It's also somewhat definitional. We're taking an established encryption scheme in both cases and constructing a new algorithm on top of it. Above where I say we're performing decryption with a key set to key^2, we could also think of it as doing Dec(key, key*m) where the key is the same but we've done some operation on the cyphertext.
> Wouldn't the information required to identify what a "word" is require the running software to be able to see breaks/periods/etc?
Yes. But that information can be encrypted and still computed on. You might suppose that since encrypted data is essentially gibberish, then multiplying or adding different blocks of it together will only generate gibberish. The insight is that it is not entirely gibberish- or else how would we be able to decrypt it? The information to identify a word still there, and you can write a program that sees breaks/periods/etc, but you won't know when it sees a break/period until decryption of the result.
> Doesn't that leak information about the cyphertext?
This question seems to be encoding an assumption that the cloud in your example is able to see the result of the search query. The key insight here is that the cloud is only able to compute the encrypted result. It can only return the encrypted result to the requester who has the private key, and can decrypt the result, and see how many words were counted.
> How does it stop someone from writing software that, for example, maps out the position of all the a's, then b's, then c's, etc in a cyphertext and MITMing it?
I'm a bit confused by the attack here. I think it is also assuming that the untrusted computing party is able to read the plaintext result of the operation.
Here's an illustrative example. Suppose my encryption scheme is Enc(key, m) = key*m = c. Suppose my decryption scheme is Dec(key, c) = c/key = m. This scheme is not secure, but pretend that it is, and that separating out key and m from c is difficult.
I want the untrusted cloud to compute m1 * m2.
I can perform Enc(key, m1) = c1, Enc(key, m2) = c2 and send c1 and c2 to the cloud to multiply.
The cloud receives c1 and c2 which are really key*m1 and key*m2, but we are assuming that the cloud can't separate these factors from the products.
The cloud returns c1*c2, which we know equals key*m1*key*m2 = key^2 * (m1*m2).
If we divide c1*c2 by (key^2) - note: this is just running our decryption algorithm with a modified key - we will get m1*m2, which is what we wanted!
For a more formal example using ElGamal Encryption (apologies for spelling ElGamal wrong in the paper) I wrote this up: https://github.com/lsnow99/elgamal/blob/main/elgamal.pdf
Credit to https://www.cs.cmu.edu/~goyal/15356/lecture_notes.pdf for definitions
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