نتایج جستجو برای: y curves

تعداد نتایج: 584642  

1998
Franz Lemmermeyer

Generalizing results of Stroeker and Top we show that the 2-ranks of the TateShafarevich groups of the elliptic curves y = (x + k)(x + k) can become arbitrarily large. We also present a conjecture on the rank of the Selmer groups attached to rational 2-isogenies of elliptic curves. 1991 Mathematics Subject Classification: 11 G 05

2004

The `th modular polynomial, φ`(x, y), parameterizes pairs of elliptic curves with an isogeny of degree ` between them. Modular polynomials provide the defining equations for modular curves, and are useful in many different aspects of computational number theory and cryptography. For example, computations with modular polynomials have been used to speed elliptic curve point-counting algorithms (...

2008
Jim Pettigrew John A G Roberts G Roberts

We consider families of biquadratic curvesB = 0 onC, defined with respect to arbitrarily many complex parameters. Due to the fact that these families include curve intersections across different parameter combinations, they represent a generalization of the non-intersecting foliations of one-parameter invariant curves associated with the QRT mapping. We use algebraic methods involving discrimin...

2010
Arzu Özkoç Ahmet Tekcan A. Özkoç A. Tekcan

In this paper, we consider some properties of the family of indefinite binary quadratic forms and elliptic curves. In the first section, we give some preliminaries from binary quadratic forms and elliptic curves. In the second section, we define a special family of indefinite forms Fi and then we obtain some properties of these forms. In the third section, we consider the number of rational poi...

Journal: :IACR Cryptology ePrint Archive 2005
Eunjeong Lee Yoonjin Lee

In recent papers [4], [9] they worked on hyperelliptic curves Hb defined by y +y = x+x+b over a finite field F2n with b = 0 or 1 for a secure and efficient pairing-based cryptosystems. We find a completely general method for computing the Tate-pairings over divisor class groups of the curves Hb in a very explicit way. In fact, Tate-pairing is defined over the entire divisor class group of a cur...

2003
FRANZ LEMMERMEYER

In this article we study the Tate-Shafarevich groups corresponding to 2-isogenies of the curve Ek : y 2 = x(x2 − k2) and construct infinitely many examples where these groups have odd 2-rank. Our main result is that among the curves Ek, where k = pl ≡ 1 mod 8 for primes p and l, the curves with rank 0 have density ≥ 1 2 .

Journal: :IACR Cryptology ePrint Archive 2004
Denis Xavier Charles Kristin E. Lauter

The l modular polynomial, φl(x, y), parameterizes pairs of elliptic curves with an isogeny of degree l between them. Modular polynomials provide the defining equations for modular curves, and are useful in many different aspects of computational number theory and cryptography. For example, computations with modular polynomials have been used to speed elliptic curve point-counting algorithms ([B...

2008
Mitsuru Kawazoe Tetsuya Takahashi

An explicit construction of pairing-friendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman. In this paper, we give other explicit constructions of pairing-friendly hyperelliptic curves. Our methods are based on the closed formulae for the order of the Jacobian of a hyperelliptic curve of type y = x + ax over a finite prime field Fp which are given by E. Furukawa, ...

2008
Dawei Chen

Consider genus g curves that admit degree d covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family Y that naturally maps into the moduli space of stable genus g curves Mg. We study the geometry of Y , and produce a combinatorial method by which to investigate its slope, irreducible components, genus and orbifold...

2004
Jan Nekovář

(0.0) Elliptic curves are perhaps the simplest 'non-elementary' mathematical objects. In this course we are going to investigate them from several perspectives: analytic (= function-theoretic), geometric and arithmetic. Let us begin by drawing some parallels to the 'elementary' theory, well-known from the undergraduate curriculum. Elementary theory This course arcsin, arccos elliptic integrals ...

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