scholarly journals Construction of elliptic curves with cyclic groups over prime fields

2006 ◽  
Vol 73 (2) ◽  
pp. 245-254 ◽  
Author(s):  
Naoya Nakazawa
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Modular Group ◽  
Modular Function ◽  
Invariant Function ◽  
Rational Points ◽  
Modular Invariant ◽  
Cyclic Groups ◽  
Prime Fields

The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.

The Sato-Tate distribution in thin families of elliptic curves over high degree extensions of finite fields

2019 ◽  
Vol 15 (03) ◽  
pp. 469-477
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Prime Fields ◽  
High Degree

Over the last two decades, there has been a wave of activity establishing the Sato-Tate kind of distribution in various families of elliptic curves over prime fields. Typically the goal here is to prove this for families which are as thin as possible. We consider a function field analogue of this question, that is, for high degree extensions of a finite field where new effects allow us to study families, which are much thinner that those typically investigated over prime fields.

scholarly journals Elliptic Curves over the Perfect Closure of a Function Field

2010 ◽  
Vol 53 (1) ◽  
pp. 87-94
Author(s):  
Dragos Ghioca
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Positive Characteristic ◽  
Rational Points ◽  
Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

scholarly journals Metrics on the Sets of Nonsupersingular Elliptic Curves in Simplified Weierstrass Form over Finite Fields of Characteristic Two

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Keisuke Hakuta
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Elliptic Curve Cryptography ◽  
Analogous Result ◽  
Interesting Property ◽  
Integer Factorization ◽  
Computational Number Theory ◽  
Prime Fields ◽  
Characteristic Two ◽  
Pairing Based Cryptography

Elliptic curves have a wide variety of applications in computational number theory such as elliptic curve cryptography, pairing based cryptography, primality tests, and integer factorization. Mishra and Gupta (2008) have found an interesting property of the sets of elliptic curves in simplified Weierstrass form (or short Weierstrass form) over prime fields. The property is that one can induce metrics on the sets of elliptic curves in simplified Weierstrass form over prime fields of characteristic greater than three. Later, Vetro (2011) has found some other metrics on the sets of elliptic curves in simplified Weierstrass form over prime fields of characteristic greater than three. However, to our knowledge, no analogous result is known in the characteristic two case. In this paper, we will prove that one can induce metrics on the sets of nonsupersingular elliptic curves in simplified Weierstrass form over finite fields of characteristic two.

ON THE NUMBER OF RATIONAL POINTS OF GENERALIZED FERMAT CURVES OVER FINITE FIELDS

2012 ◽  
Vol 08 (04) ◽  
pp. 1087-1097 ◽  
Author(s):  
STEFANIA FANALI ◽  
MASSIMO GIULIETTI
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Direct Product ◽  
Finite Fields ◽  
Classical Problem ◽  
Rational Points ◽  
Fermat Curve ◽  
Cyclic Groups ◽  
Fermat Curves ◽  
Curves Over Finite Fields

The Stöhr–Voloch approach has been largely used to deal with the classical problem of estimating the number of rational points of a Fermat curve over a finite field. The same method actually applies to any curve admitting as an automorphism group the direct product of two cyclic groups C1 and C2 of the same size k, and such that the quotient curves with respect to both C1 and C2 are rational. In this paper such a curve is called a generalized Fermat curve. Our main achievement is that of extending some known results on Fermat curves to generalized Fermat curves.

scholarly journals EMBEDDING FINITE FIELDS INTO ELLIPTIC CURVES

2019 ◽  
pp. 889
Author(s):  
Amirmehdi Yazdani Kashani ◽  
Hassan Daghigh
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Hyperelliptic Curves ◽  
Rational Points ◽  
Elliptic Curve Cryptosystems ◽  
General Elliptic ◽  
Genus 2

Many elliptic curve cryptosystems require an encoding function from a finite field Fq into Fq-rational points of an elliptic curve. We propose a uniform encoding to general elliptic curves over Fq. We also discuss about an injective case of SWU encoing for hyperelliptic curves of genus 2. Moreover we discuss about an injective encoding for elliptic curves with a point of order two over a finite field and present a description for these elliptic curves.

scholarly journals ON THE RANK OF QUADRATIC TWISTS OF ELLIPTIC CURVES OVER FUNCTION FIELDS

2006 ◽  
Vol 02 (02) ◽  
pp. 267-288 ◽  
Author(s):  
E. KOWALSKI
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Upper Bounds ◽  
Density Theorem ◽  
Chebotarev Density Theorem ◽  
Quadratic Twists ◽  
Rank 2

We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/Fq(C) over a function field over a finite field that have rank ≥ 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded.

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.

scholarly journals On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

1971 ◽  
Vol 43 ◽  
pp. 199-208 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Elliptic Curves ◽  
Cusp Form ◽  
Mellin Transform ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Function Fields ◽  
Complex Multiplication ◽  
Modular Function ◽  
Imaginary Quadratic Field

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

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