09-15-2026, 04:47 PM
The article “The Mathematics of Bitcoin” by Cyril Grunspan and Ricardo Pérez-Marco examines Bitcoin primarily as a mathematical and probabilistic system rather than as a financial asset. It explains how cryptographic hashing and Proof of Work create a decentralized consensus mechanism, then models Bitcoin mining using Poisson processes, exponential and Gamma distributions, Markov chains, martingales, and probability theory. A central problem is the “double-spend” attack: an attacker attempts to secretly build an alternative blockchain and replace confirmed transactions.
The authors derive an exact formula for the probability of such an attack succeeding after a given number of confirmations, improving on an approximation used in Satoshi Nakamoto’s original paper. They prove that, when the attacker controls less than half of the network's computing power, the probability of success decreases exponentially as the number of confirmations increases.
The second major theme is the mathematical stability and profitability of Bitcoin mining strategies. The authors show that, without Bitcoin’s periodic difficulty adjustment, the mathematically optimal strategy is simply to publish newly mined blocks immediately. They then analyze selfish mining and several more aggressive block-withholding strategies using Markov chains, martingale methods, Catalan numbers, and Dyck words. These methods reveal that Bitcoin’s difficulty-adjustment mechanism is an important potential weakness because it can make certain dishonest strategies profitable after a delay.
Finally, the paper revisits double-spending from an economic perspective and shows how an attacker must balance the probability of success against the rewards and risks of continuing the attack. Overall, the article demonstrates that Bitcoin provides a surprisingly rich application of modern probability, stochastic processes, combinatorics, and mathematical optimization.
ARTICLE [PDF]
The authors derive an exact formula for the probability of such an attack succeeding after a given number of confirmations, improving on an approximation used in Satoshi Nakamoto’s original paper. They prove that, when the attacker controls less than half of the network's computing power, the probability of success decreases exponentially as the number of confirmations increases.
The second major theme is the mathematical stability and profitability of Bitcoin mining strategies. The authors show that, without Bitcoin’s periodic difficulty adjustment, the mathematically optimal strategy is simply to publish newly mined blocks immediately. They then analyze selfish mining and several more aggressive block-withholding strategies using Markov chains, martingale methods, Catalan numbers, and Dyck words. These methods reveal that Bitcoin’s difficulty-adjustment mechanism is an important potential weakness because it can make certain dishonest strategies profitable after a delay.
Finally, the paper revisits double-spending from an economic perspective and shows how an attacker must balance the probability of success against the rewards and risks of continuing the attack. Overall, the article demonstrates that Bitcoin provides a surprisingly rich application of modern probability, stochastic processes, combinatorics, and mathematical optimization.
ARTICLE [PDF]
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