> But that's not always the model we pick, often we just assume that arithmetic is O(1) and sweep length considerations under the rug. So, I only have rudiment

> But that's not always the model we pick, often we just assume that arithmetic is O(1) and sweep length considerations under the rug. So, I only have rudimentary undergraduate knowledge about complexity theory, but my materials (e.g. in theoretical CS, cryptography and computational algebra) were always incredibly clear about the fact that arbitrary integer arithmetic isn't constant - this is why complexity classes such as weakly vs. strongly NP-complete exist. Do you have any examples of where people would treat arbitrary length integer arithmetic as constant? (I think most people just never have to use arbitrary precision arithmetic, so that's why they'd not talk about it explicitly.) You're right that computational complexity depends on the model and that this makes it nontrivial to use in practice. The theoreticians would sweep this under the rug with "these models are all polynomially equivalent to Turing machines" (which lets us formulate a conjecture such as P=NP precisely), but of course that doesn't in practice help you distinguish between a constant, linear or quadratic algorithm.