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Cryptology: Math and Codes

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Eligibility: CTY-level or Advanced CTY-level math score required

Prerequisites: Successful completion of Grade 4 Mathematics or equivalent

Course Format: Individually Paced

Course Length: Typically 3-6 months

Course Code: CMC

Course Description


Cryptology: Math and Codes introduces students to the exciting practice of making and breaking secret codes. This popular course is designed as a mathematical enrichment offering for students in grades 5-7.

Students begin with simple Caesar Ciphers, learning to encrypt and decrypt messages as well as the history behind the cipher. They will move through history and more advanced mathematical concepts to learn substitution ciphers, Vigenère ciphers, and multiplicative and affine ciphers. Students will need to put all their newly acquired knowledge to the test by finishing with public key cryptography and the modern day RSA cryptosystem. This course intersects the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptology include ATM cards, computer passwords, and electronic commerce.

Throughout the course, students will have the opportunity to encrypt and decrypt messages, invent their own ciphers, discuss relevant historical events and literature, and learn some mathematical concepts that are often not seen until college!

Mathematical topics include:

  • positive and negative numbers
  • decimals and percents
  • data analysis and probability
  • prime numbers and factorization
  • modular arithmetic
  • inverses and modular inverses
  • exponentiation

Assignments are based on a text that is purchased separately by the student.

This course does not have any synchronous class meetings, but students may schedule one-on-one virtual meetings directly with the instructor to answer questions or concerns.

Virtual classrooms, and student activities in the classroom, may be recorded and added to the course as an ongoing asset for all class students to review. Students may be invited to interact in CTY community spaces that include students and instructors and potentially specially invited guests that are not enrolled in their course. Student contributions (e.g., projects, forum posts, etc.) may remain in the course after the student completes the course. These artifacts may be preserved to showcase student work or to continue important conversations.

Materials Needed

There is a textbook purchase required for this course:

The Cryptoclub: Using Mathematics to Make and Break Secret Codes. Janet Beissinger and Vera Pless (AK Peters, 2006)

Cryptoclub textbook cover

List of Topics

Introduction to Cryptology

  • Caesar Ciphers
  • ROT13
  • Steganography
  • Sending Messages with Numbers
  • Breaking Caesar Ciphers
  • Navajo Code Talkers

Substitution Ciphers

  • Keyword Ciphers
  • Cryptography in Fiction
  • Letter Frequencies
  • Breaking Substitution Ciphers
  • Nomenclators

Vigenère Ciphers

  • Combining Caesar Ciphers
  • Transposition Ciphers
  • Cracking Vigenère Ciphers Using Key Length
  • Factorization
  • Solving Problems Using Number Theory
  • Cracking Vigenère Ciphers Using Common Factors
  • Long Keywords

Modular Arithmetic

  • Introduction to Modular Arithmetic
  • The Zimmerman Telegram
  • Applications of Modular Arithmetic
  • Check Digits

Multiplicative and Affine Ciphers

  • Multiplicative Ciphers
  • Password Security
  • Using Inverses to Decrypt
  • The Enigma Machine
  • Affine Ciphers
  • Atbash and Pigpen Ciphers

Math for Modern Cryptography

  • Finding Prime Numbers
  • Properties of Exponents
  • Raising to Powers in Modular Arithmetic
  • Numeration Systems

Public Key Cryptography

  • The RSA Cryptosystem
  • Cryptographic Hash Functions
  • Revisiting Inverses in Modular Arithmetic
  • Sending RSA Messages

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In addition to the textbook, there are interactive websites and videos in the course.

In a Caesar Cipher, the alphabet is shifted a certain number of places and each letter of a message is replaced by the corresponding shifted letter.

sailboat, beach cartoon
mountains, trail with footprints cartoon

Students have the opportunity to decrypt secret messages while exploring the mysterious island of Manalupo or the vast forest of Fiordland in a course-long scavenger hunt!

Sample Quiz

sample quiz questions Cryptology Math Codes

Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

Zoom online virtual classroom
This course uses an online virtual classroom which can be used for instructor-student communication if the student has any questions about the course or curriculum. The classroom works on standard computers with the Zoom desktop client and also tablets or handhelds that support the Zoom Mobile app. Students will need a computer with the Zoom desktop client installed to watch any recorded meetings. The Zoom desktop client and Zoom Mobile App are both available for free download.