Discussion on Matrix NTRU

In recent years the study of cryptosystem has shifted noticeably from symmetric to asymmetric key encryptions. One of the more intriguing issues of the research is NTRU encryption system, which is based on ring theory. The security of NTRU always depends on the lattices. Several studies have suggested that it is very difficult to know whether a polynomial is invertible or not. Nayak et al. introduced a new method as a matrix solution to solve the problem. However, this method is not without its flaws. In this paper, we expose the weakness regarding network security in matrix NTRU cryptosystem of Nayak et al. (2008, 2010) conscientiously, and we also propose a novel solution to this weakness. Our approach is based on the fact that some new conditions for selection of keys can increase the size of domain compared to what was shown in the previous studies and improve the strength of security against different kind of network attacks. First, we use a counter example to point out the flaw in the theorem of inverse modulo q introduced in the previous studies. Second, we prepare a new approach for inverse modulo q. The purpose of this paper is to demonstrate that our twofold selection scheme is superior to the original matrix NTRU cryptosystem and will help cryptosystems function under a safer environment by creating one public key and two private keys.