A MAC Scheme for Optical Lenses
MACs provide authentication over arbitrary bitstrings, but there is no decent hash function for arbitrary physical objects. I believe that dropping the goal of trying to be a general purpose "any physical object" MAC, you can get much closer to meeting the properties of a cryptographic MAC.
Here is a physical analogue of a MAC for optical lenses. I've imagined a world where there is a need to be able to verify the integrity of lenses sent from place to place, with analogous developments to what has happened in crypto.
There is a publicly known mechanism for validating MACs over lenses - a standard jig containing mirrors and lasers into which you place the lens, and a paper holder (lets say it's A4). Manufacturers can sell you certified jigs that are know to meet the standard.
The jig is designed such that the laser beams pass through the lens multiple times as they bounce around the mirrors, before finally shining on the paper (or hitting another non-reflective part of the jig).
The jig design is critical to the security of the system - many jigs were designed in the old days that had weaknesses allowing attackers to break the scheme. Jigs these days are typically developed by academics, then standardised by national bodies as a result of a competition comparing security, size and ease of use (speed) across the entrants.
Sender and receiver share a secret: a particular configuration of lasers and mirrors (which positions all of the mirrors and lasers are in, potentially which lasers are on or off).
There is a finite but very large set of valid configurations (which means that the jig supports lasers being in one of a set positions, not an analogue configuration). An example configuration might be specified as: Laser n; position. Mirror n; position. L1:45; L2:0; L3:90; L4:20; ... L15:0; M1:45; M2:80; ... M10:90.
(There are some weak keys, e.g. very short ones, all zeros, and more subtle issues like setups that typically lead to null MACs. The algorithm specifies constraints for constructing strong keys.)
Sender dispatches a lens (the message) to the receiver, alongside a piece of paper with some dots on it (the MAC).
The receive puts the lens into the jig (the algorithm), sets up the mirrors and lasers according to the shared secret configuration (key input), and slots the paper into its holder (MAC input).
When the lenses are on, if each dot on the paper is illuminated by a laser (and there are no lasers shining onto the paper where there are no dots), then the MAC is verified and the receiver knows that they have received the correct lens.
This is an analogue of a symmetric MAC scheme - generation follows the same process as verification, but blank paper is used and the position of the dots are recorded rather than checked.
Comparison to MAC Properties
Provided the algorithm (jig) is well defined, a small change to the lens will lead to a large change to the MAC, as each laser beam passes through the lens a number of times at multiple different points: by the time it hits the paper a small change in the lens will have a large change in the position of the spots in the paper.
(c.f. pictures or other representations of messages).
Detection not prevention
The MAC does nothing to stop an attacker swapping out or altering the lens or MAC - it just alerts the receiver to the change.
(c.f. lock boxes)
Deriving the key from example valid message:MAC pairs
An attacker cannot derive the secret key from example plaintext:MAC pairs.
(c.f. wax seals, signatures, key or combination locks that can be disassembled)
Without knowing the secret key, there is no practical way for an attacker to create a valid MAC for their lens.
(c.f. locks that open whatever key you insert / combination you put in)
Chosen Plaintext Attack
A cryptographic MAC should be resistent to attack even if the attacker has access to an oracle that will produce valid MACs for arbitrary plaintexts (Chosen Plaintext Attack).
This scheme is not as resistant as you would like under these circumstances - the null message of a non-distorting lens reveals more information about the secret key than is ideal. It doesn't immediately reveal the key provided the jig (algorithm) is well designed, but you could imagine a small set of well chosen lenses that could give the attacker a significant advantage.
So users of this scheme need to try and avoid providing attackers access to an oracle of this kind - something you try to avoid in a real cryptosystem too!
Does Not Reveal The Message
If you want to also put the lens in a strongbox and ship the MAC outside the box, the MAC doesn't reveal much useful information about the lens without knowing the key.
(c.f. manifests, photos, ...)