remains a good resource. Number Field Sieve keeps being the fastest algorithm to factor "large" integers which are known not to have small factors.
We used the information on that page in 2009, for factoring the 512-bit RSA public keys of TI-Z80 and TI-68k calculators. In 2009, factoring 512-bit RSA keys was already in the realm of solo factorization, as shown by Benjamin Moody.
In order to speed up the process of factoring remaining relevant 512-bit RSA public keys, the RSALS BOINC grid was built up by the French scene at yAronet (idea: Godzil IIRC, implementation: squalyl). That was before activity on yAronet was destroyed by the most poisonous person in the community.
For reasons I've long forgotten (why me instead of anyone else ? I know that I had some free time, and I was probably trustworthy), I got to be the backup admin of RSALS. Later, I entered the vast majority of numbers, and acted as its public face, for all of its life after the fantastic tool had outlived its initial purpose, and was repurposed to factor integers of mathematical interest
Amusingly, the move of amateurs building a BOINC grid to distribute the sieving phase of the Number Field Sieve algorithm caught the integer factoring community off-guard. They had already performed harder factorizations for years before the advent of RSALS, but never through such work distribution software.
Greg "frmky" Childers, who had already helped us post-processing some of the RSA keys, very soon created NFS@Home. RSALS and NFS@Home shared the factoring duties for several years (smaller numbers for RSALS, larger numbers for NFS@Home), before we shut down RSALS. NFS@Home broke factoring records for a public effort, though the post-processing phase of the most difficult tasks required academic-scale resources.
A piece of the RSALS infrastructure lives on NFS@Home: the Web interface made by squalyl for me, for entering numbers and sieving ranges into a database, which is read by the work unit (WU) generation system. From that Web interface, we can commandeer hundreds of GFLOPS on average, peaking above 2 TFLOPS shortly before we shut RSALS down and some people who had never heard about it before went all-out for the stats. Too bad the poor little server didn't have enough disk space for handling so much post-processing output at once, and therefore, I had to starve clients part-time... I'm sure we could have gone quite a bit higher than that
There also exist semi-turnkey resources such as Tom Ritter's cloud-and-control. If I had to factor new 512-bit RSA keys myself, and could leverage reliable hosts, I'd probably build the sieving process on a Gearman server, and Perl or Node.js/io.js scripts for queuing jobs and spawning sieving tasks.
1024-bit RSA keys are likely to be publicly factored by 5 years for now (that's what the leading researchers behind the factorization of RSA-768, in 2006-2009, predicted), but 2048-bit RSA keys aren't going to be factored in the foreseeable future, unless there's an unlikely algorithmic breakthrough. By now, NFS has remained the best generic integer factoring [algorithm] for over 20 years...