Digital Chaotic Sequence Generation via RNS Arithmetic
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Solution Overview
Problem
Chaotic communications systems face low throughput due to the limitations of analog chaos generators, which drift over time and require frequent synchronization, reducing data throughput and being impractical for high data rate applications.
Innovation Solution
A method for digitally generating chaotic sequences using residue number system (RNS) arithmetic operations and polynomial equations, with solutions iteratively computed and expressed as RNS residue values, and mapped to a weighted number system using the Chinese Remainder Theorem, to produce a chaotic sequence with improved state drift and update properties.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If analog chaotic circuits are used to generate chaotic sequences, then chaotic properties are achieved, but drift occurs over time and synchronization requires state information exchange which reduces throughput
Solution Approach 1:
The patent replaces analog chaotic circuits with a digital implementation using polynomial equations and RNS arithmetic operations. This substitution eliminates the drift problem inherent in analog circuits while maintaining chaotic properties through deterministic mathematical operations, thereby improving reliability without sacrificing throughput.
Solution Approach 2:
The patent changes the fundamental parameters of chaos generation from continuous analog values to discrete digital values using polynomial equations. By using RNS arithmetic with carefully selected moduli, the system achieves chaotic behavior through parameter transformations rather than physical circuit drift, resolving the contradiction between stability and throughput.
2Reliability
If state information is exchanged frequently between transmitter and receiver to maintain synchronization, then drift is reduced, but data throughput decreases
Solution Approach 1:
The digital chaotic generator uses deterministic polynomial equations where the same initial conditions and polynomial coefficients at both transmitter and receiver enable autonomous synchronization. The system serves itself by reproducing identical chaotic sequences without requiring state information exchange, thereby maintaining reliability while preserving throughput.
Solution Approach 2:
The patent extracts the synchronization requirement from the system by using deterministic polynomial equations with shared initial conditions. This removes the need for state information exchange entirely, as both ends independently generate identical sequences through the same mathematical operations, resolving the contradiction between synchronization accuracy and data throughput.
3Device complexity
If multiple pseudo-random number generators are used to generate chaotic-like sequences, then sequence complexity increases, but true chaotic properties are not achieved
Solution Approach 1:
The patent transforms the approach by changing from multiple pseudo-random generators to a single polynomial equation system using RNS arithmetic. This parameter change enables true chaotic properties through deterministic mathematical operations with sensitive dependence on initial conditions, achieving both complexity and authenticity simultaneously.
Solution Approach 2:
The patent combines polynomial equations with RNS arithmetic operations to create a composite mathematical system. This composite approach integrates the benefits of deterministic chaos theory with efficient digital computation, achieving true chaotic properties while maintaining sequence complexity suitable for communication applications.
4Ease of manufacture
If binary arithmetic is used to achieve digital chaos, then arithmetic precision requirements become impractical
Solution Approach 1:
The patent changes the arithmetic base from binary to residue number system with carefully selected moduli. This parameter change reduces precision requirements by working with discrete modular arithmetic operations that are inherently more suitable for digital implementation, making the system both precise and manufacturable.
Solution Approach 2:
The patent substitutes traditional binary arithmetic with RNS arithmetic operations. This substitution eliminates the precision problems of binary chaos generation by using modular arithmetic that naturally confines values within manageable ranges, improving both implementation feasibility and arithmetic precision.
Data Source
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AI summary
A method is provided for generating a chaotic sequence. The method includes selecting a plurality of polynomial equations. The method also includes using residue number system (RNS) arithmetic operations to respectively determine solutions for the polynomial equations. The solutions are iteratively computed and expressed as RNS residue values. The method further includes determining a series of digits in a weighted number system (e.g., a binary number system) based on the RNS residue values. According to an aspect of the invention, the method includes using a Chinese Remainder Theorem process to determine a series of digits in the weighted number system based on the RNS residue values. According to another aspect of the invention, the determining step comprises identifying a number in the weighted number system that is defined by the RNS residue values.