Representations of divisors on hyperelliptic curves, Gr\"obner bases and plane curves with quasi-toric relations

Abstract

In the study of hyperelliptic curve cryptography, presentations of semi-reduced divisors on a hyperelliptic curve play important roles. In this note, we give an interpretation for such presentations from view points of Gr\"obner bases. As applications, (i) we give a method to construct weak $n$-contact curves and (ii) we study plane curves satisfying quasi-toric relations of type $(2, n, 2)$