CORRELATION OF BIG ORDER POINTS SETS OF THE EDWARDS CURVES OVER PRIME FIELD

Authors

  • Анатолий Владимирович Бессалов NTUU «KPI» (Kyiv,Ukraine)
  • Оксана Валентиновна Цыганкова NTUU «KPI» (Kyiv,Ukraine)

DOI:

https://doi.org/10.18372/2410-7840.17.8327

Keywords:

elliptic curve, Edwards curve, curve order, points order, Legendre symbol, square, non-square, twisted curves,

Abstract

Modification of the addition law of an Edwards curvepoints over a prime field is offered. It ensures traditionalhorizontal symmetry of inverse points of an elliptic curve.2 theorems of properties of points co-ordinates of the bigorder points are proved. These properties generated bypoint halving, inverse of point doubling. On their basis itis possible to calculate of points order with only two operationsin the field without group operations. The theorem3 about degenerate pair of twisted curves with orderNE  p 1 is proved, if p  3mod 4and p  3mod8 ,d  2 or 1 d 2   . The statement 1 about a non-existenceof point halving for points of a maximum order andpoints of 4th order is proved. The statement 2 is provedthat at among 8 points of a set of the points lying on onecircle, 2 points have an order n , 2 points - an order 2nand 4 points - a maximum order 4n . The algorithm ofreconstruction without evaluations of all unknown pointskP of a of Edwards curve is offered, if only at 1/8 partsof points is known.

Author Biographies

Анатолий Владимирович Бессалов, NTUU «KPI» (Kyiv,Ukraine)

Dr eng (information security), professor

Оксана Валентиновна Цыганкова, NTUU «KPI» (Kyiv,Ukraine)

aspirant

References

. Edwards H.M. A normal form for elliptic curves. Bulletin of the American Mathematical Society, Volume 44, Number 3, July 2007, Pages 393-422.

. Bernstein Daniel J., Lange Tanja. Faster addition and doubling on elliptic curves. IST Programme under Contract IST–2002–507932 ECRYPT, 2007, PP. 1-20.

. Bessalov A.V. Бессалов А.В. Число изоморфизмов и пар кручения кривых Эдвардса над простым полем. Радиотехника, вып. 167, 2011. С. 203-208.

. Бессалов А.В. Деление точки на два для кривой Эдвардса над простым полем. Прикладная радио-электроника, 2013, Том 12, №2. С. 278-279.

. Бессалов А.В., Телиженко А.Б. Криптосистемы на эллиптических кривых: Учеб. пособие. – К.: ІВЦ «Політехніка», 2004. – 224с.

. Бессалов А.В. Построение кривой Эдвардса на базе изоморфной эллиптической кривой в канонической форме. Прикладная радиоэлектроника, 2014, Том 13, №3. – С.286-289.

Downloads

Published

2015-06-04

How to Cite

Бессалов, А. В., & Цыганкова, О. В. (2015). CORRELATION OF BIG ORDER POINTS SETS OF THE EDWARDS CURVES OVER PRIME FIELD. Ukrainian Information Security Research Journal, 17(1), 73–80. https://doi.org/10.18372/2410-7840.17.8327

Issue

Vol. 17 No. 1 (2015)

Section

Articles

License

The scientific journal adheres to the principles of Open Access and provides free, immediate, and permanent access to all published materials without financial, technical, or legal barriers for readers.

All articles are published in Open Access under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.

Copyright

Authors who publish their works in the journal:

  • retain the copyright to their publications;

  • grant the journal the right of first publication of the article;

  • agree to the distribution of their materials under the CC BY 4.0 license;

  • have the right to reuse, archive, and distribute their works (including in institutional and subject repositories), provided that proper reference is made to the original publication in the journal.