Further Investigations on Nonlinear Complexity of Periodic Binary Sequences

TL;DR

This work investigates how circular shifts affect the nonlinear complexity (nlc) of binary sequences and establishes explicit relations between the n-length subsequences and their periodic counterparts. It introduces structured finite-sequence classes $\mathcal{B}(n,c)$ and $\mathcal{R}(n,c)$, proving that for $c \ge \lceil n/2\rceil$ the periodic nonlinear complexity satisfies $nlc(s_n^{\infty}) = nlc(s_n) + add(s_n)$, with $add(s_n)$ capturing added terms under shifts. Leveraging these results, two algorithms are developed to generate all $n$-periodic binary sequences with any prescribed nonlinear complexity $\omega$, including special cases such as binary de Bruijn sequences. The findings provide a constructive bridge between finite-length and periodic nonlinear complexities, enabling efficient design of pseudorandom binary sequences with targeted nonlinear properties and linking to prior high-nonlinearity characterizations.

Abstract

Nonlinear complexity is an important measure for assessing the randomness of sequences. In this paper we investigate how circular shifts affect the nonlinear complexities of finite-length binary sequences and then reveal a more explicit relation between nonlinear complexities of finite-length binary sequences and their corresponding periodic sequences. Based on the relation, we propose two algorithms that can generate all periodic binary sequences with any prescribed nonlinear complexity.

Figures (5)

• Figure 1: An $m$-stage FSR with feedback function $f$
• Figure 2: Added terms in Definition \ref{['def-t']} for $t< d$
• Figure 3: The visualized description of Proposition \ref{['fini-shift-nlc']} (i)
• Figure 4: The visualized description of Example \ref{['ex1']}
• Figure 5: The description of Case (2) in Theorem \ref{['c+t']}

Theorems & Definitions (39)

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