  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.
