Noise free fully homomorphic encryption scheme over non-associative algebra

Iqra Mustafa, Hasnain Mustafa, Ahmad Taher Azar, Sheraz Aslam, Syed Muhammad Mohsin, Muhammad Bilal Qureshi, Nouman Ashraf

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


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 Zn rings. The innovation of the scheme is the compression of a 16-dimensional vector for the application of Frobenius automorphism. For sedenion, we have p16 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 Zn 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.

Original languageEnglish
Article number9134724
Pages (from-to)136524-136536
Number of pages13
JournalIEEE Access
Publication statusPublished - 2020


  • automorphism Aut(V)
  • Frobenius automorphism φ
  • fully homomorphic encryption
  • multivariate polynomial equations
  • Sedenion
  • totally isotropic subspaces


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