Noise Free Fully Homomorphic Encryption Scheme over Non-associative Algebra

Among several approaches to privacy-preserving cryptographic schemes, we have concentrated on noise-free homomorphic encryption. It is a symmetric key encryption that supports homomorphic operations on encrypted data. We present a fully homomorphic encryption (FHE) scheme based on sedenion algebra over finite $Z_{n}$ rings. The innovation of the scheme is the compression of a 16-dimensional vector for the application of Frobenius automorphism. For sedenion, we have $p^{16}$ different possibilities that create a significant bijective mapping over the chosen 16-dimensional vector that adds permutation to our scheme. The security of this scheme is based on the assumption of the hardness of solving a multivariate quadratic equation system over finite $Z_{n}$ rings. The scheme results in $256n$ multivariate polynomial equations with $256+16n$ unknown variables for $n$ messages. For this reason, the proposed scheme serves as a security basis for potentially post-quantum cryptosystems. Moreover, after sedenion, no newly constructed algebra loses its properties. This scheme would therefore apply as a whole to the following algebras, such as 32-dimensional trigintadunion.