We consider algebraic ane and projective curves of Edwards [9, 12] over a nite eld Fpn. Most cryptosystems of the modern cryptography [2] can be naturally transform into elliptic curves [11]. We research Edwards algebraic curves over a nite eld, which at the present time is one of the most promising supports of sets of points that are used for fast group operations. We nd not only a speci c set of coecients with corresponding eld characteristics, for which these curves are supersingular but also a general formula by which one can determine whether a curve Ed[Fp] is supersingular over this eld or not. The embedding degree of the supersingular curve of Edwards over Fpn in a nite eld is investigated, the eld characteristic, where this degree is minimal, was found. The criterion of supersungularity of the Edwards curves is found over Fpn. Also the generator of crypto stable sequence on an elliptic curve with a deterministic lower estimate of its period is proposed.