[pqoek]

FHE:

- I encrypt data Encrypt(key, data) = ciphertext1

- you operate on encrypted data f(ciphertext1) = ciphertext2

- I decrypt the result to obtain Decrypt(key, ciphertext2) = f(data)

for iO it's more like: I give you the algorithm to encrypt your data: some_algorithm(data) = Encrypt(key, data)

but you can't figure out what "key" is, even though you have the code and it's not some remote service

[aruss]

iO doesn't guarantee that the adversary can't figure out what "key" is. It turns out that this is possible, but it requires some very careful applications of a special kind of pseudorandom generator.