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

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.