The objective of this course is to introduce students with basic concepts, facts and algorithms concerning elliptic curves over the rational numbers and finite fields and their applications in cryptography and algorithmic number theory.

There are no formal prerequisites. It would be desirable that the students passed the course Number Theory from the undergraduate mathematics study programme as well as one of the courses Algebraic curves (theoretical mathematics programme) or Cryptography and network security (computer science and mathematics programme).

*Elliptic curves over the field of rational numbers.*
Addition of points on elliptic curves.
The Mordell-Weil group of the elliptic curve over the field of rational numbers.
Algorithms for computing the torsion group and rank.

*Elliptic curves over finite fields.*
Efficient implementation of basic operations on elliptic curves.
Elliptic curves over the field of characteristic 2.
Algorithms for cumputing the order of the group of points on elliptic curves.

*Public key cryptography.*
The idea of the public key. Cryptosystems based on factorization and
the discrete logarithm problem in a finite group. Digital signatures..

*Cryptosystems based on elliptic curves.*
Analogues of El-Gamal and DSA cryptosystem.
Comparisons with other public key cryptosystems.
The discrete logarithm problem on elliptic curves. Parameter choices in the cryptosystem.

*Other applications of elliptic curves.*
Elliptic curve factorization method by Lenstra. Primality proving using elliptic curves.

- A. Dujella:
*Teorija brojeva*, Školska knjiga, Zagreb, 2019. - A. Dujella:
*Number Theory*, Školska knjiga, Zagreb, 2021. - A. Dujella, M. Maretić:
*Kriptografija*, Element, Zagreb, 2007. - N. Koblitz:
*A Course in Number Theory and Cryptography*, Springer-Verlag, New York, 1994.

- I. Blake, G. Seroussi, N. Smart:
*Elliptic Curves in Cryptography*, Cambridge University Press, Cambridge, 1999. - A. Enge:
*Elliptic Curves and Their Applications to Cryptography. An Intoduction*, Kluwer, Boston, 1999. - D. Hankerson, A. Menezes, S. Vanstone:
*Guide to Elliptic Curve Cryptography*, Springer-Verlag, New York, 2004. - J. H. Silverman, J. Tate:
*Rational Points on Elliptic Curves*, Springer-Verlag, Berlin, 1992. - L. C. Washington:
*Elliptic Curves: Number Theory and Cryptography*, CRC Press, Boca Raton, 2008.

**Lecture notes**

(in *pdf* format; in Croatian)

**Cryptography**
- Undergraduate course

**Number Theory**
- Undergraduate course

**Algorithms for Elliptic Curves** - Graduate course (2008/2009)

**Number Theory in Cryptography** - Graduate course (2003/2004)

Recommended readings in number theory

High rank elliptic curves with prescribed torsion

Infinite families of elliptic curves with high rank and prescribed torsion

History of elliptic curves rank records