Factoring Integers with the Self-Initializing Quadratic Sieve

In 1996, we used the self initializing quadratic sieve (siqs) to set the general purpose integer factorization record for the Cunningham project. Here, we show that this algorithm is about twice as fast as the ordinary multiple polynomial quadratic sieve (mpqs). We give running times of both algorithms for 60, 70, and 80 digit numbers. These tables show the best timings we were able to get using various parameters for each of algorithms. In all cases, the best siqs times are about twice as fast as the best mpqs times. We also explain a way of distributing the block Lanczos algorithm to solve the matrix part of the quadratic sieve. Our method was tested on an IBM SP2 parallel computer. Date Acknowledgements Many of the computers used for our 116 digit factorization came from Andrew Granville's Presidential Faculty Fellows grant. Also, Red Alford, David Benson, and Carl Pomerance have generously allowed us to use their computers. Acknowledgements are also due to our system administrators, Shaheed Bacchus and Ron Rouhani, who were very helpful and also tolerant of the factoring programs. We also thank UCNS at UGA for allowing us to use their IBM SP2, and especially Alan Ferrenberg for helping us get started on the SP2.ance were especially helpful in making the thesis more readable. I have greatly beneetted from working with Arjen Lenstra at Bellcore. Arjen has also put a lot of time into helping me with my present research. His comments and criticisms have helped me strengthen the results. I am most grateful to my advisor, Carl Pomerance. Carl has spent an enormous amount of time in helping me understand the algorithms in this thesis, as well as many other mathematical problems. Working with him has greatly improved my organization, mathematical abilities, and research skills. It amazes me that a person can spend so much time with his students and still have time to do his own research. I cannot forget the University of Georgia Mathematics Department, which has given me an excellent education. Many of the professors here are among the best teachers I have ever had. The amount of time they have spent preparing for their iii iv classes and helping students is very much appreciated. I thank Red Alford, Carl Pomerance, and Robert Rumely for helping get accepted to the University and the Department. Finally, I thank my parents, who have always encouraged me to …