For those who don't know, Carl Pomerance was the guy who came up with/discovered the quadratic field sieve which for a time was the fastest factoring algorithm for large (100-ish digits) semiprimes. He has a beautiful write-up describing the algorithm and the story of its discovery:
Pomerance, Carl. “A Tale of Two Sieves.” In Biscuits of Number Theory, edited by Arthur Benjamin and Ezra Brown, 85–104. Providence, Rhode Island: American Mathematical Society, 2009. https://doi.org/10.1090/dol/034/15.
To this day the question I have about the Quadratic Seive is something like "what is being reduced in degrees of freedom by each relation added to the matrix?"
It seems like the sequence of primes in the factor base (which are quadratic residues) is very much related to the residues of the factors (they are) and somehow this it's doing in the abstract something like CRT to construct the factors. But I have never read a deeper explanation beyond the mechanics of implementation.
Pomerance, Carl. “A Tale of Two Sieves.” In Biscuits of Number Theory, edited by Arthur Benjamin and Ezra Brown, 85–104. Providence, Rhode Island: American Mathematical Society, 2009. https://doi.org/10.1090/dol/034/15.