Characterizing NTRU-Variants Using Group Ring and Evaluating their Lattice Security

The encryption scheme NTRU is designed over a quotient ring of a polynomial ring. Basically, if the ring is changed to any other ring, NTRU-like cryptosystem is constructible. In this paper, we propose a variant of NTRU using group ring, which is called GRNTRU. GR-NTRU includes NTRU as a special case. Moreover, we analyze and compare the security of GR-NTRU for several concrete groups. It is easy to investigate the algebraic structure of group ring by using group representation theory. We apply this fact to the security analysis of GR-NTRU. We show that the original NTRU and multivariate NTRU are most secure among several GR-NTRUs which we investigated.