1. komponenta

Lecture type	Total
Lectures	45

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES:
Enable students to:
- understand the basic concepts, facts and algorithms related to elliptic curves over a field of rational numbers and finite fields;
- independently apply elliptic curves in public key cryptography.

COURSE DESCRIPTION AND SYLLABUS:
1. 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.
2. Elliptic curves over finite fields. Efficient implementation of basic operations on elliptic curves.Elliptic curves over the field of characteristic 2. Algorithms for computing the order of the group of points on elliptic curves.
3. Public key cryptography. The idea of the public key. Cryptosystems based on factorization and the discrete logarithm problem in a finite group. Digital signatures.
4. Cryptosystems based on elliptic curves. Analogues of El-Gamals and the DSA cryptosystem. Comparisons of other public key cryptosystems. The discrete logarithm problem on elliptic curves. Parameter choice in the cryptosystem.
5. Other applications of elliptic curves. Elliptic curve factorization method of Lenstra. Primality proving using elliptic curves.

1. Teorija brojeva, A. Dujella, Školska knjiga, Zagreb, 2019.
2. Number Theory, A. Dujella, Školska knjiga, Zagreb, 2021.
3. Kriptografija, A. Dujella, M. Maretić, Element, Zagreb, 2007.
4. A Course in Number Theory and Cryptography, N. Koblitz, Springer-Verlag, New York, 1994.
5. Elliptic Curves in Cryptography, I. Blake, G. Seroussi, N. Smart, Cambridge University Press, Cambridge, 1999.
6. Elliptic Curves and Their Applications to Cryptography. An Introduction, A. Enge, Kluwer, Boston, 1999.
7. Guide to Elliptic Curve Cryptography, D. Hankerson, A. Menezes, S. Vanstone, Springer-Verlag, New York, 2004.
8. Rational Points on Elliptic Curves, J. H. Silverman, J. Tate, Springer-Verlag, Berlin, 1992.
9. Elliptic Curves: Number Theory and Cryptography, L. C. Washington, CRC Press, Boca Raton, 2008.
10. Cryptography and Network Security. Principles and Practice, W. Stallings, Prentice Hall, Upper Sadle River, 2005.
11. Introduction to Cryptography with Coding Theory, W. Trappe, L. C. Washington, Prentice Hall, Upper Sadle River, 2002.

Ostali izborni predmeti - Regular study - Computer Science and Mathematics
Vezani kolegiji B - Regular study - Computer Science and Mathematics

Ostali izborni predmeti - Regular study - Computer Science and Mathematics
Vezani kolegiji B - Regular study - Computer Science and Mathematics