Generating Gaussian-Distributed Pseudo-Random Numbers Using the Logistic Map on an FPGA

Core Concepts

This project develops a pseudo-random number generator (PRNG) using the logistic map, implemented in Verilog HDL on an FPGA and processes its output through a Central Limit Theorem (CLT) function to achieve a Gaussian distribution.

Abstract

The project explores the potential of chaotic systems, particularly the logistic map, for generating pseudo-random numbers. It implements the logistic map function in Verilog HDL on an FPGA and applies the Exponentially Weighted Moving Average (EWMA) technique to transform the chaotic output into a Gaussian-distributed sequence.

The system integrates additional FPGA modules for real-time interaction and visualization, including a clock generator, UART interface, XADC, and a 7-segment display driver. These components facilitate the direct display of PRNG values on the FPGA and the transmission of data to a laptop for histogram analysis, verifying the Gaussian nature of the output.

The proof of concept in Julia demonstrates the empirical ability to obtain a Gaussian distribution from the logistic map-based PRNG. The Verilog HDL implementation on the FPGA further validates this approach, showcasing the practical application of chaotic systems for generating Gaussian-distributed pseudo-random numbers in digital hardware.

Stats

The logistic map function is given by:
x_n+1 = r*x_n(1 - x_n)
The EWMA function is given by:
EWMA_t = α*r_t + (1 - α)*EWMA_t-1

Quotes

"The logistic map's appeal lies in its deterministic behaviour that, when initiated with a slight variation in initial conditions, can lead to vastly different outcomes. This sensitivity to initial conditions is a hallmark of chaotic systems and is particularly valuable in the context of PRNGs, where unpredictability is a desired feature."
"Transforming the output from chaotic systems to follow a Gaussian distribution is crucial for many applications. However, the dependency of successive outputs in chaotic systems poses a challenge to traditional methods like the CLT. Research in this area has focused on adapting the CLT to accommodate these dependencies, exploring theoretical frameworks and practical techniques to achieve normally distributed outputs from chaotic sequences."

Key Insights Distilled From

Logistic Map Pseudo Random Number Generator in FPGA

by Mateo Jalen ... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19246.pdf

Deeper Inquiries

How could the PRNG algorithm be further optimized to improve the quality and performance of the generated pseudo-random numbers?

To optimize the PRNG algorithm based on the logistic map, several strategies can be implemented:

Fine-tuning Parameters: Experiment with different values of the parameter "r" in the logistic map equation to explore a wider range of chaotic behavior. This can lead to more diverse and unpredictable sequences.
Enhanced Seed Generation: Implement a more sophisticated seed generation mechanism that incorporates multiple sources of entropy, such as sensor data from various environmental factors, to increase randomness and reduce predictability.
Parallel Processing: Utilize parallel processing techniques in the FPGA to speed up the generation of pseudo-random numbers, enabling faster computations and higher throughput.
Hardware Acceleration: Implement specific hardware accelerators within the FPGA to optimize the computation of the logistic map function, reducing latency and improving overall performance.
Dynamic Reconfiguration: Explore dynamic reconfiguration capabilities of the FPGA to adapt the PRNG algorithm on-the-fly, allowing for real-time adjustments and optimizations based on performance metrics.

What are the potential limitations or drawbacks of using the logistic map as the basis for a PRNG, and how could these be addressed?

Using the logistic map for a PRNG comes with certain limitations:

Periodicity: The logistic map can exhibit periodic behavior for certain parameter values, leading to predictable sequences. This can be addressed by carefully selecting the parameter "r" to ensure chaotic behavior.
Seed Sensitivity: The logistic map is highly sensitive to initial conditions, which can pose challenges in generating truly random numbers. Implementing a robust seed generation mechanism and incorporating external entropy sources can mitigate this issue.
Complexity: The logistic map algorithm may be computationally intensive, especially for FPGA implementations. Simplifying the algorithm or optimizing the hardware design can help address this complexity.
Statistical Properties: While the logistic map can produce chaotic sequences, ensuring that the generated numbers exhibit desired statistical properties like uniform distribution or specific correlation structures may require additional post-processing or modifications to the algorithm.

What other chaotic systems or techniques could be explored to generate pseudo-random numbers with different statistical properties, such as uniform distribution or specific correlation structures?

Henon Map: The Henon map is another chaotic system that can be used for PRNGs, offering different dynamics and statistical properties compared to the logistic map. It can generate sequences with varying statistical distributions.
Lorenz System: The Lorenz system is known for its complex behavior and sensitivity to initial conditions. By utilizing the Lorenz equations, PRNGs with unique statistical properties can be developed.
Coupled Map Lattices: Instead of a single chaotic map, using coupled map lattices where multiple maps interact can lead to more intricate and diverse pseudo-random sequences with specific correlation structures.
Quantum Random Number Generators: Exploring quantum-based random number generation techniques can provide truly random numbers based on quantum phenomena, offering a different approach to generating pseudo-random numbers with unique statistical properties.