It is clear that the negation map can be used to speed up the computation of elliptic curve discrete logarithms with the Pollard rho method. However, the random walks defined on elliptic curve points equivalence class {±P} used by Pollard rho will always get trapped in fruitless cycles. We propose an efficient alternative approach to resolve fruitless cycles. Besides the theoretical analysis, w...

Let C be a Q-curve with no complex multiplication. In this note we characterize the number fields K such that there is a curve C′ isogenous to C having all the isogenies between its Galois conjugates defined over K, and also the curves C′ isogenous to C defined over a number field K such that the abelian variety ResK/Q(C/K) obtained by restriction of scalars is a product of abelian varieties of...

We describe a variational problem on a surface under a constraintof geometrical character. Necessary and sufficient conditions for the existence ofbifurcation points are provided. In local coordinates the problem corresponds toa quasilinear elliptic boundary value problem. The problem can be consideredas a physical model for several applications referring to continuum medium andmembranes.

We prove the existence of steady 2-dimensional flows, containing a bounded vortex, and approaching a uniform flow at infinity. The data prescribed is the rearrangement class of the vorticity field. The corresponding stream function satisfies a semilinear elliptic partial differential equation. The result is proved by maximizing the kinetic energy over all flows whose vorticity fields are rearra...

Elliptic curve cryptosystems, proposed by Koblitz([8]) and Miller([11]), can be constructed over a smaller definition field than the ElGamal cryptosystems([5]) or the RSA cryptosystems([16]). This is why elliptic curve cryptosystems have begun to attract notice. There are mainly two types in elliptic curve cryptosystems, elliptic curves E over IF2r and E over IFp. Some current systems based on ...

In this paper we propose the construction of an efficient cryptographic system, based on the combination of the ElGamal Elliptic Curve Algorithm and Low Density Parity Check (LDPC) codes for mobile communication. When using elliptic curves and codes for cryptography it is necessary to construct elliptic curves with a given or known number of points over a given finite field, in order to represe...

In this chapter, we study elliptic curves defined over finite fields. Our discussion will include the Weil conjectures for elliptic curves, criteria for supersingularity and a description of the possible groups arising as E(Fq). We shall use basic algebraic geometry of elliptic curves. Specifically, we shall need the notion and properties of isogenies of elliptic curves and of the Weil pairing....

grows exponentially. Thus, monodromy groups of elliptic fibrations over P constitute a small, but still very significant fraction of all subgroups of finite index in SL(2,Z). Our goal is to introduce some structure on the set of monodromy groups of elliptic fibrations which would help to answer some natural questions. For example, we show how to describe the set of groups corresponding to ratio...

We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certiication of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the 1065-digit (2 3539 +1)=3, the rst ordinary Titanic prime.

We prove that there are at most finitely many complex λ = 0, 1 such that two points on the Legendre elliptic curve Y2 = X(X − 1)(X − λ) with coordinates X = 2, 3 both have finite order. This is a very special case of some conjectures on unlikely intersections in semiabelian schemes.