The improvements of embedded processors make future technologies including wireless sensor network and internet of things feasible. These applications firstly gather information from target field through wireless network. However, this networking process is highly vulnerable to malicious attacks including eavesdropping and forgery. In order to ensure secure and robust networking, information should be kept in secret with cryptography. Well known approach is public key cryptography and this algorithm consists of finite field arithmetic. There are many works considering high speed finite field arithmetic. One of the famous approach is Montgomery multiplication. In this study, we investigated Montgomery multiplication for public key cryptography on embedded microprocessors. This paper includes helpful information on Montgomery multiplication implementation methods and techniques for various target devices including 8-bit and 16-bit microprocessors. Further, we expect that the results reported in this paper will become part of a reference book for advanced Montgomery multiplication methods for future researchers.