In modern cryptographic systems, the messages are represented by numerical values prior to being encrypted and transmitted. The encryption processes are mathematical operations that turn the input numerical values into output numerical values. Building, analyzing, and attacking these cryptosystems requires mathematical tools. The most important of these is number theory, especially the theory of congruences. This chapter presents the basic tools needed for the rest of the book. More advanced topics such as factoring, discrete logarithms, and elliptic curves, will be treated in later chapters (Chapters 9, 10, and 21, respectively).