4-Adic Complexity of Interleaved Quaternary Sequences

TL;DR

This paper analyzes the 4-adic complexity of interleaved quaternary sequences with ideal autocorrelation, providing a general formula, bounds, and exact values for sequences based on known binary sequences, demonstrating their cryptographic strength.

Contribution

It introduces a general formula for the 4-adic complexity of interleaved quaternary sequences and evaluates it for various known binary sequences, showing high complexity levels.

Findings

The 4-adic complexity has a lower bound of (4^n - 1)

Most sequences achieve or nearly reach maximum complexity

Sequences have high resistance to rational approximation attacks

Abstract

Tang and Ding \cite{X. Tang} present a series of quaternary sequences $w(a, b)$ interleaved by two binary sequences $a$ and $b$ with ideal autocorrelation and show that such interleaved quaternary sequences have optimal autocorrelation. In this paper we consider the 4-adic complexity $FC_{w}(4)$ of such quaternary sequence $w=w(a, b)$. We present a general formula on $FC_{w}(4)$, $w=w(a, b)$. As a direct consequence, we obtain a general lower bound $FC_{w}(4)\geq\log_{4}(4^{n}-1)$ where $2n$ is the period of the sequence $w$. By taking $a$ and $b$ to be several types of known binary sequences with ideal autocorrelation ($m$-sequences, twin-prime, Legendre, Hall sequences and their complement, shift or sample sequences), we compute the exact values of $FC_{w}(4)$, $w=w(a, b)$ and show that in most cases $FC_{w}(4)$ reaches or nearly reaches the maximum value $\log_{4}(4^{2n}-1)$. Our results show that the 4-adic complexity of the quaternary sequences defined in \cite{X. Tang} are large enough to resist the attack of the rational approximation algorithm.