Constructing eliptic curves with zero trace of Frobenius endomorphism

Most cryptosystems of the modern cryptography can benaturally transform into elliptic curves. We consider Edwardsalgebraic curves over a finite field, which at the presenttime is one of the most promising supports of sets ofpoints that are used for fast group operations[1,2,14].These are found in asymmetric cryptosystems. In particular,for constructing random crypto-stable sequences. It isshown that the projective curve is not elliptic. This paperaims to find the criterion and sufficient conditions for thesupersingularity of the Edwards curve and the elliptic curvein the Montgomery form over the finite field p also a generalizationof this criterion for a finite algebraic extentionof n pF . The result obtained allows us to construct an arbitrarysupersingle curve of Edwards and Montgomery withoutdecomposing on the factors the polynomial from,which is distinguished in the formula by the defining curve.Till now it was proved that only for coefficients1 d 2, d 2   over p [10]. The set of all coefficients ofd Ewhich contribute supersingularity of d E over p isresearched in this paper. Also in purpouse of our paper iscriterion and sufficient conditions of Edwards and ellipticcurves supersingularity over n pF , viz our purpouse is researchingof the parametrs set such that whereby we get apair of cirves with Frobenius trace which is equal to zero.It was found not only the set of such coefficients and characteristicsof fields where these curves are supersingularand general formula which provids a way to check for supersingularcurve over a field n p . In this paper the resultabout supersingular curves with coefficients 1 d 2, d 2  over p obtained in [10] was generalized also formulationof Theorem 3 was refined. The same research was providedfor elliptic curve in the Montgomery form over fieldsp and n p .