A new family of binary sequences with a low correlation via elliptic curves
Lingfei Jin, Liming Ma, Chaoping Xing, Runtian Zhu
TL;DR
The paper addresses the need for binary sequences with low correlation and flexible lengths for applications in communications and cryptography. It introduces an explicit construction based on the cyclic translation group of rational points on cyclic elliptic function fields over $\mathbb{F}_{2^n}$, yielding sequences of length $2^n+1+t$ and, for gcd$(d,2^n+1+t)=1$, size $q^{d-1}-1$ with a correlation bound of $$(2d+1)\cdot 2^{(n+2)/2} + |t|,$$ along with high linear complexity. The method leverages Artin-Schreier extensions and Riemann-Roch spaces to control correlation via genus bounds, and provides an explicit algorithm with numerical results demonstrating practical effectiveness. This work broadens the parameter landscape of sequence families, enabling more flexible design for CDMA, spread-spectrum, and cryptographic applications, while maintaining rigorous performance guarantees.
Abstract
In the realm of modern digital communication, cryptography, and signal processing, binary sequences with a low correlation properties play a pivotal role. In the literature, considerable efforts have been dedicated to constructing good binary sequences of various lengths. As a consequence, numerous constructions of good binary sequences have been put forward. However, the majority of known constructions leverage the multiplicative cyclic group structure of finite fields $\mathbb{F}_{p^n}$, where $p$ is a prime and $n$ is a positive integer. Recently, the authors made use of the cyclic group structure of all rational places of the rational function field over the finite field $\mathbb{F}_{p^n}$, and firstly constructed good binary sequences of length $p^n+1$ via cyclotomic function fields over $\mathbb{F}_{p^n}$ for any prime $p$ \cite{HJMX24,JMX22}. This approach has paved a new way for constructing good binary sequences. Motivated by the above constructions, we exploit the cyclic group structure on rational points of elliptic curves to design a family of binary sequences of length $2^n+1+t$ with a low correlation for many given integers $|t|\le 2^{(n+2)/2}$. Specifically, for any positive integer $d$ with $\gcd(d,2^n+1+t)=1$, we introduce a novel family of binary sequences of length $2^n+1+t$, size $q^{d-1}-1$, correlation bounded by $(2d+1) \cdot 2^{(n+2)/2}+ |t|$, and a large linear complexity via elliptic curves.