Applicability of physical unclonable functions for public-key cryptography Can physical unclonable functions (PUFs) be used to protect secrets against physical tampering in the context of public-key cryptography?

Some solutions (i.e. ChipDNA or QuiddiKey) apparently use PUFs to protect their private key that are used to perform ECDSA. Some MIT slides about Physical Unclonable Functions and Applications mention private/public keys, implying that this is possible.

From my basic understanding of PUFs, a database of challenge-response pairs is created during manufacturing and we can authenticate a chip when it provides the correct response(s) to challenge(s) as long as challenges are not reused. How is the PUF "fingerprint" derived to a secret key?
Caveat: my understanding of Physically Unclonable Functions is partial, and my limited exposure to would-be PUF systems is old. I offer a tentative current opinion, hopping there will be constructive contradiction, and that I'll learn something new.

I know no practical system (or credible blueprint for such) where possession of a PUF can be assimilated to possession of a private key (the way we can with a Smart Card holding a private key, as long as it does resist attempts to extract the private key, as designed).

It is known how to achieve that after enrolling of a PUF with some PUF reading device (hereafter reader):


The reader+PUF can perfom a private-key operation.
When the PUF is separated from the reader, the private key can't be extracted from the reader (even if the reader is not tamper-proof), and the reader can no longer perform private-key operation.
The PUF itself can't be cloned in such a way that the unmodified reader will accept the cloned PUF (and perfom private-key operation).


However:


Moving the PUF to a new reader can only move the private key associated with the PUF to that new reader if some associated enrollment data is also moved to that new reader (e.g. thru an online connection, or with some traditional data storage moved along the PUF).
An attacker with the PUF, and said enrollment data, and the reader's full design (or the reader itself if not tamper-proof) can find the private key, then perform private-key operations without the PUF or/and simulate reader+PUF.


Some realizations (existing at least at the prototype stage) integrate the PUF, the reader, and perhaps enrollment data on the same piece of silicon. This does prevent attempts to make exact silicon-level copies, but won't resist extraction of the private key or more educated attacks.

Note: all the above holds if we replace private key by secret key.