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Exact Number of Elliptic Curves in the Canonical Form, Which are Isomorphic to Edwards Curves Over Prime Field

Cybernetics and Systems Analysis Aims and scope


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 about the properties of quadratic residues are proved, using the Gauss scheme for quadratic residues and non-residues. Based on this lemmas, the exact formulas are derived for 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.

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Correspondence to A. V. Bessalov.

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Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2015, pp. 3–12.

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Bessalov, A.V., Kovalchuk, L.V. Exact Number of Elliptic Curves in the Canonical Form, Which are Isomorphic to Edwards Curves Over Prime Field. Cybern Syst Anal 51, 165–172 (2015).

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