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What is SHA-256 (Secure Hash Algorithm 256-bit)?

Ever since the inception (and continued advancements) of network security, encryption and hashing have been the core principles for developing additional security modules. The secure hash algorithm with a digest size of 256 bits, or commonly referred as SHA-256, is one of the most widely used hashing algorithms. While there are other variants, SHA-256 has been at the forefront of real-world applications.

Bitcoin is a decentralized digital currency that uses public distributed ledger called blockchain to register the transactions. The network nodes confirm the authenticity of the transactions using “cryptography”.  Bitcoin applies SHA-256, a cryptographic hashing function that turns random input data into a 256-bit string (called as the ‘Hash’). This is a one-way function and it is easy to find the hash from an input, but the reverse is not possible.

click here to generate your own SHA-256 HASH

Implementation of Grover’s algorithm in finding the target value

In quantum computing, Grover’s search is sometimes presented as an elixir. It finds with a very high probability the unique input (value) to a black box function that produces a particular output. Hence, it is a perfect solution to finding the target value and has a quadratic quantum speedup. The equation is directly proportional to “t” (time in seconds) and “r” (Hash rate) for quantum miners running Grover’s algorithm. Compared with classical success probability of Trt/2256, the success probability of Grover’s quantum algorithm is sin²(2r_q(T/2²⁵⁶)^1/2), where “r_q” is the number of Grover’s iterations per second or the “quantum hash rate”. Solving for r (Hash rate):

r=sin²(tr_qT^1/2/2¹²⁷)2²⁵⁶/tT; where T!=0

There is a different dynamic between the classical and quantum miner because Bitcoin is designed to find a new block on average every 10 minutes (or 600 seconds). Hence, the nature of the search problem changes in this duration. For high probability of success of Grover procedure, quantum miners should run their algorithm for a time “t” before the problem changes, and then make the measurement. Meanwhile, the classical miner, during this period, has been trying as many nonces as possible. So, the quantum miner is hoping that none of the classical miners have found a solution during the Grover evolution. Since the interval between blocks follows an exponential distribution, the probability that the block is still mineable is t/600 e.

Assuming a constant cost of running a quantum computer for a given amount of time, the profitability of quantum bitcoin mining is then:

Re^-t/600sin²(2r_qt(T/2²⁵⁶)^½)-Ct

where “R” is the reward (currently equal to the price of 12.5 bitcoins plus transaction fees) and “C” is the cost of running the quantum computer.

Is quantum bitcoin mining profitable?

Let us do some estimation to determine whether quantum bitcoin mining is profitable. Assume that the cost of running a quantum computer is the same as that of a classical computer. Using the above equations, it can be determined that quantum mining will be possible at a quantum hash rate of 48 kilo-hashes/s, compared with the existing best classical hardware having 125 kilo-hashes/s.

Classical Bitcoin miners can achieve enormous hash rates because the random guess mining algorithm can be quite easily parallelized. The problem is that the quantum advantage does not exceed the factor of (2²⁵⁶/T)½, irrespective of the number of qubits.  Although there is a quantum advantage, it is not insurmountable enough that classical parallelization cannot beat it.  For a quantum computer with a slower hash rate than the minimally profitable 48 kilo-hashes/s, quantum parallelization seems to be necessary.

For example, for a quantum hash rate of 3 kilo-hashes/s, one would require1300 quantum computers to be on par with classical best mining hardware available today. Thus, profitable quantum mining would need rather fast quantum hash rates, and/or a much more significant quantum speedup. This may still happen in the future, but for now, classical mining seems difficult to beat.

Bitcoin mining and electricity

As of August 2021, the leader of the global bitcoin network hash-rate is the USA (35.4%), followed by Kazakhstan (18.1%) and Russia (11%). According to the New York Times, bitcoin mining consumes 0.5% of global electricity annually (refer to figure 2) and is seven times Google’s yearly energy consumption. To mine one bitcoin, an individual miner could take up to 5 years and consume up to 21900 kWh.

Read next part- Bitcoin Blockchain and Quantum Computer – I

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