Based on the research experience of the authors, this book provides a theoretical structure of fundamental number theory algebra knowledge supporting public-key cryptography. The authors integrate the results and thinking from more than 20 years of research and teaching, bridging the gap between math theory and crypto practice. It can be used in courses for junior/ senior undergraduate students and first year graduate students, cryptography researcher sand engineers can also benefit from this manuscript to build a more rigorous and effective way of thinking.