Integer factorization using polynomial whose roots are prime factors
Knowing $F_n(1)$ or $F_n(-1)$ gives good randomized polynomial algorithm for the factorization of $n$.

We have $|F_n(1)|=\phi(n)$ and $|F_n(-1)|=\sigma(n)$ by definition.

$\phi(n)$ is Euler's totient function and $\sigma(n)$ is the sum of divisors function.

The paper you link to proves the case for $\sigma(n)$.

The same paper cites that the case for $\phi(n)$ is known
and is derandomized assuming ERH (p. 1).

I am pretty sure both algorithms will be quite fast in practice.