Abstract. We introduce and formally define polynomial commitment schemes, and provide two efficient constructions. A polynomial commitment scheme allows a committer to commit to a polynomial with a short string that can be used by a verifier to confirm claimed evaluations of the committed polynomial. Although the homomorphic commitment schemes in the literature can be used to achieve this goal, the sizes of their commitments are linear in the degree of the committed polynomial. On the other hand, polynomial commitments in our schemes are of constant size (single elements). The overhead of opening a commitment is also constant; even opening multiple evaluations requires only a constant amount of communication overhead. Therefore, our schemes are useful tools to reduce the communication cost in cryptographic protocols. On that front, we apply our polynomial commitment schemes to four problems in cryptography: verifiable secret sharing, zero-knowledge sets, credentials and content extraction signatures.

The decentralized currency network Bitcoin is emerging as a potential new way of performing financial transactions across the globe. Its use of pseudonyms towards protecting users' privacy has been an attractive feature to many of its adopters. Nevertheless, due to the inherent public nature of the Bitcoin transaction ledger, users' privacy is severely restricted to linkable anonymity, and a few transaction deanonymization attacks have been reported thus far.In this paper we propose CoinShuffle, a completely decentralized Bitcoin mixing protocol that allows users to utilize Bitcoin in a truly anonymous manner. CoinShuffle is inspired by the accountable anonymous group communication protocol Dissent and enjoys several advantages over its predecessor Bitcoin mixing protocols. It does not require any (trusted, accountable or untrusted) third party and it is perfectly compatible with the current Bitcoin system. CoinShuffle introduces only a small communication overhead for its users, while completely avoiding additional anonymization fees and minimalizing the computation and communication overhead for the rest of the Bitcoin system.

Permissionless blockchains protocols such as Bitcoin are inherently limited in transaction throughput and latency. Current efforts to address this key issue focus on off-chain payment channels that can be combined in a Payment-Channel Network (PCN) to enable an unlimited number of payments without requiring to access the blockchain other than to register the initial and final capacity of each channel. While this approach paves the way for low latency and high throughput of payments, its deployment in practice raises several privacy concerns as well as technical challenges related to the inherently concurrent nature of payments that have not been sufficiently studied so far.In this work, we lay the foundations for privacy and concurrency in PCNs, presenting a formal definition in the Universal Composability framework as well as practical and provably secure solutions. In particular, we present Fulgor and Rayo. Fulgor is the first payment protocol for PCNs that provides provable privacy guarantees for PCNs and is fully compatible with the Bitcoin scripting system. However, Fulgor is a blocking protocol and therefore prone to deadlocks of concurrent payments as in currently available PCNs. Instead, Rayo is the first protocol for PCNs that enforces non-blocking progress (i.e., at least one of the concurrent payments terminates). We show through a new impossibility result that non-blocking progress necessarily comes at the cost of weaker privacy. At the core of Fulgor and Rayo is Multi-Hop HTLC, a new smart contract, compatible with the Bitcoin scripting system, that provides conditional payments while reducing running time and communication overhead with respect to previous approaches. Our performance evaluation of Fulgor and Rayo shows that a payment with 10 intermediate users takes as few as 5 seconds, thereby demonstrating their feasibility to be deployed in practice.

Tremendous growth in cryptocurrency usage is exposing the inherent scalability issues with permissionless blockchain technology. Payment-channel networks (PCNs) have emerged as the most widely deployed solution to mitigate the scalability issues, allowing the bulk of payments between two users to be carried out off-chain. Unfortunately, as reported in the literature and further demonstrated in this paper, current PCNs do not provide meaningful security and privacy guarantees [30], [40]. In this work, we study and design secure and privacypreserving PCNs. We start with a security analysis of existing PCNs, reporting a new attack that applies to all major PCNs, including the Lightning Network, and allows an attacker to steal the fees from honest intermediaries in the same payment path. We then formally define anonymous multi-hop locks (AMHLs), a novel cryptographic primitive that serves as a cornerstone for the design of secure and privacy-preserving PCNs. We present several provably secure cryptographic instantiations that make AMHLs compatible with the vast majority of cryptocurrencies. In particular, we show that (linear) homomorphic one-way functions suffice to construct AMHLs for PCNs supporting a script language (e.g., Ethereum). We also propose a construction based on ECDSA signatures that does not require scripts, thus solving a prominent open problem in the field. AMHLs constitute a generic primitive whose usefulness goes beyond multi-hop payments in a single PCN and we show how to realize atomic swaps and interoperable PCNs from this primitive. Finally, our performance evaluation on a commodity machine finds that AMHL operations can be performed in less than 100 milliseconds and require less than 500 bytes of communication overhead, even in the worst case. In fact, after acknowledging our attack, the Lightning Network developers have implemented our ECDSA-based AMHLs into their PCN. This demonstrates the practicality of our approach and its impact on the security, privacy, interoperability, and scalability of today's cryptocurrencies.

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