What is homomorphic encryption? Who cares?

It seems reasonable to presume that to do any sort of work on encrypted data, you should need to decrypt it first, but this is not the case.  Suppose you have two numbers a and b as well as encryption and decryption algorithms Enc and Dec.  These algorithms are said to be homomorphic in addition (substitute in multiplication throughout the following if you like) if there is a third algorithm Add(_,_) such that Dec(Add(Enc(a),Enc(b))) = a + b.

The very short story here is that homomorphic encryption is about doing work on encrypted data without needing to decrypt it, or otherwise learn about it, and still getting the right answer.

It's a big problem today how much plaintext is lying around.  There is a data breach announced almost every day and the data lost was rarely encrypted because someone needed to do work on it.  Working on encrypted data directly allows full-time encryption, and full-time encryption will allow a standard of security that ends data breaches for good.