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Fraud risk assessment within blockchain transactions

Part of: Stochastic processes General field theory

Published online by Cambridge University Press:  07 August 2019

Pierre-O. Goffard
Pierre-O. Goffard*
Affiliation:
University of California, Santa Barbara
*
*Postal address: Institut de Science Financière et d’Assurances, Université Lyon 1, 50 Avenue Tony Garnier, 69007 Lyon, France.
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Abstract

The probability of successfully spending twice the same bitcoins is considered. A double-spending attack consists in issuing two transactions transferring the same bitcoins. The first transaction, from the fraudster to a merchant, is included in a block of the public chain. The second transaction, from the fraudster to himself, is recorded in a block that integrates a private chain, exact copy of the public chain up to substituting the fraudster-to-merchant transaction by the fraudster-to-fraudster transaction. The double-spending hack is completed once the private chain reaches the length of the public chain, in which case it replaces it. The growth of both chains are modelled by two independent counting processes. The probability distribution of the time at which the malicious chain catches up with the honest chain, or, equivalently, the time at which the two counting processes meet each other, is studied. The merchant is supposed to await the discovery of a given number of blocks after the one containing the transaction before delivering the goods. This grants a head start to the honest chain in the race against the dishonest chain.

Keywords

Bitcoin blockchaindouble-spending problemrisk theoryorder statistic point processesrenewal processesboundary crossing problems

MSC classification

Primary: 60G55: Point processes
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 12E10: Special polynomials
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 443 - 467
Copyright
© Applied Probability Trust 2019 

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References

Aigner, M. (2007). A Course in Enumeration (Graduate Texts Math. 238). Springer, Berlin.Google Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities. World Scientific, Hackensack, NJ.CrossRefGoogle Scholar
Borovkov, K. and Burq, Z. (2001). Kendall’s identity for the first crossing time revisited. Electron. Commun. Prob. 6, 91–94.CrossRefGoogle Scholar
Borovkov, K. A. and Dickson, D. C. M. (2008). On the ruin time distribution for a Sparre Andersen process with exponential claim sizes. Insurance Math. Econom . 42, 1104–1108.CrossRefGoogle Scholar
Boucherie, R. J. and Boxma, O. J. (1996). The workload in the M/G/1 queue with work removal. Prob. Eng. Inf. Sci. 10, 261–277.CrossRefGoogle Scholar
Boucherie, R. J., Boxma, O. J. and Sigman, K. (1997). A note on negative customers, GI/G/1 workload, and risk processes. Prob. Eng. Inf. Sci. 11, 305–311.CrossRefGoogle Scholar
Bowden, R. Intersection of the longest increasing subsequences. https://github.com/rhysbowden/LIS.Google Scholar
Bowden, R., Keeler, H. P., Krzesinski, A. E. and Taylor, P. G. (2018). Block arrivals in the bitcoin blockchain. Preprint. Available at https://arxiv.org/abs/1801.07447.Google Scholar
Dimitrova, D. S., Ignatov, Z. G. and Kaishev, V. K. (2017). On the first crossing of two boundaries by an order statistics risk process. Risks 5, 14pp.CrossRefGoogle Scholar
Dimitrova, D. S., Kaishev, V. K. and Zhao, S. (2015). On finite-time ruin probabilities in a generalized dual risk model with dependence. Europ. J. Operat. Res. 242, 134–148.CrossRefGoogle Scholar
Dimitrova, D. S., Kaishev, V. K. and Zhao, S. (2016). On the evaluation of finite-time ruin probabilities in a dependent risk model. Appl. Math. Comput. 275, 268–286.Google Scholar
Eyal, I. and Sirer, E. G. (2014). Majority is not enough: Bitcoin mining is vulnerable. In Financial Cryptography and Data Security: 18th International Conference, Springer, Berlin, pp. 436–454.Google Scholar
Gelenbe, E., Glynn, P. and Sigman, K. (1991). Queues with negative arrivals. J. Appl. Prob. 28, 245–250.CrossRefGoogle Scholar
Göbel, J., Keeler, H. P., Krzesinski, A. E. and Taylor, P. G. (2016). Bitcoin blockchain dynamics: The selfish-mine strategy in the presence of propagation delay. Performance Evaluation 104, 23–41.CrossRefGoogle Scholar
Goffard, P.-O. (2017). Two-sided exit problems in the ordered risk model. Methodology Comput. Appl. Prob. 2pp.Google Scholar
Goffard, P.-O. (2018). Online accompaniment for “Fraud risk assessment within blockchain transactions” . Available at https://github.com/LaGauffre/FraudRiskBlockchainTransaction.Google Scholar
Goffard, P.-O. and Lefèvre, C. (2017). Boundary crossing of order statistics point processes. J. Math. Anal. Appl. 447,890–907.CrossRefGoogle Scholar
Goffard, P.-O. and Lefèvre, C. (2018). Duality in ruin problems for ordered risk models. Insurance Math. Econom . 78,44–52.CrossRefGoogle Scholar
Harrison, P. and Pitel, E. (1996). The M/G/1 queue with negative customers. Adv. Appl. Prob. 28,540–566.CrossRefGoogle Scholar
Ignatov, Z. G. and Kaishev, V. K. (2016). First crossing time, overshoot and Appell–Hessenberg type functions. Stochastics 88,1240–1260.CrossRefGoogle Scholar
Jain, G. and Sigman, K. (1996). Generalizing the Pollaczek-Khintchine formula to account for arbitrary work removal. Prob. Eng. Inf. Sci. 10,519–531.CrossRefGoogle Scholar
Lefèvre, C. and Loisel, S. (2009). Finite-time ruin probabilities for discrete, possibly dependent, claim severities. Methodology Comput. Appl. Prob. 11,425–441.CrossRefGoogle Scholar
Lefèvre, C. and Picard, P. (2011). A new look at the homogeneous risk model. Insurance Math. Econom . 49,512–519.CrossRefGoogle Scholar
Lefèvre, C. and Picard, P. (2014). Ruin probabilities for risk models with ordered claim arrivals. Methodology Comput. Appl. Prob. 16,885–905.CrossRefGoogle Scholar
Lefèvre, C. and Picard, P. (2015). Risk models in insurance and epidemics: A bridge through randomized polynomials. Prob. Eng. Inf. Sci. 29,399–420.CrossRefGoogle Scholar
Mazza, C. and Rullière, D. (2004). A link between wave governed random motions and ruin processes. Insurance Math. Econom . 35,205–222.CrossRefGoogle Scholar
Nakamoto, S. (2008). Bitcoin: A peer-to-peer electronic cash system. Available at https://bitcoin.org/bitcoin.pdf. Google Scholar
Perry, D., Stadje, W. and Zacks, S. (2002). Boundary crossing for the difference of two ordinary or compound Poisson processes. Ann. Operat. Res. 113,119–132.CrossRefGoogle Scholar
Perry, D., Stadje, W. and Zacks, S. (2005). A two-sided first-exit problem for a compound Poisson process with a random upper boundary. Methodology Comput. Appl. Prob. 7,51–62.CrossRefGoogle Scholar
Picard, P. and Lefèvre, C. (2003). On the first meeting or crossing of two independent trajectories for some counting processes. Stoch. Process. Appl. 104,217–242.CrossRefGoogle Scholar
Puri, P. S. (1982). On the characterization of point processes with the order statistic property without the moment condition. J. Appl. Prob. 19,39–51.CrossRefGoogle Scholar
Rosenfeld, M. (2014). Analysis of hashrate-based double spending. Preprint. Available at https://arxiv.org/abs/1402.2009.Google Scholar
Sapirshtein, A., Sompolinsky, Y. and Zohar, A. (2016). Optimal selfish mining strategies in bitcoin. In Financial Cryptography and Data Security: 20th International Conference, Springer, Berlin, pp. 515–532.Google Scholar
Shi, T. and Landriault, D. (2013). Distribution of the time to ruin in some Sparre-Andersen risk models. ASTIN Bull . 43,39–59.CrossRefGoogle Scholar
Van der Hofstad, R. and Keane, M. (2008). An elementary proof of the hitting time theorem. Amer. Math. Monthly 115,753–756.CrossRefGoogle Scholar