Discrete algebraic methods represent a forward-looking field whose foundations will continue to gain importance. The main idea of this textbook is to convey essential elements of discrete mathematics in order to competently assess modern developments in the information age from a mathematical perspective. It begins with a general chapter on algebraic structures, which provides the foundation for the entire book. The following chapter imparts basic knowledge in cryptography. Chapter 3 on number-theoretic algorithms is crucial for generating cryptosystems, which require large random prime numbers, for example. In Chapter 4 on prime number recognition in polynomial time, the authors present the deterministic polynomial time test by Agrawal, Kayal, and Saxena. The next chapter covers elliptic curves.