A new technique for proving uniqueness of martingale problems is introduced. The method is illustrated in the context of elliptic diffusions in Rd.

In elliptic curve cryptosystems, scalar multiplications performed on the curves have much effect on the efficiency of the schemes, and many efficient methods have been proposed. In particular, recoding methods of the scalars play an important role in the performance of the algorithm used. For integer radices, non-adjacent form (NAF) and its generalizations (e.g., generalized non-adjacent form (...

This paper introduces fast algorithms for performing group operations on twisted Edwards curves, pushing the recent speed limits of Elliptic Curve Cryptography (ECC) forward in a wide range of applications. Notably, the new addition algorithm uses 8M for suitably selected curve constants. In comparison, the fastest point addition algorithms for (twisted) Edwards curves stated in the literature ...

In the underlying finite field arithmetic of an elliptic curve cryptosystem, field multiplication is the next computational costly operation other than field inversion. We present two novel algorithms for efficient implementation of field multiplication and modular reduction used frequently in an elliptic curve cryptosystem defined over GF (2). We provide a complexity study of the two algorithm...

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...