Double Spend Races

Abstract

We correct the double spend race analysis given in Nakamoto's foundational Bitcoin article and give a closed-form formula for the probability of success of a double spend attack using the Regularized Incomplete Beta Function. We give a proof of the exponential decay on the number of confirmations, often cited in the literature, and find an asymptotic formula. Larger number of confirmations are necessary compared to those given by Nakamoto. We also compute the probability conditional to the known validation time of the blocks. This provides a finer risk analysis than the classical one.To the memory of our beloved teacher André Warusfel who taught us how to have fun with the applications of mathematics.Date: February 9th 2017. 2010 Mathematics Subject Classification. 68M01, 60G40, 91A60, 33B20.

The Mathematics of Bitcoin

“…Rosenfeld observes in [4] that the negative binomial variable seems to be a better approximation. We proved in [3] that this was indeed the case and we could find the exact formula for the double spend probability after z confirmations (z is the classical notation used by Nakamoto for the number of confirmations). where I x (a, b) is the incomplete regularized beta function…”

“…(1−λ) 3 We can plot the parameter regions where each startegy is the best one ( Figure 3). The Catalan numbers appear naturally in the computations.…”

“…[3], 2017). After z confirmations, the probability of success of a double spend by attackers with a relative hashrate of 0 < q < 1/2 is P(z) = I 4pq (z, 1/2)…”

On Cyber Risk Management of Blockchain Networks: A Game Theoretic Approach

“…For the transaction-oriented cyber services built upon permissionless blokchains, the designer of cyber-insurance also faces the similar issues. Nevertheless, recent studies on the mechanisms of doublespending attacks [25], [26], [27] have shed light upon the possible approaches in analytically assessing the risks of this fundamental threat on the blockchain systems. For example, in [25], the authors proposed a new protocol which requires the consensus nodes, i.e., block miners, to confirm transactions only if the inputs of the transactions have not been spent, hence preventing users from double-spending their funds.…”

“…For example, in [25], the authors proposed a new protocol which requires the consensus nodes, i.e., block miners, to confirm transactions only if the inputs of the transactions have not been spent, hence preventing users from double-spending their funds. Based on the characteristics of intentional forking in double-spending attacks, the authors in [26], [27] introduced the methods to estimate the probability of successful double-spending attacks by analyzing the investments in computing resource of both the blockchain maintainers and attackers. With these studies, it is now possible to estimate the probability of successful double spending and evaluate the potential risks transferred to the cyber-insurers.…”