In this paper, we propose a new McEliece public-key cryptosystem based on polar codes with non-permutation equivalent. Due to the fact that polar code is a decreasing monomial code, any permutation of columns of its generator matrix can be found. Thus, any McEliece cryptosystem based on only the permutation equivalent of the standard polar code seems to be insecure against some structural attacks. In contrast, our proposed scheme is able to resist against these attacks and achieves good security level against the generic decoding attacks.