A New Approach Towards Fully Homomorphic Encryption Over Geometric Algebra

Abstract

The ability to compute on encrypted data in a meaningful way is a subject of increasing interest in both academia and industry. The type of encryption that allows any function to be evaluated on encrypted data is called fully homomorphic encryption, or FHE, and is one promising way to achieve secure computation. The problem was first stated in 1978 by Rivest et al. and first realized by Gentry in 2009, but remains an open problem since an FHE scheme that is both efficient and secure is yet to be presented. Most of the prominent FHE schemes follow Gentry's blueprint which concentrates the efforts of researchers on very similar algebraic structures and noise management techniques. The intrinsic complexity of these schemes results in the similar shortfalls that they share in efficiency. We introduce the application of Geometric Algebra (GA) to encryption in conjunction with p-adic arithmetic and a modified version of the Chinese Remainder Theorem and we demonstrate an efficient, noise-free, symmetric-key FHE scheme. We focus the security analysis on demonstrating that our FHE scheme is not linearly decryptable. Further, we discuss a practical approach for generalizing different types of algebraic structures in the geometric product space of two dimensions, which allows us to export GA operations to other algebras and vice-versa. Our construction supports a variety of applications, from homomorphic obfuscation to general purpose FHE computations.