Fully Homomorphic Encryption

Abstract

Homomorphic encryption is a very useful tool with a number of attractive applications. However, the applications are limited by the fact that only one operation is possible (usually addition or multiplication in the plaintext space) to be able to manipulate the plaintext by using only the ciphertext. What would really be useful is to be able to utilize both addition and multiplication simultaneously. This would permit more manipulation of the plaintext by modifying the ciphertext. In fact, this would allow one without the secret key to compute any efficiently computable function on the plaintext when given only the ciphertext. In this chapter, we introduce fully homomorphic encryption (FHE) techniques, which allow one to evaluate both addition and multiplication of plaintext, while remaining encrypted. The concept of FHE was introduced by Rivest [14] under the name privacy homomorphisms. The problem of constructing a scheme with these properties remained unsolved until 2009, when Gentry [6] presented his breakthrough result. His scheme allows arbitrary computation on the ciphertexts and it yields the correct result when decrypted. This chapter begins with an introduction of FHE model and definitions, followed by the construction of FHE scheme over integers.