Number-theoretic methods in cryptography
Description: The discipline is one of the main profiling components in the training of specialists in the field of information security. Within the framework of the discipline, questions are considered about the number-theoretic principles of constructing cryptographic systems with a symmetric and asymmetric key, mathematical methods for calculating the reliability, stability of cryptographic systems, methods for constructing mathematical models of protected information, ciphers, cryptographic systems and cryptographic protocols.
Amount of credits: 5
Пререквизиты:
- Discrete Mathematics
Course Workload:
Types of classes | hours |
---|---|
Lectures | 15 |
Practical works | 30 |
Laboratory works | |
SAWTG (Student Autonomous Work under Teacher Guidance) | 30 |
SAW (Student autonomous work) | 75 |
Form of final control | Exam |
Final assessment method | Written exam |
Component: University component
Cycle: Base disciplines
Goal
- study of the mathematical foundations of cryptography, mathematical methods for constructing cryptographic systems, ciphers and protocols, and methods for calculating their reliability (cryptostrength).
Objective
- a clear understanding of the need and importance of mathematical training for a competitive specialist;
- familiarization with the basics of classical and modern number theory, which have practical applications for solving some problems of the professional sphere;
- familiarization of students with mathematical methods for calculating the reliability of cryptographic systems.
Learning outcome: knowledge and understanding
- the basic properties of modulo comparisons, algorithms for checking numbers for primality and constructing large prime numbers; number decomposition algorithms
Learning outcome: applying knowledge and understanding
- correctly apply the apparatus of mathematical analysis and number-theoretic methods in solving professional problems
Learning outcome: formation of judgments
- logically correctly, reasonably and clearly build oral and written speech in the language of instruction,
Learning outcome: communicative abilities
- prepare and edit texts for professional purposes, publicly present their own and well-known scientific results, conduct discussions.
Learning outcome: learning skills or learning abilities
- have the skills to develop effective algorithms for solving applied problems; be ready for use in professional work of mathematical methods and tools
Teaching methods
- lectures and online lectures, practical exercises using slides and other multimedia tools.
Assessment of the student's knowledge
Teacher oversees various tasks related to ongoing assessment and determines students' current performance twice during each academic period. Ratings 1 and 2 are formulated based on the outcomes of this ongoing assessment. The student's learning achievements are assessed using a 100-point scale, and the final grades P1 and P2 are calculated as the average of their ongoing performance evaluations. The teacher evaluates the student's work throughout the academic period in alignment with the assignment submission schedule for the discipline. The assessment system may incorporate a mix of written and oral, group and individual formats.
Period | Type of task | Total |
---|---|---|
1 rating | Устный опрос | 0-100 |
ИДЗ | ||
Теоретический опрос | ||
Промежуточный контроль | ||
2 rating | Устный опрос | 0-100 |
ИДЗ | ||
Теоретический опрос | ||
Промежуточный контроль | ||
Total control | Exam | 0-100 |
The evaluating policy of learning outcomes by work type
Type of task | 90-100 | 70-89 | 50-69 | 0-49 |
---|---|---|---|---|
Excellent | Good | Satisfactory | Unsatisfactory |
Evaluation form
The student's final grade in the course is calculated on a 100 point grading scale, it includes:
- 40% of the examination result;
- 60% of current control result.
The final grade is calculated by the formula:
FG = 0,6 | MT1+MT2 | +0,4E |
2 |
Where Midterm 1, Midterm 2are digital equivalents of the grades of Midterm 1 and 2;
E is a digital equivalent of the exam grade.
Final alphabetical grade and its equivalent in points:
The letter grading system for students' academic achievements, corresponding to the numerical equivalent on a four-point scale:
Alphabetical grade | Numerical value | Points (%) | Traditional grade |
---|---|---|---|
A | 4.0 | 95-100 | Excellent |
A- | 3.67 | 90-94 | |
B+ | 3.33 | 85-89 | Good |
B | 3.0 | 80-84 | |
B- | 2.67 | 75-79 | |
C+ | 2.33 | 70-74 | |
C | 2.0 | 65-69 | Satisfactory |
C- | 1.67 | 60-64 | |
D+ | 1.33 | 55-59 | |
D | 1.0 | 50-54 | |
FX | 0.5 | 25-49 | Unsatisfactory |
F | 0 | 0-24 |
Topics of lectures
- Криптография
- Наибольший общий делитель
- Простые и составные числа
- Критерий взаимной простоты
- Сравнимость целых чисел по модулю данного натурального числа
- Кольцо вычетов по модулю данного числа
- Малая теорема Ферма и её следствие
- Группа подстановок
- Неприводимые многочлены над полем вычетов по модулю простого числа
- Сравнения первого порядка и системы сравнений
- Сравнения высших порядков
- Символы Лежандра и Якоби
- Псевдослучайные последовательности над конечным полем, их применение в криптографии
- Факторизация чисел
- Числа Кармайкла
Key reading
- И.М. Виноградов Основы теории чисел. – М.: Наука, 2021. – 402 с.
- А.А. Бухштаб. Теория чисел. — С.-Пб. Лань, 2020, 384 с.
- Н. Н. Осипов. Теория чисел. — Красноярск: Изд. Сибирского федерального университета, 2008 г. 117 с.
- В.М. Фомичев. Дискретная математика и криптология. – М.: Диалог МИФИ, 2003. – 400 с.
- В.А. Романьков Введение в криптографию. Курс лекций. – М.: Форум, 2012. - 240 с.
- О.Н. Жданов, К.К. Елемесов. Сборник задач по криптографическим методам защиты информации: Учеб. пособие. – Алматы: КазНТУ имени К. И. Сатпаева, 2014. – 73 с.
- Е.Г. Кукина, В.А. Романьков Введение в криптографию. Сборник задач и упражнений. – Омск: Изд-во ОмГУ, 2013.
- Қ.Ә. Əбдіқалықов. Криптографияның негіздері: Оқулық. Алматы. 2012 ж. - 184 бет
- ОРАЗБАЕВ Б. М. Сандар теориясы. Алматы. "Мектеп" 1979. - 393 бет
Further reading
- В.И. Нечаев. Элементы криптографии, основы теории защиты информации, – М.: Высшая школа, 1999. –172 с.
- Гашков С.Б., Чубариков В.Н. Арифметика. Алгоритмы. Сложность вычислений. – М.: Высшая школа. – 320 с.
- А.В. Рожков, О.В. Ниссенбаум. Теоретико-числовые методы в криптографии: Учебное пособие. – Тюмень: Изд-во ТюмГУ, 2007.
- Панкратова И.А. Теоретико-числовые методы криптографии: Учебное пособие. -Томск: Томский государственный университет, 2009. - 120 с.
- Рябко Б.Я., Фионов А.Н. Криптографические методы защиты информации: - М. Горячая линия -Телеком,р 2005.- 229с.