Article

Article title DENSITY OF THE CANONICAL ELLIPTIC CURVES HAVING A PROPERTY OF THE ISOMORPHISM TO AN EDWARDS FORM
Authors A.V. Bessalov, A.A. Dikhtenko, O.V. Tsygankova
Section SECTION IV. METHODS AND MEANS OF CRYPTOGRAPHY AND STEGANOGRAPHY
Month, Year 02, 2014 @en
Index UDC 681.3.06
DOI
Abstract Edwards curves are the most interesting in terms of practical applications. The ease of programming and the best performance are their main advantages before other known forms of elliptic curves representation. A problem of determining the precise number of canonical elliptic curves which are isomorphic to Edwards curves over a prime field is posed in the paper. An approach is proposed by authors in order to solve the problem. The approach is based on substitution of canonical curve parameters by a parameters pair, where – is the cubic equation unique root in a field. As a result the conditions for existence canonical curves isomorphic to Edwards curves over a prime field are found. Additionally two lemmas are proved in a quadratic residues theory, which is constructed on the Gauss scheme. The precise formulas are given in the paper for counting the number of elliptic curves with non-zero and parameters which are isomorphic to Edwards curves. It is proved that density of such curves for large fields is close to 0,25.

Download PDF

Keywords Canonical elliptic curve; Edwards curve; twisted curve; curve parameters; isomorphism; quadratic residue; quadratic non-residue.
References 1. Edwards H.M. A normal form for elliptic curves // Bulletin of the American Mathematical Society. – July 2007. – Vol. 44, № 3. – P. 393-422.
2. Bernstein Daniel J., Lange Tanja. Faster addition and doubling on elliptic curves. IST Programme under Contract IST–2002–507932 ECRYPT, 2007. – P. 1-20.
3. Бессалов А.В. Число изоморфизмов и пар кручения кривых Эдвардса над простым полем // Радиотехника. – 2011. – Вып. 167. – С. 203-208.
4. Бессалов А.В., Гурьянов А.И., Дихтенко А.А. Кривые Эдвардса почти простого порядка над расширениями малых простых полей // Прикладная радиоэлектроника. – 2012. – Т. 11, № 2. – С. 225-227.
5. Бессалов А.В., Дихтенко А.А. Криптостойкие кривые Эдвардса над простыми полями // Прикладная радиоэлектроника. – 2013. – Т. 12, № 2. – С. 285-291.
6. Дэвенпорт Г. Высшая арифметика: введение в теорию чисел: Пер. с англ. / Под ред. Ю.В. Линника. – М.: Наука, 1965. – 176 с.
7. Бессалов А.В., Телиженко А.Б. Криптосистемы на эллиптических кривых: Учеб. пособие. – Киев: ІВЦ «Політехніка», 2004. – 224 с.

Comments are closed.