The structure and enumeration of periodic binary sequences with high nonlinear complexity

TL;DR

This work addresses the problem of understanding and counting $n$-periodic binary sequences with high nonlinear complexity $nlc \ge \left\lfloor \tfrac{3n}{4} \right\rfloor$. It develops a rigorous framework of finite-length representative sequences, proves that these representatives are unique for high complexity, and derives the precise structure of such periodic sequences. Building on this, the authors obtain an exact enumeration formula for the number of $n$-periodic sequences meeting the high-complexity criterion and illustrate the approach with concrete calculations. The results provide a detailed picture of the distribution of nonlinear complexity in periodic binary sequences and have implications for assessing randomness in pseudorandom sequences and stream-cipher design.

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.

Figures (4)

• Figure 1: The visualized description of Case (1)
• Figure 2: The visualized description of Case (2)
• Figure 3: The visualized description of Subcase (3.1)
• Figure 4: The visualized description of Subcase (3.2) and Subcase (3.3)

Theorems & Definitions (42)

• Definition 1
• Lemma 1
• Lemma 2
• Definition 2
• Definition 3
• Lemma 3
• Definition 4
• Lemma 4
• Lemma 5
• Lemma 6