scholarly journals Goppa codes over Edwards curves

2021 ◽  
Author(s):  
Giuseppe Filippone
Keyword(s):  
Singular Points ◽  
Goppa Codes ◽  
Goppa Code ◽  
Edwards Curves ◽  
Geometric Goppa Code ◽  
Generating Matrix ◽  
Edwards Curve

Abstract Given an Edwards curve, we determine a basis for the Riemann-Roch space of any divisor whose support does not contain any of the two singular points. This basis allows us to compute a generating matrix for an algebraic-geometric Goppa code over the Edwards curve.

scholarly journals Counting Irreducible Polynomials of Degree r over Fqn and Generating Goppa Codes Using the Lattice of Subfields of Fqnr

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Kondwani Magamba ◽  
John A. Ryan
Keyword(s):  
Irreducible Polynomial ◽  
Goppa Codes ◽  
Irreducible Polynomials ◽  
Goppa Code

The problem of finding the number of irreducible monic polynomials of degree r over Fqn is considered in this paper. By considering the fact that an irreducible polynomial of degree r over Fqn has a root in a subfield Fqs of Fqnr if and only if (nr/s,r)=1, we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of Fqnr . We also use the lattice of subfields of Fqnr to determine if it is possible to generate a Goppa code using an element lying in a proper subfield of Fqnr.

scholarly journals CALCULATION OF PARAMETERS OF CRYPTIC CRIVIAE EDWARDS OVER THE FIELDS OF CHARACTERISTICS 5 AND 7

2018 ◽  
pp. 94-104 ◽  
Author(s):  
Anatoliy V. Bessalov
Keyword(s):  
Prime Order ◽  
Discrete Logarithm ◽  
Problem Solution ◽  
Characteristic P ◽  
System Parameters ◽  
Calculation Of Parameters ◽  
Edwards Curves ◽  
Primitive Polynomials ◽  
Prime Fields ◽  
Edwards Curve

The method of search of cryptographic strong elliptic curves in the Edwards form (where parameter d is non square in the field) over the extended finite fields of small characteristics p ≠ 2.3 is proposed. For these curves is performed the completeness of the points addition law, so they are called as complete Edwards curve. In the first stage over a small prime fields and we find the parameters d of complete Edwards curves who have minimum orders . For both curves we obtain the same values d = 3, which are non square in the fields and . Next with help recurrent formulae for both curves we calculated the orders (where n is odd) of these curves over the extended fields with prime degrees of extension m within known cryptographic standards (with the same bit-length field module 200 ... 600 bits). The calculated values n are tested on primelity. The extensions m, which provide a psevdoprime order 4n of curve with a prime value n, are selected. This provides the highest cryptographic stability of curve by the discrete logarithm problem solution. As a result, over the fields of the characteristic p = 5 we obtain two curves with degrees of expansion m = 181 and m = 277, and over the fields of the characteristic p = 7 one curve with the degree m = 127. For them, the corresponding large prime values of n are determined. The next stage is the calculation of other system-parameters of cryptographic systems based on complete Edwards curves. over the fields of characteristics 5 and 7. The arithmetic of extended fields is based on irreducible primitive polynomials P (z) of degree m. The search and construction of polynomial tables P (z) (for 10 different polynomials for each value m, respectively, for the values of the characteristics p = 5 and p = 7) has been performed. On the basis of each polynomial according to the developed method, the coordinates of the random point P of the curve are calculated. The possible order of this point is the value of 4n, 2n or n. The double doubling of this point is the coordinates and for 30 different generators G = 4P cryptosystems that have a prime order n. The set of parameters that satisfy the standard cryptographic requirements and can be recommended in projecting cryptosystems is obtained.

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 EXTENDED SUBCLASS OF CYCLIC SEPARABLE GOPPA CODES

2016 ◽  
Vol 15 (5) ◽  
pp. 6776-6784
Author(s):  
Ajay Sharma ◽  
O. P. VINOCHA
Keyword(s):  
Cyclic Codes ◽  
Goppa Codes ◽  
Goppa Code

In 2013[4]a new subclass of cyclic Goppa code with Goppa polynomial of degree 2 is presented by Bezzateev and Shekhunova. They proved that this subclass contains all cyclic codes of considered length. In the present work we consider a Goppa polynomial of degree three and proved that the subclass generated by this polynomial represent a cyclic, reversible and separable Goppa code.

scholarly journals Low-Cost, Low-Power FPGA Implementation of ED25519 and CURVE25519 Point Multiplication

Information ◽  
2019 ◽  
Vol 10 (9) ◽  
pp. 285 ◽  
Author(s):  
Mohamad Ali Mehrabi ◽  
Christophe Doche
Keyword(s):  
Low Power ◽  
Low Cost ◽  
Fpga Implementation ◽  
Large Integer ◽  
Modular Multiplication ◽  
Point Multiplication ◽  
Low Area ◽  
Edwards Curves ◽  
Digital Signature Algorithm ◽  
Edwards Curve

Twisted Edwards curves have been at the center of attention since their introduction by Bernstein et al. in 2007. The curve ED25519, used for Edwards-curve Digital Signature Algorithm (EdDSA), provides faster digital signatures than existing schemes without sacrificing security. The CURVE25519 is a Montgomery curve that is closely related to ED25519. It provides a simple, constant time, and fast point multiplication, which is used by the key exchange protocol X25519. Software implementations of EdDSA and X25519 are used in many web-based PC and Mobile applications. In this paper, we introduce a low-power, low-area FPGA implementation of the ED25519 and CURVE25519 scalar multiplication that is particularly relevant for Internet of Things (IoT) applications. The efficiency of the arithmetic modulo the prime number 2 255 - 19 , in particular the modular reduction and modular multiplication, are key to the efficiency of both EdDSA and X25519. To reduce the complexity of the hardware implementation, we propose a high-radix interleaved modular multiplication algorithm. One benefit of this architecture is to avoid the use of large-integer multipliers relying on FPGA DSP modules.

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 .

scholarly journals Point Compression and Coordinate Recovery for Edwards Curves over Finite Field

2014 ◽  
Vol 52 (2) ◽  
Author(s):  
Benjamin Justus
Keyword(s):  
Finite Field ◽  
Scalar Multiplication ◽  
Computational Approaches ◽  
Recovery Algorithm ◽  
Edwards Curves ◽  
Addition Chain ◽  
Point Compression ◽  
Edwards Curve

AbstractWe present two computational approaches for the purpose of point compression and decompression on Edwards curves over the finite field Fp where p is an odd prime. The proposed algorithms allow compression and decompression for the x or y affine coordinates. We also present a x-coordinate recovery algorithm that can be used at any stage of a differential addition chain during the scalar multiplication of a point on the Edwards curve.

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 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.

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