Ah, I've got a response for your decryption concern now. Your decryption fails to work in the ring of polynomials, among other things.

It doesn't prevent equality tests in a single encrypted domain. But in a single encrypted domain, two ciphertexts for the same plaintext (i.e. including an extra block for obfuscation/resolution is cheating) are the same anyways. The unnecessary worry is about equality across different encrypted domains. I should hope the kernel is trivial, else we have collisions, and this indicates a different sort of a problem with our encryption scheme.

Your decryption concern is more interesting, but it takes me out of my depth for the moment. I think there's a flaw to be found in that line of reasoning though. Perhaps someone more knowledgeable reading this thread can provide a proper explanation?

I'm an idiot. There a modifier for HTML retard?

"[username]I like apples" or [username] XOR "I like apples"

ah... sorry. failed to properly proofread my comment. That should be "...can produce another encryptoed block of "I like apples" or XOR "I like apples" or..."

It doesn't allow you to compare data encrypted by different ciphers. It allows particular "multiplication" and "addition" operations on data encrypted by the same cipher. If I like apples, and have a username that starts with "S", someone with both encrypted blocks can produce another encrypted block of "I like apples" or XOR "I like apples" or something more computationally complicated along those lines. So, it does not allow the sort of attack you're proposing.

This article needs some clarification. In particular, a lot of the worried comments here show a lack of understanding of the word "homomorphic".

Here's a very simplified example of a homomorphism. I define a function
f(x) = 3x
This function is a homomorphism on numbers under addition. Its image "preserves" the addition operation. What I mean more precisely is
f(a) + f(b) = f(a + b)
That's pretty easy to verify for the function I've given.

Homomorphic encryption is interested in an encryption function f() that preserves useful computational operations. If we take my example as a very very simplified encryption then, say I have two numbers, 6, and 15, and I lack the computational power to do addtion, but I can encrypt my data with my key--3. (I'm generalizing my function to be multiplication by a key. And yes, for some reason I have the computational power to do multiplication. Humor me). I can encrypt my data, f(6) = 18 and f(15) = 45, and pass these to you, and ask you do do addtion for me. You'll do the addition, get 63, and pass this result to me, which I can then decrypt, which yields 21.

Now, my encryption here is very simple and very, very weak, but if you're willing to suspend disbelief, you'll note that the information I've allowed you to handle does not reveal either my inputs or my outputs. (In fact, with the particular numbers I've chosen, you might guess that my key is 9 instead of 3, (though relying on lucky choices or constraining myself to choices which have this property make my scheme rather useless))

If you generalize this to strong encryption and more useful computational operations, you begin to see how homomorphic encryption can be useful. One should note that, no, homomorphic encryption will not be a drop-in replacement for other forms of encryption. (Sending encrypted emails with homormorphic encryption would be unwise. An attacker can modify the data (though, if my understanding is correct, only with other data encrypted with the same key)) Homomorphic encryption simply fills a need that the other forms do not serve.

Hopefully you now also see how the article's use of the word "analysis" can be rather misleading. In particular, one of the earlier comments notes that it might be useful in allowing you to determine if different people's encrypted information is identical. By my understanding, homomorphic encryption would not allow this.

In any case, if my explanation is not enough, here's the wikipedia article.

What?! No! The heart of science is not fitting hypotheses to data. That's the sort of dangerous fallacy that produces Aristotle's "science", and in fact what dangerous fundamentalists thrive on. The thing that sets science apart is rigorous, repeatable empirical testing of not previously observed predictions. Not to say that the hypothesis in the article isn't exciting, but the already raising it up as a shining example of scientific triumph starts down a path I find terrifying.