In this paper, we present a new design of Finite Alphabet (FA) Message Passing (MP) decoders using only integer operations. We utilize Discrete Density Evolution with a multidimensional Lookup-Table (mLUT) design for Variable Node (VN) updates to consider all input messages jointly for reducing the information loss compared to the frequent sequential LUT design approaches. In the proposed Minimum-Integer Computation (MIC) decoder we allow for different bit-widths for node operations and message exchanges between nodes. The mLUT operations for VN updates are replaced by low complexity signed integer additions and threshold operations, and the Check Node (CN) updates simplify to a minimum search over integers. For a (816,406) regular LDPC code, we show that our 3-bit MIC decoder achieves the communication performance of the corresponding mLUT decoder and outperforms a 4-bit state-of-the-art Min-Sum (MS) decoder. We show that the node implementations on a 22 nm FD-SOI technology yield an improved area and energy efficiency over the respective MS implementation. To the best of our knowledge, this is the first time that an implementation improvement for the VNs and CNs is shown when using FA MP.