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The 4-adic complexity of quaternary sequences with low autocorrelation and high linear complexity

Feifei Yan, Pinhui Ke, Lingmei Xiao

TL;DR

The paper addresses the problem of estimating the $4$-adic complexity of quaternary sequences with low autocorrelation and high linear complexity generated via the inverse Gray mapping. Using cyclotomic class theory and gcd analysis of the generating polynomial with $4^p-1$, the authors derive explicit lower bounds on the $4$-adic complexity for the four sequence classes, showing that the gcd factors are limited to small primes and numbers of the form $1+kp$ while satisfying modular constraints. These results demonstrate that the sequences have large $4$-adic complexity, hence robust resistance to the rational approximation algorithm (RAA) and favorable properties for cryptographic and communication applications. The work extends understanding of $4$-adic complexity for quaternary sequences beyond ideal autocorrelation cases and provides a framework for analyzing similar constructions from inverse Gray mappings.

Abstract

Recently, Jiang et al. proposed several new classes of quaternary sequences with low autocorrelation and high linear complexity by using the inverse Gray mapping (JAMC, \textbf{69} (2023): 689--706). In this paper, we estimate the 4-adic complexity of these quaternary sequences. Our results show that these sequences have large 4-adic complexity to resist the attack of the rational approximation algorithm.

The 4-adic complexity of quaternary sequences with low autocorrelation and high linear complexity

TL;DR

The paper addresses the problem of estimating the -adic complexity of quaternary sequences with low autocorrelation and high linear complexity generated via the inverse Gray mapping. Using cyclotomic class theory and gcd analysis of the generating polynomial with , the authors derive explicit lower bounds on the -adic complexity for the four sequence classes, showing that the gcd factors are limited to small primes and numbers of the form while satisfying modular constraints. These results demonstrate that the sequences have large -adic complexity, hence robust resistance to the rational approximation algorithm (RAA) and favorable properties for cryptographic and communication applications. The work extends understanding of -adic complexity for quaternary sequences beyond ideal autocorrelation cases and provides a framework for analyzing similar constructions from inverse Gray mappings.

Abstract

Recently, Jiang et al. proposed several new classes of quaternary sequences with low autocorrelation and high linear complexity by using the inverse Gray mapping (JAMC, \textbf{69} (2023): 689--706). In this paper, we estimate the 4-adic complexity of these quaternary sequences. Our results show that these sequences have large 4-adic complexity to resist the attack of the rational approximation algorithm.
Paper Structure (6 sections, 16 theorems, 26 equations, 1 table)

This paper contains 6 sections, 16 theorems, 26 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cyclotomic classes and cyclotomic numbers
  4. Definition of sequences
  5. 4-Adic complexity of quaternary sequences
  6. Conclusions

Key Result

Lemma 2.1

w18 Let $p=x^{2}+4y^{2}$, $x,y\in\mathbb{Z}$ and $x\equiv1\pmod4$. When $p\equiv5\pmod8$, the relationship and formula of these cyclotomic numbers are shown in Table 1, where $y$ is two possible values, depending on the choice of the primitive element $\theta$ of $\mathbb{F}_{p}$.

Theorems & Definitions (30)

  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • ...and 20 more