Continued fraction factorization

In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning that it is suitable for factoring any integer n, not depending on special form or properties. It was described by D. H. Lehmer and R. E. Powers in 1931,^[1] and developed as a computer algorithm by Michael A. Morrison and John Brillhart in 1975.^[2]

The continued fraction method is based on Dixon's factorization method. It uses convergents in the regular continued fraction expansion of

{\sqrt {kn}},\qquad k\in \mathbb {Z^{+}}

Since this is a quadratic irrational, the continued fraction must be periodic (unless n is square, in which case the factorization is obvious).

It has a time complexity of $O\left(e^{\sqrt {2\log n\log \log n}}\right)=L_{n}\left[1/2,{\sqrt {2}}\right]$ , in the O and L notations.^[3]

References

^ Lehmer, D.H.; Powers, R.E. (1931). "On Factoring Large Numbers". Bulletin of the American Mathematical Society. 37 (10): 770–776. doi:10.1090/S0002-9904-1931-05271-X.
^ Morrison, Michael A.; Brillhart, John (January 1975). "A Method of Factoring and the Factorization of F₇". Mathematics of Computation. 29 (129). American Mathematical Society: 183–205. doi:10.2307/2005475. JSTOR 2005475.
^ Pomerance, Carl (December 1996). "A Tale of Two Sieves" (PDF). Notices of the AMS. Vol. 43, no. 12. pp. 1473–1485.