Additive Proofs of Knowledge - A New Notion For Non-Interactive Proofs

In this paper, we study the opacity property of verifiably encrypted signatures (VES) of Boneh et al. (proposed in Eurocrypt 2003). Informally, opacity implies that although some given aggregate signatures can verified, no useful information about the individual signatures is leaked. However, the very fact that an aggregate signature can be verified leaks certain information - that the individual signature is indeed well-formed. Apart from this, is there any other information leaked? In this paper, we show that there is absolutely no other information leaked about the individual signatures when the aggregation contains only two signatures. In more formal terms, we show that VES are Zero-Knowledge (ZK). We then extend the ZK property of VES to propose efficient Additive Non-Interactive Witness-Indistinguishable (A-NIWI) proofs. Intuitively an A-NIWI proof can be considered as a Proof of Knowledge (PoK) of another A-NIWI proof