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The main purpose of this paper is to illustrate the fundamental concepts behind the NTRU public key cryptosystem can be extended to a broader algebra than Dedekind domains and the NTRU underlying ring may be replaced by a non-commutative or even nonassociative algebra. To cross the border of Dedekind or Euclidean domains, we prove that it is possible to extend NTRU to the algebra of polynomials with coefficients in the non-commutative ring of quaternions as well as the non-associative octonions algebra (a power-associative and alternative algebra of dimension eight over a principal ideal domain). We also demonstrate that the security of the proposed nonassociative cryptosystem relies on the intractability of shortest vector problem in a certain type of lattice. The least advantage of the non-associativity of the underlying algebra is that the resulting lattice is not fully classified under Convolutional Modular Lattice (CML). To the best of our knowledge, no non-associative public key cryptosystem based on non-associative algebra has been proposed in the literature.

NTRU-Like Public Key Cryptosystems beyond Dedekind Domain up to Alternative Algebra