Linear Complexity of Binary Interleaved Sequences of Period 4n

TL;DR

This paper analyzes the linear complexity of binary interleaved sequences with period 4n, providing a formula and demonstrating sequences with maximum linear complexity, which is significant for cryptographic applications.

Contribution

It presents a general formula for the linear complexity of interleaved binary sequences of period 4n and constructs sequences achieving maximum linear complexity.

Findings

Linear complexity of such sequences is at most 2n+2.

Sequences interleaved with known ideal autocorrelation sequences can reach maximum linear complexity.

The 2-adic complexity of these sequences reaches the maximum value.

Abstract

Binary periodic sequences with good autocorrelation property have many applications in many aspects of communication. In past decades many series of such binary sequences have been constructed. In the application of cryptography, such binary sequences are required to have larger linear complexity. Tang and Ding \cite{X. Tang} presented a method to construct a series of binary sequences with period 4$n$ having optimal autocorrelation. Such sequences are interleaved by two arbitrary binary sequences with period $n\equiv 3\pmod 4$ and ideal autocorrelation. In this paper we present a general formula on the linear complexity of such interleaved sequences. Particularly, we show that the linear complexity of such sequences with period 4$n$ is not bigger than $2n+2$. Interleaving by several types of known binary sequences with ideal autocorrelation ($m$-sequences, Legendre, twin-prime and Hall's sequences), we present many series of such sequences having the maximum value $2n+2$ of linear complexity which gives an answer of a problem raised by N. Li and X. Tang \cite{N. Li}. Finally, in the conclusion section we show that it can be seen easily that the 2-adic complexity of all such interleaved sequences reaches the maximum value $\log_{2}(2^{4n}-1)$.