I admit that the name "UDVSP" confuses people as they expect that "UDVSP" provides
"non-transfereability privacy", as defined by Steinfeld et al (in their Asiacrypt 03 paper), meaning that there 
must not be any way to distinguish a real designated verifier and a simulated one. The "UDVSP" does not 
provide this property: it is correct that in a "UDVSP" protocol, the "designated" verifier cannot play the role 
of a prover and convince an honest verifier through the same protocol after some interactions, but the 
"designated" verifier can convince another person, who does not necessarily execute the honest verifier's 
algorithm according to the protocol, that he has been involved in a "UDVSP" protocol as a verifier where 
the prover was successful, and leak information in this way, which contradicts Steinfeld et al's definition of 
non-transferability privacy. - However, we did warn readers in this paper that "UDVSP" does not provide 
explicit authentication of the verifier's identity, which is provided in the non-interactive designated signatures 
in the literature. (Please take a look at the end of Section 1.2.)

To put it another way, the "UDVSP" definitions guarantee that prover's ability to prove knowledge of a 
signature is not transfered to the verifier, however they do not guarantee that the verifier will not have the ability 
to convince a third party that the message has been actually signed by the signer. 

So in our recent ASIACCS paper "Concurrently-Secure Credential Ownership Proofs", we abandon the 
term "UDVSP" but use "Credential Ownership Proof (COP)" instead.