We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields. We prove some conditional results for the p′-part on it, and prove the p′-part of the conjecture for constant or isoconstant Abelian schemes, in particular the p′-part for (1) relative elliptic curves with good reduction or ...

In this paper, we present the results of computing the Tate pairing using a supersingular elliptic curve defined over a prime field. The aim of this work is to demonstrate the feasibility of the primitives of identity based cryptosystem for application in embedded processors such as a smartcard. The most computationally intensive operation in an Identity Based Protocol is the calculation of a p...

This paper addresses the discrete logarithm problem in elliptic curve cryptography. In particular, we generalize the Menezes, Okamoto, and Vanstone (MOV) reduction so that it can be applied to some non-supersingular elliptic curves (ECs); decrypt Frey and Rück (FR)’s idea to describe the detail of the FR reduction and to implement it for actual elliptic curves with finite fields on a practical ...

Constructing non-supersingular elliptic curves for pairing-based cryptosystems have attracted much attention in recent years. The best previous technique builds curves with ρ = lg(q) / lg(r) ≈ 1 (k = 12) and ρ = lg(q) / lg(r) ≈ 1.25 (k = 24). When k > 12, most of the previous works address the question by representing r(x) as a cyclotomic polynomial. In this paper, we propose a method to find m...

In this paper, we introduce a new approach to the generation of binary sequences by applying trace functions to elliptic curves over GF (2). We call these sequences elliptic curve pseudorandom sequences (EC-sequence). We determine their periods, distribution of zeros and ones, and linear spans for a class of EC-sequences generated from supersingular curves. We exhibit a class of EC-sequences wh...

Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by OD to supersingular elliptic curves in characteristic p. In the algorithm we first determine an explicit constant Dp so that |D| > Dp implies that the map is necessarily surjective and then we compu...

Let k be a field of positive characteristic p. Question: Does every twisted form of μp over k occur as subgroup scheme of an elliptic curve over k? We show that this is true for most finite fields, for local fields and for fields of characteristic p ≤ 11. However, it is false in general for fields of characteristic p ≥ 13, which is related to the fact that the Igusa curves are not rational in t...

The supersingular K3 surface X in characteristic 2 with Artin invariant 1 admits several genus 1 fibrations (elliptic and quasi-elliptic). We use a bijection between fibrations and definite even lattices of rank 20 and discriminant 4 to classify the fibrations, and exhibit isomorphisms between the resulting models of X. We also study a configuration of (−2)-curves on X related to the incidence ...

We provide a relation between the $\mu $-invariants of dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from cyclotomic ${\mathbb Z}_p$-extension to Z}_p^2$-extension over an imaginary quadratic field.

In this paper, we give an explicit description of the de Rham 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 ...