This book intends to provide material for a graduate course on computational commutative algebra and algebraic geometry, highlighting potential applications in cryptography. Also, the topics in this book could form the basis of a graduate course that acts as a segue between an introductory algebra course and the more technical topics of commutative algebra and algebraic geometry. This book contains a total of 124 exercises with detailed solutions as well as an important number of examples that illustrate definitions, theorems, and methods. This is very important for students or researchers who are not familiar with the topics discussed. Experience has shown that beginners who want to take their first steps in algebraic geometry are usually discouraged by the difficulty of the proposed exercises and the absence of detailed answers. Therefore, exercises (and their solutions) as well as examples occupy a prominent place in this course. This book is not designed as a comprehensive reference work, but rather as a selective textbook. The many exercises with detailed answers make it suitable for use in both a math or computer science course. Sample Chapter(s) Preface Chapter 1: Gröbner bases over arithmetical rings Request Inspection Copy Contents: Introduction Gröbner Bases Over Arithmetical Rings Varieties, Ideals, and Gröbner Bases Finite Fields and Field Extensions Algorithms for Cryptography Algebraic Plane Curves Elliptic Curves Index Bibliography Readership: Postgraduates, advanced undergraduates, academics and researchers of mathematics.