The cryptographic community has, of late, shown much inventiveness in the creation of powerful new IBE-like primitives that go beyond the basic IBE notion and extend it in many new directions. Virtually all of these “super-IBE” schemes rely on bilinear pairings for their implementation, which they tend to use in a surprisingly small number of different ways: three of them as of this writing.

What is interesting is that, among the three main frameworks that we know of so far, one has acted as a veritable magnet for the construction of many of these “generalized IBE” primitives, whereas the other two have not been nearly as fruitful in that respect. This refers to the Commutative Blinding framework defined by the Boneh-Boyen [Bscr ][Bscr ]₁ IBE scheme from 2004.

The aim of this chapter is to try to shed some light on this approach's popularity, first by comparing its key properties with those of the competing frameworks, and then by providing a number of examples that illustrate how those properties have been used.