Computational trapdoor where the problem is tractable for both parties but easier for one Usually the sort of trapdoors which are talked about are designed such as to make the computation intractable for one party and tractable for the other.

But what if one party merely has a big advantage in computation over the other? Can you use it to create a construct in cryptography?

For example, suppose you have a way to construct a large number such that using it as an operand in multiplication is very cheap for you: O(kN) for a k you can somewhat control (by randomly trying many primes looking for such a quality and then multiplying them masking the quality). The best others can do seems to be O(N*logN*loglogN) nowadays.

How can you use this computational trapdoor? The multiplication is tractable for both parties but one has an advantage.

Is there a way with multiplication to amplify a linear advantage into an exponential or polynomial one?
Russell Impaglizzio wrote "A personal view of average case complexity" roughly 25 years ago. In this work, he describes "five worlds" that we could live in with regards to the $P\stackrel{?}{=} NP$ problem. In two of these worlds one can do cryptography, they are roughly summarized as:


Minicrypt: One-way functions exist
Cryptomania: Trapdoor one-way functions exist


To be clear, each of the five worlds describes a possibility of the resolution of the $P$ vs $NP$ problem (and various related problems). So if "we are in minicrypt but not cryptomania", it means that trapdoor one-way functions don't exist, but one-way functions do exist.

What about other cryptographic primitives? There has been a decent amount of effort to show that (roughly) "Minicrypt = private-key cryptography", and "Cryptomania = public-key cryptography". There's actually some really interesting recent work showing that there's more to the story than just this [1], but that's a separate topic.

So if we aren't in cryptomania, can we do public key encryption? Strictly speaking no (that's the point), but we can still hope for something akin to what you're describing --- while our protocols may not be impossible for a poly-time adversary to break, we can "have a computational gap" between honest parties and dishonest parties (although the gap is some polynomial). One of the initial public key schemes was actually of this form (Merkle's Puzzles). There has been some recent work revisiting this (seemingly motivated by "fine-grained complexity"), see Fine-grained Cryptography and Public-key Cryptography in the Fine-grained Setting.



[1] See Minicrypt primitives with algebraic structure and applications. Essentially all minicrypt primitives have been known to be equivalent for a few decades, but cryptomania primitives are not all known to be equivalent. There appears to be ways in which they aren't equivalent, meaning they break up into separate "continents" within the "world" of minicrypt.