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In this paper, we define and discuss {\phi}-cyclic code, which may beregarded as a general form of the ordinary cyclic code. As applications, weexplain how to extend two public key encryption schemes, one is McEliece andNiederriter's cryptosystem of which is based on error-correcting code theory.Another one is NTRU public key cryptosystem of which is based on polynomialring theory. The main purpose of this paper is to give a more generalconstruction of NTRU based on ideal matrices and q-ary lattice theory.

A Generalization of Cyclic Code and Applications to Public Key Cryptosystems