Delegatability of designated verifier signature

There are only two possible signers in designated verifier signature(DVS) scheme, thus anyone else can not know who is the real signer according to signature/message pairs. Lipmaa et al. discovered delegatability attack on almost all existing designated verifier signature according to original definition of DVS, and later Li et al. subdivided the delegation and defined verifier-only delegatability. Here, we point out the formal definition of verifier-only delegatability is not reasonable and redefine it. Meanwhile we show ZFI DVS scheme is not verifier-only delegatable, but is delegatable, and show ZJ DVS scheme is verifier-only delegatable scheme. We present notion of signer-only delegatability and put forward general construction from verifier-only delegatable DVS scheme to signer-only delegatable DVS scheme, vice verse. We using ZJ DVS scheme as example to show how to construct signer-only delegatable scheme. Finally we classify delegatable DVS scheme into both signer and designated verifier delegatable scheme, verifier-only delegatable scheme and signer-only scheme.