Cryptography Stack Exchange is a question and answer site for software developers, mathematicians and others interested in cryptography. It only takes a minute to sign up.
Sign up to join this community
Homomorphic cryptography is a kind of cryptography that allows you to do special math operations on the ciphertext, and the maths performed are identical to the obvious ones. For example, one person can combine two cached numbers, and decode the result, the sum will be two numbers.
In addition to the other answers: A secure secret-key homomorphic encryption scheme can be used to create a public-key encryption scheme.
If we consider encryption to be control of read and write abilities on data, then it's easy to see how this is the case.
With normal secret-key encryption:
With normal public-key encryption:
With homomorphic secret-key encryption:
This overlaps with the above definition for public-key encryption. While someone without the secret key cannot create a ciphertext for a given plaintext directly, they can do so if they are given pre-existing ciphertexts with known plaintext by the key holder.
Assume the existence of a secure, secret-key, partially homomorphic encryption algorithm that supports integer addition on ciphertexts.
For the sake of simplicity, we will also assume that noise levels and resultant corruption of plaintext are not an issue (this is a real issue that many homomorphic encryption schemes suffer from).
A private key is simply a secret key $d$ for the secret-key encryption algorithm.
A public key $\mathbf{e}$ of is a vector of encryptions of $1$ of length $n$ created using the private key with the symmetric encryption algorithm.
The public key operation generates a vector $\mathbf{r}$ such that $$m = \sum_{i=0}^{n}r_i$$ (secret splitting of $m$ into $n$ shares) and computes $$c \leftarrow \mathbf{er}$$
The inner product of $\mathbf{e}$ and $\mathbf{r}$.
The private key operation applies the decryption procedure from the secret-key cryptosystem to $c$ to recover $$1 * r_0 + 1 * r_1 + 1 * r_2 + \dots + 1 * r_n$$
Which is of course simply equal to $$m = r_0 + r_1 + r_2 + \dots + r_n$$
There are other possible public-key operations, e.g. based on subset-sum problems and/or LWE.
* Only the user with the secret key can create meaningful ciphertexts; An arbitrary block of bits of correct size is a valid ciphertext, but picking such a block of bits arbitrarily is unlikely to yield a meaningful plaintext with overwhelming probability
Homomorphic encryption lets you encrypt data, do operations/calculations with it such that when you decrypt the result, it's as if the same operations were done on the plaintext instead of the cyphertext.
The use of this is that untrusted parties can do computation on sensitive data.
This lets you use the computing power of people you might not trust with your data, like using publically available cloud computing services on your personal financial data.
Another usage case if it ever gets more applicable is in video games.
Some type of games (specifically real-time strategy games) is programmed such that every player needs to know about every other player's data due to being a deterministic simulation. Homomorphic encryption would allow deterministic simulation without players being able to cheat and get info about what other players are doing.
First, I just copy-paste the introduction of homomorphic encryption from Wikipedia here
Homomorphic encryption is a form of encryption that allows computation on ciphertexts, generating an encrypted result which, when decrypted, matches the result of the operations as if they had been performed on the plaintext. The purpose of homomorphic encryption is to allow computation on encrypted data.
For its applications (which demonstrate its purpose), Gentry said the following for its strong version fully homomorphic encryption in his famous thesis:
Fully homomorphic encryption has numerous applications. For example, it enables private queries to a search engine – the user submits an encrypted query and the search engine computes a succinct encrypted answer without ever looking at the query in the clear. It also enables searching on encrypted data – a user stores encrypted files on a remote file server and can later have the server retrieve only files that (when decrypted) satisfy some boolean constraint, even though the server cannot decrypt the files on its own. More broadly, fully homomorphic encryption improves the efficiency of secure multiparty computation.
Finally, homomorphic encryption is only part of secure voting systems. See here: Homomorphic encryption used for e-voting?
