Suppose , we want to find prime factors of a huge number $N$ , say $N=3^{3^{3^3}}+2$. We can assume that we can find easily $N\mod p$ for some positive integer $p$ (as it is the case in the example) , but $N$ is too big to handle numbers with approximately this site directly.
Can we do any better than to apply the simple trial division ? The methods I am aware of all need numbers not much smaller than $N$ as the Pollard-Rho or the p-1-method.
But maybe , I missed some method that allows to find , say , $20$ digit factors with a reasonable amount of effort. Any references or ideas ?