Homomorphic encryption is characterized by several important properties that enable computations to be performed on encrypted data without decryption. These properties are crucial for understanding how homomorphic encryption works:

1. Homomorphism: The term “homomorphic” itself signifies the primary property of these encryption schemes. A homomorphism is a mathematical operation that preserves the structure of the data being operated on. In the context of homomorphic encryption, it means that operations performed on the ciphertext produce results that, when decrypted, are equivalent to the operations performed on the plaintext.
2. Additivity (Additive Homomorphism): In additively homomorphic encryption schemes, you can perform addition operations on ciphertexts without decrypting them first. Mathematically, if you have encrypted values `C(a)` and `C(b)`, you can compute `C(a + b)` without knowledge of `a` and `b`. When you decrypt the result, you get `a + b`.
3. Multiplicativity (Multiplicative Homomorphism): In multiplicatively homomorphic encryption schemes, you can perform multiplication operations on ciphertexts without decryption. If you have encrypted values `C(a)` and `C(b)`, you can compute `C(a * b)` without revealing `a` or `b`. After decryption, you obtain `a * b`.
4. Operation Composition: Some advanced homomorphic encryption schemes support more complex operations, such as addition and multiplication in the same computation, as well as other mathematical functions. This allows for a wide range of computations to be performed on encrypted data.
5. Public Key Encryption: Homomorphic encryption schemes are typically implemented using public key cryptography. They involve two keys: a public key for encrypting data and a private key for decrypting the results of computations. The public key is used to encrypt data, while the private key is used to decrypt the results after computations.
6. Security Properties: Homomorphic encryption schemes must also exhibit the standard security properties of encryption, such as semantic security (indistinguishability of ciphertexts) and resistance to chosen plaintext attacks.

These properties allow data to be encrypted, outsourced to a server or cloud service, and processed while still in its encrypted form. The server can perform computations on the ciphertexts without gaining access to the plaintext data, preserving the privacy and security of sensitive information.

The combination of homomorphic properties with public key cryptography makes homomorphic encryption a powerful tool for privacy-preserving computations in scenarios where data confidentiality is paramount.