One Parameter Family of Elliptic Curves and the Equation $$x^4+y^4+kx^2y^2=z^2$$

2021 ◽  
pp. 26-36
Author(s):  
Stoyan Apostolov ◽  
Miroslav Stoenchev ◽  
Venelin Todorov
Keyword(s):  
Elliptic Curves ◽  
Parameter Family

scholarly journals The cyclicity of a class of quadratic reversible centers defining elliptic curves

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guilin Ji ◽  
Changjian Liu
Keyword(s):  
Elliptic Curves ◽  
Linear Combination ◽  
Asymptotic Expansions ◽  
Parameter Family ◽  
Reversible System ◽  
Abelian Integrals ◽  
Number Of Zeros ◽  
Period Annulus ◽  
New Criteria

<p style='text-indent:20px;'>In this paper, the cyclicity of period annulus of an one-parameter family quadratic reversible system under quadratic perturbations is studied which is equivalent to the number of zeros of any nontrivial linear combination of three Abelian integrals. By the criteria established in [<xref ref-type="bibr" rid="b28">28</xref>] and the asymptotic expansions of Abelian integrals, we obtain that the cyclicity is two when the parameter in <inline-formula><tex-math id="M1">\begin{document}$ (-\infty,-2)\cup[-\frac{8}{5},+\infty) $\end{document}</tex-math></inline-formula>. Moreover, we develop new criteria which combined with the asymptotic expansions of Abelian integrals show that the cyclicity is three when the parameter belongs to <inline-formula><tex-math id="M2">\begin{document}$ (-2,-\frac{8}{5}) $\end{document}</tex-math></inline-formula>.</p>

Elliptic Curves

1997 ◽  
Author(s):  
Henry McKean ◽  
Victor Moll
Keyword(s):  
Elliptic Curves

The Joint Universality for L-Functions of Elliptic Curves

2004 ◽  
Vol 9 (4) ◽  
pp. 331-348
Author(s):  
V. Garbaliauskienė
Keyword(s):  
Elliptic Curves ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

PERSPECTIVES OF ELLIPTICAL CRYPTOGRAPHY TO ENSURE THE INTEGRITY AND CONFIDENTIALITY OF INFORMATION

2019 ◽  
Author(s):  
Anna ILYENKO ◽  
Sergii ILYENKO ◽  
Yana MASUR
Keyword(s):  
Information Systems ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Computational Efficiency ◽  
Digital Signature ◽  
Discrete Logarithm ◽  
Algebraic Groups ◽  
Schnorr Signature ◽  
Stability Of Algorithms

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.

Structured digital multisignature algorithm based on elliptic curves

2009 ◽  
Vol 29 (1) ◽  
pp. 158-160
Author(s):  
He-gang FU ◽  
Ying CHEN
Keyword(s):  
Elliptic Curves
Sign in / Sign up

Export Citation Format

Share Document