How to compare performances of lattice-based and pairing-based IBE schemes 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)
...
It is very hard to give a concrete, "apples-to-apples" comparison of lattice-based and pairing-based IBE schemes.  There are many reasons: the research surrounding concrete secure parameters for LWE is still evolving, efficient implementations of operations used in lattice-based IBE (e.g., discrete Gaussian sampling) are still works in progress, one can consider more efficient ring-based analogues, etc. etc.

However, it is possible to give meaningful asymptotic comparisons, at least up to poly-logarithmic factors, under reasonable hardness conjectures about lattices and pairings (so that we're normalizing to equivalent security levels).

To your specific questions: no, the dimension $n$ for LWE and the bit size $\log |G|$ are not equivalent.  However, they are asymptotically close, up to polylog factors, in the following sense: for conjectured $2^\lambda$ security, we can set $\log |G| = O(\lambda)$ and $n = \tilde{O}(\lambda)$.  For pairings, this implies that all key sizes are $O(\lambda)$.  For lattices it implies:


master public keys of size $\tilde{O}(n^2)$ [plain LWE], or $\tilde{O}(n)$ [ring LWE].
user secret keys of size $\tilde{O}(n \cdot \ell)$ for $\ell$-bit messages [plain LWE], or $\tilde{O}(n)$ [ring LWE] for $\ell = O(n)$.
encryption time of $\tilde{O}(n^2)$ [plain LWE], or $\tilde{O}(n)$ [ring LWE].
etc. etc.


In short, for ring-LWE all relevant sizes and runtimes can be made $\tilde{O}(n)$, though the hidden polylog factors can be significant in practice.