Fully Homomorphic Encryption Over Exterior Product Spaces

摘要

In this work I propose a new symmetric fully homomorphic encryption powered by Exterior Algebra and Product Spaces, more specifically by Geometric Algebra as a mathematical language for creating cryptographic solutions, which is organized and presented as the Enhanced Data-Centric Homomorphic Encryption - EDCHE, invented by Dr. Carlos Paz de Araujo, Professor and Associate Dean at the University of Colorado Colorado Springs, in the Electrical Engineering department. Given GA as mathematical language, EDCHE is the framework for developing solutions for cryptology, such as encryption primitives and sub-primitives. In 1978 Rivest et al introduced the idea of an encryption scheme able to provide security and the manipulation of encrypted data, without decrypting it. With such encryption scheme, it would be possible to process encrypted data in a meaningful way. In 2009, Craig Gentry proposed the first fully homomorphic encryption scheme based on ideal lattices. Although his original solution has faced some important improvements, Gentry's scheme is currently impracticable.;Geometric Algebra is a field of mathematics already explored in many areas of Computer Science, Physics, Electrical Engineering, to cite a few. However, this is the first time that Geometric Algebra is proposed as the main building block of cryptographic solutions. Numbers are represented as multivectors, the special object for data representation, and through the GA operations (and some of its extensions), a series of transformations are performed, providing the mathematical under-determinacy required for encryption.;Since the mathematics that empowers EDCHE is naturally homomorphic, EDCHE is illustrated as an intrinsically fully homomorphic encryption, allowing the required homomorphic additive and multiplicative properties and adding special capabilities such as comparison, sorting and searching.