Over the last two or three decades, elliptic curves have been playing an increasingly significant role not only in number theory but also in related ﬁelds such as cryptography. For instance, in the 1980s, elliptic curves started being used in cryptography and elliptic curve techniques were developed for factorization and primality testing. This techniques leads into the discrete logarithm problem on elliptic curves. The goal of the present book is to improve and invent the theory of elliptic curves assuming only modest backgrounds in elementary number theory and in groups and ﬁelds, approximately what would be covered in a strong undergraduate or beginning postgraduate abstract number theory course. The book provides an introduction to both the cryptographic side and the number theoretic side of elliptic curves.