Also i didn't understand what you mean by part private key. So in the example of "1qwertyuiop" VanitySearch shows this will take more than 150 years, and the other user wanted to split the effort. But my question is since this is just a partial public key there's no guarantee it will find the exact public key he's looking for in 150 years. It might find a key with last few digits different from what he wants, the only way to get the exact public key is to submit the full public key but then that will change from 150 year estimate to millions.
You pass the public key and a prefix through a vanity search program, and then that gives you one of the private keys needed. My understanding is that to combine two private keys, you first convert them into large numbers, and then compute something called a Lagrange interpolation polynomial (
source) at 0. And this has parameters x and y which are contained inside each private key, and the x's I know are inputs to this polynomial because I read the code, and y's are solutions of a split-key equation that uses a different polynomial.
It's complicated math, and I'll have to go over it for a few days before I fully understand it.