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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.
Register for this course and see more details by visiting:
OpenCourser.com/course/o2gnws/mathematical
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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.
Read more
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.
Read more
Modular Exponentiation
A more in-depth understanding of modular exponentiation is crucial to understanding cryptographic mathematics. In this module, we will cover the square-and-multiply method, Eulier's Totient Theorem and Function, and demonstrate the use of discrete logarithms. After completing this module you will be able to understand some of the fundamental math requirement for cryptographic algorithms. You will also have a working knowledge of some of their applications.
Chinese Remainder Theorem
The modules builds upon the prior mathematical foundations to explore the conversion of integers and Chinese Remainder Theorem expression, as well as the capabilities and limitation of these expressions. After completing this module, you will be able to understand the concepts of Chinese Remainder Theorem and its usage in cryptography.
Primality Testing
Finally we will close out this course with a module on Trial Division, Fermat Theorem, and the Miller-Rabin Algorithm. After completing this module, you will understand how to test for an equality or set of equalities that hold true for prime values, then check whether or not they hold for a number that we want to test for primality.
Good to know
Good to know
Know what's good ,
what to watch for , 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
Crypto foundations for beginners
Learners say this engaging course is very informative and provides a solid foundation in the mathematical concepts behind cryptography. Despite some drawbacks, such as a fast pace and lack of sufficient examples, students widely appreciate the well-organized content and practical approach that has helped them gain a deeper understanding and appreciation for the subject.
Concepts are well-applied to real-world examples
"Very interesant course. A very practical approach to modular arithmetics."
Organized and informative
"The course content and the assignments were quite meticulously designed and delivered efficiently."
"Introduction assez complète aux mathématiques nécessaires à la cryptologie, avec des exemples précis en fin de cours autour de l'algorithme RSA."
Provides a strong foundation in cryptography
"It was done very well"
"This course provided me a better insight into the mathematical foundations of crytpography."
"This course does a really nice job introducing you to the basics of math for cryptography"
Not enough examples are provided
"It seems almost abandoned. The additional material linked are simply wikipedia pages and the video lectures simply talk through the subjects with very little explanation or examples."
"You end up scrubbing through the videos to find the two seconds that you need to answer quiz questions."
Content is delivered at a fast pace
"The instructor goes through content too fast."
"Not enough practices are given before the assignment."
"Overall quality is not ideal for the new learner."
Some concepts are not explained clearly
"The teaching was not at all good, he was just reading it."
"Lessons are OK, but you need to watch in x1.25-x1.5 speed, to be honest."
"To whom it may concern, I love cryptography since it is an interesting subject. I enrolled for this course seeking from basics. but some concepts were not explained well."
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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.
Identify mentors in the field of cryptography
Show steps
Build connections and gain insights from experts.
Show steps
Show steps
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
Show steps
Review basic mathematical concepts that underpin cryptographic algorithms.
Browse courses on
Modular Arithmetic
Show steps
Show steps
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
Show steps
Gain a comprehensive understanding of cryptography and coding theory to supplement the course content.
View
Modern Cryptography
on Amazon
Show steps
Show steps
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
Show steps
Deepen understanding of modular exponentiation, which is essential for understanding public-key cryptography.
Show steps
Show steps
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
Show steps
Enhance understanding through peer interaction and knowledge exchange.
Show steps
Show steps
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
Show steps
Gain a deeper understanding of the Chinese Remainder Theorem and its applications in cryptography.
Show steps
Show steps
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
Show steps
Apply cryptographic principles to build a working encryption algorithm.
Show steps
Show steps
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
Show steps
Gain practical experience in implementing cryptographic algorithms and contribute to the community.
Show steps
Show steps
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
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.
Cryptographer
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.
Security Consultant
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.
Information Security Analyst
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.
Cybersecurity Engineer
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.
Software Developer
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.
Data Scientist
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.
Mathematician
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.
Computer Scientist
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.
Actuary
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.
Financial Analyst
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.
Operations Research 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.
Statistician
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.
Teacher
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.
Writer
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.
For more career information including salaries, visit:
OpenCourser.com/course/o2gnws/mathematical
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.
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.
Comprehensive reference on applied cryptography, and it covers a wide range of topics, including the mathematical foundations of cryptography.
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.
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.
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.
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.
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.
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.
Provides a comprehensive overview of classical cryptography techniques, as well as the mathematical background.
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.
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.
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.
Save
Provides a thorough introduction to abstract algebra, which is essential for understanding the mathematical foundations of cryptography.
Provides a solid foundation in number theory, which is essential for understanding the mathematical foundations of cryptography.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/o2gnws/mathematical
Share
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Effort
This is Course 2 in a 4-course specialization. Estimated workload: 15-hours per week.
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
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اُردُو,
فارسی
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tiếng Việt
پښتو
آذربایجان دیلی
This course belongs to a
collection
called
Introduction to Applied Cryptography. It's comprised of four courses:
- Asymmetric Cryptography and Key Management
- Mathematical Foundations for Cryptography (viewing)
- Classical Cryptosystems and Core Concepts
- Symmetric Cryptography
Good to know
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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