Better than Advertised Security for Non-interactive Threshold Signatures

We give a unified syntax, and a hierarchy of definitions of security of increasing strength, for non-interactive threshold signature schemes. These are schemes having a single-round signing protocol, possibly with one prior round of message-independent pre-processing. We fit FROST1 and BLS, which are leading practical schemes, into our hierarchy, in particular showing they meet stronger security definitions than they have been shown to meet so far. We also fit in our hierarchy a more efficient version FROST2 of FROST1 that we give. These definitions and results, for simplicity, all assume trusted key generation. Finally, we prove the security of FROST2 with key generation performed by an efficient distributed key generation protocol.