No. Not only is there not a generic way to build a cryptosystem based on the decision problem, there isn't a single known decision problem that:
- is NP-Complete, and
- we can build cryptography from
"Any cryptography" may seem somewhat vague, but it can be made specific fairly easily --- most symmetric-key primitives (One-way Functions, Pseudorandom Functions/Generators, and Unpredictable functions) are known to be equivalent, in the sense that if you have one (say a OWF), you can easily build the others, and vice versa.
Why hasn't there been cryptography built from from this assumption? A basic reason is the "trivial theorem" that:
If there exists a secure (OWF/PRG/PRF/UF), then $P\neq NP$
This is because for any reasonable notion of "security", the problem of breaking an OWF/PRG/PRF/UF is:
- Easily in NP ("guess" the secret key, then whatever the security notion is trivializes)
- Not in P (otherwise an efficient adversary could break it, so it would not be "secure").
Beyond that, there is the (related) question of
How do I leverage knowledge of a specific hard problem to build cryptography?
The answer is somewhat complex. If you can build a standard cryptographic primitive (a PRG/PRF/OWF/UF, or a trapdoor OWF), then things become very easy very quickly.
Beyond that, I would point out that if you can build "standard primitives" that are "homomorphic", things also become fairly straightforward.
But the general story is still being worked out --- lattice-based encryption was first proposed perhaps 25 years ago, but it took until roughly a decade ago for lattice-based signatures to be developed.
This is to say that knowing how to develop (public-key) encryption didn't lead to signatures "for free" (despite PKE being a "harder primitive to build" in a certain formal sense).
There has been some progress on understanding what generic constructions one gets from the construction of certain primitives (this is what the paper I linked does), but it is all fairly recent.
