We introduce elliptic analogues to the Bernoulli ( resp. Euler) numbers and functions. The first aim of this paper is to state and prove that our elliptic Bernoulli and Euler functions satisfied Raabe’s formulas (cf. Theorems 3.1.1, 3.2.1). We define two kinds of elliptic Dedekind-Rademacher sums, in terms of values of our elliptic Bernoulli (resp. Euler) functions. The second aim of this paper...

Elliptic curve cryptography (ECC) is an alternative to traditional techniques for public key cryptography. It offers smaller key size without sacrificing security level. In a typical elliptic curve cryptosystem, elliptic curve point multiplication is the most computationally expensive component. So it would be more attractive to implement this unit using hardware than using software. In this pa...

A theoretical foundation for a generalization of the elliptic difference Painlevé equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transforma...

1 I n t r o d u c t i o n Elliptic curves can be applied to public-key cryptosystems, and as such several schemes have been proposed [3, 4, 5, 6, 9, 11]. There are two typical elliptic curve cryptosystems: E1Gamal-type scheme [4, 11] and RSA-type schemes [3, 5, 6]. The security of the EIGamal-type elliptic curve cryptosystem is based on the difficulty of solving a discrete logarithm over ellipt...

We study the Seiberg-Witten curves for N = 2 SUSY gauge theories arising from type IIA string configurations with two orientifold sixplanes. Such theories lift to elliptic models in M-theory. We express the M-theory background for these models as a nontrivial elliptic fibration over C. We discuss singularities of this surface, and write the Seiberg-Witten curve for several theories as a subvari...

This paper presents explicit equations over Q for 32 extremal semistable elliptic K3 surfaces. They are realized as pull-back of non-semi-stable extremal rational elliptic surfaces via base change. Together with work of J. Top and N. Yui which exhibited the same procedure for the semi-stable extremal rational elliptic surfaces, this exhausts this approach to produce extremal semi-stable ellipti...

The security of most elliptic curve cryptosystems is based on the intractability of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Such a problem turns out to be computationally unfeasible when elliptic curves are suitably chosen. This paper provides an algorithm to obtain cryptographically good elliptic curves from a given one. The core of such a procedure lies on the usage of successi...