Averages of elliptic curve constants

Averages of elliptic curve constants

Let E be an elliptic curve defined over the rational numbers and for a prime p of good reduction for E, let Ep denote the reduction of E modulo p. There are various conjectured asymptotics for functions which count good primes p up to x for which the reduced curve Ep has certain properties. In this paper we will focus on three such questions, although our methods are applicable to a wider range...

Efficient elliptic curve cryptosystems

Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...

Seshadri constants on the self - product of an elliptic curve

The purpose of this paper is to study Seshadri constants on the self-product E × E of an elliptic curve E. We provide explicit formulas for computing the Seshadri constants of all ample line bundles on the surfaces considered. As an application, we obtain a good picture of the behaviour of the Seshadri function on the nef cone.

Lipschitz Constants to Curve Complexes

We determine the asymptotic behavior of the optimal Lipschitz constant for the systole map from Teichmüller space to the curve complex.

Curve Selection in Elliptic Curve Cryptography