Time-based primitives like time-lock puzzles (TLP) are finding widespread use in practical protocols, partially due to the surge of interest in the blockchain space where TLPs and related primitives are perceived to solve many problems. Unfortunately, the security claims are often shaky or plainly wrong since these primitives are used under composition. One reason is that TLPs are inherently not UC secure and time is tricky to model and use in the UC model. On the other hand, just specifying standalone notions of the intended task, left alone correctly using standalone notions like non-malleable TLPs only, might be hard or impossible for the given task. And even when possible a standalone secure primitive is harder to apply securely in practice afterwards as its behavior under composition is unclear. The ideal solution would be a model of TLPs in the UC framework to allow simple modular proofs. In this paper we provide a foundation for proving composable security of practical protocols using time-lock puzzles and related timed primitives in the UC model. We construct UC-secure TLPs based on random oracles and show that using random oracles is necessary. In order to prove security, we provide a simple and abstract way to reason about time in UC protocols. Finally, we demonstrate the usefulness of this foundation by constructing applications that are interesting in their own right, such as UC-secure two-party computation with output-independent abort.

In this paper we present ALBATROSS, a family of multiparty randomness generation protocols with guaranteed output delivery and public verification that allows to trade off corruption tolerance for a much improved amortized computational complexity. Our basic stand alone protocol is based on publicly verifiable secret sharing (PVSS) and is secure under in the random oracle model under the decisional Diffie-Hellman (DDH) hardness assumption. We also address the important issue of constructing Universally Composable randomness beacons, showing two UC versions of Albatross: one based on simple UC NIZKs and another one based on novel efficient ``designated verifier'' homomorphic commitments. Interestingly this latter version can be instantiated from a global random oracle under the weaker Computational Diffie-Hellman (CDH) assumption. An execution of ALBATROSS with $n$ parties, out of which up to $t=(1/2-\epsilon)\cdot n$ are corrupt for a constant $\epsilon>0$, generates $\Theta(n^2)$ uniformly random values, requiring in the worst case an amortized cost per party of $\Theta(\log n)$ exponentiations per random value. We significantly improve on the SCRAPE protocol (Cascudo and David, ACNS 17), which required $\Theta(n^2)$ exponentiations per party to generate one uniformly random value. This is mainly achieved via two techniques: first, the use of packed Shamir secret sharing for the PVSS; second, the use of linear $t$-resilient functions (computed via a Fast Fourier Transform-based algorithm) to improve the randomness extraction.

Homomorphic universally composable (UC) commitments allow for the sender to reveal the result of additions and multiplications of values contained in commitments without revealing the values themselves while assuring the receiver of the correctness of such computation on committed values. In this work, we construct essentially optimal additively homomorphic UC commitments from any (not necessarily UC or homomorphic) extractable commitment, while the previous best constructions require oblivious transfer. We obtain amortized linear computational complexity in the length of the input messages and rate 1. Next, we show how to extend our scheme to also obtain multiplicative homomorphism at the cost of asymptotic optimality but retaining low concrete complexity for practical parameters. Moreover, our techniques yield public coin protocols, which are compatible with the Fiat-Shamir heuristic. These results come at the cost of realizing a restricted version of the homomorphic commitment functionality where the sender is allowed to perform any number of commitments and operations on committed messages but is only allowed to perform a single batch opening of a number of commitments. Although this functionality seems restrictive, we show that it can be used as a building block for more efficient instantiations of recent protocols for secure multiparty computation and zero knowledge non-interactive arguments of knowledge.