Mathematics for Cryptology

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Information Security

Synopsis:

Students become familiar with parts of mathematics necessary for deeper understanding of the methods used in

symmetric and asymmetric cryptography. They learn the mathematical principles on which security of encryption

systems, cryptanalysis methods, cryptography over elliptic curves, and quantum cryptography are based.

Requirements:

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

Syllabus of lectures:

1. General Algebra: Group, ring, eld, vector space.

2. Extension of nite elds and choice of their bases.

3. (2) Algebraic equations: Grobner bases.

4. (2) Solving algebraic equations over nite elds.

5. Discrete logarithm: Die-Hellman key exchange, ElGamal encryption system.

6. Discrete logarithm: Babystep-giantstep algorithm, Pollard's rho method.

7. Discrete logarithm: Pohlig-Hellman algorithm.

8. Elliptic curves over real numbers and Galois elds.

9. Factoring using elliptic curves, the MOV algorithm.

10. Quantum computing: foundations of quantum mechanics, qubit and operations with it.

Syllabus of tutorials:

Examples of various mathematical structures will be discussed.

Study Objective:

Study materials:

1. Hostein, J. - Pipher,J. - Silverman, J. H. An Introduction to Mathematical Cryptography. Springer, 2008. ISBN 978-1441926746.

2. Lidl, R. - Niederreiter, N. Finite Fields. Cambridge University Press, 2008. ISBN 978-0521065672.

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

4. Nielsen, M. A. - Chuang, I. L. Quantum Computation and Quantum Information. Cambridge University Press, 2011. ISBN 978-1107002173.

Note:

Further information:

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

No time-table has been prepared for this course

The course is a part of the following study plans: