Just to answer your query here: integer factorization is still suspected to be outside of P, NP-complete and co-NP-complete (recently some have hinted that it may in fact be NP-complete, I cannot track the link to the email discussion atm). No proof for either of those claims exists yet.

Discrete logarithm has been found to be in BQP (Shor's algorithm), and therefore not NP-complete, since BQP is suspected to be disjoint from NP-complete. No proof of this claim is available, either.