caa a22 4500 605519374 CHVBK 20210128100730.0 cr unu---uuuuu 210128e20150301xx s 000 0 eng 10.1007/s10559-015-9709-x doi (NATIONALLICENCE)springer-10.1007/s10559-015-9709-x Exact Number of Elliptic Curves in the Canonical Form, Which are Isomorphic to Edwards Curves Over Prime Field [Elektronische Daten] [A. Bessalov, L. Kovalchuk] 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. Springer Science+Business Media New York, 2015 canonical elliptic curve nationallicence Edwards curve nationallicence twist curve nationallicence curve parameters nationallicence isomorphism nationallicence quadratic residue nationallicence quadratic non-residue nationallicence Bessalov A. Institute of Physics and Technology, National Technical University of Ukraine "Kyiv Polytechnic Institute”, Kyiv, Ukraine aut Kovalchuk L. Institute of Special Communication and Information Security, National Technical University of Ukraine "Kyiv Polytechnic Institute”, Kyiv, Ukraine aut Cybernetics and Systems Analysis Springer US; http://www.springer-ny.com 51/2(2015-03-01), 165-172 1060-0396 51:2<165 2015 51 10559 https://doi.org/10.1007/s10559-015-9709-x text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10559-015-9709-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bessalov A. Institute of Physics and Technology, National Technical University of Ukraine "Kyiv Polytechnic Institute”, Kyiv, Ukraine aut NATIONALLICENCE 700 1- Kovalchuk L. Institute of Special Communication and Information Security, National Technical University of Ukraine "Kyiv Polytechnic Institute”, Kyiv, Ukraine aut NATIONALLICENCE 773 0- Cybernetics and Systems Analysis Springer US; http://www.springer-ny.com 51/2(2015-03-01), 165-172 1060-0396 51:2<165 2015 51 10559