RET at UC

Research Experience for Teachers (2016-2017)

Number Theory and Internet Security

Jim Prugh's Poster
Author: Jim Prugh
Unit Title: Number Theory and Internet Security
Grade Level: HS
Subject: Mathematics
Estimated Duration: 2 weeks
Unit Activities:

Pre/Post Test:
Pre-Test  
Post-Test  
Keywords:
  • Prime Number
  • Quadratic Formula
  • Algorithm
  • Binary
  • Encryption
  • Decryption
  • Cipher
  • Frequency
  • Padding
  • Divisor

The Big Idea (including global relevance)

The importance of number theory in internet security.

Essential Questions

How is number theory applied in internet encryption?


The Hook

For the Hook, Students are shown a 10-minute clip on the Enigma Machine, and a discussion follows that addresses the following questions:

  • What was it used for?
  • Why was it so effective?
  • What was its flaw?
  • How was it defeated?

The Challenge

Students will design a secure method to exchange a message.


Guiding Questions

  • What is number theory?
  • What is the role of number theory in internet security?
  • Can I share a secret without it being intercepted and decrypted?
  • How can I protect my message in case it is intercepted?
  • How can I decrypt an intercepted message?
  • What are the different types of encryption methods?


ACS (Real world applications; career connections; societal impact)

Threats to internet security are prevalent on our everyday lives.

The unit is introduced with a 4-minute clip about Stuxnet and the hacking of the Iranian nuclear facilities. Activity 4 is designed to teach students the value using encryption to secure sensitive information. The unit closes with a clip on ethics in encryption and a reading and quiz over ethical issues surrounding internet security.

Mathematician, systems analyst, computer programmer, law enforcement (FBI, NSA, CIA). Arrangements will be made to bring a speaker to class to share internet security and law enforcement.

Misconceptions

  • A calculator is necessary for difficult computations.
  • Most of the information sent across the internet is secure.
  • It is impossible to securely send a message.
  • The mathematics behind encryption is very complicated.

Unit Lessons and Activities

  • Lesson 1: Introduce number theory
    Lesson 1 will focus on giving students an understanding of what number theory is as well as the importance of internet security. Students will investigate algorithms and number theory.
    • Activity 1: Introduce the big idea – Number Theory in Internet Security. Lesson begins with a video about Stuxnet, followed by a discussion of the implications. Students will generate their own challenge that will address the essential question. This is followed with a lesson on basic number theory. - 1 90 minute block
    • Activity 2: Investigate algorithms (Long division, Fermat’s Little Theorem with primes, Babylonian algorithm for square roots, Euclid’s theorem for divisors, The phi function) - 1 90 minute block
  • Lesson 2:Encryption
    Lesson 2 will focus on giving students an understanding of modular arithmetic, along with an exploration of the Diffie-Hellman algorithm as well as gaining an understanding of encryption. Students will send encrypted messages to one another.
    • Activity 3: Teach modular arithmetic using an activity that involves the Diffie-Hellman algorithm. - 2 50 min periods
    • Activity 4: Encryption activity - 2 block periods. The RSA lesson will take 1 regular class day
  • Evidence of CBL: lessons 1 through 4, activities 1 through 4
  • Evidence of EDP: lesson 4, activity 4


Additional Resources