[stephencanon]

My analysis holds for any implementation for which you can perform O(1) addition and multiplication of integers of some fixed-size. That's just about as platform-independent as you can get, but it still captures the real complexity of the operation in a way that handwavy "pretend you can do arbitrary arithmetic in O(1)" analyses do not.

This isn't simply my viewpoint on the subject; even the very theoretical textbook by Papadimitriou and Vazirani takes this approach to the question (see exercise 0.4 and note that everything is carried out in terms of the complexity of arbitrary-size integer multiplication M(n)).