# The 2-adic complexity of a class of binary sequences with almost optimal autocorrelation

## TL;DR

This paper analyzes the 2-adic complexity of a class of binary sequences with near-optimal autocorrelation, providing lower bounds and demonstrating their suitability for cryptographic applications.

## Contribution

The paper determines the 2-adic complexity of sequences constructed from almost difference sets, extending previous autocorrelation analyses and establishing bounds relevant for cryptography.

## Key findings

- 2-adic complexity is at least N - log2(sqrt(N+1))
- Complexity reaches N - 1 in many cases
- Sequences are resistant to rational approximation algorithms

## Abstract

Pseudo-random sequences with good statistical property, such as low autocorrelation, high linear complexity and large 2-adic complexity, have been applied in stream cipher. In general, it is difficult to give both the linear complexity and 2-adic complexity of a periodic binary sequence. Cai and Ding \cite{Cai Ying} gave a class of sequences with almost optimal autocorrelation by constructing almost difference sets. Wang \cite{Wang Qi} proved that one type of those sequences by Cai and Ding has large linear complexity. Sun et al. \cite{Sun Yuhua} showed that another type of sequences by Cai and Ding has also large linear complexity. Additionally, Sun et al. also generalized the construction by Cai and Ding using $d$-form function with difference-balanced property. In this paper, we first give the detailed autocorrelation distribution of the sequences was generalized from Cai and Ding \cite{Cai Ying} by Sun et al. \cite{Sun Yuhua}. Then, inspired by the method of Hu \cite{Hu Honggang}, we analyse their 2-adic complexity and give a lower bound on the 2-adic complexity of these sequences. Our result show that the 2-adic complexity of these sequences is at least $N-\mathrm{log}_2\sqrt{N+1}$ and that it reach $N-1$ in many cases, which are large enough to resist the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).