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#### 100% online

Start instantly and learn at your own schedule.

#### Approx. 16 hours to complete

Suggested: 4 weeks, 2-5 hours/week...

#### English

Subtitles: English, Greek

### Skills you will gain

Number TheoryCryptographyModular Exponentiation

#### 100% online

Start instantly and learn at your own schedule.

#### Approx. 16 hours to complete

Suggested: 4 weeks, 2-5 hours/week...

#### English

Subtitles: English, Greek

### Syllabus - What you will learn from this course

Week
1
4 hours to complete

## 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 videos (Total 90 min), 4 readings, 13 quizzes
10 videos
Divisibility6m
Remainders9m
Problems6m
Divisibility Tests5m
Division by 212m
Binary System11m
Modular Arithmetic12m
Applications7m
Modular Subtraction and Division11m
Python Code for Remainders5m
Slides1m
Slides1m
Slides1m
12 practice exercises
Divisibility15m
Remainders10m
Division by 45m
Four Numbers10m
Division by 10110m
Properties of Divisibility10m
Divisibility Tests8m
Division by 24m
Binary System8m
Modular Arithmetic8m
Remainders of Large Numbers10m
Modular Division10m
Week
2
4 hours to complete

## 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 videos (Total 78 min), 4 readings, 7 quizzes
7 videos
Euclid’s Algorithm15m
Extended Euclid’s Algorithm10m
Least Common Multiple8m
Diophantine Equations: Examples5m
Diophantine Equations: Theorem15m
Modular Division12m
Greatest Common Divisor: Code15m
Extended Euclid's Algorithm: Code10m
Slides1m
Slides10m
7 practice exercises
Greatest Common Divisor10m
Tile a Rectangle with Squares20m
Least Common Multiple10m
Least Common Multiple: Code15m
Diophantine Equations15m
Diophantine Equations: Code20m
Modular Division: Code20m
Week
3
4 hours to complete

## 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 videos (Total 91 min), 4 readings, 6 quizzes
14 videos
Prime Numbers3m
Integers as Products of Primes3m
Existence of Prime Factorization2m
Euclid's Lemma4m
Unique Factorization9m
Implications of Unique Factorization10m
Remainders7m
Chinese Remainder Theorem7m
Many Modules5m
Fast Modular Exponentiation10m
Fermat's Little Theorem7m
Euler's Totient Function6m
Euler's Theorem4m
Slides10m
Slides10m
Fast Modular Exponentiation7m
Slides10m
5 practice exercises
Integer Factorization20m
Remainders8m
Chinese Remainder Theorem: Code15m
Fast Modular Exponentiation: Code20m
Modular Exponentiation8m
Week
4
5 hours to complete

## Cryptography

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 videos (Total 67 min), 4 readings, 2 quizzes
9 videos
Many Messages7m
RSA Cryptosystem14m
Simple Attacks5m
Small Difference5m
Insufficient Randomness7m
More Attacks and Conclusion5m
Slides10m
Randomness Generation10m
Slides and External References10m
2 practice exercises
RSA Quiz: Code2h
RSA Quest - Quiz6m
4.6
28 Reviews

## 50%

started a new career after completing these courses

## 40%

got a tangible career benefit from this course

### Top reviews from Number Theory and Cryptography

By 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.

By 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.

## Instructors

### Alexander S. Kulikov

Visiting Professor
Department of Computer Science and Engineering

### Michael Levin

Lecturer
Computer Science

Associate Professor
Computer Science Department

## About University of California San Diego

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National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communicamathematics, engineering, and more. Learn more on www.hse.ru...

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Discrete Math is needed to see mathematical structures in the object you work with, and understand their properties. This ability is important for software engineers, data scientists, security and financial analysts (it is not a coincidence that math puzzles are often used for interviews). We cover the basic notions and results (combinatorics, graphs, probability, number theory) that are universally needed. To deliver techniques and ideas in discrete mathematics to the learner we extensively use interactive puzzles specially created for this specialization. To bring the learners experience closer to IT-applications we incorporate programming examples, problems and projects in our courses....