Homomorphic properties Clause Samples

Homomorphic properties. ‌ Important features of the Paillier scheme in general are the additive and multi- plicative homomorphic properties. Above mentioned abilities allow computing addition and multiplication of two ciphertexts. The computation behaves just as if it were operations over the corresponding plaintexts. The mathematical expression of these properties are as follows: • homomorphic addition of plaintexts, where decryption of the product of two ciphers (or a product of a cipher and a generator 𝑔 raised to the power of the plaintext) results in the addition of the two plaintexts: – 𝐷(𝐸(𝑚1, 𝑟1) · 𝐸(𝑚2, 𝑟2) mod 𝑛2 = 𝑚1 + 𝑚2 mod 𝑛, – 𝐷(𝐸(𝑚1, 𝑟1) · 𝑔𝑚2 mod 𝑛2 = 𝑚1 + 𝑚2 mod 𝑛, • homomorphic multiplication of plaintexts, where decryption of a ciphertext raised to the power of plaintext results to the multiplication of the two plain- texts: – 𝐷(𝐸(𝑚1, 𝑟1)𝑚2 mod 𝑛2 = 𝑚1𝑚2 mod 𝑛, – 𝐷(𝐸(𝑚2, 𝑟2)𝑚1 mod 𝑛2 = 𝑚1𝑚2 mod 𝑛, – more generally, where 𝑘 is a constant: 𝐷(𝐸(𝑚1𝑚𝑟1)𝑘 mod 𝑛2 = 𝑘𝑚1 mod 𝑛. However, there is no known way to compute an encryption of the product of 𝑚1 and 𝑚2 without knowing the private key 𝑠𝑘. It is because of the Paillier characteristic of the encryption of the two messages.
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Related to Homomorphic properties

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