision, it introduces a certain sequence_iteration+denominator_buffer randomness. To verify the quality of the algorithm, let’s denominator_buffer *= sequence_iteration analyze the results of testing the obtained sequences using If (numerator_buffer “1) >= denominator_buffer: series (3) with the NIST statistical test suite: sequence[i-th] = 1 numerator_buffer = (numerator_buffer “1” — Table 1 denominator_buffer Test results of the algorithm for generating pseudorandom If numerator_buffer == 0: sequences based on the calculation of the Taylor series using denominator_buffer = 1 the NIST statistical test package Otherwise: Statistical Test p-value Pass Rate Status sequence[i-th] = 0 Frequency 0.000199 7/10 Failed numerator_buffer “=1 Block Frequency 0.000000 10/10 Failed Cumulative Sums 0.008879 8/10 Pass n+ = 2 Runs 0.000000 0/10 Failed Longest Run 0.000000 1/10 Failed Output: sequence[] Rank 0.000000 0/10 Failed FFT 0.000000 0/10 Failed Non-Overlapping 0.000000 0/10 Failed After testing the improved algorithm using the NIST Template statistical test suite, we obtained the following results: Overlapping Template 0.000000 2/10 Failed Analyzing the test results, we can conclude that the Universal 0.017912 7/10 Failed Approximate Entropy 0.000000 0/10 Failed algorithm generates sequences that meet the statistical Random Excursions – 3/3 Pass standards for pseudorandom sequences. Let’s move on to Random Excursions – 3/3 Pass comparing the performance between the basic version and Variant the improved one. Serial 0.066882 9/10 Pass Comparison of the Improved Algorithm with the Basic Linear Complexity 0.739918 10/10 Pass Version: As can be seen, in some key aspects, the obtained Technical Specifications of the Computer: sequence does not meet the statistical standards of CPU: AMD Ryzen 5 4500U with Radeon Graphics (6) @ randomness. This is caused by technical limitations because, 2.375GHz at high n values, n(n+1) overflows the variable and starts to OS: Linux acquire a certain pattern, losing the aspect of randomness. RAM: 16 GB This issue can be prevented by using libraries for Sequence Generation Performance: working with large numbers (for example, the GNU 10,000 bits generation: Multiple Precision Arithmetic Library [58] or bn from Basic: 0m0.003 seconds OpenSSL [59]) or by optimizing the algorithm itself, such as Improved: 0m0.002 seconds by changing the series we iterate. To save computational 1,000,000 bits generation: resources, we will focus on the latter option. Basic: 0m0.105 seconds Let’s analyze the series iteration in the given algorithm. Improved: 0m0.081 seconds For each iteration, the value of n(n+1) is calculated, which 82 100,000,000 bits generation: From these results, we see that the improved algorithm Basic: 0m4.401 seconds consistently performs faster than the basic version at all Improved: 0m4.300 seconds tested sequence lengths, demonstrating its efficiency. This shows significant advantages, especially when the Table 2 algorithm scales to larger data sizes, suggesting that the Test results of an improved pseudorandom sequence modifications made to simplify the series calculation generation algorithm based on the calculation of the Taylor contribute to a reduction in computational time while series using the NIST statistical test suite maintaining or enhancing the randomness quality of the Statistical Test p-value Pass Rate Status sequences. Frequency 0.122325 10/10 Pass As can be seen from the figure, at any sequence length, Block Frequency 0.739918 10/10 Pass the improved algorithm shows better performance, and Cumulative Sums 0.066882 10/10 Pass Runs 0.008879 10/10 Pass considering that it also passes the NIST statistical tests, this Longest Run 0.534146 9/10 Pass indicates its significant advantage over the basic algorithm. Rank 0.350485 10/10 Pass This improvement not only enhances efficiency but also FFT 0.534146 10/10 Pass ensures that the algorithm maintains robust statistical Non-Overlapping 0.739918 10/10 Pass properties, making it highly effective for applications Template Overlapping Template 0.066882 10/10 Pass requiring high-quality pseudorandom sequences. Universal 0.213309 10/10 Pass Approximate Entropy 0.350485 9/10 Pass Random Excursions – 2/2 Pass Random Excursions – 2/2 Pass Variant Serial 0.739918 10/10 Pass Linear Complexity 0.534146 10/10 Pass Figure 1: Shows a comparison of the performance between the basic and improved algorithms. The blue color represents the basic version, and the orange color represents the improved version 5. Conclusion distribution of pseudorandom numbers, leading to its improvement. The main conclusions of our research include: Algorithm Improvement: The basic algorithm has Algorithm Development: A new algorithm based been improved, which provides better performance and on the Taylor series has been proposed that provides the improved statistical characteristics of the generated generation of pseudorandom sequences. This approach sequences. Optimization of the algorithm allows for is based on the numerical properties of the natural significantly reducing the computational complexity, logarithm of number 2 (ln 2), which is mathematically making it effective for use in real-world applications stable and accurate. Using ln 2 to initialize the generator where computation time is a critical parameter. allows achieving a high degree of randomness in the The results of this research are an important step created sequences. towards improving the reliability and quality of Algorithm Analysis: A detailed analysis of the pseudorandom number generators. The proposed developed algorithm was conducted, which includes approach may find wide application in various fields checking its statistical characteristics and testing for such as cryptography, numerical modeling, simulations, compliance with NIST requirements. Testing showed and other numerical methods that require high-quality that the algorithm could not initially provide a uniform randomness and computational efficiency. 83 Furthermore, the improved algorithm proposed in this in Information and Telecommunication Systems, paper can be used to create new generators or to vol. 2746 (2020) 23–32. enhance existing solutions, for example through [9] V. Sokolov, P. Skladannyi, H. Hulak, Stability optimization of calculations or application of new Verification of Self-Organized Wireless Networks generation methods. 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