Factoring assuming smoothness of some numbers

Question

I have came across a lot of factorization methods and most of them seem to assume smoothness of some numbers.

For example

1. When $p-1$ is smooth
2. When $|E(\mathbb{F}_p)|$ is smooth. (Elliptic curve factorization)
3. Smoothness of prime ideals in Number field sieves.

I want to know whether any other notions are known to be equivalent to factoring like smoothness of $p+1$ or $p^2+1$ ?

Answer 1

See my paper with Eric Bach, "Factoring with cyclotomic polynomials", where we show that if the cyclotomic polynomial $\Phi_k(p)$ is $B$-smooth for any $p$ dividing $N$, then we can factor $N$ in time polynomial in $\log N$ and $k$ and $B$. In particular this gives a $(p+1)$-method (see the earlier work of Williams) and $(p^2+1)$ method.

http://www.ams.org/journals/mcom/1989-52-185/S0025-5718-1989-0947467-1/S0025-5718-1989-0947467-1.pdf