Fully Homomorphic Public-Key Encryption with Three Ciphertexts

Fully homomorphic encryption is the cryptosystem which supports both addition and multiplication. The existence of the cryptosystems which have efficiency and fully homomorphism would bring great practical implications in the outsourcing of private computations, for instance, in the context of cloud computing.In this book I propose the fully homomorphic public-key encryption (FHPKE) with composite number modulus based on the discrete logarithm assumption (DLA) and computational Diffie¿Hellman assumption (CDH) of multivariate polynomials on octonion ring over the finite ring Z/rZ. The scheme has three ciphertexts corresponding to one plaintext and is immune from ¿m and - m attack¿. To break this cryptosystem it is thought that we probably need to solve the multivariate algebraic equations that is equal to solving the NP complete problem. Then it is thought that this system is immune from the attacks by the quantum computers.