Let me see if I understand correctly - First I create a private key under Windows using Puttygen and after I copy this private key to the Ubuntu machine I use again Puttygen to generate the private and public keys out of the private keys I created earlier with PuttyGen under Windows.

Are you sure that is the way, cause that's how you explained it. The program for generating keys under windows is Puttygen and it can create either a private or public keys or both.