A New Family of Binary Sequences With Low Correlation via Elliptic Curves

In the realm of modern digital communication, cryptography, and signal processing, binary sequences with good 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. This approach has paved a new way for constructing good binary sequences. Motivated by the above constructions, we exploit the cyclic group structure of rational points of elliptic curves to design a family of binary sequences of length $2^{n}+1+t$ with low correlation for many given integers $|t|\leqslant 2^{(n+2)/2}$ . 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 large linear complexity via elliptic curves.