Bootstrapping BGV ciphertexts with a wider choice of p and q

Abstract

We describe a method to bootstrap a packed BGV ciphertext which does not depend (as much) on any special properties of the plaintext and ciphertext moduli. Prior “efficient” methods such as that of Gentry et al. (PKC 2012) required a ciphertext modulus q which was close to a power of the plaintext modulus p. This enables our method to be applied in a larger number of situations. Also unlike previous methods our depth grows only as O(log p + log log q) as opposed to the log q of previous methods. Our basic bootstrapping technique makes use of a representation of the group Zq+ over the finite field Fp (either based on polynomials or elliptic curves), followed by polynomial interpolation of the reduction mod p map over the coefficients of the algebraic group. This technique is then extended to the full BGV packed ciphertext space, using a method whose depth depends only logarithmically on the number of packed elements. This method may be of interest as an alternative to the method of Alperin-Sheriff and Peikert (CRYPTO 2013). To aid efficiency we utilize the ring/field switching technique of Gentry et al. (SCN 2012, JCS 2013).