Blockchain Consensus — Proof of Work, Difficulty and the 51% Attack

Proof of work: expensive to make, trivial to check

A block header bundles the previous block's hash, a Merkle root committing to every transaction, a timestamp, a difficulty target, and a 32-bit nonce. A block is valid only if double-SHA-256 of that header falls below the target — and there is no shortcut to finding a qualifying nonce beyond brute-force guessing, with expected trials equal to 2²⁵⁶ divided by the target. This asymmetry — hard to produce, instant to verify — is the entire foundation of proof-of-work security: the current Bitcoin network runs at roughly 700 EH/s (7×10²⁰ hashes per second) and consumes on the order of 100 TWh a year purely to keep that puzzle expensive to solve.

Difficulty adjustment: a self-correcting clock

Bitcoin targets a 10-minute average block interval. Every 2,016 blocks — about two weeks — the protocol rescales the target by the ratio of actual elapsed time to the expected 20,160 minutes, capped at a factor of 4 up or 0.25 down per retarget: blocks arriving faster than expected shrink the target (harder puzzle), blocks arriving slower grow it (easier puzzle). This negative feedback loop is remarkably robust — when China banned mining in 2021 and network hash rate dropped 50% overnight, difficulty adjusted down over three retargets and block times stabilised again within about six weeks, with no coordination required beyond the protocol rule itself.

Block valid iff:  SHA256(SHA256(header)) < target
new_target = old_target × (actual_time / expected_time)   [capped ×4 / ×0.25]
P(attacker with fraction q catches up from z blocks) = (q/p)^z   if q < p
  z=6, q=0.10 → P ≈ 0.10^6 ≈ 0.0001%

Byzantine fault tolerance without knowing who's honest

The Byzantine Generals Problem (Lamport, 1982) asks how N distributed parties reach agreement when up to f of them might lie — classical theory needs N ≥ 3f+1 honest participants, which assumes you can count identities. Bitcoin's 2008 breakthrough sidesteps identity entirely: instead of counting votes, it attaches a real computational cost to each one. A block is a "vote" that cost genuine energy to produce, so an attacker needs to control more than 50% of total network hash power to reliably out-race the honest chain — the 51% attack. Nodes follow the longest-chain rule, always treating the chain with the most cumulative proof of work as valid, and this probabilistic consensus converges exponentially fast: the probability an attacker with hash-power fraction q catches up from z blocks behind is (q/p)^z when q < p, so at q = 0.10 and the conventional z = 6 confirmations, the chance of reversal is roughly 0.0001%.

Why honest mining is the rational choice

Mining is ultimately a game-theoretic equilibrium, not just cryptography. Miners earn a block reward (3.125 BTC post-2024 halving, halving again every 210,000 blocks toward a 21-million-coin cap) plus transaction fees, which become the dominant incentive as rewards shrink over time. Sustaining 51% of hash power would cost on the order of $10 billion in hardware and roughly $20 million a day in electricity — far more than any realistic double-spend payoff — so the Nash equilibrium is honest mining: cheating is strictly less profitable than cooperating, which is what lets a trustless, permissionless network run without any central authority. (Not every chain uses this exact mechanism — Ethereum switched to proof-of-stake in 2022, replacing computational cost with staked collateral that gets slashed for misbehaviour, cutting energy use by roughly 99.95% while keeping the same "expensive to attack" logic.)

Frequently asked questions

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Everything above runs in your browser — open Blockchain Consensus, adjust hash rate, difficulty and attacker share, and watch miners race to beat the target while forks resolve into a single longest chain. Nothing is installed, nothing is uploaded.