Intelligent Security Control Based on the New Criterion of Edwards and Montgomery Curves, Isogenous of These Curves Supersingularity

Abstract

It is well known supersingular curves due to pairing of Weil and pairing of Tate are used in identity-based cryptosystems so we find criterion of supersingularity of Montgomery and Edwards curves. We consider the algebraic affine and projective curves of Edwards over the finite field ${{\text {F}}_{{{p}^{n}}}}$. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. The criterions of the supersingularity of Montgomery and Edwards curves are found. In this paper, we extend our previous research into those Edwards algebraic curves over a finite field, and we construct birational isomorphism of them with cubic in Weierstrass normal form. One class of twisted Edwards is researched too. 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 ${{E}_{d}}[{{\mathbb {F}}_{p}}]$ is supersingular over this field. The method proposed has complexity $\mathcal {O}\left(p\log _{2}^{2}p \right)$. 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 $\mathcal {O}(\log _{2}^{8}{{p}^{n}})$ and $\mathcal {O}(\log _{2}^{4}{{p}^{n}})$, respectively. The embedding degree of the supersingular curve of Edwards over ${{\mathbb {F}}_{{{p}^{n}}}}$ in a finite field is additionally investigated. Singular points of twisted Edwards curve are completely described. Due existing the birational isomorphism between twisted Edwards curve and elliptic curve in Weierstrass normal form the result about order of this curve over finite field is extended on cubic in Weierstrass normal form. Also it is considered minimum degree of an isogeny (distance) between curves of this two classes when such isogeny exists. We extend the existing isogenous of elliptic curves.