cperciva's comment there also generalizes to cryptographic protocols which sometimes require batch inversion (i.e. given a1, ..., an \in F, find out all a1^{-1}

cperciva's comment there also generalizes to cryptographic protocols which sometimes require batch inversion (i.e. given a1, ..., an \in F, find out all a1^{-1}, ..., an^{-1}; e.g. when converting a bunch of EC points from projective to affine coordinates), and is there known as "Montgomery's trick" (separate from Montgomery modular multiplication). That is, you trade n inversions for 3(n-1) multiplications and one inversion.