Number Theory and Cryptography

Go to Course: https://www.coursera.org/learn/number-theory-cryptography

Introduction

### Course Review: Number Theory and Cryptography on Coursera #### Overview In an age where digital security is paramount, the course "Number Theory and Cryptography" offers an insightful journey into the mathematical underpinnings of cryptography. While renowned mathematician G. H. Hardy once labeled number theory as contrived and seemingly useless, advancements in technology have revealed its critical applications, especially in the field of secure communications. This course draws attention to the paradigm shift that number theory underwent, most notably with the development of the RSA algorithm—a cryptographic standard that protects countless online transactions today. #### Course Structure The course is elegantly structured into a series of modules that guide learners through fundamental concepts in number theory and their direct applications in cryptography. Below is a detailed breakdown of the course syllabus: 1. **Modular Arithmetic** - This week introduces the basic operations on integers and focuses on the intriguing concept of division with a remainder. It lays the groundwork for more advanced computational and cryptographic constructions by exploring arithmetic on remainders. 2. **Euclid's Algorithm** - Learners delve into Euclid's algorithm, discovering its power in finding the greatest common divisor (GCD) and its sophisticated applications, such as solving Diophantine equations and calculating modular inverses, crucial for understanding cryptographic systems. 3. **Building Blocks for Cryptography** - This module shifts gears toward cryptography, particularly focusing on the RSA algorithm. It explains the importance of unique integer factorization, the Chinese Remainder Theorem, and fast modular exponentiation. Students will grasp these concepts, preparing them for both the implementation of RSA and understanding potential vulnerabilities in cryptographic systems. 4. **Cryptography** - The final module encompasses the historical context of cryptography, particularly its development during the World Wars. Students learn about the necessity for secure communications, especially in an era of rampant espionage. The RSA cryptosystem is explored in depth, enabling learners to implement cryptographic techniques and even engage in attempts to deconstruct poorly executed RSA implementations as part of an interactive cryptographic quest. #### Learning Outcomes Participants in this course will finish with a solid understanding of not only number theory but also its critical role in contemporary cryptography. They will be able to: - Apply modular arithmetic and Euclid’s algorithm to solve real-world problems. - Understand and implement the RSA encryption algorithm. - Recognize the historical evolution of cryptography and the ongoing significance of securing digital communications. #### Recommendation I wholeheartedly recommend "Number Theory and Cryptography" on Coursera for students, professionals, or anyone with an interest in mathematics, computer science, or cybersecurity. The course balances theory and practical application, making complex topics more accessible through clear explanations and interactive tasks. Whether you are looking to strengthen your mathematical foundation, understand the principles behind cryptography, or apply these concepts in practical scenarios, this course provides comprehensive coverage that is both engaging and educational. Plus, the course is designed in a way that encourages learners to think critically about security issues in our increasingly digital world. In conclusion, enroll in "Number Theory and Cryptography" and equip yourself with essential skills that are not just useful but vital in today's tech-centric society!

Syllabus

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.

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.

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.

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!