Masahiro Yagisawa Knihy

The book introduces a novel method for signcryption on the octonion ring, merging key agreement protocols and digital signature schemes into a single operation. Building on the foundational concepts of public key cryptography established by Diffie and Hellman, this approach enhances security by leveraging the complexity of solving high-degree multivariate algebraic equations, making it resilient against Gröbner basis attacks. It ensures message confidentiality, non-repudiation, unforgeability, forward secrecy, public verifiability, and encrypted message authentication, particularly benefiting cloud computing applications.

Fully homomorphic encryption(FHE) 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. But FHE scheme which runs fast for practical use is not proposed until now because the scheme requires huge complexity. This book, therefore, provides the new fully homomorphic encryption scheme without "bootstrapping" which runs fast for practical use. The system runs on octonion ring over the finite field Fq. The security of this system is based on the computational difficulty to solve the multivariate algebraic equations. 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. This book is written for graduate students to active researchers.