“The techniques we describe in that paper made pairing-based cryptography a viable technology.”

The concept around which our work centers, that of bilinear pairings, was initially of theoretical interest only and not amenable to practical deployment. The techniques we described in this paper helped make pairing-based cryptography a viable technology. The flexibility of pairings for devising cryptosystems with novel, surprising properties, allied to the possibility of efficiently implementing such systems—as demonstrated by our efforts—gave rise to an exponential growth in the number of published research works in this area over the past several years.

We'd say it's the latter rather than the former. Miller's algorithm was already known but remained as a theoretical tool before the publication of our methods.

As a result of this paper, computer security algorithms and protocols with new, extremely useful features—notably identity-based cryptography—could be implemented in actual practice.

My interest was awakened by the need for extremely compact digital signature schemes on two simultaneous but distinct fronts (watermarking methods for my Ph.D. thesis, and a Brazilian government request for a digital tax payment authentication mechanism subject to severe technical constraints). Ben was the co-author of a paper describing a compact signature scheme based on bilinear pairings; he implemented the scheme to show it was practical and hence was naturally interested in ways to speed up pairings. Hae was my Ph.D. advisor. Mike was driven by a consideration of implementational details and optimizations, and, as he likes to discover by doing, in the course of refining the implementation he found lots of effective optimizations.