Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Collatz-Weyl Generators: High Quality and High Throughput Parameterized Pseudorandom Number Generators

Tomasz R. Działa

TL;DR

This work introduces Collatz-Weyl Generators (CWGs), a family of high-throughput, uniform PRNGs built from generalized Collatz mappings combined with Weyl sequences to avoid short cycles and guarantee long periods. The authors formalize a general scheme and provide concrete implementations (CWG64, CWG128-64, CWG128) along with initialization and parallel-stream capabilities, and they demonstrate strong statistical performance across standard test suites. They establish theoretical period bounds and practical throughput advantages, including cryptographic considerations and potential applications. Overall, CWGs offer a scalable, low-footprint approach for parallel random number generation with robust uniformity and independence properties, suitable for both standard and cryptographic contexts.

Abstract

We introduce the Collatz-Weyl Generators, a family of uniform pseudorandom number generators (PRNGs) which are based on generalized Collatz mappings, derived from the Collatz conjecture and Weyl sequences. The high-quality statistical properties of our generators is demonstrated by the fact that they pass stringent randomness tests used by the research and standardization community. The proposed Collatz-Weyl Generators have a number of important properties, including solid mathematical foundations, enablement of high throughput and low latency implementation, small code and/or ASIC size, enablement of producing multiple independent streams and potential of support of cryptographic applications.

Collatz-Weyl Generators: High Quality and High Throughput Parameterized Pseudorandom Number Generators

TL;DR

This work introduces Collatz-Weyl Generators (CWGs), a family of high-throughput, uniform PRNGs built from generalized Collatz mappings combined with Weyl sequences to avoid short cycles and guarantee long periods. The authors formalize a general scheme and provide concrete implementations (CWG64, CWG128-64, CWG128) along with initialization and parallel-stream capabilities, and they demonstrate strong statistical performance across standard test suites. They establish theoretical period bounds and practical throughput advantages, including cryptographic considerations and potential applications. Overall, CWGs offer a scalable, low-footprint approach for parallel random number generation with robust uniformity and independence properties, suitable for both standard and cryptographic contexts.

Abstract

We introduce the Collatz-Weyl Generators, a family of uniform pseudorandom number generators (PRNGs) which are based on generalized Collatz mappings, derived from the Collatz conjecture and Weyl sequences. The high-quality statistical properties of our generators is demonstrated by the fact that they pass stringent randomness tests used by the research and standardization community. The proposed Collatz-Weyl Generators have a number of important properties, including solid mathematical foundations, enablement of high throughput and low latency implementation, small code and/or ASIC size, enablement of producing multiple independent streams and potential of support of cryptographic applications.
Paper Structure (16 sections, 1 theorem, 14 equations, 16 figures, 3 tables)

This paper contains 16 sections, 1 theorem, 14 equations, 16 figures, 3 tables.

Table of Contents

  1. INTRODUCTION
  2. COLLATZ-WEYL GENERATORS
  3. Generalized Collatz Sequences
  4. General Scheme
  5. Algorithms
  6. PERIOD
  7. UNIFORMITY
  8. BIRTHDAY PROBLEM
  9. MULTIPLE INDEPENDENT STREAMS
  10. Initialization
  11. STATISTICAL TESTS
  12. Methodology
  13. Results
  14. Birthday Test
  15. COMPARATIVE TESTS OF THROUGHPUT
  16. ...and 1 more sections

Key Result

Theorem 1

Let $x$ be a random but not necessarily uniform bit stream, and let $weyl$ be a uniform but not necessarily random bit stream. Then, $x \oplus weyl$ is a uniform bit stream.

Figures (16)

  • Figure 1: Block diagram of n-bit Collatz-Weyl Generator.
  • Figure 2: Implementation of the "generate” operation of the CWG64 generator in C++.
  • Figure 3: Implementation of the “generate” operation of the CWG128-64 generator in C++.
  • Figure 4: Implementation of the “generate” operation of the CWG128 generator in C++.
  • Figure 5: Implementation of the CWG64 initializer in C++.
  • ...and 11 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof