In this book, a glance into modern mathematics is presented for the undergraduate student. Some of the necessary tools for the understanding of the arithmetic of elliptic curves over finite fields and their applications to modern cryptography are described. The goal of this book is not to provide all correct answers to the questions about modern mathematics, but rather to show many connections between the mathematics and its applications in practical cryptographic goals. Elements of abstract algebra, number theory, and affine and projective geometry are introduced and developed, and their interplay is exploited. Algebra and geometry are combined to characterize congruent numbers via rational points on the unit circle, and group law for the set of points on an elliptic curve arises from geometric intuition provided by Bézout’s theorem as well as the construction of projective space. The structure of the unit group of the integers modulo a prime explains RSA encryption, Pollard’s method of factorization, Diffie-Hellman key exchange, and ElGamal encryption, while the group of points of an elliptic curve over a finite field motivates Lenstra’s elliptic curve factorization method and elliptic curves cryptography. The only real prerequisite for this book is a course on one-variable calculus; other necessary mathematical topics are introduced on-the-fly. Numerous exercises guide further the exploration.