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 F<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i>^<i>n</i></sub>, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> is a prime and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> 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 F<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i>^<i>n</i></sub>, and firstly constructed good binary sequences of length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pⁿ</i> + 1 via cyclotomic function fields over F<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i>^<i>n</i></sub> for any prime <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> [8], [10]. 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<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup>+1+<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> with low correlation for many given integers |<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>| ⩽ 2<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(<i>n</i>+2)/2</sup>. Specifically, for any positive integer <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> with gcd(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>; 2<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup>+1+<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>) = 1, we introduce a novel family of binary sequences of length 2<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup>+1+<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>, size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>d</i>−1</sup> − 1, correlation bounded by (2<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> + 1) · 2<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(<i>n</i>+2)/2</sup> + |<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>|, and large linear complexity via elliptic curves.