Processing Encrypted Data Using Homomorphic Encryption

Fully Homomorphic Encryption (FHE) was initially introduced as a concept shortly after the development of the RSA cryptosystem, by Rivest et al. [54]. Although long sought after, the first functional scheme was only proposed over thirty years later by Gentry [34, 35] in 2009. The same blueprint to construct FHE has been followed in all subsequent work. First a scheme is constructed which can evaluate arithmetic circuits of a limited depth, a so-called Somewhat Homomorphic Encryption (SHE) scheme. If the complexity of the circuits which the SHE scheme can evaluate is slightly more than the complexity of the decryption circuit for the SHE scheme, then (by placing a SHE encryption of the scheme’s private key inside the public key) one can bootstrap the SHE scheme into a FHE scheme. This bootstrapping operation is obtained by homomorphically evaluating the decryption circuit on input of the ciphertext to be bootstrapped and the encryption of the secret key.