Why does Bitcoin have 128 bits of security?

It is commonly said that Bitcoin has 128 bits of security because there is an algorithm that can find the private key of a given public key in an average of 2^128 steps.

However, this assumes that the attacker knows the public key associated with the funds, so this attack is only possible with P2PK outputs or with addresses that have already been spent from.

In the case of an unspent P2PKH or P2WPKH, the attacker only knows the 160-bit address. Therefore, they must first brute force to find the public key associated with the given address. For each attempt, they have a 1 in 2^160 chance of succeeding, meaning the security here is effectively 160 bits. Once the attacker finds the public key, they can then apply the algorithm to find the private key.

So doesn't this imply 160 bits of security? Even if the attacker immediately found the private key corresponding to a public key, they would still need to perform a 160-bit brute-force attack first.



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