Efficient implimentation of NTRU over all one polynomial ring with CVMA

It is shown that public key cryptosystems based on discrete logarithm probrem can be solved if the quantum computer and Shor´s algorithm are realized. Thus a new cryptosystem called post-quantum cryptosystem so as not to be broken by quantum computer is needed. NTRU is proposed by Hoffstein et al. in 1998. It is one of post-quantum cryptosystem. It is based on problems on lattice for which there are no efficient algorithms to solve. In NTRU, using convolution polynomial ring as Z_q[X]/(Xⁿ-1). However, (X-1), that is a trivial factor of Xⁿ-1 sometimes make problems. Thus we consider a variant using a quotient polynomial ring such as Z_q[X]/(Xⁿ + X^x-1 +^...+X + 1) and CVMA: Cyclic Vector Multiplication Algorithm.