What kind of assumptions usually go into the design of block ciphers?

What are some standard assumptions made in showing the security of a block cipher?
For example, is it commonly assumed that $P\not=NP$? To this end, are there any block ciphers whose security does not rest on the assumption that $P\not=NP$, and if so what are the assumptions?
Furthermore, do there exist any block ciphers that are provably secure under some set of assumptions?

The existence of a secure block cipher would prove that one-way functions exist, and the existence of one-way functions would prove $\text{P} \ne \text{NP}$. That is, if they are not equivalent, then it might be possible, at least in theory, to design an ideal block cipher (but we might not know how). But even if $\text{P} = \text{NP}$ is proven, a non-secure block cipher might still be safe in practice, if the proof is non-constructive.