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We consider algebraic affine and projective curves of Edwards \cite{E,SkOdProj} over a finite field $\text{F}_{p^n}$. Most cryptosystems of themodern cryptography \cite{SkBlock} can be naturally transform into ellipticcurves \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 ofpoints that are used for fast group operations \cite{Bir}. New method ofcounting Edwards curve order over finite field was constructed. It can beapplied to order of elliptic curve due to birational equivalence betweenelliptic curve and Edwards curve. We find not only a specific set ofcoefficients with corresponding field characteristics, for which these curvesare supersingular but also a general formula by which one can determine whethera curve $E_d[\mathbb{F}_{p^n}]$ is supersingular over this field or not. Theembedding degree of the supersingular curve of Edwards over $\mathbb{F}_{p^n}$in a finite field is investigated, the field characteristic, where this degreeis minimal, was found. The criterion of supersungularity of the Edwards curvesis found over $\mathbb{F}_{p^n}$. Also the generator of crypto stable sequenceon an elliptic curve with a deterministic lower estimate of its period isproposed.

Edwards curve point counting method and supersingular Edwards and Montgomery curves