Browser-based Key Generation and interaction with the Browser's Key/Certificate Store

Imagine the following scenario: You need to get a key (in the asymmetric case the user's public key) from a user visiting your website and want the browser to remember the private part without bothering the user with lengthy import procedures. To be honest, in practice you don't even want the user to deal with cryptographic details - details that many users are not able to know and do not know any