Structure-Preserving Signatures (SPSs) are a useful tool for the design of modular cryptographic protocols. Recent series of works have shown that by limiting the message space of those schemes to the set of Diffie-Hellman (DH) pairs, it is possible to circumvent the known lower bounds in the Type-3 bilinear group setting thus obtaining the shortest signatures consisting of only 2 elements from the shorter source group. It has been shown that such a variant yields efficiency gains for some cryptographic constructions, including attribute-based signatures and direct anonymous attestation. Only the cases of signing a single DH pair or a DH pair and a vector from \Zp have been considered. Signing a vector of group elements is required for various applications of SPSs, especially if the aim is to forgo relying on heuristic assumptions. Example applications where it is required to sign a vector of group elements include group, attribute-based and proxy signatures, and k-times anonymous authentication.
An open question is whether such an improved lower bound also applies to signing a vector of ℓ>1 messages. We answer this question negatively for schemes existentially unforgeable under an adaptive chosen-message attack (EUF-CMA) whereas we answer it positively for schemes existentially unforgeable under a random-message attack (EUF-RMA) and those which are existentially unforgeable under a combined chosen-random-message attack (EUF-CMA-RMA). The latter notion is a leeway between the two former notions where it allows the adversary to adaptively choose part of the message to be signed whereas the remaining part of the message is chosen uniformly at random by the signer.
Another open question is whether strongly existentially unforgeable under an adaptive chosen-message attack (sEUF-CMA) schemes with 2-element signatures exist. We answer this question negatively, proving it is impossible to construct sEUF-CMA schemes with 2-element signatures even if the signature consists of elements from both source groups. On the other hand, we prove that sEUF-RMA and sEUF-CMA-RMA schemes with 2-element (unilateral) signatures are possible by giving constructions for those notions.