Elliptic curve factoring method via FFTs with division polynomials
Abstract
In 1985, H. W. Lenstra, Jr. discovered a new factoring method, the Elliptic Curve Method (ECM), which efficiently finds 20- to 30- digit prime factors. On January 15, 2001, C. P. Schnorr proposed an idea to apply division polynomials for ECM. In this thesis, we modify and implement the idea in detail, analyze the complexity of the algorithm and the probability of success. We first extend the concept of division polynomial to a univariate case with the parameter a in the Weierstrass form y2 = x3 + ax + b as the variable. We generalize a result about the degree of this implied univariate division polynomial. Then we discover an algorithm to efficiently generate the m-th univariate division polynomial and determine the complexity of the algorithm. We then present the main algorithm of this thesis. We demonstrate in both algebraic and geometric ways the sufficient conditions for the algorithm to be successful. We analyze the probability of success and prove the related results. To demonstrate the structure of the main algorithm, we divide it to several parts and introduce every part in detail. Then we propose and prove a theorem about the complexity of the main algorithm. We also present an optimization of the main algorithm for a family of numbers with specific form. This family is of great interest as well. We show that for this family, the optimal algorithm can factor numbers even faster and also remove the memory restriction of the multiple m of the point on the elliptic curves, which is also the index of the implied division polynomial being used. At the end, we address some issues related to the implementation of the algorithm. With the algorithm, we can find some 40-digit primes on a 1.86GHz 1066FBS PC with 4GB DDR2 SDRAM at 533MHz. It also finds some 50-digit primes. Now we are trying the algorithm on 60-digit primes and we hope that the algorithm will find some 70-digit primes.
Degree
Ph.D.
Advisors
Wagstaff, Purdue University.
Subject Area
Mathematics
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