Supersingularity of elliptic and Edwards curves over $F_{p^n}$

Abstract

We consider the algebraic curves which have form of Edwards and elliptic curves with coordinates in $F_ p^n $. These curves are most effective support for a cyclic group of points which have many applications now. In this paper it was found conditions of supersingularity for Edwards curve and same case of twisted Edwards and Montgomery curves over $F_ p^n $ where $p\equiv 3 mod 4$. There was proved that projective twisted Edwards curve is not elliptic. Some properties of this curve were investigated. Conditions of minimal cofactor of twisted Edwards curve were founded. Construction of a curve with given order and minimal cofactor was made.