Are integers in C assumed to be handled by a hardware spec or handled in software?

By integer, I am referring to the primitive "int" The underlying idea being that if integers in C are not hardware dependent would it be a violation of standard to have gcc implement different integer handlers. This way you could have your traditional 32 bit int handler, and then you could also have handlers for 256 bit integers, or maybe even dynamic sized ints. I do know about long and long long, but I'm not sure if those are independent of that hardware provided, and I'd like to just specify the size/type of "int" handler to use when building a toolchain.

I also understand the dangers of doing this (because building building 256 bit integers with a 32 bit integer compiler would be bad!), but for those bits of code that require something like a gmp library, I ... more.

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