homomorphically

Homomorphically refers to a property of certain mathematical operations or functions, particularly in the context of cryptography and abstract algebra. An operation or function is homomorphic if it preserves the structure of the data it operates on. This means that performing an operation on two elements and then applying the function to the result yields the same outcome as applying the function to each element individually and then performing the operation on those results.

In simpler terms, if you have a function 'f' and an operation '⊕', a homomorphic function satisfies the property f(a ⊕ b) = f(a) ⊕' f(b), where '⊕'' might be a different, corresponding operation in the transformed space. For example, in abstract algebra, a homomorphism is a map between two algebraic structures that preserves their operations.

A significant application of homomorphic properties is found in homomorphic encryption. This is a type of encryption that allows computations to be performed on ciphertext, producing an encrypted result that, when decrypted, matches the result of operations performed on the original plaintext. This is extremely useful for privacy-preserving cloud computing, where sensitive data can be processed without being decrypted first. There are different types of homomorphic encryption, including partially homomorphic encryption (PHE), which supports certain operations like addition or multiplication, and fully homomorphic encryption (FHE), which supports both addition and multiplication, enabling arbitrary computations.