Towards Constructing Fully Homomorphic Encryption without Ciphertext Noise from Group Theory

Abstract

In CRYPTO 2008, one year earlier than Gentry's pioneering "bootstrapping" technique on constructing the first fully homomorphic encryption (FHE) scheme, Ostrovsky and Skeith III had suggested a completely different approach towards achieving FHE. Namely, they showed that the NAND operator can be realized in some non-commutative groups; consequently , in combination with the NAND operator realized in such a group, homomorphically encrypting the elements of the group will yield an FHE scheme. However, no observations on how to homomorphically encrypt the group elements were presented in their paper, and there have been no follow-up studies in the literature based on their approach. The aim of this paper is to exhibit more clearly what is sufficient and what seems to be effective for constructing FHE schemes based on their approach. First, we prove that it is sufficient to find a surjective homomorphism π : G → G between finite groups for which bit operators are realized in G and the elements of the kernel of π are indistinguishable from the general elements of G. Secondly, we propose new methodologies to realize bit operators in some groups, which enlarges the possibility of the group G to be used in our framework. Thirdly, we give an observation that a naive approach using matrix groups would never yield secure FHE due to an attack utilizing the "linearity" of the construction. Then we propose an idea to avoid such "linearity" by using combinatorial group theory, and give a prototypical but still incomplete construction in the sense that it is "non-compact" FHE, i.e., the ciphertext size is unbounded (though the ciphertexts are noise-free as opposed to the existing FHE schemes). Completely realizing FHE schemes based on our proposed framework is left as a future research topic.