Number Theory and Cryptography from Coursera

A prominent expert in the number theory Godfrey Hardy described it in the beginning of 20th century as one of the most obviously useless branches of Pure Mathematics”. Just 30 years after his death, an algorithm for encryption of secret messages was developed using achievements of number theory. It was called RSA after the names of its authors, and its implementation is probably the most frequently used computer program in the world nowadays. Without it, nobody would be able to make secure payments over the internet, or even log in securely to e-mail and other personal services. In this course we will start with the basics of the number theory and get to cryptographic protocols based on it. By the end, you will be able to apply the basics of the number theory to encrypt and decrypt messages, and to break the code if one applies RSA carelessly. You will even pass a cryptographic quest!

As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students.

What's inside

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.

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.

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.

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!

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Explores number theory, which is standard in cryptography industry

Taught by Alexander S. Kulikov and Michael Levin, who are recognized for their work in number theory

Develops skills in modular arithmetic, cryptography, and encryption, which are core skills for IT professionals and high school students

Builds a strong foundation for beginners in number theory and cryptography

May require prior knowledge of basic math, such as squares and fractions, as well as basic programming in Python

Focuses on RSA encryption, which is still widely used but may have some limitations

Reviews summary

Number theory and cryptography

Learners say this is a foundational course for students new to number theory and cryptography. It's well received, building on previous courses in the specialization. Key features include interactive lectures, video content, practice problems, code exercises, and a final RSA puzzler that ties together key concepts. However some students mention that the course is difficult and that the proofs could be explained more clearly.

Overall, this course is largely positive

"This course was fantastic."

"Great course! Instructional material is well put together with a nice balance of reading, videos, practice problems, quizzes, and coding exercises"

"Bravo! Oh man, the "RSA Quest" was the most amazing thing ever!"

Learners especially enjoyed the RSA final project.

"RSA puzzler as a "final exam" is worth a special mention -- the instructors manage to tie together the presented number theory and RSA cryptosystem concepts and practice in a very hands-on way, scavenger hunt style. Fun!"

"The best course I've taken on Courses – and I've completed 20+ MOOCs!"

Some learners found the last chapters difficult

"In all honesty, this was my least favorite course in this specialization so far."

"This module was the worst module in the whole specialization course."

"The last chapter was way too challenging compared to the previous ones."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Number Theory and Cryptography with these activities:

Practice Python Programming

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Ensures you have a strong foundation in Python, which is necessary for implementing cryptographic algorithms in the course.

Browse courses on Python

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Review Python syntax and data structures.
Solve coding exercises on Python platforms.

Review Modular Arithmetic Concepts

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Refreshes your knowledge of modular arithmetic, which is essential for understanding more advanced topics in the course.

Browse courses on Modular Arithmetic

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Review notes or textbooks on modular arithmetic.
Solve practice problems involving modular arithmetic.

Discuss Cryptographic Concepts with Classmates

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Fosters collaboration, deepens understanding through peer-to-peer learning, and expands perspectives on the subject matter.

Browse courses on Cryptography

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Find a study partner or group.
Discuss specific cryptographic concepts or problems.
Share insights and learn from each other's perspectives.

Five other activities

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Show all eight activities

Read 'The Code Book' by Simon Singh

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Provides a comprehensive overview of the history and applications of number theory and cryptography, helping you build a strong foundation for the course.

View Fermat's Enigma on Amazon

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Read and understand the first three chapters of the book.
Summarize the key concepts and ideas presented in the book.

Follow tutorials on RSA Encryption and Decryption

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Strengthens your understanding of the practical implementation of cryptographic algorithms covered in the course.

Browse courses on RSA

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Find tutorials on RSA encryption and decryption algorithms.
Follow the tutorials to implement an RSA encryption and decryption program.

Solve Cryptographic Puzzles

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Sharpens your problem-solving skills and enhances your ability to apply number theory concepts to real-world scenarios.

Browse courses on Number Theory

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Find websites or books that offer cryptographic puzzles.
Attempt to solve the puzzles using the concepts learned in the course.

Create a Visual Explanation of the RSA Algorithm

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Solidifies your understanding of RSA and improves your ability to communicate complex concepts clearly.

Browse courses on RSA

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Choose a visual format (e.g., flowchart, diagram).
Map out the steps of the RSA algorithm in the chosen format.
Create the visual explanation.

Connect with Cryptography Experts

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Provides access to valuable insights, guidance, and support from experienced professionals in the field.

Browse courses on Cryptography

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Attend industry events or online forums.
Reach out to experts in your network or through professional organizations.

Career center

Learners who complete Number Theory and Cryptography will develop knowledge and skills that may be useful to these careers:

Cryptographer

Cryptographers use their knowledge of mathematics and computer science to design and implement cryptographic systems. This course may be useful for Cryptographers because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and developing cryptographic algorithms. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to cryptography.

See salaries and explore the career path for Cryptographer

Information Security Analyst

Information Security Analysts use their knowledge of computer security and information technology to protect organizations from cyber attacks. This course may be useful for Information Security Analysts because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and mitigating cyber threats. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to network security and data protection.

See salaries and explore the career path for Information Security Analyst

Security Analyst

Security Analysts use their knowledge of computer security and information technology to protect organizations from cyber attacks. This course may be useful for Security Analysts because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and mitigating cyber threats. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to network security and data protection.

See salaries and explore the career path for Security Analyst

Network Security Engineer

Network Security Engineers use their knowledge of computer security and networking to protect organizations from cyber attacks. This course may be useful for Network Security Engineers because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and mitigating cyber threats. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to network security.

See salaries and explore the career path for Network Security Engineer

Cybersecurity Engineer

Cybersecurity Engineers use their knowledge of computer security and information technology to protect organizations from cyber attacks. This course may be useful for Cybersecurity Engineers because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and mitigating cyber threats. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to cybersecurity.

See salaries and explore the career path for Cybersecurity Engineer

Software Engineer

Software Engineers use their knowledge of computer science and programming to design, develop, and maintain software systems. This course may be useful for Software Engineers because it provides a foundation in number theory and cryptography, which are essential concepts for secure software development. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to software security.

See salaries and explore the career path for Software Engineer

Privacy Analyst

Privacy Analysts use their knowledge of privacy laws and regulations to ensure that organizations comply with privacy requirements. This course may be useful for Privacy Analysts because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and implementing privacy measures. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to privacy.

See salaries and explore the career path for Privacy Analyst

Compliance Officer

Compliance Officers use their knowledge of laws and regulations to ensure that organizations comply with all applicable requirements. This course may be useful for Compliance Officers because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and implementing compliance measures. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to compliance.

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Data Protection Officer

Data Protection Officers use their knowledge of data protection laws and regulations to ensure that organizations comply with data protection requirements. This course may be useful for Data Protection Officers because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and implementing data protection measures. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to data protection.

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Information Technology Manager

Information Technology Managers use their knowledge of information technology to manage and maintain the IT systems of organizations. This course may be useful for Information Technology Managers because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and managing IT systems. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to IT security.

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Computer Scientist

Computer Scientists use their knowledge of computer science and mathematics to design, develop, and implement computer systems. This course may be useful for Computer Scientists because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and developing computer systems. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to computer science.

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Risk Manager

Risk Managers use their knowledge of risk management principles and practices to identify, assess, and mitigate risks for organizations. This course may be useful for Risk Managers because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and managing risks. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to risk management.

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Auditor

Auditors use their knowledge of accounting and auditing principles and practices to examine and evaluate the financial records of organizations. This course may be useful for Auditors because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and auditing financial records. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to auditing.

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Forensic Accountant

Forensic Accountants use their knowledge of accounting and auditing principles and practices to investigate financial crimes. This course may be useful for Forensic Accountants because it provides a foundation in number theory and cryptography, which are essential concepts for understanding and investigating financial crimes. The course also covers topics such as modular arithmetic, Euclid's algorithm, and RSA encryption, which are all relevant to forensic accounting.

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Data Scientist

Data Scientists use their knowledge of mathematics, statistics, and computer science to extract insights from data. This course may be useful for Data Scientists because it provides a foundation in number theory, which is essential for understanding the statistical models and algorithms used in data science. The course also covers topics such as modular arithmetic and RSA encryption, which are relevant to data security and privacy.

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