Reed-Solomon (RS) codes are widely used in storage systems to provide high data reliability. The existing RS codes are constructed over finite field with field size larger than the code length that have high encoding/decoding complexity. In this paper, we propose a new construction for RS codes over a cyclic ring instead of finite field. Our new RS codes only incurs XOR and cyclic-shift operation in the encoding/decoding processes, and therefore have lower encoding/decoding complexity than the existing RS codes. In addition, we show that we can employ the efficient Reed-Muller (RM) transform in our new RS codes that reduces the encoding/decoding complexity. Moreover, we implement our new RS codes by C++ and show that our new RS codes have better encoding/decoding performance than the existing RS codes for the evaluated parameters.