The 4-Adic Complexity of A Class of Quaternary Cyclotomic Sequences with Period 2p

TL;DR

This paper determines the 4-adic complexity of certain quaternary cyclotomic sequences with period 2p, showing they reach maximum complexity under specific conditions, which is important for cryptographic sequence design.

Contribution

It introduces a method using quadratic Gauss sums to compute the 4-adic complexity of quaternary sequences, filling a gap in the cryptography literature.

Findings

4-adic complexity reaches maximum if 5 does not divide p-2

Complexity is close to maximum otherwise

Method based on quadratic Gauss sums

Abstract

In cryptography, we hope a sequence over $\mathbb{Z}_m$ with period $N$ having larger $m$-adic complexity. Compared with the binary case, the computation of 4-adic complexity of knowing quaternary sequences has not been well developed. In this paper, we determine the 4-adic complexity of the quaternary cyclotomic sequences with period 2$p$ defined in [6]. The main method we utilized is a quadratic Gauss sum $G_{p}$ valued in $\mathbb{Z}_{4^N-1}$ which can be seen as a version of classical quadratic Gauss sum. Our results show that the 4-adic complexity of this class of quaternary cyclotomic sequences reaches the maximum if $5\nmid p-2$ and close to the maximum otherwise.