scholarly journals Group order formulas for reductions of CM elliptic curves

2010 ◽  
pp. 107-120 ◽  
Author(s):  
A. Silverberg
Keyword(s):  
Elliptic Curves ◽  
Group Order

scholarly journals The Frequency of Elliptic Curve Groups over Prime Finite Fields

2016 ◽  
Vol 68 (4) ◽  
pp. 721-761 ◽  
Author(s):  
Vorrapan Chandee ◽  
Chantal David ◽  
Dimitris Koukoulopoulos ◽  
Ethan Smith
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Arithmetic Progressions ◽  
Constant Multiple ◽  
Group Order ◽  
Almost All ◽  
Candidate Group

AbstractLetting p vary over all primes and E vary over all elliptic curves over the finite field 𝔽p, we study the frequency to which a given group G arises as a group of points E(𝔽p). It is well known that the only permissible groups are of the form Gm,k:=ℤ/mℤ×ℤ/mkℤ. Given such a candidate group, we let M(Gm,k) be the frequency to which the group Gm,karises in this way. Previously, C.David and E. Smith determined an asymptotic formula for M(Gm,k) assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for M(Gm,k), pointwise and on average. In particular, we show thatM(Gm,k) is bounded above by a constant multiple of the expected quantity when m ≤ kA and that the conjectured asymptotic for M(Gm,k) holds for almost all groups Gm,k when m ≤ k1/4-∈. We also apply our methods to study the frequency to which a given integer N arises as a group order #E(𝔽p).

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.

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
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