The structure and enumeration of periodic binary sequences with high nonlinear complexity
🤖 AI Summary
This study investigates the structure and enumeration of binary sequences of period $n$ whose nonlinear complexity is at least $3n/4$, as a means to assess their randomness. By integrating feedback shift register models, combinatorial methods, and structural analysis of sequences, the work provides the first complete characterization of the intrinsic structure of such high nonlinear complexity sequences and derives an exact formula for their count. This result fills a critical gap in the theoretical understanding of periodic sequences with high nonlinear complexity and establishes a foundational framework for the construction and analysis of high-quality pseudorandom sequences in cryptographic applications.
Application Category
📝 Abstract
Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the length of the shortest feedback shift registers that can generate a given sequence. In this paper, the structure of n-periodic binary sequences with nonlinear complexity larger than or equal to 3n/4 is characterized. Based on their structure, an exact enumeration formula for the number of such periodic sequences is determined.
Problem
Research questions and friction points this paper is trying to address.
nonlinear complexity
periodic binary sequences
sequence enumeration
feedback shift registers
randomness
Innovation
Methods, ideas, or system contributions that make the work stand out.
nonlinear complexity
periodic binary sequences
structure characterization
exact enumeration
feedback shift registers
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