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We propose Cornucopia, a protocol framework for distributed randomness beacons combining accumulators and verifiable delay functions. Cornucopia generalizes the Unicorn protocol, using an accumulator to enable efficient verification by each participant that their contribution has been included. The output is unpredictable as long as at least one participant is honest, yielding a scalable distributed randomness beacon with strong security properties. Proving this approach secure requires developing a novel property of accumulators, insertion security, which we show is both necessary and sufficient for Cornucopia-style protocols. We show that not all accumulators are insertion-secure, then prove that common constructions (Merkle trees, RSA accumulators, and bilinear accumulators) are either naturally insertion-secure or can be made so with trivial modifications. 

We consider the problem of secret leader election with accountability. Secret leader election protocols counter adaptive adversaries by keeping the identities of elected leaders secret until they choose to reveal themselves, but in existing protocols this means it is impossible to determine who was elected leader if they fail to act. This opens the door to undetectable withholding attacks, where leaders fail to act in order to slow the protocol or bias future elections in their favor. We formally define accountability (in weak and strong variants) for secret leader election protocols. We present three paradigms for adding accountability, using delay-based cryptography, enforced key revelation, or threshold committees, all of which ensure that after some time delay the result of the election becomes public. The paradigm can be chosen to balance trust assumptions, protocol efficiency, and the length of the delay before leaders are revealed. Along the way, we introduce several new cryptographic tools including re-randomizable timed commitments and timed VRFs. 

Zero-knowledge range proofs (ZKRPs) allow a prover to convince a verifier that a secret value lies in a given interval. ZKRPs have numerous applications: from anonymous credentials and auctions, to confidential transactions in cryptocurrencies. At the same time, a plethora of ZKRP constructions exist in the literature, each with its own trade-offs. In this work, we systematize the knowledge around ZKRPs. We create a classification of existing constructions based on the underlying building techniques, and we summarize their properties. We provide comparisons between schemes both in terms of properties as well as efficiency levels, and construct a guideline to assist in the selection of an appropriate ZKRP for different application requirements. Finally, we discuss a number of interesting open research problems.