scholarly journals Edwards Curve Points Counting Method and Supersingular Edwards and Montgomery Curves. (Cryptosystems, Cryptology and Theoretical Computer Science)

2020 ◽  
Vol 17 ◽  
Keyword(s):  
Finite Field ◽  
Computer Science ◽  
Elliptic Curves ◽  
Algebraic Curves ◽  
Theoretical Computer Science ◽  
Theoretical Computer ◽  
Edwards Curves ◽  
Projective Curves ◽  
Supersingular Curves ◽  
Edwards Curve

In this paper, an algebraic affine and projective curves of Edwards [3, 9] over the finite field Fpn . In the theory of Cryptosystems, Cryptology and Theoretical Computer Science it is well known that many modern cryptosystems [11] can be naturally transformed into elliptic curves [5]. Here we study Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve over a finite field. It should be noted that this method can be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. We not only find a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve [ ] d p E F is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over pn F in a finite field is investigated and the field characteristic, where this degree is minimal, is found. A birational isomorphism between the Montgomery curve and the Edwards curve is also constructed. A one-to-one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over pn F .

scholarly journals SUPERSINGULAR EDWARDS CURVES AND EDWARDS CURVE POINTS COUNTING METHOD OVER FINITE FIELD

2020 ◽  
pp. 68-88
Author(s):  
Ruslan Skuratovskii
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Algebraic Curves ◽  
Discrete Logarithm ◽  
Edwards Curves ◽  
Projective Curves ◽  
Birational Equivalence ◽  
Supersingular Curves ◽  
Edwards Curve ◽  
Embedding Degree

We consider problem of order counting of algebraic affine and projective curves of Edwards [2, 8] over the finite field $F_{p^n}$. The complexity of the discrete logarithm problem in the group of points of an elliptic curve depends on the order of this curve (ECDLP) [4, 20] depends on the order of this curve [10]. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve over a finite field. It should be noted that this method can be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. We not only find a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve $E_d [F_p]$ is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over $F_{p^n}$ in a finite field is investigated and the field characteristic, where this degree is minimal, is found. A birational isomorphism between the Montgomery curve and the Edwards curve is also constructed. A one-to-one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over $F_{p^n}$.

scholarly journals Elliptic and Edwards Curves Order Counting Method

2021 ◽  
Vol 15 ◽  
pp. 52-62
Author(s):  
Ruslan Skuratovskii ◽  
Volodymyr Osadchyy
Keyword(s):  
Normal Form ◽  
Finite Field ◽  
Elliptic Curves ◽  
Algebraic Curves ◽  
Basic Algorithm ◽  
Edwards Curves ◽  
Projective Curves ◽  
Edwards Curve ◽  
Embedding Degree ◽  
Fast Arithmetic

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.

scholarly journals Criterions of Supersinguliarity and Groups of Montgomery and Edwards Curves in Cryptography

2021 ◽  
Vol 19 ◽  
pp. 709-722
Author(s):  
Ruslan Skuratovskii ◽  
Volodymyr Osadchyy
Keyword(s):  
Normal Form ◽  
Finite Field ◽  
Elliptic Curves ◽  
Algebraic Curves ◽  
Minimum Degree ◽  
Basic Algorithm ◽  
Edwards Curves ◽  
Edwards Curve ◽  
Birational Isomorphism ◽  
Twisted Edwards Curve

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. 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 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. 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.

scholarly journals The Order of Edwards and Montgomery Curves

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Digital Signature ◽  
Discrete Logarithm ◽  
Log P ◽  
Edwards Curves ◽  
Digital Signature Algorithm ◽  
Supersingular Curves ◽  
Edwards Curve

The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA) [2]. It is well known that the problem of discrete logarithm is NP-hard on group on elliptic curve (EC) [5]. The orders of groups of an algebraic affine and projective curves of Edwards [3, 9] over the finite field Fpn is studied by us. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve [F ] d p E over a finite field Fp . It should be noted that this method can be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. The method we have proposed has much less complexity 22 O p log p at not large values p in comparison with the best Schoof basic algorithm with complexity 8 2 O(log pn ) , as well as a variant of the Schoof algorithm that uses fast arithmetic, which has complexity 42O(log pn ) , but works only for Elkis or Atkin primes. We not only find a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve [F ] d p E is supersingular over this field or not. The symmetric of the Edwards curve form and the parity of all degrees made it possible to represent the shape curves and apply the method of calculating the residual coincidences. A birational isomorphism between the Montgomery curve and the Edwards curve is also constructed. A oneto- one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over F pn .

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