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arXiv:2110.00878 · Blockchain Research and Applications

Conditions for advantageous quantum Bitcoin mining

Robert R. Nerem, Daya Ram Gaur

Published Apr 21, 2023• 15 citations
arXivHTMLPDF

Abstract

Our aim is to determine conditions for quantum computing technology to give rise to security risks associated with quantum Bitcoin mining. Specifically, we determine the speed and energy efficiency a quantum computer needs to offer an advantage over classical mining. We analyze the setting in which the Bitcoin network is entirely classical except for a single quantum miner who has a small hash rate compared to that of the network. We develop a closed-form approximation for the probability that the quantum miner successfully mines a block, with this probability dependent on the number of Grover iterations the quantum miner applies before making a measurement. Next, we show that for a quantum miner that is “peaceful”, this success probability is maximized if the quantum miner applies Grover iterations for 16 minutes before measuring, which is surprising as the network mines blocks every 10 minutes on average. Using this optimal mining procedure, we show that the quantum miner outperforms a classical computer in efficiency (cost per block) if the condition Q < Crb is satisfied, where Q is the cost of a Grover iteration, C is the cost of a classical hash, r is the quantum miner's speed in Grover iterations per second, and b is a factor that attains its maximum if the quantum miner uses our optimal mining procedure. This condition lays the foundation for determining when quantum mining, and the known security risks associated with it, will arise.

DOI
10.1016/j.bcra.2023.100141

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Our aim is to determine conditions for quantum computing technology to give rise to security risks associated with quantum Bitcoin mining. Specifically, we determine the speed and energy efficiency a quantum computer needs to offer an advantage over classical mining. We analyze the setting in which the Bitcoin network is entirely classical except for a single quantum miner who has a small hash rate compared to that of the network. We develop a closed-form approximation for the probability that the quantum miner successfully mines a block, with this probability dependent on the number of Grover iterations the quantum miner applies before making a measurement. Next, we show that for a quantum miner that is “peaceful”, this success probability is maximized if the quantum miner applies Grover iterations for 16 minutes before measuring, which is surprising as the network mines blocks every 10 minutes on average. Using this optimal mining procedure, we show that the quantum miner outperforms a classical computer in efficiency (cost per block) if the condition Q < Crb is satisfied, where Q is the cost of a Grover iteration, C is the cost of a classical hash, r is the quantum miner's speed in Grover iterations per second, and b is a factor that attains its maximum if the quantum miner uses our optimal mining procedure. This condition lays the foundation for determining when quantum mining, and the known security risks associated with it, will arise.
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Bitcoin miners equipped with purpose-built quantum computers that execute Grover’s algorithm on classical candidate blocks, measuring the resulting quantum state to yield a potentially valid proof-of-work, pose a threat to Bitcoin’s security against 51% attacks. I aim to assess this threat by employing a game-theoretic framework inspired by the Lee-Ray-Santha quantum races model, extending it to better reflect how quantum miners would behave in practice. These extensions address some of the simplifications of prior works, where quantum miners performed only one quantum measurement (which does not maximise their resources), and were assumed to employ a peaceful strategy, wherein the quantum miners discard the resources already invested into searching upon receiving a broadcast of a newly-mined block (which is not enforceable). I consider a setting in which the quantum miners perform multiple quantum measurements between successive blocks, employ Sattath’s Aggressive Quantum Mining Strategy (AQMS), and must allocate limited quantum resources across these measurements. When employing Sattah’s AQMS, a quantum miner halts their execution of Grover’s algorithm and measures the resulting quantum state upon receiving a broadcast of a newly mined block, attempting to create a temporary fork and thereby decreasing the effective hash rate required for a 51% attack. In this novel setting, I derive the payoff matrices for the quantum miners and compute optimal quantum mining strategies that correspond to Nash equilibria. I simulate the deployment of these optimal quantum mining strategies within the Bitcoin network and estimate their effect on Bitcoin’s security against a 51% attack. I determine how two quantum miners should optimally allocate their quantum resources based on the network difficulty. Overall, I find that even when behaving optimally, two quantum miners cannot create enough temporary forks to render the Bitcoin network vulnerable to this quantum threat, given realistic constraints on their quantum resources. This thesis contributes to the growing body of research on the potential threats posed by scalable quantum computers, using the case of two quantum miners as a step toward understanding how quantum miners in general could affect Proof-of-Work-based cryptocurrencies like Bitcoin.
Bitcoin already faces a quantum threat through Shor attacks on elliptic-curve signatures. This paper isolates the other component that public discussion often conflates with it: mining. Grover's algorithm halves the exponent of brute-force search, promising a quadratic edge to any quantum miner of Bitcoin. Exactly how large that edge grows depends on fault-tolerant hardware. No prior study has costed that hardware end to end. We build an open-source estimator that sweeps the full attack surface: reversible oracles for double-SHA-256 mining and RIPEMD-based address preimages, surface-code factory sizing, fleet logistics under Nakamoto-consensus timing, and Kardashev-scale energy accounting. A parametric sweep over difficulty bits b, runtime caps, and target success probabilities reveals a sharp transition. At the most favourable partial-preimage setting (b = 32, 2^224 marked states), a superconducting surface-code fleet still requires about 10^8 physical qubits and about 10^4 MW. That load is comparable to a large national grid. Tightening to Bitcoin's January 2025 mainnet difficulty (b about 79) explodes the bill to about 10^23 qubits and about 10^25 W, approaching the Kardashev Type II threshold. These numbers settle a narrower question than "Is Bitcoin quantum-secure?" Once Grover mining is lifted from asymptotic query counts to fault-tolerant physical cost, practical quantum mining collapses under oracle, distillation, and fleet overhead. To push mining into non-trivial consensus effects, one must invoke astronomical quantum fleets operating at energy scales that lie far above present-day civilization.
Abstract This paper explores the impact of quantum computing on the efficiency of blockchain mining, focusing on three cryptographic hash algorithms: SHA-256, BLAKE2b, and Keccak-256. Through simulations, we compare the performance of these algorithms in both classical and quantum environments, with a specific emphasis on the effects of increasing qubit counts. The study also examines the potential of Grover’s algorithm to accelerate mining times and improve overall mining efficiency. Our findings indicate significant reductions in mining times when using quantum computing, particularly with the application of Grover’s algorithm. While classical mining techniques are widely used today, the introduction of quantum computing presents both opportunities and challenges for the future of blockchain technology. This paper provides valuable insights into how quantum advancements could reshape blockchain mining, highlighting the need for continued exploration of quantum-resistant cryptographic methods.
The key cryptographic protocols used to secure the internet and financial transactions of today are all susceptible to attack by the development of a sufficiently large quantum computer. One particular area at risk is cryptocurrencies, a market currently worth over 100 billion USD. We investigate the risk posed to Bitcoin, and other cryptocurrencies, by attacks using quantum computers. We find that the proof-of-work used by Bitcoin is relatively resistant to substantial speedup by quantum computers in the next 10 years, mainly because specialized ASIC miners are extremely fast compared to the estimated clock speed of near-term quantum computers. On the other hand, the elliptic curve signature scheme used by Bitcoin is much more at risk, and could be completely broken by a quantum computer as early as 2027, by the most optimistic estimates. We analyze an alternative proof-of-work called Momentum, based on finding collisions in a hash function, that is even more resistant to speedup by a quantum computer. We also review the available post-quantum signature schemes to see which one would best meet the security and efficiency requirements of blockchain applications.
This paper studies the effect of quantum computers on Bitcoin mining. The shift in computational paradigm towards quantum computation allows the entire search space of the golden nonce to be queried at once by exploiting quantum superpositions and entanglement. Using Grover’s algorithm, a solution can be extracted in time O(2256/t), where t is the target value for the nonce. This is better using a square root over the classical search algorithm that requires O(2256/t) tries. If sufficiently large quantum computers are available for the public, mining activity in the classical sense becomes obsolete, as quantum computers always win. Without considering quantum noise, the size of the quantum computer needs to be ≈104 qubits.