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Прибл. 16 часа на выполнение

Предполагаемая нагрузка: 4 weeks, 2-5 hours/week...


Субтитры: Английский, Греческий

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Number TheoryCryptographyModular Exponentiation

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Прибл. 16 часа на выполнение

Предполагаемая нагрузка: 4 weeks, 2-5 hours/week...


Субтитры: Английский, Греческий

Программа курса: что вы изучите

4 ч. на завершение

Modular Arithmetic

In this week we will discuss integer numbers and standard operations on them: addition, subtraction, multiplication and division. The latter operation is the most interesting one and creates a complicated structure on integer numbers. We will discuss division with a remainder and introduce an arithmetic on the remainders. This mathematical set-up will allow us to created non-trivial computational and cryptographic constructions in further weeks.

10 видео ((всего 90 мин.)), 4 материалов для самостоятельного изучения, 13 тестов
10 видео
Divisibility Tests5мин
Division by 212мин
Binary System11мин
Modular Arithmetic12мин
Modular Subtraction and Division11мин
4 материала для самостоятельного изучения
Python Code for Remainders5мин
12 практического упражнения
Division by 45мин
Four Numbers10мин
Division by 10110мин
Properties of Divisibility10мин
Divisibility Tests8мин
Division by 24мин
Binary System8мин
Modular Arithmetic8мин
Remainders of Large Numbers10мин
Modular Division10мин
4 ч. на завершение

Euclid's Algorithm

This week we'll study Euclid's algorithm and its applications. This fundamental algorithm is the main stepping-stone for understanding much of modern cryptography! Not only does this algorithm find the greatest common divisor of two numbers (which is an incredibly important problem by itself), but its extended version also gives an efficient way to solve Diophantine equations and compute modular inverses.

7 видео ((всего 78 мин.)), 4 материалов для самостоятельного изучения, 7 тестов
7 видео
Euclid’s Algorithm15мин
Extended Euclid’s Algorithm10мин
Least Common Multiple8мин
Diophantine Equations: Examples5мин
Diophantine Equations: Theorem15мин
Modular Division12мин
4 материала для самостоятельного изучения
Greatest Common Divisor: Code15мин
Extended Euclid's Algorithm: Code10мин
7 практического упражнения
Greatest Common Divisor10мин
Tile a Rectangle with Squares20мин
Least Common Multiple10мин
Least Common Multiple: Code15мин
Diophantine Equations15мин
Diophantine Equations: Code20мин
Modular Division: Code20мин
4 ч. на завершение

Building Blocks for Cryptography

Cryptography studies ways to share secrets securely, so that even eavesdroppers can't extract any information from what they hear or network traffic they intercept. One of the most popular cryptographic algorithms called RSA is based on unique integer factorization, Chinese Remainder Theorem and fast modular exponentiation. In this module, we are going to study these properties and algorithms which are the building blocks for RSA. In the next module we will use these building blocks to implement RSA and also to implement some clever attacks against RSA and decypher some secret codes.

14 видео ((всего 91 мин.)), 4 материалов для самостоятельного изучения, 6 тестов
14 видео
Prime Numbers3мин
Integers as Products of Primes3мин
Existence of Prime Factorization2мин
Euclid's Lemma4мин
Unique Factorization9мин
Implications of Unique Factorization10мин
Chinese Remainder Theorem7мин
Many Modules5мин
Fast Modular Exponentiation10мин
Fermat's Little Theorem7мин
Euler's Totient Function6мин
Euler's Theorem4мин
4 материала для самостоятельного изучения
Fast Modular Exponentiation7мин
5 практического упражнения
Integer Factorization20мин
Chinese Remainder Theorem: Code15мин
Fast Modular Exponentiation: Code20мин
Modular Exponentiation8мин
5 ч. на завершение


Modern cryptography has developed the most during the World War I and World War II, because everybody was spying on everybody. You will hear this story and see why simple cyphers didn't work anymore. You will learn that shared secret key must be changed for every communication if one wants it to be secure. This is problematic when the demand for secure communication is skyrocketing, and the communicating parties can be on different continents. You will then study the RSA cryptosystem which allows parties to exchange secret keys such that no eavesdropper is able to decipher these secret keys in any reasonable time. After that, you will study and later implement a few attacks against incorrectly implemented RSA, and thus decipher a few secret codes and even pass a small cryptographic quest!

9 видео ((всего 67 мин.)), 4 материалов для самостоятельного изучения, 2 тестов
9 видео
One-time Pad4мин
Many Messages7мин
RSA Cryptosystem14мин
Simple Attacks5мин
Small Difference5мин
Insufficient Randomness7мин
Hastad's Broadcast Attack8мин
More Attacks and Conclusion5мин
4 материала для самостоятельного изучения
Many Time Pad Attack10мин
Randomness Generation10мин
Slides and External References10мин
2 практического упражнения
RSA Quiz: Code
RSA Quest - Quiz6мин
Рецензии: 28Chevron Right


начал новую карьеру, пройдя эти курсы


получил значимые преимущества в карьере благодаря этому курсу

Лучшие отзывы о курсе Number Theory and Cryptography

автор: PWNov 22nd 2018

I was really impressed especially with the RSA portion of the course. It was really well explained, and the programming exercise was cleverly designed and implemented. Well done.

автор: LJan 2nd 2018

A good course for people who have no basic background in number theory , explicit clear explanation in RSA algorithm. Overall,a good introduction course.



Alexander S. Kulikov

Visiting Professor
Department of Computer Science and Engineering

Michael Levin

Computer Science

Vladimir Podolskii

Associate Professor
Computer Science Department

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