Bulletproofs++: Next Generation Confidential Transactions via Reciprocal Set Membership Arguments

Zero-knowledge proofs are a cryptographic cornerstone of privacy-preserving technologies such as Confidential Transactions (CT), which aims at hiding monetary amounts in cryptocurrency transactions. Due to its asymptotically logarithmic proof size and transparent setup, most state-of-the-art CT protocols use the Bulletproofs (BP) zero-knowledge proof system for set membership proofs such as range proofs. However, even taking into account recent efficiency improvements, BP comes with a serious overhead in terms of concrete proof size as well as verifier running time and thus puts a large burden on practical deployments of CT and its extensions. In this work, we introduce Bulletproofs++ (BP++), a drop-in replacement for BP that improves its concrete efficiency and compactness significantly. As for BP, the security of BP++ relies only on the hardness of the discrete logarithm problem in the random oracle model, and BP++ retains all features of Bulletproofs including transparent setup and support for proof aggregation, multi-party proving and batch verification. Asymptotically, BP++ range proofs require only $O(n / \log n)$ group scalar multiplications compared to $O(n)$ for BP and BP+. At the heart of our construction are novel techniques for permutation and set membership, which enable us to prove statements encoded as arithmetic circuits very efficiently. Concretely, a single BP++ range proof to establish that a committed value is in a 64-bit range (as commonly required by CT) is just 416 bytes over a 256-bit elliptic curve, 38\% smaller than an equivalent BP and 27\% smaller than BP+. When instantiated using the secp256k1 curve as used in Bitcoin, our benchmarks show that proving is about 5 times faster than BP and verification is about 3 times faster than BP. When aggregating 32 range proofs, proving and verification are about 9.5 times and 5.5 times faster, respectively

An Empirical Analysis of Privacy in the Lightning Network

Payment channel networks, and the Lightning Network in particular, seem to offer a solution to the lack of scalability and privacy offered by Bitcoin and other blockchain-based cryptocurrencies. Previous research has focused on the scalability, availability, and crypto-economics of the Lightning Network, but relatively little attention has been paid to exploring the level of privacy it achieves in practice. This paper presents a thorough analysis of the privacy offered by the Lightning Network, by presenting several attacks that exploit publicly available information about the network in order to learn information that is designed to be kept secret, such as how many coins a node has available or who the sender and recipient are in a payment routed through the network.Comment: 26 pages, 5 figure