You may not mean it this way, but for any reader who might be mistaken:
P == NP => integer factorization can be done in polynomial time with a classic computer but not vice versa.
In fact, many people believe P != NP && integer factorization can be done in polynomial time since FACTOR lies in this awkward space in the intersection of NP and co-NP. (Most other problem in this space are also in P, but we think P != NP, so...)
Now of course, if you put in quantum computers into the mix, Shor's algorithm can break both RSA and ECDSA in polynomial time.
P == NP => integer factorization can be done in polynomial time with a classic computer but not vice versa. In fact, many people believe P != NP && integer factorization can be done in polynomial time since FACTOR lies in this awkward space in the intersection of NP and co-NP. (Most other problem in this space are also in P, but we think P != NP, so...)
Now of course, if you put in quantum computers into the mix, Shor's algorithm can break both RSA and ECDSA in polynomial time.