Suppose I have encrypted messages sent to a device authenticated with a MAC keyed with a strong shared cryptographic key. This device can only process so many messages per unit of time. This cryptographic key is renewed every few seconds: a new key is independently created and the previously one is destroyed.
Does this actually have any effect on the difficulty of an attacker to forge a message with a valid MAC? I originally thought it did since any computational effort spent for a key would be wasted once the key is renewed, but it occurred to me that it's not really possible to attack a MAC in a structured way (otherwise it would be broken); tags are just tried at random until a valid one is found, independently of the key.
Am I right in saying that constantly changing the key does not help with increasing the strength of a MAC? If not, can I shorten the MAC for these messages and achieve equivalent security against forgery to a larger MAC length due to the use of short-lived keys?