In this paper, we present a study of elliptic curves, focusing on theirunderlying mathematical concepts, properties and applications in numbertheory. We begin by introducing elliptic curves and their unique features,discussing their algebraic structure, and exploring their group law, pro-viding examples and geometric interpretations. The core of our studyfocuses on the Elliptic Curve Method (ECM) for integer factorization.We present the motivation behind ECM and compare it to Pollard’s (p-1) method. A discussion on pseudocurves and the choice of an ellipticcurve and bound B is provided. We also address the differences betweenECM and Pollard’s (p-1) method and propose optimization techniques forECM, including the calculation of the least common multiple (LCM) ofthe first B integers using the Sieve of Eratosthenes.

<p>This thesis aims at implementing methods for factorisation of large numbers. Seeing that there is no deterministic algorithm for finding the prime factors of a given number, the task proves rather difficult. Luckily, there have been developed some effective probabilistic methods since the invention of the computer so that it is now possible to factor numbers having about 200 decimal digits. This however consumes a large amount of resources and therefore, virtually all new factorisations are achieved using the combined power of many computers in a distributed system. </p><p>The nature of the distributed system can vary. The original goal of the thesis was to develop a client/server system that allows clients to carry out a portion of the overall computations and submit the result to the server. </p><p>Methods for factorisation discussed for implementation in the thesis are: the quadratic sieve, the number field sieve and the elliptic curve method. Actually implemented was only a variant of the quadratic sieve: the multiple polynomial quadratic sieve (MPQS).</p>

This thesis aims at implementing methods for factorisation of large numbers. Seeing that there is no deterministic algorithm for finding the prime factors of a given number, the task proves rather difficult. Luckily, there have been developed some effective probabilistic methods since the invention of the computer so that it is now possible to factor numbers having about 200 decimal digits. This however consumes a large amount of resources and therefore, virtually all new factorisations are achieved using the combined power of many computers in a distributed system. The nature of the distributed system can vary. The original goal of the thesis was to develop a client/server system that allows clients to carry out a portion of the overall computations and submit the result to the server. Methods for factorisation discussed for implementation in the thesis are: the quadratic sieve, the number field sieve and the elliptic curve method. Actually implemented was only a variant of the quadratic sieve: the multiple polynomial quadratic sieve (MPQS).