Elliptic curve cryptosystems are expected to be a next standard of public-key cryptosystems. A security level of elliptic curve cryptosystems depends on a difficulty of a discrete logarithm problem on elliptic curves. The security level of a elliptic curve cryptosystem which has a public-key of 160-bit is equivalent to that of a RSA system which has a public-key of 1024-bit. We propose an ellip...

The Elliptic Curve Discrete Logarithm Problem(ECDLP) is known to be an exponential time problem except the cases of smooth curves, supersingular curves and anomalous curves. Recently, several new methods to solve ECDLP on a prime eld were proposed. All of them try to solve ECDLP on a prime eld by lifting a given elliptic curve to low rank elliptic curves de ned over the rationals. In this exten...

The security of elliptic curve cryptosystem depends on the choice of an elliptic curve on which cryptographic operations are performed. Schoof’s algorithm is used to define a secure elliptic curve, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as Atkin-Elkies’s m...

To every non-singular elliptic curve (complex torus) we assign a C∗algebra Tθ = {u, v | vu = e uv} known as noncommutative torus. It is shown that morphisms of elliptic curves generate Morita equivalence of the corresponding noncommutative tori. Real number θ we call projective curvature attached to the elliptic curve. It is proved that projective curvatures of isomorphic elliptic curves are mo...

A rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using Jacobi elliptic functions. To illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators, we first investigate the exact solution of the non-homogenous, undamped Duffing equation. Then, we introduce first and second order...

In this paper, we give an explicit description of the complex and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic ...

Signcryption is a new cryptographic approach which provides authentication and encryption simultaneously in a single logical step. The aim is to reduce the cost of signature-then-encryption approach. This cost includes computational cost and communication cost. Furthermore some signcryption schemes are based on RSA while some are based on elliptic curve. This paper provides a critical review of...

that is both analytic (as a mapping of complex manifolds) and algebraic: addition of points in E(C) corresponds to addition in C modulo the lattice L. This correspondence between lattices and elliptic curves over C is known as the Uniformization Theorem; we will spend this lecture and part of the next proving it. To make the correspondence explicit, we need to specify the map Φ from C/L and an ...

This paper describes synthesizable VDHL implementation of elliptic curve Point Multiplication. Elliptic curves used for ECC are defined over mathematical structures called Galois fields. Based on the theory of ECC, this paper has carried out Modular addition/subtraction, EC Point doubling/addition, Modular multiplicative inversion, EC point multiplier, projective to affine coordinates conversio...

This paper is concerned with the construction of extremal elliptic K3 surfaces. It gives a complete treatment of those fibrations which can be derived from rational elliptic surfaces by easy manipulations of their Weierstrass equations. In particular, this approach enables us to find explicit equations for 38 semi-stable extremal elliptic K3 fibrations, 32 of which are indeed defined over Q. Th...