Does the ability to factor in polynomial time give you smooth numbers in the number field sieve?

Question

I have read that despite strong connections between prime factorization and DLP an algorithm for the former does not imply the latter directly. But I was reading about the number field sieve and it seemed like the bottleneck was identifying smooth norms. Wouldn't an ultrafast prime factorization algorithm achieve that?

Answer 1

Being able to test for smooth norms quickly (e.g. polynomial time) is insufficient to quickly detect smooth norms in a large selection of candidates as the test must still be applied to a large number of candidates. Thus even if smooth norms could be identified in time $O(1)$, the log-asymptotic complexity of the number field sieve is unaffected. In fact sieving for smooth norms allows one to test $n$ candidates for $y$-smoothness in time and memory roughly $n\log\log y$ so that the parallelised smoothness test only takes an average $\log\log y$ work per candidate.

The bottleneck lies in the fact that $n$ cannot be too small or we will not find enough relations to produce a solvable linear algebra system. The cost to test the $n$ candidate norms is at best a second order effect (for example, the log asymptotic cost of the number field sieve is unchanged whether we test using sieving at cost $O(\log\log y)$ per candidate or ECM at cost $\exp((1+o(1))\sqrt{2\log y\log\log y})$ per candidate).