Improved Fully Homomorphic Encryption over the Integers with Shorter Public Keys

Abstract

Fully homomorphic encryption (FHE) is a “holy grail” of cryptography. However, it is not yet adopted in practice because no known scheme is efficient. In this paper, we mainly focus on how to reduce the public key sizes in FHE. Based on Dijk et al.’s FHE scheme (DGHV) and Gentry’s fully homomorhpic technology, we propose two schemes with shorter public keys by encrypting with a combination of the public key elements to replace a linear form in DGHV scheme and a quadratic form in Coron et al.’s scheme. Compared with DGHV scheme and Coron et al.’s scheme, our schemes can greatly reduce the public key sizes, which make our schemes more efficient. Furthermore, we prove the security of our schemes based on difficulties of the approximate greatest common divisor (AGCD) problem and the sparse subset sum problem.