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Cybernetics And Systems Analysis
International Theoretical Science Journal
UDC 519.686.2
A.V. Bessalov, L.V. Kovalchuk

THE EXACT NUMBER OF ELLIPTIC CURVES IN THE CANONICAL FORM, WHICH ARE ISOMORPHIC TO EDWARDS CURVES OVER THE PRIME FIELD

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.



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Бессалов Анатолий Владимирович,
доктор техн. наук, профессор Физико-технического института НТТУ «Киевский политехнический институт»,
e-mail: lv_kov_crypto@mail:ru.

Ковальчук Людмила Васильевна,
доктор техн. наук, профессор Института специальной связи и защиты информации НТТУ «Киевский политехнический институт»,
e-mail: bessalov@ukr.net.

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