# A lower bound on the 2-adic complexity of Ding-Helleseth generalized cyclotomic sequences of period $p^n$

## TL;DR

This paper establishes a lower bound on the 2-adic complexity of Ding-Helleseth generalized cyclotomic sequences, demonstrating they have high complexity, which is beneficial for cryptographic security.

## Contribution

The paper proves that Gauss periods for certain cyclotomic classes are zero and derives a new lower bound on the 2-adic complexity of these sequences.

## Key findings

- Gauss periods for classes D0 and D1 are zero for n ≥ 2
- Lower bound on 2-adic complexity is p^n - p^{n-1} - 1
- The complexity exceeds half the period, indicating strong cryptographic properties

## Abstract

Let $p$ be an odd prime, $n$ a positive integer and $g$ a primitive root of $p^n$. Suppose $D_i^{(p^n)}=\{g^{2s+i}|s=0,1,2,\cdots,\frac{(p-1)p^{n-1}}{2}\}$, $i=0,1$, is the generalized cyclotomic classes with $Z_{p^n}^{\ast}=D_0\cup D_1$. In this paper, we prove that Gauss periods based on $D_0$ and $D_1$ are both equal to 0 for $n\geq2$. As an application, we determine a lower bound on the 2-adic complexity of a class of Ding-Helleseth generalized cyclotomic sequences of period $p^n$. The result shows that the 2-adic complexity is at least $p^n-p^{n-1}-1$, which is larger than $\frac{N+1}{2}$, where $N=p^n$ is the period of the sequence.