Elliptic Curve Cyrptography has gained a lot of significance in recent times. This is mostly due to the small key sizes associated with Elliptic Curve Cryptograhic systems. This paper presents a study of various algorithms for performing underlying field arithmetic and point representation useful in software implementations of Elliptic curve cyptography over prime fileds as well as binary fields.

In 1999, Smart has shown how to solve in linear time ECDLP for elliptic curves of trace 1 defined over a prime finite field Fp, the so-called anomalous elliptic curves. In this article, we show how to construct such cryptographically weak curves for primes p of industrial length, using complex multiplication theory.

Mazur and Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curve overQwith the order of vanishing of Mazur-Tate elements, which are analogues of Stickelberger elements. Under some relatively mild assumptions, we prove this conjecture. Our strategy of the proof is to study divisibility of certain derivatives of Kato’s Euler system. CONTENTS

and repeated indices indicate summation from 1 to n. The functions a'(x, u, p), a(x, u, p) are defined in QX£ n + 1 . If furthermore for any ikf>0, the ratio of the maximum to minimum eigenvalues of [a(Xy u, p)] is bounded in ÛX( — M, M)XE, Qu is called uniformly elliptic. A solution of the Dirichlet problem Qu = Q, u—<f)(x) on <50 is a C(n)P\C(O) function u(x) satisfying Qu = 0 in £2 and agree...

In this paper we examine the underlying hard problems in asymmetric pairings, their precise relationships and how they affect a number of existing protocols. Furthermore, we present a new model for the elliptic curve groups used in asymmetric pairings, which allows both an efficient pairing and an efficient computable isomorphism.

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and ĥ be the canonical height on E. Bounds for the difference h − ĥ are of tremendous theoretical and practical importance. It is possible to decompose h − ĥ as a weighted sum of continuous bounded functions Ψυ : E(Kυ) → R over the set of places υ of K. A standard method for bounding h− ĥ, (due to L...

Although this formula can be obtained by a limiting process from (0.1), it was found before [FS] by the paper of Kiepert [K]. If we set y(u) = 12℘ ′(u) and x(u) = ℘(u), then we have an equation y(u) = x(u)+ · · · , that is a defining equation of the elliptic curve to which the functions ℘(u) and σ(u) are attached. Here the complex number u and the coordinate (x(u), y(u)) correspond by the equality

Although this formula can be obtained by a limiting process from (0.1), it was found before [FS] by the paper of Kiepert [K]. If we set y(u) = 12℘ ′(u) and x(u) = ℘(u), then we have an equation y(u) = x(u)+ · · · , that is a defining equation of the elliptic curve to which the functions ℘(u) and σ(u) are attached. Here the complex number u and the coordinate (x(u), y(u)) correspond by the equality

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p > 3, that the trace of Frobenius of any elliptic curve over Fp, whose jinvariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of ...

We use elliptic divisibility sequences to describe a method for estimating the global canonical height of an algebraic point on an elliptic curve. This method requires almost no knowledge of the number field or the curve, is simple to implement, and requires no factorization. The method is ideally suited to searching for algebraic points with small height, in connection with the elliptic Lehmer...