Boolean functions - NMMB331

• Title: Booleovské funkce
• Guaranteed by: Department of Algebra (32-KA)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2020
• Semester: winter
• E-Credits: 3
• Hours per week, examination: winter s.:2/0, Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: taught
• Language: English
• Teaching methods: full-time
• Guarantor: doc. Faruk Göloglu, Dr. rer. nat.
• Teacher(s): doc. Faruk Göloglu, Dr. rer. nat.
• Class: M Mgr. MMIB; M Mgr. MMIB > Povinně volitelné
• Classification: Mathematics > Algebra

Annotation

English

The course is devoted to vectorial nonlinear Boolean functions.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (09.05.2018)

Czech

Kurz se zabývá nelineárními vektorovými booleovskými funkcemi.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (09.05.2018)

Aim of the course

This course is on the mathematical analysis of cryptographically important

(Boolean) functions on vector spaces over finite fields

and their relationship to combinatorics, algebra and geometry.

Optimal Boolean functions that are invulnerable to differential, linear,

algebraic and correlation attacks are discussed. These include bent and

perfect nonlinear functions.

The relevant combinatorial objects that we will study include Latin squares,

Hadamard matrices, difference sets, orthogonal arrays and block

designs as well as projective planes from finite geometry and algebraic

field-like structures such as finite semifields and quasifields.

Last update: Göloglu Faruk, doc., Dr. rer. nat. (01.10.2024)

Course completion requirements

English

Grading:

• Final, written/oral (60%)

• 3 homeworks (40%)

Last update: Göloglu Faruk, doc., Dr. rer. nat. (01.10.2024)

Literature

English

Textbooks: Lecture notes will be provided every week after lectures. The

following textbooks are going to be used.

• Hall, Marshall, Jr.: Combinatorial theory. Second edition John Wiley

& Sons, Inc., New York, 1986. xviii+440 pp.

• van Lint, J. H., Wilson, R. M.: A course in combinatorics. Second

edition Cambridge University Press, Cambridge, 2001. xiv+602 pp.

• Hou, Xiang-dong: Lectures on finite fields. Grad. Stud. Math., 190

American Mathematical Society, Providence, RI, 2018. x+229 pp.

• Stinson, Douglas R.: Combinatorial designs and cryptography, revis-

ited. 50 years of combinatorics, graph theory, and computing, 335–357.

Discrete Math. Appl. (Boca Raton) CRC Press, Boca Raton, FL, 2020.

Last update: Göloglu Faruk, doc., Dr. rer. nat. (01.10.2024)

Requirements to the exam

English

Students have to pass final test/oral exam based on the material covered in the lectures.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (09.05.2018)

Czech

Zkouška má ústní formu. Její požadavky odpovídají obsahu přednesené látky.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (10.06.2019)

Syllabus

English

Course Outline:

i. Introduction (1 week)

ii. Bent and perfect nonlinear functions (2–3 weeks)

iii. Finite fields and polynomials over finite fields (1 week)

iv. Notions of equivalence for Boolean functions (1 week)

v. Bent functions, Hadamard matrices and difference sets (2 weeks)

vi. LFSRs (linear feedback shift registers), pseudo-noise sequences

(1 week)

vii. Difference sets in cyclic groups (1 week)

viii. Correlation immunity, orhogonal arrays (1 week)

ix. Latin squares, secret sharing (1 week)

x. Möbius inversion, algebraic immunity (1 week)

xi. Projective planes, planar functions, semifields, quasifields (1–2

weeks)

Last update: Göloglu Faruk, doc., Dr. rer. nat. (01.10.2024)