Edwards Curve Points Counting Method and Supersingular Edwards and Montgomery Curves. (Cryptosystems, Cryptology and Theoretical Computer Science)

Abstract

In this paper, an algebraic affine and projective curves of Edwards [3, 9] over the finite field Fpn . In the theory of Cryptosystems, Cryptology and Theoretical Computer Science it is well known that many modern cryptosystems [11] can be naturally transformed into elliptic curves [5]. Here we study Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve over a finite field. It should be noted that this method can be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. We not only find a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve [ ] d p E F is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over pn F in a finite field is investigated and the field characteristic, where this degree is minimal, is found. A birational isomorphism between the Montgomery curve and the Edwards curve is also constructed. A one-to-one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over pn F .