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Discouraging pool block withholding attacks in Bitcoin


The existence of mining pools in Bitcoin enables the miners to gain more stable reward. However, it is proved that the pools are vulnerable for security attacks. A strategic pool manager has strong incentive to launch pool block withholding attack by sending some of her miners to infiltrate the other pools. The infiltrating miners try to find (partial) proof-of-work solutions but discard the solution that can actually create blocks. As it is hard to recognize malicious miners,these miners still get reward in the infiltrated pools. In this work, we revisit the game-theoretic model for pool block withholding attacks and propose a revised approach to reallocate the reward to the miners. Instead of proportionally allocating the reward to all miners, a pool manager deducts a fraction from the reward to award the miner who actually mined the block. Accordingly, we prove that, under our scheme, for any number of mining pools, no-pool-attacks is always a Nash equilibrium. Moreover, with only two minority mining pools, no-pool-attacks is the unique Nash equilibrium

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  1. 1.

  2. 2.

    Bitcoin hashrate distribution. Accessed June 14, 2019.

  3. 3.


  1. Alkalay-Houlihan C, Shah N (2019) The pure price of anarchy of pool block withholding attacks in bitcoin mining. In: The Thirty-Third AAAI Conference on Artificial Intelligence. pages 1724–1731. AAAI Press

  2. Babaioff M , Dobzinski S, Oren S, Zohar A (2012) On bitcoin and red balloons. In: Proceedings of the 13th ACM conference on electronic commerce. pages 56–73. ACM, 2012

  3. Bag S, Ruj S, Sakurai K (2017) Bitcoin block withholding attack: analysis and mitigation. IEEE Trans Inf Forensics Secur 12(8):1967–1978

    Article  Google Scholar 

  4. Bag S, Sakurai K (2016) Yet another note on block withholding attack on bitcoin mining pools. In: Information security: 19th international conference. 9866: 167–180

  5. Carlsten M, Kalodner HA, Matthew-Weinberg S, Narayanan A (2016) On the instability of bitcoin without the block reward. In: Proceedings of the 2016 ACM SIGSAC conference on computer and communications security. pages 154–167

  6. Chen X , Papadimitriou CH, Roughgarden T (2019) An axiomatic approach to block rewards. In: Proceedings of the 1st ACM conference on advances in financial technologies. pages 124–131

  7. Courtois NT, Bahack L (2014) On subversive miner strategies and block withholding attack in bitcoin digital currency. CoRR

  8. Eyal I (2015) The miner’s dilemma. In: Symposium on security and privacy. 89–103

  9. Eyal I, Sirer EG (2014) Bitcoin mining is vulnerable. Commun ACM 61(7):95–102

    Article  Google Scholar 

  10. Haghighat AT, Shajari M (2019) Block withholding game among bitcoin mining pools. Future Gener 97:482–491

    Article  Google Scholar 

  11. Heilman E, Kendler A, Zohar A, Goldberg S (2015) Eclipse attacks on bitcoin’s peer-to-peer network. In: 24th USENIX security symposium. pages 129–144. USENIX Association

  12. Johnson B, Laszka A, Grossklags J, Vasek M, Moore T (2014) Game-theoretic analysis of ddos attacks against bitcoin mining pools. Financl Cryptograph Data Secur FC 8438:72–86

    Google Scholar 

  13. Lewenberg Y, Bachrach Y, Sompolinsky Y, Zohar A, Rosenschein JS (2015) Bitcoin mining pools: a cooperative game theoretic analysis. In: Proceedings of the 2015 international conference on autonomous agents and multiagent systems. pages 919–927. ACM

  14. Luu L, Saha R, Parameshwaran I, Saxena P, Hobor A (2015) On power splitting games in distributed computation: the case of bitcoin pooled mining. IACR Cryptol ePrint Arch 2015:155

    Google Scholar 

  15. Mousavinejad IE, Yang F, Han Q-L, Vlacic Ljubo B (2018) A novel cyber attack detection method in networked control systems. Trans Cybernd 48(11):3254–3264

    Article  Google Scholar 

  16. Nakamoto S (2008) Bitcoin: a peer-to-peer electronic cash system

  17. Rosenfeld M (2011) Analysis of bitcoin pooled mining reward systems. arXiv preprint arXiv:1112.4980, abs/1112.4980

  18. Schrijvers O, Bonneau J, Boneh D, Roughgarden T (2016) Incentive compatibility of bitcoin mining pool reward functions. In: Financial cryptography and data security: 20th international conference. 9603: 477–498

  19. Tosh DK, Shetty S, Liang X, Kamhoua CA, Kwiat KA, Njilla L (2017) Security implications of blockchain cloud with analysis of block withholding attack. In: Proceedings of the 17th IEEE/ACM international symposium on cluster, cloud and grid computing. 458–467. IEEE Computer Society/ACM

  20. Wang Q, Chen Y (2021) The tight bound for pure price of anarchy in an extended miner’s dilemma game. CoRR

  21. Wu D, Liu X-D, Yan X, Peng R, Li G (2019) Equilibrium analysis of bitcoin block withholding attack: a generalized model. Reliab Eng Syst Saf 185:318–328

    Article  Google Scholar 

  22. Yuan Q, Wei P, Jia K, Xue H (2020) Analysis of blockchain protocol against static adversarial miners corrupted by long delay attackers. Sci China Inf Sci 63(3):1

    MathSciNet  Article  Google Scholar 

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This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 61832003, 61872334), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA27000000), the Hong Kong Polytechnic University (Grant No. P0034420).

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Correspondence to Xiaohan Shan.

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Chen, Z., Li, B., Shan, X. et al. Discouraging pool block withholding attacks in Bitcoin. J Comb Optim (2021).

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  • Bitcoin
  • Mining pool
  • Block withholding attack
  • Nash equilibrium