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Accelerating true orbit pseudorandom number generation using Newton's method

Asaki Saito, Akihiro Yamaguchi

TL;DR

The paper tackles the prohibitive $O(N^2)$ time of true orbit pseudorandom number generation based on the Bernoulli map over quadratic algebraic integers by integrating Newton's method to compute exact binary expansions from rational approximants. It proves a sufficient condition for the first $N$ bits to match between the irrational root $\alpha$ and its Newton-based approximation, and shows the overall worst-case complexity reduces to $O\big(N\,g(N)\big)$, where $g(N)$ reflects large-integer multiplication cost. The authors present a practical algorithm that iteratively refines the approximation, demonstrates linear memory scaling, and provides extensive numerical validation, including a long sequence that passes all RabbitFile tests in TestU01. The results indicate a substantial speedup over prior true orbit generators while preserving high statistical quality, enabling efficient, high-precision PRNG suitable for rigorous empirical testing and simulations.

Abstract

The binary expansions of irrational algebraic numbers can serve as high-quality pseudorandom binary sequences. This study presents an efficient method for computing the exact binary expansions of real quadratic algebraic integers using Newton's method. To this end, we clarify conditions under which the first $N$ bits of the binary expansion of an irrational number match those of its upper rational approximation. Furthermore, we establish that the worst-case time complexity of generating a sequence of length $N$ with the proposed method is equivalent to the complexity of multiplying two $N$-bit integers, showing its efficiency compared to a previously proposed true orbit generator. We report the results of numerical experiments on computation time and memory usage, highlighting in particular that the proposed method successfully accelerates true orbit pseudorandom number generation. We also confirm that a generated pseudorandom sequence successfully passes all the statistical tests included in RabbitFile of TestU01.

Accelerating true orbit pseudorandom number generation using Newton's method

TL;DR

The paper tackles the prohibitive time of true orbit pseudorandom number generation based on the Bernoulli map over quadratic algebraic integers by integrating Newton's method to compute exact binary expansions from rational approximants. It proves a sufficient condition for the first bits to match between the irrational root and its Newton-based approximation, and shows the overall worst-case complexity reduces to , where reflects large-integer multiplication cost. The authors present a practical algorithm that iteratively refines the approximation, demonstrates linear memory scaling, and provides extensive numerical validation, including a long sequence that passes all RabbitFile tests in TestU01. The results indicate a substantial speedup over prior true orbit generators while preserving high statistical quality, enabling efficient, high-precision PRNG suitable for rigorous empirical testing and simulations.

Abstract

The binary expansions of irrational algebraic numbers can serve as high-quality pseudorandom binary sequences. This study presents an efficient method for computing the exact binary expansions of real quadratic algebraic integers using Newton's method. To this end, we clarify conditions under which the first bits of the binary expansion of an irrational number match those of its upper rational approximation. Furthermore, we establish that the worst-case time complexity of generating a sequence of length with the proposed method is equivalent to the complexity of multiplying two -bit integers, showing its efficiency compared to a previously proposed true orbit generator. We report the results of numerical experiments on computation time and memory usage, highlighting in particular that the proposed method successfully accelerates true orbit pseudorandom number generation. We also confirm that a generated pseudorandom sequence successfully passes all the statistical tests included in RabbitFile of TestU01.

Paper Structure

This paper contains 11 sections, 4 theorems, 34 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Newton's method
  4. Conditions for binary expansions to match or not
  5. Algorithm
  6. Computational complexity
  7. Experiments
  8. Computation time
  9. Memory usage
  10. Statistical testing
  11. Quadratic true orbit generator and its computational complexity

Key Result

proposition thmcounterproposition

The elements of $I_{b}$ are distributed almost uniformly in the unit interval for sufficiently large $|b|$.

Figures (2)

  • Figure 1: Results of computation times for the proposed method using Newton's method (red circles) and the quadratic true orbit generator (blue diamonds)
  • Figure 2: Results of memory usages for the proposed method using Newton's method (red circles) and the quadratic true orbit generator (blue squares and green diamonds). The red circles and blue squares represent the averages of 100 samples, whereas the green diamonds represent the values obtained from a single sample.

Theorems & Definitions (5)

  • proposition thmcounterproposition: SaitoChaos2016SaitoPreprint
  • proposition thmcounterproposition: SaitoPreprint
  • proposition thmcounterproposition
  • proof
  • corollary thmcountercorollary