Fast Batched Asynchronous Distributed Key Generation

We present new protocols for threshold Schnorr signatures that work in an asynchronous communication setting, providing robustness and optimal resilience. These protocols provide unprecedented performance in terms of communication and computational complexity. In terms of communication complexity, for each signature, a single party must transmit a few dozen group elements and scalars across the network (independent of the size of the signing committee). In terms of computational complexity, the amortized cost for one party to generate a signature is actually less than that of just running the standard Schnorr signing or verification algorithm (at least for moderately sized signing committees, say, up to 100). For example, we estimate that with a signing committee of 49 parties, at most 16 of which are corrupt, we can generate 50,000 Schnorr signatures per second (assuming each party can dedicate one standard CPU core and 500 Mbs of network bandwidth to signing). Importantly, this estimate includes both the cost of an offline precomputation phase (which just churns out message independent "presignatures") and an online signature generation phase. Also, the online signing phase can generate a signature with very little network latency (just one to three rounds, depending on how throughput and latency are balanced). To achieve this result, we provide two new innovations. One is a new secret sharing protocol (again, asynchronous, robust, optimally resilient) that allows the dealer to securely distribute shares of a large batch of ephemeral secret keys, and to publish the corresponding ephemeral public keys. To achieve better performance, our protocol minimizes public-key operations, and in particular, is based on a novel technique that does not use the traditional technique based on "polynomial commitments". The second innovation is a new algorithm to efficiently combine ephemeral public keys contributed by different parties (some possibly corrupt) into a smaller number of secure ephemeral public keys. This new algorithm is based on a novel construction of a so-called "super-invertible matrix" along with a corresponding highly-efficient algorithm for multiplying this matrix by a vector of group elements. As protocols for verifiably sharing a secret key with an associated public key and the technology of super-invertible matrices both play a major role in threshold cryptography and multi-party computation, our two new innovations should have applicability well beyond that of threshold Schnorr signatures.