Where is Double hashing performed in Bitcoin? where in the Bitcoin Protocol is SHA256(SHA256(x)) performed and why?
Bitcoin uses double hashing almost everywhere it hashes in one of two variants:


RIPEMD160(SHA256(x)) called Hash160 which produces a 160 bit output


hashing the public key to generate part of a Bitcoin addresses

SHA256(SHA256(x)) called Hash256 which produces a 256 bit output


generating the checksum in a Bitcoin address
hashing the block in a merkle tree
linking transaction outputs and inputs
hash of the block header (and thus the proof of work and the link to the previous block)



It seems like Satoshi chose Hash256 whenever collisions are a problem, and Hash160 when only (multi target) second pre-images matter. This is consistent with a goal of achieving 128 bits of security.

You need a 2*n bit hash to achieve n bit collision resistance, and you need a t*n bit hash to achieve n bit second pre-image resistance. If we assume a conservative 4 billion targets, and a 128 bit security level, this leads to 256 bit hashes for collision resistance and 160 bit hashes for multi-target second-preimages.

So why does he hash twice? I suspect it's in order to prevent length-extension attacks.

SHA-2, like all Merkle-Damgard hashes suffers from a property called "length-extension". This allows an attacker who knows H(x) to calculate H(x||y) without knowing x. This is usually not a problem, but there are some uses where it totally breaks the security. The most relevant example is using H(k||m) as MAC, where an attacker can easily calculate a MAC for m||m'. I don't think Bitcoin ever uses hashes in a way that would suffer from length extensions, but I guess Satoshi went with the safe choice of preventing it everywhere.

To avoid this property, Ferguson and Schneier suggested using SHA256d = SHA256(SHA256(x)) which avoids length-extension attacks. This construction has some minor weaknesses (not relevant to bitcoin), so I wouldn't recommend it for new protocols, and would use HMAC with constant key, or truncated SHA512 instead.

Some related reading:


Hashing or encrypting twice to increase security? on crypto.SE
Cryptographic reasoning for double-hash? on bitcointalk