A Brief History of Factorization Techniques

1. A Brief History of Factorization Techniques March 9th, 2006

2. Factoring is Hard • 1874 – English Economist W. Stanley Jevons conjectured that no one would ever factor 8,616,460,799. • Proved wrong by Bancroft Brown in 1925 • 96,079 x 89,681 • 1903 – F. N. Cole Presents at a meeting of the American Mathematical Society • Without uttering a single word he demonstrated that the 67th Mersenne number is composite • 1977 – Martin Gardner publishes a challenge to factor a 129 digit number. • Ron Rivest estimated it would take more than “40 quadrillion years” • The number was factored in 1994 after eight months of work on hundreds of machines by Atkins, Graff, Lenstra and Leyland

3. Historical Techniques Used • Pre-calculated Tables • 1659 Rahn – factors for n <= 24,000 • 1668 John Pell – factors for n <= 100,000 • 1776 Felkel – factors for n <= 408,000 • 19th Century tables existed for n <= 10,000,000 • Early Algorithms • Trial Division • Fermat’s Algorithm • Euler’s Algorithm

4. Mechanical Aids • 1925 – Mechanical Calculators in the US support Multiply and Divide • 1927 – Bicycle Chain Sieve • 1932 – Photoelectric Number Sieve • 1945 – ENIAC • 1952 - SWAC

5. A Sampling of Modern Algorithms • Special Purpose • Pollard rho method • Pollard’s p-1 method • Elliptic Curve method • Fermat’s method • General Purpose • Quadratic Sieve • Number Field Sieve

6. Future of Factoring • Iterative Improvements • Algorithmic Improvements • Moore’s Law • Taking advantage of Distributed Computing • Quantum Computers • Shor’s Algorithm • IBM Demonstration