Generalized NTRU Algorithms on Algebraic Rings

The NTRU (Number Theory Research Unit) is a prominent post-quantum public key cryptography algorithm and a current focus of research. Although many NTRU variants have been proposed, a comprehensive generalization of these variants is still lacking. To address this issue, we propose the G-NTRU (Generalized NTRU), an algorithmic framework for NTRU variants on algebraic rings. This framework generalizes specific ring extensions to more general algebraic structures, allowing the unification of various NTRU variants. We then analyze the key properties of the G-NTRU, including correctness, lattice-based characteristics, and security. The semantic security of the G-NTRU is demonstrated through the introduction of the G-AGCD (Generalized Approximate Greatest Common Divisor). To validate the generality of the G-NTRU framework, we introduce the CNTRU (Complex Number NTRU), a variant of the NTRU over the ring of complex numbers. The CNTRU shares the same properties as the G-NTRU, further confirming the versatility of the G-NTRU in studying NTRU variants over algebraic rings.