# Algorithmic Number Theory and Cryptography (CS 303)

Announcments  Lectures  Programs  Course Resources   Assignments and Solutions  Grading Policy
Course Description
Covers fundamental algorithms for integer arithmetic, greatest common divisor calculation, modular arithmetic, and other number theoretic computations. Algorithms are derived, implemented and analyzed for primality testing and integer factorization. Applications to cryptography are explored including symmetric and public-key cryptosystems. A cryptosystem will be implemented and methods of attack investigated.
Course Goals
To be able to implement and analyze algorithms for integer factorization and primality testing. To be able to use a system like Maple to explore concepts and theorems from number theory. To understand fundamental algorithms from symmetric key and public key cryptography.
Course Objectives
• To understand fundamental number theoretic algorithms such as the Euclidean algorithm, the Chinese Remainder algorithm, binary powering, and algorithms for integer arithmetic.
• To understand fundamental algorithms for symmetric key and public key cryptography.
• To understand the number theoretic foundations of modern cryptography and the principles behind their security.
• To implement and analyze cryptographic and number theoretic algorithms.
• To be able to use Maple to explore mathematical concepts and theorems.
Audience
Undergraduate computer science, computer engineering, mathematics, and students interested in security, cryptography and applied number theory. The course will cover both the underlying mathematical theory and practice as algorithms will be implemented and analyzed (the Maple computer algebra system will be used for implementation of algorithms and exploration of concepts). For computer science students, the course counts towards the numeric and symbolic computing and computer and network security tracks.
Prerequisites
Undergraduate data structures course (CS 260)
Courses in linear algebra (MATH 201 or equivalent), discrete mathematics (MATH 221 or equivalent).
Instructor
Jeremy Johnson
Office: 139 University Crossings
phone: (215) 895-2893
e-mail: jjohnson AT cs DOT drexel DOT edu
office hours: M 2-4 and W 2-4 (UC 139). Additional hours by appointment.
TA
Ken Fox
Office Hours: F 12-2 (CS Student Resource Center)
e-mail: kwf26@drexel.edu
Meeting Time
MW 9:00-10:20 in Univ. Crossings 153
Course Mailing List
See Piazza Discussion Forum for announcements and to ask questions and discuss course material.

Please use Piazza for questions and discussions related to the course. If you know the answer to someone's question, please feel free to jump in, as long as well it is not an answer to a homework problem. You should check the discussion and announcements regularly. Please do NOT post answers to homework.
Textbook
1. Wade Trappe and Lawrence Washington, Introduction to Cryptography with Coding Theory, 2nd Edition, Prentice Hall, 2006.
2. Every student must have access to Maple. Course notes will be provided on the web page as Maple worksheets that can not be read without Maple. Maple is available in the CS labs as well as Drexel labs, and is available for free to Drexel students as part of the campus site license.

### Topics

1. Maple Computer Algebra System
2. Integer and polynomial arithmetic
3. Euclidean algorithm and continued fractions
4. Modular Arithmetic, Fermat's theorem, Chinese Remainder Theorem
5. Symmetric key cryptosystems (DES, AES)
6. Public-key cryptosystems (RSA, El Gamel)
7. Side chanel attacks
8. Primality testing
9. Integer factorization
10. Cryptographic protocols
11. Digital cash
12. Homomorphic encryption

1. Class Participation (25%)
2. Homework Assignments (75%)
Assignments will be returned on a regular basis to provide feedback to students. All students must do their own work. Any violation of this will result in a zero grade for the assignment. A second violation will lead to an F for the course.

Grades are based on a curve with the mean normalized to a B provided the mean performance shows competency of the material.

### Resources

Reference Books
1. Maple Getting Started Guide.
2. Maple Users Manual.
3. Maple Introductory Programming Guide.
Web Pages

### Look Here for Important Announcements

See Piazza Discussion Forum.

### Lectures

This list is subject to change.
1. Week 1 (Introduction - Chapter 2)
2. Week 2 (Introduction to Modern Cryptography and Basic Number Theory - Chapter 3 and 6 )
3. Week 3 (Modular Arithmetic, Linear Algebra and Matrices - Chapter 2 and 3)
1. (Univ. Holiday) Jan. 18, 2016
2. Jan. 20, 2016 (Linear Algebra and Hill Ciphers)
4. Week 4 (RSA - Chapter 6 )
5. Week 5 (Attacks on RSA - Chapter 6)
6. Week 6 (Symmetric Key Encryption - Chapters 4 and 5)
1. (Maple Lab) Feb. 8, 2016
2. Feb. 10, 2016 (Data Encryption Standard)
7. Week 7 (Diffie-Hellman Key Exchange - Chapters 7 and 9)
8. Week 8 (Primality Testing - Chapter 6)
9. Week 9 (Probabalistic Encryption - Chapter 3 and 8)
10. Week 10 (Integer Factorization - Chapter 6)

### Programs and Worksheets

• Lab 1 - lab1.mw - practice Maple lab on list processing, shift ciphers, modular arithmetic, and affine cyphers.
• Lab 2 - lab2.mw - practice Maple lab on linear algebra, modular matrices and Hill ciphers.
• Lab 3 - lab3.mw - practice Maple lab on the RSA algorithm.
• Lab 4 - lab4.mw - practice Maple lab on RSA attacks.
• Lab 5 - lab5.mw - practice Maple lab on Probabalistic Primality testing.
• Lab 6 - lab6.mw - practice Maple lab on integer factorization (the quadratic sieve).

### Assignments

• Assignment 1 (Vigenere Cipher) - assign1.mw (15%) - Due Mon. Jan. 25 at midnight.
• Assignment 2 (Hill Ciphers and RSA) - assign2.mw (15%) - Due Mon. Feb. 8 at midnight.
• Assignment 3 (CRT and RSA attacks) - assign3.mw (15%) - Due Tue. Feb. 23 at midnight.
• Assignment 4 (Primitive roots, El Gamal and Coin Flipping) - assign4.mw (15%) - Due Tue. Mar. 8 at midnight.
• Assignment 5 (Discrete Logarithms and Digital Cash) - assign5.mw (15%) - Due Wed. Mar. 16 at midnight.