Efficient fully homomorphic encryption from RLWE with an extension to a threshold encryption scheme

• We present an efficient fully homomorphic encryption (FHE) from RLWE.
• We get the FHE scheme with re-linearization and modulus reduction techniques.
• We extend the FHE scheme to the threshold fully homomorphic encryption scheme.

In this paper, we present an effective fully homomorphic encryption (FHE) from ring learning with errors (RLWE) assumption without using Gentry’s standard squashing and bootstrapping techniques. Our FHE scheme is to modify the recent FHE scheme of Brakerski. We use the re-linearization technique to reduce the length of ciphertext considerably, and use the modulus reduction technique to manage the noise level and decrease the decryption complexity without introducing additional assumptions. Furthermore, with the key-homomorphic property, we extend our FHE scheme to a threshold fully homomorphic encryption (TFHE), which allows parties to cooperatively decrypt a ciphertext without learning anything but the plaintext. The TFHE scheme can be protected from related-key attacks, as long as we add extra smudging noise during sensitive operations.