I am also puzzled by this example of utility. It would be easy for the identity provider to separately sign all properties, assign each a unique identifier, and then return them to the users to use the signed identity properties as needed.
* The actors are: Individuals I_1, ..., I_n, businesses B_1, ..., B_m, and a central authority A.
* An individual I_j wants to prove to business B_k that A attests that DateOfBirth(I_j) >= 20060902, and that I_j is in possession of a private key, where A attests that it has verified the linkage of the corresponding public key to I_j.
* I_j doesn't want to provide any information about I_j's identity except for the DateOfBirth(I_j) >= 20060902 to B_k. That means, for example, I_j doesn't want to reveal to B_k their ordinal j, nor a single public key that is used everywhere. This means, for example, B_k shouldn't be able to collude with B_{k+1} to combine facts separately provided to the two businesses and build a profile of I_j.
* I_j also doesn't want A to be able to collect information about the fact they provided information to B_k specifically.
With a ZKP, it is possible for a solution like:
* I_j generates a keypair P_1 (private) / p_1 (public) and proves their identity out-of-band to A. A gives them a certificate C_1 typing p_1 to their ordinal j and their date of birth.
* I_j generates a new keypair P_2/p_2 just for dealing with B_k.
* I_j generates a signed certificate C_2 using P_1, tying p_2 to their ordinal j.
* I_j generates a ZKP that there exists a (private input) certificate C_1 signed by A's public key, and that certificate meets the constraint DateOfBirth(I_j) >= 20060902, and there exists a (private input) public key p_1 which is referenced in C_1, and there exists a (private input) certificate C_2 signed by p_1, and that certificate references (public input) public key p_2, and sends the ZKP to B_k.
Now instead you could imagine a solution where A generates certificates for I_j, but that has some downsides:
* The properties to be signed might vary over time. The date of birth cutoff certainly would, and different businesses might want different properties.
* Just one certificate per property isn't enough, because the certificate identifies which public key it relates to. That allows B_k and B_{k+1} to work out they have the same customer I_j. With the ZKP solution, the customer gives a different public key to each. You could work around this by having A provide lots of certificates upfront (inefficient), or by generating certificates on demand (but the A is needed to be involved online in the transaction, and it risks leaking information to A).
So the ZKP solution is, in many ways, simpler in that it removes a lot of constraints while implementing the desired properties, but there are other workarounds if you don't have it.
