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Definition
An encryption mechanism E is called homomorphic basically if it preserves certain algebraic structure between the plaintext space and the ciphertext space, where the encryption key is fixed. For example, if the product of any two ciphertexts is equal to a ciphertext of the sum of the two corresponding plaintexts: \(E({m}_{1}) {_\ast} E({m}_{2}) = E({m}_{1} + {m}_{2})\), where it is understood that all these encryptions use the same key.
Background
Homomorphic properties are natural for number-theoretic public key cryptosystems like ElGamal, Paillier, and also plain RSA. Homomorphic cryptosystems are routinely used in the design of privacy-protecting cryptographic schemes, such as secret ballot election schemes and sealed bid auction schemes. A typical use is to compute a random re-encryption c′ = E(m, r′) of a given probabilistic encryption c = E(m, r) such that ciphertexts c and c′ contain the same plaintext m, but the...
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Schoenmakers, B. (2011). Homomorphic Encryption. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_870
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