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. 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.

Note:

Further information:

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

No time-table has been prepared for this course

The course is a part of the following study plans:

• Master specialization Computer Security, in Czech, 2020 (PS)
• Master specialization Design and Programming of Embedded Systems, in Czech, 2020 (elective course)
• Master specialization Computer Systems and Networks, in Czech, 202 (elective course)
• Master specialization Management Informatics, in Czech, 2020 (elective course)
• Master specialization Software Engineering, in Czech, 2020 (elective course)
• Master specialization System Programming, in Czech, version from 2020 (elective course)
• Master specialization Web Engineering, in Czech, 2020 (elective course)
• Master specialization Knowledge Engineering, in Czech, 2020 (elective course)
• Master specialization Computer Science, in Czech, 2020 (elective course)
• Mgr. programme, for the phase of study without specialisation, ver. for 2020 and higher (VO)
• Master Specialization Digital Business Engineering, 2023 (VO)
• Master specialization System Programming, in Czech, version from 2023 (elective course)
• Master specialization Computer Science, in Czech, 2023 (elective course)