Classical Block Withholding (BWH) attacks have been extensively studied in block-dependent reward schemes, where pool members are compensated upon a block discovery within the pool. However, most contemporary mining pools operate under share-based schemes, wherein participants are paid immediately upon submission of valid shares. In this paper, we analyze BWH under Pay-Per-Share (PPS) and Full-PPS (FPPS) schemes for Nakamoto-style blockchains and prove that these mechanisms are not incentive compatible -- contrary to claims in prior literature. Under PPS/FPPS, the optimal strategy for a BWH attacker is the All-out Attack (AoA): the adversary allocates its entire hashpower toward the victim pool, submitting only partial Proof-of-Work shares (pPoW) while withholding all valid blocks, i.e., full Proof-of-Work (fPoW). Prior to the first difficulty adjustment, the adversary incurs negligible loss from withheld fPoWs. After the adjustment reduces block difficulty, the adversary either generates more pPoWs per unit time or, if pPoW difficulty is held fixed, earns a higher reward per share, in both cases achieving a relative gain of $\fracα{1-α}$ over pre-adjustment rates, where $α$ is the adversarial hashpower fraction. Honest miners benefit at the same rate as the adversary per unit hashpower, while the victim pool operator bears all losses, paying out-of-pocket for pPoW submissions without receiving fPoW compensation in return. Finally, advanced BWH variants such as Fork After Withholding (FAW) yield no additional profit under PPS/FPPS.
Privacy-preserving cryptocurrency exchanges alter what the pricing mechanism observes about order flow. We derive the unique linear Kyle equilibrium when a committed Bayesian market maker observes order flow perturbed by independent Gaussian privacy noise. The price-impact coefficient and informed-trader strategy rescale by reciprocal factors of the privacy parameter (one down, one up), so their product is invariant. A welfare decomposition then identifies a closed-form per-period transfer from the protocol's LP pool to traders -- the "privacy subsidy", the break-even fee any privacy-aggregated exchange must charge. The result is the single-period closed-form privacy-noise analog of Loss-Versus-Rebalancing (Milionis et al. 2022). The primary application is shielded AMMs with explicit additive-noise injection (e.g., differential privacy); related designs (batched swaps, sealed-bid auctions, oracle-pegged crossings) require separate frameworks that we leave to future work.
We consider the block withholding attacks on pools, more specifically the state-of-the-art Power Adjusting Withholding (PAW) attack. We propose a generalization called Temporary PAW (T-PAW) where the adversary withholds a fPoW from pool mining at most $T$-time even when no other block is mined. We show that PAW attack corresponds to $T\to\infty$ and is not optimal. In fact, the extra reward of T-PAW compared to PAW improves by an unbounded factor as adversarial hash fraction $α$, pool size $β$ and adversarial network influence $γ$ decreases. For example, the extra reward of T-PAW is 22 times that of PAW when an adversary targets a pool with $(α,β,γ)=(0.05,0.05,0)$. We show that honest mining is sub-optimal to T-PAW even when there is no difficulty adjustment and the adversarial revenue increase is non-trivial, e.g., for most $(α,β)$ at least $1\%$ within $2$ weeks in Bitcoin even when $γ=0$ (for PAW it was at most $0.01\%$). Hence, T-PAW exposes a significant structural weakness in pooled mining-its primary participants, small miners, are not only contributors but can easily turn into potential adversaries with immediate non-trivial benefits.
Taki E. M. Abedesselam, Fabio Giacomelli, Francesco Pasquale, Michele Salvi
We study a random process over graphs inspired by the way payments are executed in the Lightning Network, the main layer-two solution on top of Bitcoin. We first prove almost tight upper and lower bounds on the time it takes for a payment failure to occur, as a function of the number of nodes and the edge capacities, when the underlying graph is complete. Then, we show how such a random process is related to the edge-betweenness centrality measure and we prove upper and lower bounds for arbitrary graphs as a function of edge-betweenness and capacity. Finally, we validate our theoretical results by running extensive simulations over some classes of graphs, including snapshots of the real Lightning Network.
Partha S. Dey, Aditya S. Gopalan, Vijay G. Subramanian
We investigate the time to consensus in Nakamoto blockchains. Specifically, we consider two competing growth processes, labeled \emph{honest} and \emph{adversarial}, and determine the time after which the honest process permananetly exceeds the adversarial process. This is done via queueing techniques. The predominant difficulty is that the honest growth process is subject to \emph{random delays}. In a stylized Bitcoin model, we compute the Laplace transform for the time to consensus and verify it via simulation.
Mining blocks in a blockchain using the \textit{Proof-of-Work} consensus protocol involves significant risk, as network participants face continuous operational costs while earning infrequent capital gains upon successfully mining a block. A common risk mitigation strategy is to join a mining pool, which combines the computing resources of multiple miners to provide a more stable income. This article examines a Pay-per-Share (PPS) reward system, where the pool manager can adjust both the share difficulty and the management fee. Using a simplified wealth model for miners, we explore how miners should allocate their computing resources among different mining pools, considering the trade-off between risk transfer to the manager and management fees.
Bitcoin's (BTC) Difficulty Adjustment Algorithm (DAA) has been a source of vulnerability for incentive attacks such as selfish mining, block withholding and coin hopping strategies. In this paper, first, we rigorously study the short-term revenue change per hashpower of the adversarial and honest miners for these incentive attacks. To study the long-term effects, we introduce a new efficiency metric defined as the revenue/cost per hashpower per time for the attacker and the honest miners. Our results indicate that the short-term benefits of intermittent mining strategies are negligible compared to the original selfish mining attack, and in the long-term, selfish mining provides better efficiency. We further demonstrate that a coin hopping strategy between BTC and Bitcoin Cash (BCH) relying on BTC DAA benefits the loyal honest miners of BTC in the same way and to the same extent per unit of computational power as it does the hopper in the short-term. For the long-term, we establish a new boundary between the selfish mining and coin hopping attack, identifying the optimal efficient strategy for each parameter. For block withholding strategies, it turns out, the honest miners outside the pool profit from the attack, usually even more than the attacker both in the short-term and the long-term. Moreover, a Power Adjusting Withholding (PAW) attacker does not necessarily observe a profit lag in the short-term. In other words, even without a difficulty adjustment, a PAW attacker makes profits. It has been long thought that the profit lag of selfish mining is among the main reasons why such an attack has not been observed in practice. We show that such a barrier does not apply to PAW and relatively small pools are at an immediate threat.
Decentralized data-feed systems enable blockchain-based smart contracts to access off-chain information by aggregating values from multiple oracles. To improve accuracy, these systems typically use an aggregation function, such as majority voting, to consolidate the inputs they receive from oracles and make a decision. Depending on the final decision and the values reported by the oracles, the participating oracles are compensated through shared rewards. However, such incentive mechanisms are vulnerable to mirroring attacks, where a single user controls multiple oracles to bias the decision of the aggregation function and maximize rewards. This paper analyzes the impact of mirroring attacks on the reliability and dependability of majority voting-based data-feed systems. We demonstrate how existing incentive mechanisms can unintentionally encourage rational users to implement such attacks. To address this, we propose a new incentive mechanism that discourages Sybil behavior. We prove that the proposed mechanism leads to a Nash Equilibrium in which each user operates only one oracle. Finally, we discuss the practical implementation of the proposed incentive mechanism and provide numerical examples to demonstrate its effectiveness.
In this paper, we present the Tokenized Sovereign Debt Conversion Mechanism (TSDCM), a smart-contracted instrument that, upon meeting both debt-to-GDP and GDP-growth thresholds, automates the retirement of sovereign debt. TSDCM initiates the conversion of a portion of outstanding bonds into performance-linked tokens by integrating a two-state regime-switching jump-diffusion framework into decentralized protocols. We prove finite-time activation and expected debt reduction through new propositions, establish the existence and uniqueness of the underlying stochastic processes, and introduce a main theorem that ensures a strict decline in expected debt levels. With significant tail-risk mitigation, calibration using IMF data and MATLAB Monte Carlo simulations shows a 20-25% decrease in expected debt-to-GDP ratios over a ten-year period. A transparent and incentive-aligned route to sustainable sovereign debt management is provided by TSDCM.
Extreme volatility, nonlinear dependencies, and systemic fragility are characteristics of cryptocurrency markets. The assumptions of normality and centralized control in traditional financial risk models frequently cause them to miss these changes. Four components-volatility stress testing, stablecoin hedging, contagion modeling, and Monte Carlo simulation-are integrated into this paper's modular simulation framework for crypto portfolio risk analysis. Every module is based on mathematical finance theory, which includes stochastic price path generation, correlation-based contagion propagation, and mean-variance optimization. The robustness and practical relevance of the framework are demonstrated through empirical validation utilizing 2020-2024 USDT, ETH, and BTC data.
The cryptocurrency market presents both significant investment opportunities and higher risks relative to traditional financial assets. This study examines the tail behavior of daily returns for two leading cryptocurrencies, Bitcoin and Ethereum, using seven-parameter estimates from prior research, which applied the Generalized Tempered Stable (GTS) distribution. Quantile-quantile (Q-Q) plots against the Normal distribution reveal that both assets exhibit heavy-tailed return distributions. However, Ethereum consistently shows a greater frequency of extreme values than would be expected under its Bitcoin-modeled counterpart, indicating more pronounced tail risk.
Although modern blockchains almost universally produce blocks at fixed intervals, existing models still lack an analytical formula for the loss-versus-rebalancing (LVR) incurred by Automated Market Makers (AMMs) liquidity providers in this setting. Leveraging tools from random walk theory, we derive the following closed-form approximation for the per block per unit of liquidity expected LVR under constant block time: \[ \overline{\mathrm{ARB}}= \frac{\,σ_b^{2}} {\,2+\sqrt{2π}\,γ/(|ζ(1/2)|\,σ_b)\,}+O\!\bigl(e^{-\mathrm{const}\tfracγ{σ_b}}\bigr)\;\approx\; \frac{σ_b^{2}}{\,2 + 1.7164\,γ/σ_b}, \] where $σ_b$ is the intra-block asset volatility, $γ$ the AMM spread and $ζ$ the Riemann Zeta function. Our large Monte Carlo simulations show that this formula is in fact quasi-exact across practical parameter ranges. Extending our analysis to arbitrary block-time distributions as well, we demonstrate both that--under every admissible inter-block law--the probability that a block carries an arbitrage trade converges to a universal limit, and that only constant block spacing attains the asymptotically minimal LVR. This shows that constant block intervals provide the best possible protection against arbitrage for liquidity providers.
Theoretical guarantees for double spending probabilities for the Nakamoto consensus under the $k$-deep confirmation rule have been extensively studied for zero/bounded network delays and fixed mining rates. In this paper, we introduce a ruin-theoretical model of double spending for Nakamoto consensus under the $k$-deep confirmation rule when the honest mining rate is allowed to be an arbitrary function of time including the block delivery periods, i.e., time periods during which mined blocks are being delivered to all other participants of the network. Time-varying mining rates are considered to capture the intrinsic characteristics of the peer to peer network delays as well as dynamic participation of miners such as the gap game and switching between different cryptocurrencies. Ruin theory is leveraged to obtain the double spend probabilities and numerical examples are presented to validate the effectiveness of the proposed analytical method.
Abstract This article introduces a blockchain-based insurance scheme that integrates parametric and collaborative elements. A pool of investors, referred to as surplus providers, locks funds in a smart contract, enabling blockchain users to underwrite parametric insurance contracts. These contracts automatically trigger compensation when predefined conditions are met. The collaborative aspect is embodied in the generation of tokens, which are distributed to surplus providers. These tokens represent each participant’s share of the surplus and grant voting rights for management decisions. The smart contract is developed in Solidity, a high-level programming language for the Ethereum blockchain, and deployed on the Sepolia testnet, with data processing and analysis conducted using Python. In addition, open-source code is provided and main research challenges are identified, so that further research can be carried out to overcome limitations of this first proof of concept.
The paper proposes and implements a methodology to fit a seven-parameter Generalized Tempered Stable (GTS) distribution to financial data. The nonexistence of the mathematical expression of the GTS probability density function makes the maximum likelihood estimation (MLE) inadequate for providing parameter estimations. Based on the function characteristic and the fractional Fourier transform (FRFT), we provide a comprehensive approach to circumvent the problem and yield a good parameter estimation of the GTS probability. The methodology was applied to fit two heavily tailed data (Bitcoin and Ethereum returns) and two peaked data (S\&P 500 and SPY ETF returns). For each index, the estimation results show that the six-parameter estimations are statistically significant except for the local parameter, $μ$. The goodness-of-fit was assessed through Kolmogorov-Smirnov, Anderson-Darling, and Pearson's chi-squared statistics. While the two-parameter geometric Brownian motion (GBM) hypothesis is always rejected, the GTS distribution fits significantly with a very high p-value; and outperforms the Kobol, Carr-Geman-Madan-Yor, and Bilateral Gamma distributions.
We present three very simple variants of the classic Heads or Tails game using chips, each of which contributes to our understanding of the Bitcoin protocol. The first variant addresses the issue of temporary Bitcoin forks, which occur when two miners discover blocks simultaneously. We determine the threshold at which an honest but temporarily ``Byzantine'' miner persists in mining on their fork to save his orphaned blocks. The second variant of Heads or Tails game is biased in favor of the player and helps to explain why the difficulty adjustment formula is vulnerable to attacks of Nakamoto's consensus. We derive directly and in a simple way, without relying on a Markov decision solver as was the case until now, the threshold beyond which a miner without connectivity finds it advantageous to adopt a deviant mining strategy on Bitcoin. The third variant of Heads or Tails game is unbiased and demonstrates that this issue in the Difficulty Adjustment formula can be fully rectified. Our results are in agreement with the existing literature that we clarify both qualitatively and quantitatively using very simple models and scripts that are easy to implement.
Financial markets, particularly cryptocurrency markets, are characterized by high volatility and sudden price jumps, making it essential to develop models that can capture these dynamics effectively. In this paper, our focus lies on the Merton’s jump diffusion model, employing jump processes characterized by the compound Poisson process. Our primary objective is to forecast the drift and volatility of the model using a variety of methodologies. We adopt an approach that involves implementing different drift, volatility, and jump terms within the model through various machine learning techniques, traditional methods, and statistical methods on price-volume data. Additionally, we introduce a path-dependent Monte Carlo simulation to model cryptocurrency prices, taking into account the volatility and unexpected jumps in prices. The results indicate that incorporating jump processes significantly improves forecasting accuracy, especially in volatile markets. Our findings highlight the effectiveness of combining machine learning and traditional methods for more robust predictions.
This paper proposes a novel recursive polynomial commitment scheme (PCS) and a new polynomial interactive oracle proof (PIOP) protocol, which compile into efficient and transparent zk-SNARKs (zero-knowledge succinct non-interactive arguments of knowledge). The Ethereum blockchain utilizes zero-knowledge Rollups (ZKR) to improve its scalability (the ability to handle a large number of transactions), and ZKR uses zk-SNARKs to validate transactions. The currently used zk-SNARKs rely on a trusted setup ceremony, where a group of participants uses secret information about transactions to generate the public parameters necessary to verify the zk-SNARKs. This introduces a security risk into Ethereum's system. Thus, researchers have been developing transparent zk-SNARKs (which do not require a trusted setup), but those are not as efficient as non-transparent zk-SNARKs, so ZKRs do not use them. In this research, I developed LUMEN, a set of novel algorithms that generate transparent zk-SNARKs that improve Ethereum's security without sacrificing its efficiency. Various techniques were creatively incorporated into LUMEN, including groups with hidden orders, Lagrange basis polynomials, and an amortization strategy. I wrote mathematical proofs for LUMEN that convey its completeness, soundness and zero-knowledgeness, and implemented LUMEN by writing around $8000$ lines of Rust and Python code, which conveyed the practicality of LUMEN. Moreover, my implementation revealed the efficiency of LUMEN (measured in proof size, proof computation time, and verification time), which surpasses the efficiency of existing transparent zk-SNARKs and is on par with that of non-transparent zk-SNARKs. Therefore, LUMEN is a promising solution to improve Ethereum's security while maintaining its efficiency.
Abstract Distributed ledgers, including blockchain and other decentralized databases, are designed to store information online where all trusted network members can update the data with transparency. The dynamics of a ledger’s development can be mathematically represented by a directed acyclic graph (DAG). In this paper, we study a DAG model that considers batch arrivals and random delay of attachment. We analyze the asymptotic behavior of this model by letting the arrival rate go to infinity and the inter-arrival time go to zero. We establish that the number of leaves in the DAG, as well as various random variables characterizing the vertices in the DAG, can be approximated by its fluid limit, represented as the solution to a set of delayed partial differential equations. Furthermore, we establish the stable state of this fluid limit and validate our findings through simulations.
Blockchain and other decentralized databases, known as distributed ledgers, are designed to store information online where all trusted network members can update the data with transparency. The dynamics of ledger's development can be mathematically represented by a directed acyclic graph (DAG). One essential property of a properly functioning shared ledger is that all network members holding a copy of the ledger agree on a sequence of information added to the ledger, which is referred to as consensus and is known to be related to a structural property of DAG called one-endedness. In this paper, we consider a model of distributed ledger with sequential stochastic arrivals that mimic attachment rules from the IOTA cryptocurrency. We first prove that the number of leaves in the random DAG is bounded by a constant infinitely often through the identification of a suitable martingale, and then prove that a sequence of specific events happens infinitely often. Combining those results we establish that, as time goes to infinity, the IOTA DAG is almost surely one-ended.
Mohammad M. Jalalzai, Alexander Mozeika, Marcin P. Pawlowski, Ganesh Narayanaswamy
We present Carnot, a leader-based Byzantine Fault Tolerant (BFT) consensus protocol that is responsive and operates under the partially synchronous model. Responsive BFT consensus protocols exhibit wire-speed operation and deliver instantaneous finality, thereby addressing a fundamental need in distributed systems. A key challenge in scaling these protocols has been the computational complexity associated with authenticator verification. We demonstrate that Carnot effectively addresses this bottleneck by adeptly streamlining the verification and aggregation of $O(log(N))$ authenticators per node. This notable advancement marks a substantial improvement over the prevailing $O(N)$ state-of-the-art approaches. Leveraging this inherent property, Carnot demonstrates its capacity to seamlessly scale to networks comprising tens to hundreds of thousands of nodes. We envision Carnot as a critical stride towards bridging the gap between classical BFT consensus mechanisms and blockchain technology.
The practical Byzantine fault tolerant (PBFT) consensus protocol is one of the basic consensus protocols in the development of blockchain technology. At the same time, the PBFT consensus protocol forms a basis for some other important BFT consensus protocols, such as Tendermint, Streamlet, HotStuff, and LibraBFT. In general, the voting nodes may always fail so that they can leave the PBFT-based blockchain system in a random time interval, making the number of timely available voting nodes uncertain. Thus, this uncertainty leads to the analysis of the PBFT-based blockchain systems with repairable voting nodes being more challenging. In this paper, we develop a novel PBFT consensus protocol with repairable voting nodes and study such a new blockchain system using a multi-dimensional Markov process and the first passage time method. Based on this, we provide performance and reliability analysis, including throughput, availability, and reliability, for the new PBFT-based blockchain system with repairable voting nodes. Furthermore, we provide an approximate algorithm for computing the throughput of the new PBFT-based blockchain system. We employ numerical examples to demonstrate the validity of our theoretical results and illustrate how the key system parameters influence performance measures of the PBFT-based blockchain system with repairable voting nodes. We hope the methodology and results developed in this paper will stimulate future research endeavors and open up new research trajectories in this field.