### Homomorphic Cryptography

Informally, a cryptographic function $C(x)$ is homomorphic if it allows to peform computations on its input $x$ without inverting $C$. The most studied homomorphic cryptography problem in the area of lattice cryptography is that of Fully Homomorphic Encryption, i.e., the construction of secure encryption schemes $E$ such that, given the encryption of a message $E(m)$ and the description of a function $f$, one can compute the encryption $E(f(m))$ of the result of applying $f$ to the message, without knowing the decryption key. There is so much work on fully homomorphic encryption that we list it on a separate page. Here are some additional homomorphic cryptography problems that have been solved using lattices.

Homomorphic signature schemes are digital signature schemes that allow some limited form of forgery, e.g., given a signature $S(m)$ on a message $m$, one should be able to derive a signagure $S(f(m))$ on a related message. Notice that homomorphic signature schemes are not secure digital signatures because they allow some form of forgery.

1. Homomorphic signatures for polynomial functions
(Boneh & Freeman - Eurocrypt 2010)

Key-homomorphic encryption differs from (fully) homomorphic encryption in that it allows to operate homomorphically on encryption key rather than the encrypted message.

1. Fully Key-Homomorphic Encryption, Arithmetic Circuit ABE and Compact Garbled Circuits
(Boneh, Gentry, Gorbunov, Halevi, Nikolaenko, Segev, Vaikuntanathan & Vinayagamurthy - EuroCrypt 2014)