Universally Composable Interactive and Ordered Multi-signatures

Multi-signatures allow a given set of parties to cooperate in order to create a digital signature whose size is independent of the number of signers. At the same time, no other set of parties can cre ate such a signature. While non-interactive multi-signatures are known (e.g. BLS from pairings), many popular multi-signature schemes such as MuSig2 (which are constructed from pairing-free discrete logarithm-style assumptions) require interaction. Such interactive multi-signatures have recently found practical applications e.g. in the cryptocurrency space.
Motivated by classical and emerging use cases of such interactive multi-signatures, we introduce the first systematic treatment of inter active multi-signatures in the universal composability (UC) framework. Along the way, we revisit existing game-based security notions and prove that constructions secure in the game-based setting can easily be made UC secure and vice versa.
In addition, we consider interactive multi-signatures where the sign ers must interact in a fixed pattern (so-called ordered multi-signatures). Here, we provide the first construction of ordered multi-signatures based on the one-more discrete logarithm assumption, whereas the only other previously known construction required pairings. Our scheme achieves a stronger notion of unforgeability, guaranteeing that the adversary can not obtain a signature altering the relative order of honest signers. We also present the first formalization of ordered multi-signatures in the UC framework and again show that our stronger game-based definitions are equivalent to UC security.