Payment channels are the most prominent solution to the blockchain scalability problem. We introduce the problem of network design with fees for payment channels from the perspective of a Payment Service Provider (PSP). Given a set of transactions, we examine the optimal graph structure and fee assignment to maximize the PSP's profit. A customer prefers to route transactions through the PSP's network if the cheapest path from sender to receiver is financially interesting, i.e., if the path costs less than the blockchain fee. When the graph structure is a tree, and the PSP facilitates all transactions, the problem can be formulated as a linear program. For a path graph, we present a polynomial time algorithm to assign optimal fees. We also show that the star network, where the center is an additional node acting as an intermediary, is a near-optimal solution to the network design problem.Keywords: blockchain · layer 2 · channels · lightning protocol 1 IntroductionScaling the transaction throughput on blockchain systems, such as Bitcoin [12] and Ethereum [1], is a fundamental problem and an active research direction [4]. Many solutions have been proposed, in particular sharding [8,10], sidechains [3] and channels [5,13,2,9]. Channels seem to be the most promising solution since they allow transactions to occur securely off-chain, and use the blockchain only for resolving disputes. We study the problem from the viewpoint of a Payment Service Provider (PSP). The PSP wants to establish an alternative payment network for customers to execute transactions. We assume a PSP can open a channel between two parties without acting as an intermediate node; this can be done using three-party channels. The two parties and the PSP join a three-party channel funded only by the PSP who then loans money to the other parties. We assume that the PSP will eventually get his money back in fiat currency as he provides a service similar to credit cards (the risk lies to the PSP). Furthermore, the PSP signs each new state if and only if the fees have the correct value. This way he enforces the fee assignment on the channels. Initially, a PSP will compete with the blockchain: customers only prefer the alternative network if the total fees cost less than the blockchain. We introduce the network design problem for the PSP, whose goal is to decide the graph structure and the fee assignments in order to maximize its profit.

The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on highdimensional spaces and present a new randomized algorithm which efficiently computes approximate rnets with respect to Euclidean distance. For any fixed > 0, the approximation factor is 1 + and the complexity is polynomial in the dimension and subquadratic in the number of points. The algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSHbased) construction of Eppstein et al. [EHS15] in terms of complexity, by reducing the dependence on , provided that is sufficiently small. Our method does not require LSH but, instead, follows Valiant's [Val15] approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the (1 + )-approximate kth nearest neighbor distance in time subquadratic in the size of the input.

We prove Bitcoin is secure under temporary dishonest majority. We assume the adversary can corrupt a specific fraction of parties and also introduce crash failures, i.e., some honest participants are offline during the execution of the protocol. We demand a majority of honest online participants on expectation. We explore three different models and present the requirements for proving Bitcoin's security in all of them: we first examine a synchronous model, then extend to a bounded delay model and last we consider a synchronous model that allows message losses.

Payment networks were introduced to address the limitation on the transaction throughput of popular blockchains. To open a payment channel one has to publish a transaction on-chain and pay the appropriate transaction fee. A transaction can be routed in the network, as long as there is a path of channels with the necessary capital. The intermediate nodes on this path can ask for a fee to forward the transaction. Hence, opening channels, although costly, can benefit a party, both by reducing the cost of the party for sending a transaction and by collecting the fees from forwarding transactions of other parties. This trade-off spawns a network creation game between the channel parties. In this work, we introduce the first game theoretic model for analyzing the network creation game on blockchain payment channels. Further, we examine various network structures (path, star, complete bipartite graph and clique) and determine for each one of them the constraints (fee value) under which they constitute a Nash equilibrium, given a fixed fee policy. Last, we show that the star is a Nash equilibrium when each channel party can freely decide the channel fee. On the other hand, we prove the complete bipartite graph can never be a Nash equilibrium, given a free fee policy.

Sharding distributed ledgers is the most promising on-chain solution for scaling blockchain technology. In this work, we define and analyze the properties a sharded distributed ledger should fulfill. More specifically, we show that a sharded blockchain cannot be scalable under a fully adaptive adversary, but it can scale up to O(n/logn) under an epoch-adaptive adversary. This is possible only if the distributed ledger employs a checkpoint process at the end of each epoch. Our model builds upon and extends the Bitcoin backbone protocol by defining consistency and scalability. Consistency encompasses the need for atomic execution of cross-shard transactions to preserve safety, whereas scalability encapsulates the speedup a sharded system can gain in comparison to a non-sharded system. Lastly, we analyze existing sharded blockchains and either show their correctness (OmniLedger, RapidChain) under our model or pinpoint where they fail to balance the consistency and scalability requirements (Elastico, Monoxide).

Off-chain protocols (channels) are a promising solution to the scalability and privacy challenges of blockchain systems. Current proposals, however, require synchrony assumptions to preserve the safety of a channel, leaking to an adversary the exact amount of time needed to control the network for a successful attack. In this paper, we introduce Brick, the first off-chain construction that remains secure under network asynchrony and concurrently provides correct incentives. The core idea is to incorporate the conflict resolution process within the off-chain channel by introducing a rational watchtower committee. Hence, if a party wants to close a channel unilaterally, it can only get the committee's approval for the last valid state. Brick provides sub-second latency because it does not employ heavy-weight consensus. Instead, Brick uses consistent broadcast to announce updates and close the channel, a light-weight abstraction that is powerful enough to preserve safety and liveness to honest parties. Furthermore, we consider the permissioned model of blockchains, where the additional property of auditability might be desired for regulatory purposes. We introduce Brick+, an off-chain construction that provides auditability on top of Brick without conflicting with its privacy guarantees. We formally define the properties our state channel construction should fulfill, and prove that both Brick and Brick+ satisfy them. We also design incentives for Brick such that honest and rational behavior aligns for both the committee and the parties of a channel.1 This censoring ability is encompassed by the chain-quality property [19] of blockchain systems which is rightly bound to the synchrony of the network. 2 Liveness states that a transaction will be eventually included on-chain as long as it is provided to all honest parties for a long enough time period [19].

Payment networks, also known as channels, are a most promising solution to the throughput problem of cryptocurrencies. In this paper we study the design of capital-efficient payment networks, offline as well as online variants. We want to know how to compute an efficient payment network topology, how capital should be assigned to the individual edges, and how to decide which transactions to accept. Towards this end, we present a flurry of interesting results, basic but generally applicable insights on the one hand, and hardness results and approximation algorithms on the other hand.

Off-chain protocols constitute one of the most promising approaches to solve the inherent scalability issue of blockchain technologies. The core idea is to let parties transact on-chain only once to establish a channel between them, leveraging later on the resulting channel paths to perform arbitrarily many peer-to-peer transactions off-chain. While significant progress has been made in terms of proof techniques for offchain protocols, existing approaches do not capture the game-theoretic incentives at the core of their design, which led to overlooking significant attack vectors like the Wormhole attack in the past. This work introduces the first game-theoretic model that is expressive enough to reason about the security of off-chain protocols. We advocate the use of Extensive Form Games -EFGs and introduce two instances of EFGs to capture security properties of the closing and the routing of the Lightning Network. Specifically, we model the closing protocol, which relies on punishment mechanisms to disincentivize the uploading on-chain of old channel states, as well as the routing protocol, thereby formally characterizing the Wormhole attack, a vulnerability that undermines the fee-based incentive mechanism underlying the Lightning Network.