Mathematical Cryptography

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The lecture sets mathematical foundations of modern cryptography (RSA, El-Gamal, elliptic curve cryptography). Related algorithms for primality testing, number factorisation and discrete logarithm are treated as well.

Requirements:

Syllabus of lectures:

1. Introduction into cryptography. Basic notions of number theory.

2. Counting modulo n. Complexity of operations in Z_n.

3. RSA cryptosystem and attacks on it.

4. Abelian groups.

5. Element order in a group, cyclic groups.

6. Structure of Z_n^* groups.

7. Discrete logarithm, Diffie-Hellman protocol.

8. Elliptic curves, discrete logarithm on an elliptic curve.

9. Generating of random primes, probability algoritms.

10. Primality tests, Carmichael numbers.

11. Factorisation using the Euler function.

12. Subexponential algorithms for discrete logarithm.

13. Subexponential algorithms for factorisation, a quadratic sieve.

13. Quantum computing and satefy of cryptosystems.

Syllabus of tutorials:

Study Objective:

Study materials:

[1] V.Shoup, A Computational introduction to number theory and algebra, Cambridge University Press, 2008, http://shoup.net/ntb/

[2] D.Boneh, Twenty Years of Attacks on the RSA Cryptosystem. https://crypto.stanford.edu/~dabo/papers/RSA-survey.pdf

[3] D.Hankerson, A.J.Menezes, S.Vanstone, Guide to elliptic curve cryptography, Springer, 2004.

Note:

Further information:

http://math.feld.cvut.cz/gollova/mkr.html

No time-table has been prepared for this course

The course is a part of the following study plans:

• Open Informatics - Cyber Security (compulsory course of the specialization)