  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 $(2, n, 2)$
