Question 2.3.4. 
What are the best factoring methods in use today?

Factoring is a very active field of research among mathematicians and computer scientists; the best factoring algorithms are mentioned below with some references and their big-O asymptotic efficiencies (see Question 2.3.1). (O-notation refers to the upper bound on the asymptotic running time of an algorithm [CLR90]).  For textbook treatment of factoring algorithms, see [Knu81] [Kob94] [LL90] [Bre89]; for a detailed explanation of big-O notation, see [CLR90].  

Factoring algorithms come in two flavors, special purpose and general purpose; the efficiency of the former depends on the unknown factors, whereas the efficiency of the latter depends on the number to be factored. Special-purpose algorithms are best for factoring numbers with small factors, but the numbers used for the modulus in the RSA system do not have any small factors. Therefore, general-purpose factoring algorithms are the more important ones in the context of cryptographic systems and their security.

Special-purpose factoring algorithms include the Pollard rho method [Pol75], with expected running time O(p), and the Pollard p - 1 method [Pol74], with running time O(p'), where p' is the largest prime factor of p - 1. The Pollard p + 1 method is also a special purpose factoring algorithm, with running time O(p'), where p' is the largest prime factor of p+1.  All of these take an amount of time that is exponential in the size of p, the prime factor that they find; thus these algorithms are too slow for most factoring jobs. The elliptic curve method (ECM) [Len87] is superior to these; its asymptotic running time is O(e^{(2ln p ln ln p)}). The ECM is often used in practice to find factors of randomly generated numbers; it is not fast enough to factor a large RSA modulus.

The best general-purpose factoring algorithm today is the Number Field Sieve (NFS) [BLP94] [BLZ94], which runs in time approximately O(e^{1.9223(ln n1/3)(ln ln n2/3)}). Previously, the most widely used general-purpose algorithm was the Multiple Polynomial Quadratic Sieve (MPQS) [Sil87], which has running time O(e^{(ln n ln ln n)}). Recent improvements to the Number Field Sieve make the NFS more efficient than MPQS in factoring numbers larger than about 115 digits [DL95]; MPQS is better for small integers. RSA-129 (see Question 2.3.6) was factored using a variation of MPQS. It is now estimated that if the NFS had been used, it would have taken one quarter of the time.

Clearly, NFS will overtake MPQS as the most widely used factoring algorithm, as the size of the numbers being factored increases from about 130 digits (the current threshold of general numbers which can be factored) to 140 or 150 digits.  A "general number" is one with no special form that might make it easier to factor; RSA moduli are created to be general numbers. Note that a 512-bit number has about 155 digits.

Numbers with up to 155 digits or more that have a special form are easier to factor than general numbers [LLM93]. The Cunningham Project [BLS88] keeps track of the factorizations of numbers with these special forms and maintains a "10 Most Wanted" list of desired factorizations. Also, a good way to survey current factoring capability is to look at recent results of the RSA Factoring Challenge (see Question 2.3.6).

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