How to calculate the work in PoW (proof-of-work) in the Bitcoin blockchain

Abstract

When we consider the PoW (proof-of-work) in the Bitcoin blockchain, how is the work calculated? How does this work convert to energy quantities? This paper demonstrates that in the Bitcoin blockchain, "Proof-of-Work" (PoW) is not a complex calculus equation, but rather a probabilistic brute-force search. Miners repeatedly run block header data through a cryptographic hash function, tweaking variables until they output a number that meets a strict network threshold. In the Bitcoin blockchain, Proof-of-Work (PoW) is a probabilistic brute-force search where miners repeatedly run block headers through a double SHA-256 hash function to find an output below a global target threshold. The mathematical "work" is quantified by the network Difficulty (D), requiring roughly D × 2³² expected hashes per block. To convert this cryptographic effort into physical energy, the global network hashrate is first derived by dividing total block hashes by Bitcoin’s 10-minute target block time (600 seconds). This computational rate is then bridged to the physical world using hardware efficiency—measured in Joules per Terahash (J/TH)—multiplied by operational time. Because modern semiconductor ASICs operate roughly seven orders of magnitude above the absolute thermodynamic limits outlined by Landauer's principle, nearly all electricity consumed by this cryptographic pipeline directly converts into waste heat. The calculation of this work, how it translates mathematically to network metrics, and how those metrics convert into physical energy quantities is the discussion of this paper.<b>Part 1: How the "Work" is Calculated</b><b>1. The Hashing Puzzle (Double SHA-256)</b>A miner constructs a block header containing transaction data, a timestamp, the hash of the previous block, and a changing variable called a nonce. They pass this header through the SHA-256 algorithm twice:<br>H(x) = SHA-256(SHA-256(Block Header))The resulting output is a 256-bit unsigned integer, typically represented as a 64-character hexadecimal string.<b>2. The Target (</b><b>T</b><b>)</b>The network enforces a global threshold called the Target (T). For a block to be accepted, the hash output interpreted as a massive 256-bit integer must satisfy:<br>Hash Output ≤ T<br>Because the output of a cryptographic hash function is completely random and uniformly distributed, miners cannot predict the output. Finding a valid hash is essentially a Bernoulli trial (like rolling a die with an astronomical number of sides).<b>3. Mathematical Definition of Difficulty (D)</b>Because the Target T is a massive 256-bit number that changes every 2,016 blocks, Bitcoin uses a human-readable metric called Difficulty (D), scaled relative to a baseline "genesis" target (T_max).<br>T_max = 0x00000000FFFF0000000000000000000000000000000000000000000000000000The difficulty formula is D = T_max/TAs the network gains more miners, T drops (becomes smaller), making hashes harder to find, which increases D.<br>The expected number of hashes E[hashes] required to find a valid block at a given difficulty is proportional to D:E[hashes] = D × 2³² × T/T_max (scaled to baseline expectations)<br>More simply, the total expected hashes per block is roughly:Expected Hashes ≈ D × 4.295 × 10⁹<b>Part 2: From Computational Work to Energy Quantities</b>Energy consumption is a byproduct of hardware efficiency operating over a span of time to execute these hash attempts. There is no direct algorithmic conversion from a hash to Joules in the protocol code; instead, the conversion bridges cryptographic operations and thermodynamic hardware efficiency.<b>Step 1: Calculate Total Network Hashrate (H</b>_{<strong>net</strong>}<b>)</b>The global hashrate represents the total number of hashes computed per second across all active machines globally. It is derived directly from the current difficulty (D) and Bitcoin's target block time (t = 600 seconds or 10 minutes):<br>Hashes per block = D × 2³²<br>Network Hashrate (H_net) = D × 2³²/600 [hashes/second or H/s]<b>Step 2: Factor in Hardware Efficiency (EF)</b>ASIC (Application-Specific Integrated Circuit) miners dominate Bitcoin mining. Their electrical efficiency is measured in Joules per Terahash (J/TH) or Watts per Gigashash. Let the aggregate hardware efficiency of the network be denoted as EF (expressed in Joules per Hash, J/H):EF = Total Power Consumption (Watts)/Hashrate (H/s)<b>Step 3: Energy Derivation Formula</b>To calculate the total energy consumed by the entire Bitcoin network over a specific timeframe (e.g., 1 second, 1 day, or 1 year), we multiply the network hashrate by the hardware efficiency and time (t):<br>Energy (E) = H_net × EF × Δ tSubstituting H_net into the equation:<br>E = (D · 2³²/600) × EF × Δ t<br>For example, assume a network difficulty (D) of roughly 80 × 10¹² (80 trillion). Also, assume an average fleet hardware efficiency (EF) of 25 Joules per Terahash (25 × 10⁻¹² J/H). Calculate energy consumed over 1 day (Δ t = 86,400 seconds):Hashes/sec = 80 × 10¹² × 4,294,967,296/600 ≈ 5.72 × 10²⁰ H/sPower (Watts) = (5.72 × 10²⁰ H/s) × (2.5 × 10⁻¹¹ J/H) ≈ 14,300,000,000 W = 14.3 GWEnergy over 1 day = 14.3 GW × 24 hours ≈ 343.2 GWhThe summary of the conversion pipeline may be expressed as<br>Target (T) ⟶ Difficulty (D) ⟶ Network Hashrate (H_net) ⟶× Hardware Efficiency (J/H)⟶ Power (Watts) ⟶× Time⟶ Energy (Joules/kWh)<b>Part 3: Thermodynamic Limits and Efficiency Bounds (Landauer's Principle)</b>To fully connect cryptographic work to physical energy, we can look at the theoretical minimum energy required by the laws of physics to perform computation.<b>1. Landauer's Principle</b>Landauer's principle establishes the minimum possible amount of energy required to erase or irreversibly manipulate a bit of information at a given temperature (T_temp):<br>E_min = k_<em>B</em> T_temp ln(2)k_<em>B</em> is the Boltzmann constant (1.380649 × 10⁻²³ J/K).T_temp is the absolute temperature of the environment (e.g., 300 K).For a single bit modification at room temperature, this absolute thermodynamic floor is roughly 2.8 × 10⁻²¹ Joules per bit.<b>2. Comparing SHA-256 to the Thermodynamic Limit</b>A single SHA-256 calculation involves processing a 512-bit message block through 64 rounds of complex logical operations (bitwise additions, rotations, and shifts), manipulating hundreds of thousands of bits cumulatively.Theoretical minimum energy per hash: Factoring in the sheer number of bit operations inside SHA-256, even a reversibly ideal computer would require thousands of bit manipulations, putting a strict physical floor on a single hash well above Landauer's limit (roughly on the order of 10⁻¹⁹ to 10⁻¹⁸ Joules per hash under optimal theoretical conditions).Actual ASIC efficiency: Modern state-of-the-art ASIC miners (like the Bitmain Antminer S21 series) operate around 15 to 20 J/TH (1.5 × 10⁻¹¹ Joules per hash).Comparing real-world hardware (10⁻¹¹ J/H) to absolute physical limits (10⁻¹⁸ J/H) reveals that current silicon-based semiconductor technology is roughly 7 orders of magnitude away from theoretical thermodynamic efficiency—meaning nearly all energy put into Bitcoin mining converts directly into waste heat.<b>Part 4: Complete Comprehensive Master Equation</b>Combining all components into a single macro-equation, the total daily electrical energy (E_day) consumed by the global Bitcoin network can be calculated directly from the network's current Difficulty (D) and the average hardware efficiency fleet-wide (EF_avg in J/TH):E_day = (D · 2³²/600) × (EF_avg × 10⁻¹²) × 86,400<br>Where:<br>D · 2³² / 600 yields the Network Hashrate (hashes/sec).EF_avg × 10⁻¹² scales Joules-per-Terahash down to Joules-per-Hash.86,400 converts seconds into one full day.This mathematical coupling ensures that as network security (Difficulty D) scales up over time to attract more capital and hashpower, energy consumption scales linearly with it, modulated only by the parallel improvement rate of semiconductor manufacturing efficiency (EF_avg).To recap the end-to-end framework:The Work: Quantified by the difficulty D and scaled via 2³² to determine total expected hashes per block.The Hashrate: Derived by dividing total hashes per block by the target 10-minute block time (600 seconds).The Energy Conversion: Bridged physically using the hardware's efficiency metric (Joules per Terahash, or J/TH) multiplied over time.The Physical Bound: Bounded by thermodynamic limits like Landauer's principle, explaining why modern ASICs produce the massive amounts of waste heat characteristic of the Bitcoin network.