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The subject matter of a course is described in various texts.

NIE-MKY Mathematics for Cryptology Extent of teaching: 3P+1C

Instructor: Jureček M., Lórencz R. Completion: Z,ZK

Department: 18106 Credits: 5 Semester: L

Annotation:
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.

Lecture syllabus:

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.

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

Literature:

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.

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

Informace o předmětu a výukové materiály naleznete na https://courses.fit.cvut.cz/MI-MKY/

The course is also part of the following Study plans:

Study Plan Study Branch/Specialization Role Recommended semester

NIE-TI.21 Computer Science 2021 VO 2

NIE-NPVS.21 Design and Programming of Embedded Systems 2021 V 2

NIE-PB.21 Computer Security 2021 PS 2

NIE-PSS.21 Computer Systems and Networks 2021 V 2

NIE-SI.21 Software Engineering 2021 V 2

Page updated 26. 4. 2024, semester: Z/2020-1, L/2021-2, L/2019-20, L/2022-3, Z/2019-20, L/2020-1, L/2023-4, Z/2022-3, Z/2021-2, Z/2023-4, Z/2024-5, Send comments to the content presented here to Administrator of study plans Design and implementation: J. Novák, I. Halaška

NIE-MKY	Mathematics for Cryptology			Extent of teaching:	3P+1C
Instructor:	Jureček M., Lórencz R.			Completion:	Z,ZK
Department:	18106	Credits:	5	Semester:	L

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.

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.