A factorization method for elliptic BVP

Random Walk Algorithm for Estimating the Derivatives of Solution to the Elliptic BVP

Sobolev gradients: a nonlinear equivalent operator theory in preconditioned numerical methods for elliptic PDEs

Solution methods for nonlinear boundary value problems form one of the most important topics in applied mathematics and, similarly to linear equations, preconditioned iterative methods are the most efficient tools to solve such problems. For linear equations, the theory of equivalent operators in Hilbert space has proved an efficient organized framework for the study of preconditioners [6, 9], ...

Improving Lenstra’s Elliptic Curve Method

In this paper we study an important algorithm for integer factorization: Lenstra’s Elliptic Curve Method. We first discuss how and why this method works and then draw from various research papers to demonstrate how it can be improved. In order to achieve this, we take a look at the torsion subgroup of elliptic curves and review methods for how to generate elliptic curves with prescribed torsion.

Elliptic Curve Method for Integer Factorization on Parallel Architectures

The elliptic curve method (ECM) for integer factorization is an algorithm that uses the algebraic structure of the set of points of an elliptic curve for factoring integers. The running time of ECM depends on the size of the smallest prime divisor of the number to be factored. One of its main applications is the co-factorization step in the number field sieve algorithm that is used for assessin...

Fpga Implementation of Elliptic Curve Method for Factorization

The security of the most popular asymmetric cryptographic scheme RSA depends on the hardness of factoring large numbers. The best known method for factorization large integers is the General Number Field Sieve (GNFS). One important step within the GNFS is the factorization of midsize numbers for smoothness testing, an efficient algorithm for which is the Elliptic Curve Method (ECM). We present ...