We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We prove that the values on points of order small with respect to their degree of rational functions on an elliptic curve have high order. We discuss several special cases, including an old construction of Wiedemann, giving the first non-trivial estimate for the ...

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

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...