Sybil attacks, 51% attacks & economic security

Identity itself is free in every case — the difference is entirely in what backs it. Fake keypairs cost nothing; fake hash power and fake stake don't exist, so the attacker has to buy the real thing.

Creating fake identities costs nothing — that bar sits at zero no matter how many you generate. The bars that actually cost something are the ones backed by a real, scarce resource: hash power you have to buy and run, or stake you have to actually own. Identity is free; influence isn't.

1. A Sybil attack is fake identities, not fake resources

   Generating $n$ keypairs costs $\text{sybilCost}(n) = n \times \text{costPerIdentity}$. In a naive peer-to-peer network, $\text{costPerIdentity} = 0$, so this is $0$ regardless of $n$ — a thousand "nodes" for free.

2. Proof-of-work and proof-of-stake price influence, not identity count

   $\text{attackCost}(\text{share}, \text{totalValue}) = \text{share} \times \text{totalValue}$. It doesn't matter whether an attacker splits their acquired resource across 1 identity or 10,000 — the cost only depends on how much of the total resource they end up controlling.

3. The 51% threshold is where an attacker can start overriding the honest majority

   Below 50% share, honest nodes' combined resource still outweighs the attacker's on every rule that relies on majority agreement (longest-chain, majority stake-weighted voting). At or above it, the attacker's resource can start winning those votes.

4. A security budget turns 'expensive' into 'deterred'

   An attack is priced out once its cost exceeds what the attacker is rationally willing to spend — not because the attack becomes impossible, but because it stops being worth it.

• share — the fraction of the network's total resource (hash power or stake) an attacker needs.
• totalValue — the resource's total value across the whole honest network; a bigger network is a more expensive target.

The cost is linear in share and independent of how the attacker packages it — one identity holding 30% of the resource costs exactly the same as a thousand identities splitting it.

Slide the attacker's targeted share and watch both bars grow linearly — the cost formula doesn't care what mechanism backs the resource, only how much of it there is in total and what fraction is being bought.

Fixed scenario: total proof-of-work resource value $1,000,000; total staked value $2,000,000.

1. A Sybil attack, priced

   1,000 fake identities in a naive P2P network: sybilCost(1000) = 1000 × 0, which is $0.

2. A 51% proof-of-work attack

   attackCost(0.51, 1000000) $= 0.51 \times 1{,}000{,}000 = 510{,}000$, i.e. $510,000, against the $1,000,000 proof-of-work total.

3. A 51% proof-of-stake attack

   attackCost(0.51, 2000000) $= 0.51 \times 2{,}000{,}000 = 1{,}020{,}000$, i.e. $1,020,000, against the $2,000,000 staked total — twice as expensive here, simply because this scenario's total staked value is twice the proof-of-work total.

4. Against a $500,000 security budget

   • Proof-of-work: $510,000 is more than $500,000 → priced out
   • Proof-of-stake: $1,020,000 is more than $500,000 → priced out (even pricier)
   • Sybil: $0 never exceeds any budget → never priced out — exactly why identity alone was never the thing worth defending