Mathematics for Cryptology
Code  Completion  Credits  Range  Language 

NIMKY  Z,ZK  5  3P+1C  Czech 
 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 (BILIN, BIZDM, NIMPI).
 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, p1 method, Fermat factorization.
7. Factorization: sieve methods.
8. Discrete logarithm: PohligHellman algorithm, Babystep giantstep algorithm, Pollard rho method.
9. Discrete logarithm: Index calculus.
10. Elliptic curves over real numbers and Galois fields.
11. ECDLP, elliptic curve factorization.
12. Latticebased 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 9781466570269.
2. Hoffstein, J.  Pipher, J.  Silverman, J. H. : An Introduction to Mathematical Cryptography. Springer, 2008. ISBN 9781441926746.
3. Lidl, R.  Niederreiter, H. : Finite Fields. Cambridge University Press, 2008. ISBN 9780521065672.
4. Menezes, A. J.  van Oorschot, P. C.  Vanstone, S. A. : Handbook of Applied Cryptography. CRC Press, 1996. ISBN 0849385237.
 Note:
 Further information:
 https://courses.fit.cvut.cz/NIMKY/
 No timetable 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)