Edwards curve point counting method and supersingular Edwards and Montgomery curves

Abstract

We consider algebraic affine and projective curves of Edwards \cite{E, SkOdProj} over a finite field $\text{F}_{p^n}$. Most cryptosystems of the modern cryptography \cite{SkBlock} can be naturally transform into elliptic curves \cite{Kob}. We research Edwards algebraic curves over a finite field, which at the present time is one of the most promising supports of sets of points that are used for fast group operations \cite{Bir}. New method of counting Edwards curve order over finite field was constructed. It can be applied to order of elliptic curve due to birational equivalence between elliptic curve and Edwards curve. We find not only a specific set of coefficients with corresponding field characteristics, for which these curves are supersingular but also a general formula by which one can determine whether a curve $E_d[\mathbb{F}_{p^n}]$ is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over $\mathbb{F}_{p^n}$ in a finite field is investigated, the field characteristic, where this degree is minimal, was found. The criterion of supersungularity of the Edwards curves is found over $\mathbb{F}_{p^n}$. Also the generator of crypto stable sequence on an elliptic curve with a deterministic lower estimate of its period is proposed.Comment: Subject: Algebraic Geometry and Number Theory. The order of projective Edwards curve over Fpn. Conference. CAIT-Odessa. (2018) pp. 87-9