[mdpopescu]

Chapter 1 explains the usage of the protocols. Quoting from the book, on page 33:

"The authentication mechanism is such that the receiver not only authenticates the message, but also demonstrates a property of the attributes encoded into its certified key pair. The receiver has full control over which property is demonstrated: it can be any satisfiable proposition from proposition logic, where the atomic propositions are relations that are linear in the encoded attributes. Any other information about the attributes remains unconditionally hidden." (Emphasis mine.)

I will admit that the math was not something I studied in depth (and I definitely didn't check the proofs!), since crypto is at best a hobby for me and not my main job.