I try to compare the performances (cost of Enc, Dec, ... size of keys, ciphertexts, ...) of IBE schemes using lattices (LWE hardness assumption) or pairing (Diffie-Hellman hardness assumption).
I've observed that performances of lattice-based schemes are often expressed in terms of $n$ (the dimension of the lattice) whereas pairing problems use the bit size $|G|$ of the elements of the underlying group(s) $G$ (and the cost of basic operations such as exponentiation or pairing).
Are these 2 quantities "equivalent"? For instance, is it relevant to say:
- keys of length $O(n)$ (in a lattice scheme) and $O(|G|)$ (in a pairing scheme) will have roughly the same size in practice
- encryption of cost $O(n^2)$ (in a lattice scheme) is quicker than $O(|G|^3)$ (in a pairing scheme)