In this paper, we deal with c-secure codes in a fingerprinting scheme, which encode user ID to be embedded into the contents. If a pirate copy appears, c-secure codes allow the owner of the contents to trace the source of the illegal redistribution under collusion attacks. However, when dealing in practical applications, most past proposed codes are failed to obtain a good efficiency, i.e. their codeword length are too large to be embedded into digital contents. In this paper, we propose a construction method of c-secure CRT codes based on polynomials over finite fields and it is shown that the codeword length in our construction is shorter than that of Muratani's scheme. We compare the codeword length of our construction and that of Muratani's scheme by numerical experiments and present some theoretical results which supports the results obtained by numerical experiments. As a result, we show that our construction is especially efficient in respect to a large size of any coalition c. Furthermore, we discuss the influence of the random error on the traceability and formally define the Weak IDs in respect to our construction.