Finding a trapdoor function for the 3-partition problem
As Yuval says, it's an open question whether one can build a public-key cryptosystem whose security is based on the assumption that $\textsf{P} \ne \textsf{NP}$.  There is no known public-key cryptosystem with that property, though many cryptographers have tried to find one.  So, you're trying to solve a problem that is likely to be extremely hard to solve.

Part of the challenge is that NP-complete problems are generally hard, but that's not enough for cryptography.  Public-key cryptography also requires that there be a "trapdoor" that makes them easy.  So, you'll need to find a way to encode a secret trapdoor in the 3-partition instance that makes it easy to solve the 3-partition problem if you know the trapdoor -- but where you can prove that the existence of the trapdoor doesn't make the problem any easier for people who don't know the trapdoor.  It's not clear how to do that.  There are ways to embed a trapdoor, but then they require choosing 3-partition instances with a particular structure/format, and then solving those special instances might be easy (might be doable in polynomial time).

You might be interested in knapsack cryptography (see the knapsack tag on Crypto.SE), Impagliazzo's five worlds, this question on Crypto.SE, and Impagliazzo and Rudich's black-box separation between one-way functions and key exchange, which provides some kind of evidence that it's likely not going to be easy to do what you want (e.g., there's no black-box way to convert a NP-complete problem to a public-key encryption algorithm).