This textbook provides a fundamental introduction to a mathematically demanding area of modern cryptography that is increasingly gaining practical significance. The relevant facts about elliptic curves and public-key cryptography are explained in detail. Only minimal prior knowledge is required, making the text accessible to students of mathematics and computer science from the 5th semester onwards, as well as to practitioners. Contents include: public-key cryptography, elliptic curves, elliptic curves over finite fields, the discrete logarithm problem for elliptic curves, practical consequences, appendix: mathematical foundations, bibliography.