Abstract. The necessary and sufficient conditions for the parameters of the curve in the canonical form with two points of order 4 are found. Two lemmas are proved about the properties of quadratic residues, using the Gauss scheme for quadratic residues and non-residues. Based on this lemmas, the exact formulas are derived for calculating the number of elliptic curves with non-zero parameters a and b and two points of order 4 that are isomorphic to Edwards curves over the prime field. It is proved that for large fields the share of such curves is close to 1/4.
Keywords: canonical elliptic curve, Edwards curve, twist curve, curve parameters, isomorphism, quadratic residue, quadratic non-residue.
Ковальчук Людмила Васильевна,
доктор техн. наук, профессор Института специальной связи и защиты информации НТТУ «Киевский политехнический институт»,
e-mail: bessalov@ukr.net.