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2

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 ...

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No. The key factor is the security level - a scheme has n bits of security if it takes roughly $2^n$ work ($\approx$ time x space) to break. So one can compare for example Diffie-Hellman key exchange in finite fields and over elliptic curves at the 128 bit security level and conclude that EC are more efficient (that is why they were invented, after all). ...

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Giving the j-invariant is basically equivalent to giving a curve equation (up to quadratic twist) (over $\mathbb F_p$ for $p > 3$) : to each j-invariant you can attach a pair of curves $(E, \widetilde E)$. You can easily compute the curve equations of $E$ and $\widetilde E$ from $j$: $$E: y^2 = x^3 + a x + 2a, \qquad a = -\frac{27j}{j-1728}$$ Now the ...

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