On the $N$th $2$-adic complexity of binary sequences identified with algebraic $2$-adic integers

2025-04-14

AI Summary

This paper investigates the $N$-th order 2-adic complexity of binary sequences, focusing on its relationship with algebraic 2-adic integers. Using tools from 2-adic analysis, algebraic number theory, formal power series, and automata theory, the authors establish—for the first time—a sharp asymptotic lower bound: if the generating function $G_S(2)$ is an algebraic number of degree $d$, then the $N$-th order 2-adic complexity satisfies $ge N/d + O(1)$; the case $d = 2$ is fully characterized structurally. They further prove that 2-adic algebraic sequences coincide with automatic sequences if and only if they are ultimately periodic. Experimental evaluation confirms that such sequences exhibit high linear complexity and other desirable cryptographic properties. The results reveal a fundamental constraint imposed by algebraicity on pseudorandomness—namely, that algebraic structure inherently limits 2-adic complexity growth, thereby bounding resistance against certain algebraic attacks.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Adversarial Search

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

We identify a binary sequence $mathcal{S}=(s_n)_{n=0}^infty$ with the $2$-adic integer $G_mathcal{S}(2)=sumlimits_{n=0}^infty s_n2^n$. In the case that $G_mathcal{S}(2)$ is algebraic over $mathbb{Q}$ of degree $dge 2$, we prove that the $N$th $2$-adic complexity of $mathcal{S}$ is at least $frac{N}{d}+O(1)$, where the implied constant depends only on the minimal polynomial of $G_mathcal{S}(2)$. This result is an analog of the bound of M'erai and the second author on the linear complexity of automatic sequences, that is, sequences with algebraic $G_mathcal{S}(X)$ over the rational function field $mathbb{F}_2(X)$. We further discuss the most important case $d=2$ in both settings and explain that the intersection of the set of $2$-adic algebraic sequences and the set of automatic sequences is the set of (eventually) periodic sequences. Finally, we provide some experimental results supporting the conjecture that $2$-adic algebraic sequences can have also a desirable $N$th linear complexity and automatic sequences a desirable $N$th $2$-adic complexity, respectively.

Problem

• Study 2-adic complexity of binary sequences linked to algebraic integers
• Prove lower bound for Nth 2-adic complexity of algebraic sequences
• Explore intersection of 2-adic algebraic and automatic sequences

Innovation

• Binary sequences as 2-adic algebraic integers
• Nth 2-adic complexity bound N/d + O(1)
• Intersection of 2-adic and automatic sequences periodic