Abstract. Although they have been studied for a long time, distributed signature protocols have garnered renewed interest in recent years in view of novel applications
to topics like blockchains. Most recent works have focused on distributed versions of
ECDSA or variants of Schnorr signatures; however, and in particular, little attention has
been given to constructions based on post-quantum secure assumptions like the hardness of lattice problems. A few lattice-based threshold signature and multi-signature
schemes have been proposed in the literature, but they either rely on hash-and-sign
lattice signatures (which tend to be comparatively inefficient), use expensive generic transformations, or only come with incomplete security proofs. In this paper, we
construct several lattice-based distributed signing protocols with low round complexity
following the Fiat–Shamir with Aborts (FSwA) paradigm of Lyubashevsky (Asiacrypt
2009). Our protocols can be seen as distributed variants of the fast Dilithium-G signature scheme and the full security proof can be made assuming the hardness of module
SIS and LWE problems. A key step to achieving security (unexplained in some earlier papers) is to prevent the leakage that can occur when parties abort after their first
message—which can inevitably happen in the Fiat–Shamir with Aborts setting. We
manage to do so using homomorphic commitments. Exploiting the similarities between
FSwA and Schnorr-style signatures, our approach makes the most of observations from
recent advancements in the discrete log setting, such as Drijvers et al.’s seminal work
on two-round multi-signatures (S&P 2019). In particular, we observe that the use of
commitment not only resolves the subtle issue with aborts, but also makes it possible
to realize secure two-round n-out-of-n distributed signing and multi-signature in the
plain public key model, by equipping the commitment with a trapdoor feature. The construction of suitable trapdoor commitment from lattices is a side contribution of this
paper.
