Is it possible to affect the output of HMAC w/ SHA-512 by providing it with a custom message? Consider an HMAC algorithm that uses SHA-512 with an unknown private key. Is it possible to affect its output, either partially or totally, by providing it with carefully crafted messages?

For the purposes of this question, consider that the application "updates" its state using our hand-crafted message plus a string that changes on each request which is also known to us. Additionally, consider that we are able to ask the service to sign for us an arbitrary number of messages.
Consider an HMAC algorithm that uses SHA512 with an unknown private key. Is it possible to affect its output, either partially or totally, by providing it with carefully crafted messages?


Sure, however you will get totally unrelated HMAC output even if you change (or add or remove) even a single bit from the HMAC input. So each bit flips with a 50% certainty. Creating "crafted messages" doesn't have any effect on this property. 

If you are looking for a specific output then you won't be able to get it, assuming that at least, say 64, bits of HMAC output is kept (if that's the case you might try and brute force). Similarly for other properties, the outcome of HMAC is totally dependent on chance if you don't know the key and /or input message.