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Mathematical Foundations for Cryptography

4.6 Filled star Filled star Filled star Filled star Half star
Based on 323 ratings
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William Bahn, Richard White, and Sang-Yoon Chang

Welcome to Course 2 of Introduction to Applied Cryptography. In this course, you will be introduced to basic mathematical principles and functions that form the foundation for cryptographic and cryptanalysis methods. These principles and functions will be helpful in understanding symmetric and asymmetric cryptographic methods examined in Course 3 and Course 4. These topics should prove especially useful to you if you are new to cybersecurity. It is recommended that you have a basic knowledge of computer science and basic math skills such as algebra and probability.

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What's inside

Syllabus

Integer Foundations
Building upon the foundation of cryptography, this module focuses on the mathematical foundation including the use of prime numbers, modular arithmetic, understanding multiplicative inverses, and extending the Euclidean Algorithm. After completing this module you will be able to understand some of the fundamental math requirement used in cryptographic algorithms. You will also have a working knowledge of some of their applications.
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Traffic lights

Read about what's good
what should give you pause
and possible dealbreakers
Concepts are supported by examples, which helps learners understand real-world usage
Topics are broken down into discrete units that build upon one another
Modular exponentiation is a complex topic that is covered in a clear and concise manner
The importance of primality testing is explained in detail
The audience should have a basic understanding of algebra and probability

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Reviews summary

Essential math for applied cryptography

According to learners, this course provides the essential mathematical foundations necessary for understanding cryptography. Many found the material challenging but rewarding, highlighting that it effectively prepares them for subsequent courses in the series. While some reviewers praised the clarity of explanations for complex topics like modular arithmetic and primality testing, others noted the material could be quite dense and requires a strong background in prerequisite math (algebra and probability) to fully grasp. The course focuses heavily on theory, which is crucial for the field, but might not offer many hands-on applied exercises.
Views differ on lecture clarity.
"The instructor explained complex topics very clearly."
"Some lectures were difficult to follow and felt rushed."
"I had to rely on external resources sometimes to fully understand."
Requires solid prior math background.
"Make sure you have a solid background in algebra and probability before starting this."
"The prerequisites are real; don't underestimate the math level required."
"I struggled because my math skills were rusty, contrary to the stated prerequisites."
Good preparation for next courses.
"This course is absolutely necessary if you plan to take the next ones in the series."
"It lays a strong theoretical groundwork for applied crypto."
"After completing this, I felt much more prepared for Course 3."
Builds the core math needed for crypto.
"This course provides the essential mathematical foundations necessary for understanding cryptography."
"Crucial to understand the concepts here before moving onto applied cryptography."
"I learned the fundamental math requirements for cryptographic algorithms."
Math concepts can be quite difficult.
"The math concepts were challenging, but crucial to understand."
"I found the material dense and required extra effort to grasp."
"Some topics like discrete logarithms were particularly difficult to follow."

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 Mathematical Foundations for Cryptography with these activities:
Identify mentors in the field of cryptography
Build connections and gain insights from experts.
Show steps
  • Attend industry events and conferences.
  • Reach out to professionals on LinkedIn or other networking platforms.
  • Seek guidance and advice from professors or researchers.
Revisit modular arithmetic and primality testing
Review basic mathematical concepts that underpin cryptographic algorithms.
Browse courses on Modular Arithmetic
Show steps
  • Review the concept of modular arithmetic and practice finding remainders.
  • Explore divisibility tests for common numbers (2, 3, 5, and 9).
  • Learn different methods for primality testing, such as trial division and the Sieve of Eratosthenes.
Read "Introduction to Cryptography with Coding Theory" by W. Mao
Gain a comprehensive understanding of cryptography and coding theory to supplement the course content.
View Modern Cryptography on Amazon
Show steps
  • Read the book and take notes on key concepts.
  • Work through the practice problems and exercises.
  • Research additional topics of interest related to cryptography and coding theory.
Five other activities
Expand to see all activities and additional details
Show all eight activities
Solve practice problems on modular exponentiation
Deepen understanding of modular exponentiation, which is essential for understanding public-key cryptography.
Show steps
  • Practice calculating modular exponentiation using the square-and-multiply algorithm.
  • Solve problems involving Euler's Totient Theorem and Euler's Function.
  • Explore applications of modular exponentiation in cryptography.
Participate in a study group for the course
Enhance understanding through peer interaction and knowledge exchange.
Show steps
  • Find a study group or create one with fellow students.
  • Meet regularly to discuss the course material, ask questions, and solve problems together.
  • Take turns leading the discussions and presenting on specific topics.
Create a presentation on the Chinese Remainder Theorem
Gain a deeper understanding of the Chinese Remainder Theorem and its applications in cryptography.
Show steps
  • Research the Chinese Remainder Theorem and its mathematical properties.
  • Develop a presentation that explains the theorem and its applications.
  • Present the presentation to a group or online forum.
Develop a simple encryption algorithm
Apply cryptographic principles to build a working encryption algorithm.
Show steps
  • Design the encryption algorithm, including the key generation and encryption/decryption process.
  • Implement the algorithm in a programming language.
  • Test the algorithm with different inputs and keys.
  • Analyze the security of the algorithm and identify potential vulnerabilities.
Contribute to an open-source cryptographic library
Gain practical experience in implementing cryptographic algorithms and contribute to the community.
Show steps
  • Choose an open-source cryptographic library to contribute to.
  • Identify an issue or feature to work on.
  • Implement a solution and submit a pull request.
  • Collaborate with other contributors and maintain the contribution.

Career center

Learners who complete Mathematical Foundations for Cryptography will develop knowledge and skills that may be useful to these careers:
Cryptanalyst
A Cryptanalyst is a person who studies and analyzes cryptosystems in order to crack them and discover their weaknesses. The Mathematical Foundations for Cryptography course would be of great help to a Cryptanalyst as it covers the mathematical principles and functions that form the foundation for cryptographic and cryptanalysis methods. The course also covers specific topics such as modular exponentiation and primality testing, which are essential for understanding and breaking cryptosystems.
See salaries and explore the career path for Cryptanalyst
Cryptographer
A Cryptographer is a person who designs and develops cryptosystems and protocols to protect information from unauthorized access or disclosure. The Mathematical Foundations for Cryptography course would be of great help to a Cryptographer as it provides a solid understanding of the mathematical principles and functions that form the foundation for cryptographic methods. The course also covers specific topics such as modular exponentiation and primality testing, which are essential for developing secure and robust cryptosystems.
See salaries and explore the career path for Cryptographer
Security Consultant
A Security Consultant is a person who provides advice and guidance to organizations on how to protect their information systems from unauthorized access or disclosure. The Mathematical Foundations for Cryptography course would be of great help to a Security Consultant as it provides a solid understanding of the mathematical principles and functions that form the foundation for cryptographic methods. The course also covers specific topics such as modular exponentiation and primality testing, which are essential for understanding and evaluating the security of cryptosystems.
See salaries and explore the career path for Security Consultant
Information Security Analyst
An Information Security Analyst is a person who monitors and analyzes an organization's information systems to identify and mitigate security risks. The Mathematical Foundations for Cryptography course would be of great help to an Information Security Analyst as it provides a solid understanding of the mathematical principles and functions that form the foundation for cryptographic methods. The course also covers specific topics such as modular exponentiation and primality testing, which are essential for understanding and evaluating the security of cryptosystems.
See salaries and explore the career path for Information Security Analyst
Cybersecurity Engineer
A Cybersecurity Engineer is a person who designs, implements, and maintains cybersecurity systems to protect an organization's information systems from unauthorized access or disclosure. The Mathematical Foundations for Cryptography course would be of great help to a Cybersecurity Engineer as it provides a solid understanding of the mathematical principles and functions that form the foundation for cryptographic methods. The course also covers specific topics such as modular exponentiation and primality testing, which are essential for developing and maintaining secure cybersecurity systems.
See salaries and explore the career path for Cybersecurity Engineer
Software Developer
A Software Developer is a person who designs, develops, and maintains software applications. The Mathematical Foundations for Cryptography course may be of help to a Software Developer who is working on developing secure software applications. The course provides a solid understanding of the mathematical principles and functions that form the foundation for cryptographic methods, which can be used to protect software applications from unauthorized access or disclosure.
See salaries and explore the career path for Software Developer
Data Scientist
A Data Scientist is a person who uses data to solve business problems. The Mathematical Foundations for Cryptography course may be of help to a Data Scientist who is working on developing secure data analysis methods. The course provides a solid understanding of the mathematical principles and functions that form the foundation for cryptographic methods, which can be used to protect data from unauthorized access or disclosure.
See salaries and explore the career path for Data Scientist
Mathematician
A Mathematician is a person who studies mathematics. While the Mathematical Foundations for Cryptography course is not directly related to mathematics, it does cover some mathematical topics such as modular exponentiation and primality testing. This knowledge may be of help to a Mathematician who is interested in working in the field of cryptography.
See salaries and explore the career path for Mathematician
Computer Scientist
A Computer Scientist is a person who studies computer science. While the Mathematical Foundations for Cryptography course is not directly related to computer science, it does cover some mathematical topics such as modular exponentiation and primality testing. This knowledge may be of help to a Computer Scientist who is interested in working in the field of cryptography.
See salaries and explore the career path for Computer Scientist
Actuary
An Actuary is a person who uses mathematics to assess risk and uncertainty. While the Mathematical Foundations for Cryptography course is not directly related to actuarial science, it does cover some mathematical topics such as modular exponentiation and primality testing. This knowledge may be of help to an Actuary who is interested in working in the field of risk management.
See salaries and explore the career path for Actuary
Financial Analyst
A Financial Analyst is a person who uses financial data to make investment decisions. While the Mathematical Foundations for Cryptography course is not directly related to financial analysis, it does cover some mathematical topics such as modular exponentiation and primality testing. This knowledge may be of help to a Financial Analyst who is interested in using mathematical methods to analyze financial data.
See salaries and explore the career path for Financial Analyst
Operations Research Analyst
An Operations Research Analyst is a person who uses mathematical methods to solve business problems. While the Mathematical Foundations for Cryptography course is not directly related to operations research, it does cover some mathematical topics such as modular exponentiation and primality testing. This knowledge may be of help to an Operations Research Analyst who is interested in using mathematical methods to solve business problems.
See salaries and explore the career path for Operations Research Analyst
Statistician
A Statistician is a person who uses statistics to collect, analyze, and interpret data. While the Mathematical Foundations for Cryptography course is not directly related to statistics, it does cover some mathematical topics such as modular exponentiation and primality testing. This knowledge may be of help to a Statistician who is interested in using mathematical methods to analyze data.
See salaries and explore the career path for Statistician
Teacher
A Teacher is a person who teaches students. While the Mathematical Foundations for Cryptography course is not directly related to teaching, it does cover some mathematical topics such as modular exponentiation and primality testing. This knowledge may be of help to a Teacher who is interested in teaching mathematics or computer science.
See salaries and explore the career path for Teacher
Writer
A Writer is a person who writes books, articles, or other written content. While the Mathematical Foundations for Cryptography course is not directly related to writing, it does cover some mathematical topics such as modular exponentiation and primality testing. This knowledge may be of help to a Writer who is interested in writing about mathematics or computer science.
See salaries and explore the career path for Writer

Reading list

We've selected 14 books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Mathematical Foundations for Cryptography.
Cover image
Introduction to Cryptography with Coding Theory
Save
This popular textbook on coding theory and cryptography, which covers many of the mathematical principles and functions that form the foundation for cryptographic and cryptanalysis methods.
Introduction to Cryptography with Coding Theory
Hardcover
$$$$
Introduction to Cryptography: With Coding Theory by...
Paperback
$$$
Introduction to Cryptography with Coding Theory
Kindle Edition
$$$
Cover image
Handbook of Applied Cryptography
Save
Comprehensive reference on applied cryptography, and it covers a wide range of topics, including the mathematical foundations of cryptography.
Handbook of Applied Cryptography (Discrete...
Hardcover
$$$$
Handbook of Applied Cryptography (Discrete...
Kindle Edition
$$$
Cover image
Foundations of Cryptography: Volume 1, Basic Tools
Save
Provides a comprehensive overview of the basic tools used in cryptography, and it valuable resource for anyone who wants to learn more about the mathematical foundations of cryptography.
Foundations of Cryptography: Volume 1, Basic Tools
Kindle Edition
$$
Cover image
Cryptography and Secure Communication
Save
Provides a comprehensive overview of coding theory and cryptography, and it valuable resource for anyone who wants to learn more about the mathematical foundations of cryptography.
Cryptography and Secure Communication
Kindle Edition
$$$$
Cover image
Applied Cryptography
Save
Provides a comprehensive overview of applied cryptography, and it valuable resource for anyone who wants to learn more about the mathematical foundations of cryptography.
Applied Cryptography: Protocols, Algorithms and...
Hardcover
$$$
Applied Cryptography: Protocols, Algorithms, and...
Hardcover
$$$$
Applied Cryptography: Protocols, Algorithms, and...
Paperback
$$
Applied Cryptography: Protocols, Algorithms, and...
Paperback
$$$$
Applied Cryptography: Protocols, Algorithms and...
Kindle Edition
$$
Cover image
Cryptography and Network Security: Principles and...
Save
Provides a comprehensive overview of algebra for cryptographers, and it valuable resource for anyone who wants to learn more about the mathematical foundations of cryptography.
Cryptography and Network Security: Principles and...
Paperback
$$$
Cover image
A Classical Introduction to Modern Number Theory
Save
Provides a comprehensive overview of advanced number theory with applications, and it valuable resource for anyone who wants to learn more about the mathematical foundations of cryptography.
A Classical Introduction to Modern Number Theory...
Hardcover
$$$
Cover image
Introduction to Modern Cryptography
Save
Provides a comprehensive overview of the mathematical foundations of computer security, and it covers many of the mathematical principles and functions that form the foundation for cryptographic and cryptanalysis methods. This book introduces concepts and tools from number theory, algebra, and probability theory that are essential for understanding.
Introduction to Modern Cryptography: Third Edition...
Hardcover
$$$
Introduction to Modern Cryptography (Chapman &...
Hardcover
$$$
Introduction to Modern Cryptography (Chapman &...
Kindle Edition
$$$
Cover image
Introduction to Quantum Algorithms
Save
Provides a comprehensive overview of classical cryptography techniques, as well as the mathematical background.
Introduction to Quantum Algorithms (Pure and...
Paperback
$$$
Cover image
Cryptography and Network Security: Principles and...
Save
Provides a comprehensive overview of cryptography and network security, and it covers many of the mathematical principles and functions that form the foundation for cryptographic and cryptanalysis methods.
Cryptography and Network Security: Principles and...
Paperback
$$$
Cover image
Cryptology and Network Security
Save
Provides a comprehensive overview of cryptology and network security, and it covers many of the mathematical principles and functions that form the foundation for cryptographic and cryptanalysis methods.
Cryptology and Network Security: 21st International...
Paperback
$$$
Cryptology and Network Security: 21st International...
Kindle Edition
$$$
Cover image
The Art of Computer Programming, Volume 4B
Save
Provides a comprehensive overview of sequences and their applications, and it valuable resource for anyone who wants to learn more about the mathematical foundations of cryptography.
Slipcase for The Art of Computer Programming,...
Kindle Edition
$$$
Cover image
Abstract Algebra
Save
Provides a thorough introduction to abstract algebra, which is essential for understanding the mathematical foundations of cryptography.
Abstract Algebra
Paperback
Check
Abstract Algebra
Kindle Edition
Check
Cover image
An Introduction to the Theory of Numbers
Save
Provides a solid foundation in number theory, which is essential for understanding the mathematical foundations of cryptography.
An Introduction to the Theory of Numbers
Hardcover
$$$$

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Course image
Rating
4.6 Filled star Filled star Filled star Filled star Half star
Effort
Approx. 14 hours
Level
Introductory
Via
Coursera
Institution
University of Colorado System
Instructors
William Bahn
Richard White
Sang-Yoon Chang
Language
English
Transcripts
Español Français Português Deutsch Italiano 中文 العربية 한국어 日本語 हिंदी Nederlands Svenska Pусский Türkçe Polski Ελληνικά українська мова magyar nyelv Bahasa Indonesia ไทย қазақша اُردُو, فارسی বাংলা tiếng Việt پښتو آذربایجان دیلی
This course belongs to a collection called Introduction to Applied Cryptography. It's comprised of four courses:
  1. Asymmetric Cryptography and Key Management
  2. Mathematical Foundations for Cryptography (viewing)
  3. Classical Cryptosystems and Core Concepts
  4. Symmetric Cryptography
Traffic lights
Read about what's good
what should give you pause
and possible dealbreakers
Concepts are supported by examples, which helps learners understand real-world usage
Topics are broken down into discrete units that build upon one another
Modular exponentiation is a complex topic that is covered in a clear and concise manner
The importance of primality testing is explained in detail
The audience should have a basic understanding of algebra and probability
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