Elliptic and Edwards Curves Order Counting Method

We consider the algebraic affine and projective curves of Edwards over the finite field Fpn. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. In this paper, we extend our previous research into those Edwards algebraic curves over a finite field. We propose a novel effective method of point counting for both Edwards and elliptic curves. In addition to finding a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, we also find a general formula by which one can determine whether or not a curve Ed[Fp] is supersingular over this field. The method proposed has complexity O ( p log2 2 p ) . This is an improvement over both Schoof’s basic algorithm and the variant which makes use of fast arithmetic (suitable for only the Elkis or Atkin primes numbers) with complexities O(log8 2 pn) and O(log4 2 pn) respectively. The embedding degree of the supersingular curve of Edwards over Fpn in a finite field is additionally investigated. Due existing the birational isomorphism between twisted Edwards curve and elliptic curve in Weierstrass normal form the result about order of curve over finite field is extended on cubic in Weierstrass normal form.

Improved divisor arithmetic on generic hyperelliptic curves

The divisor class group of a hyperelliptic curve defined over a finite field is a finite abelian group at the center of a number of important open questions in algebraic geometry, number theory and cryptography. Many of these problems lend themselves to numerical investigation, and as emphasized by Sutherland [14, 13], fast arithmetic in the divisor class group is crucial for their efficiency. Besides, implementations of these fundamental operations are at the core of the algebraic geometry packages of widely-used computer algebra systems such as Magma and Sage.