Mathematics for Cryptology

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Information Security

Synopsis:

Students will gain deeper knowledge of algebraic procedures solving the most important mathematical problems concerning the security of ciphers. In particular, the course focuses on the problem of solving a system of polynomial equations over a finite field, the problem of factorization of large numbers and the problem of discrete logarithm. The problem of factorization will also be solved on elliptic curves. Students will further become familiar with modern encryption systems based on lattices.

Requirements:

Good knowledge of algebra, linear algebra, and basics of number theory (BI-LIN, BI-ZDM, NI-MPI).

Syllabus of lectures:

1. Groups - basic properties

2. Factor groups, cyclic groups

3. Ideals in rings

4. Factor rings

5. Polynomial rings

6. Extension of finite fields

7. Solving algebraic equations over finite bodies: relinearization, XL and XSL algorithms

8. Gröbner's bases, Buchberger's algorithm

9. Factorization: Pollard's rho method, p-1 method, Fermat factorization.

10. Factorization: network methods.

11. Discrete logarithm: Pohlig-Hellman algorithm, Babystep-giantstep algorithm, Pollard's rho method.

12. Discrete logarithm: Index calculus.

13. Elliptic curves - basic properties

14. Elliptic curves over real numbers and Galois fields.

15. ECDLP, factorization using elliptic curves.

16. Menezes-Okamoto-Vanston algorithm

17. Latice-based cryptography, GGH encryption system.

18. Orthogonalization and reduction, NTRU encryption system.

Syllabus of tutorials:

Examples of various mathematical structures,, and algorithms will be discussed.

Study Objective:

Study materials:

1. Katz, J. - Lindell, Y. : Introduction to modern cryptography. CRC press, 2014. ISBN 978-1466570269.

2. Hoffstein, J. - Pipher, J. - Silverman, J. H. : An Introduction to Mathematical Cryptography. Springer, 2008. ISBN 978-1441926746.

3. Lidl, R. - Niederreiter, H. : Finite Fields. Cambridge University Press, 2008. ISBN 978-0521065672.

4. Menezes, A. J. - van Oorschot, P. C. - Vanstone, S. A. : Handbook of Applied Cryptography. CRC Press, 1996. ISBN 0-8493-8523-7.

Note:

Further information:

https://courses.fit.cvut.cz/NIE-MKY/

No time-table has been prepared for this course

The course is a part of the following study plans:

• Master specialization Software Engineering, in English, 2021 (elective course)
• Master specialization Computer Security, in English, 2021 (PS)
• Master specialization Computer Systems and Networks, in English, 2021 (elective course)
• Master specialization Design and Programming of Embedded Systems, in English, 2021 (elective course)
• Master specialization Computer Science, in English, 2021 (VO)