What I want to do is deterministically derive a secure key from an RSA private key, such that the RSA private key cannot be derived from the secure key.
The first approach that came to mind was simply using the private key (or some fixed part of it) as the seed for a PRNG.
Since the output of AES is random, another idea is to use AES to generate the secure key, using the first 256 bits of the RSA private key as the AES key, and encrypting some known data.
Can anyone verify either approach, or otherwise suggest an alternative way to do this?
EDIT I left out the details of why I wanted to do this, to keep it focussed, but I guess it may be useful to know more:
I have an installer for a piece of server software that will run as part of a cluster. During installation, the installer will have access to a 'deployment' certificate - an X509 certificate that is used to sign certificates for each instance of the server (so, a CA certificate). Each server will also need to have access to a shared secret, which will be used to encrypt certain data that is shared between the servers (using AES). My thinking is that securely deriving this shared secret from the deployment certificate private key during installation is a low friction way to do it.