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

In this paper, we show that the fundamental concepts behind the Ntrū cryptosystem can be extended to a broader algebra than Dedekind domains. Also, we present an abstract and generalized algorithm for constructing a Ntrū-like cryptosystem such that the underlying algebra can be non-commutative or even non-associative. To prove the main claim, we show that it is possible to generalize Ntrū over non-commutative Quaternions (algebra in the sense of Cayley-Dikson, of dimension four over an arbitrary principal ideal domain) as well as non-associative Octonions (a power-associative and alternative algebra of dimension eight over a principal ideal domain). Given the serious challenges ahead of non-commutative/non-associative algebra in quaternionic or octonionic lattices, the proposed cryptosystems are more resistant to lattice-based attacks when compared to Ntrū. Concisely, this paper is making an abstract image of the mathematical base of Ntrū in such a way that one can make a similar cryptosystem based on various algebraic structures with the goal of better security against lattice attack and/or more capability for protocol design.