Mathematics for Cryptology

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Code Completion Credits Range Language
NI-MKY Z,ZK 5 3P+1C Czech
Garant předmětu:
Róbert Lórencz
Martin Jureček
Martin Jureček, Olha Jurečková, Ivana Trummová
Department of Information Security

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.


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

Syllabus of lectures:

1. Ideals in rings, factor rings, rings of polynomials.

2. Finite field extensions and choice of bases in them.

3. Solving algebraic equations over finite fields: relinearization, XL, and XSL algorithms.

4. Solving algebraic equations over finite fields: conversion to the SAT problem and methods solving this problem.

5. Groebner bases, Buchberger algorithm, Faugere F4 algorithm.

6. Factorization: Pollard rho method, p-1 method, Fermat factorization.

7. Factorization: sieve methods.

8. Discrete logarithm: Pohlig-Hellman algorithm, Baby-step giant-step algorithm, Pollard rho method.

9. Discrete logarithm: Index calculus.

10. Elliptic curves over real numbers and Galois fields.

11. ECDLP, elliptic curve factorization.

12. Lattice-based cryptography, GGH encryption system.

13. 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.

Further information:
Time-table for winter semester 2023/2024:
Time-table is not available yet
Time-table for summer semester 2023/2024:
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The course is a part of the following study plans:
Data valid to 2023-11-30
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