Classical integer factorization · Pollard → ECM → QS
Race seven real factoring algorithms against one BigInt N, watch each one succeed or
give up, and see the exact structural property of N that decided it — a small factor, a
narrow |p − q|, a smooth p ± 1, a lucky curve, or nothing at all.
There is no single "factoring algorithm" — there is a family of them, each waiting for a different mistake. Every rule in RSA key generation exists to close one of these doors. Shor closes none of them, because it needs no door.