Elliptic Curves, the Finiteness Theorem of Shafarevič

1995 ◽  
pp. 1-9
Author(s):  
Rolf-Peter Holzapfel
Keyword(s):  
Elliptic Curves ◽  
Finiteness Theorem

scholarly journals Critically separable rational maps in families

2012 ◽  
Vol 148 (6) ◽  
pp. 1880-1896 ◽  
Author(s):  
Clayton Petsche
Keyword(s):  
Fixed Point ◽  
Elliptic Curves ◽  
Number Field ◽  
Rational Maps ◽  
Finiteness Theorem ◽  
Good Reduction ◽  
Rational Map ◽  
Arithmetic Complexity ◽  
Special Case

AbstractGiven a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich’s theorem for elliptic curves. We also define the minimal critical discriminant, a global object which can be viewed as a measure of arithmetic complexity of a rational map. We formulate a conjectural bound on the minimal critical discriminant, which is analogous to Szpiro’s conjecture for elliptic curves, and we prove that a special case of our conjecture implies Szpiro’s conjecture in the semistable case.

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

Algorithm for directly computing 7P elliptic curves and its application

2013 ◽  
Vol 33 (7) ◽  
pp. 1870-1874
Author(s):  
Zhongxi LAI ◽  
Zhanjun ZHANG ◽  
Dongya TAO
Keyword(s):  
Elliptic Curves

Elliptic Curves. (MN-40), Volume 40

2018 ◽  
Author(s):  
Anthony W. Knapp
Keyword(s):  
Elliptic Curves
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