When the factorization of RSA-768 was announced in December 2009: the sieving took about 24 months and the matrix step took 119 days (4 months). So sieving took about 6 times as long. This is despite them over-sieving, meaning they spent longer on the sieving so that the matrix step would be easier.
When the factorization of RSA-220 was announced in May 2016: the sieving was done from May to September 2014 (5 months), but the authors say code developments were done between October 2014 and January 2016 to make the matrix step easier. After that they only spent 48 hours on the $\texttt{lingen}$ part of the matrix step, and a few days in the program $\texttt{mksol}$.
These dates, timings, and program names come from here. But the specific code developments that were made between 2014 to 2016 remain a mystery to me. Has there been a bottleneck surpassed in the matrix step, or is it still considered the most cumbersome part of the factorization process?
A related question, regarding the precise cost of the matrix step, which doesn't seem to be given anywhere, can be found here.
