Homomorphic

Homomorphic is an adjective used to describe a property of a mapping between algebraic structures that preserves their operation. In mathematics, a function f between two structures is called a homomorphism if it preserves the relevant operation, for example f(xy) = f(x)f(y) for a group, or f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b) for rings. The term highlights that structural relationships are maintained under the mapping.

In cryptography, the term is most often encountered in reference to homomorphic encryption. A homomorphic encryption scheme allows computations to be performed on ciphertexts such that decrypting the result yields the same outcome as if the operations had been performed on the plaintexts. This enables processing of encrypted data without exposing it. Homomorphic schemes are categorized as partially homomorphic (supporting a single type of operation, such as addition or multiplication) or fully homomorphic (supporting arbitrary computations, i.e., any circuit or program).

Examples of partially homomorphic schemes include RSA and ElGamal, which are multiplicatively homomorphic, and Paillier, which is additively homomorphic. Fully homomorphic encryption (FHE) schemes, which support arbitrary computations on encrypted data, were first constructed concretely by Craig Gentry in 2009; since then, numerous improvements have reduced some of the practical barriers.

Applications of homomorphic encryption include secure cloud computing, privacy-preserving data analysis, secure voting, and machine learning on encrypted data. Limitations remain significant: current FHE schemes incur substantial computational and storage overhead, and operations on ciphertexts are far slower than on plaintexts. Ongoing research aims to broaden practical deployment and reduce costs while preserving data privacy.