What does Shor's algorithm tell us about the complexity class of RSA and the DLP?
That tells us those problems are in BQP.
This answer
describes a way of reducing factoring to SAT.
More elaborate approaches will give reductions from ​ RSA or ECC ​ to SAT.
On the other hand, if there is a polynomial-time reduction

from an NP-hard problem to ​ RSA or ECC ​ ​ then ​ ​ ​ BQP ∩ UP ∩ coUP ​ = ​ NP ​ .