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. American Mathematical Society. 29 (129): 183–205. doi:10.2307/2005475. JSTOR 2005475.
↑ Pomerance, Carl (December 1996). "A Tale of Two Sieves" (PDF). Notices of the AMS. 43 (12). pp. 1473–1485.

Number-theoretic algorithms

Primality tests	AKS test APR test Baillie–PSW Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin

Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Wheel factorization

Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks' square forms Trial division Shor's

Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's

Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve

Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's

Modular square root	Cipolla Pocklington's Tonelli–Shanks

Other algorithms	Chakravala Cornacchia LLL Integer square root Modular exponentiation Schoof's

Italics indicate that algorithm is for numbers of special forms

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