Date of Thesis
Spring 2026
Description
Integer factorization, the problem of finding a nontrivial factor of a composite integer N=pq for large primes p,q, particularly at the size of RSA moduli, is a notoriously difficult challenge that takes classical methods such as Trial Division and Fermat’s Algorithm trillions of years to solve. This thesis studies four probabilistic algorithms that exploit algebraic group structures to achieve significantly better, subexponential efficiency for certain classes of N: Pollard’s p-1, Williams’ p+1, Lenstra’s Elliptic Curve Method, and Pell’s Conic Method. In each case, the algorithm operates on a group over ℤ/Nℤ that decomposes, via the Chinese Remainder Theorem, into corresponding groups over ℤ/pℤ and ℤ/qℤ, with success determined by the smoothness of the relevant group order over ℤ/pℤ. Building on this framework, we develop Pell’s Conic Method as a natural extension of the three better-known algorithms and compare their theoretical and practical behavior. Finally, experimental results confirm most theoretical results while also reveal discrepancies that suggest directions for future work.
Access Type
Honors Thesis
Degree Type
Bachelor of Arts
Major
Mathematics
First Advisor
Dr. Jen Berg
Second Advisor
Dr. Nathan C. Ryan
Recommended Citation
Cao, Phuong, "Group-based Integer Factorization: Theory and Performance" (2026). Honors Theses. 780.
https://digitalcommons.bucknell.edu/honors_theses/780