Algorithmic Number Theory and Cryptography (CS 300)

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: 100C University Crossings
phone: (215) 895-2669
e-mail: jjohnson AT cs DOT drexel DOT edu
office hours: R2-4. Additional hours by appointment.

TA

Lingchuan Meng

Office: Univ. Crossings 147 (CS Student Resource Center)
e-mail: lingchuan DOT meng AT drexel DOT edu
office hours: TF 2:00-4:00.

Meeting Time

TR 12:30-2:00 in Univ. Crossings 153

Course Mailing List

BbVista will be used for discussion and announcements

Please use the discussion groups 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. David M. Bressoud, Factorization and Primality Testing, Springer, 1989.
2. Every student must have access to Maple version 11. 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.
3. Students will be required to read several additional papers.

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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. Coin flipping protocols (Blum)
8. Primality testing
9. Algorithms for integer factorization

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Grading

1. Class Participation (15%)
2. Three Homework assignments (45%)
3. Two Quizes (40%)

Resources

Reference Books

1. Maple Getting Started Guide.
2. Maple Users Manual.
3. Maple Introductory Programming Guide.
4. Maple Advanced Programming Guide.

Web Pages

1. Waterloo Maple
2. Maple Student Center
3. Maple Essentials
4. Maple Programming
5. Maple Application Center
6. SymbolicNet -- Symbolic Mathematical Computation Information Center
7. The Prime Pages
8. GIMPS: The Great Internet Mersenne Prime Search

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Look Here for Important Announcements

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Lectures

1. Week 1 (Chapter 1)
1. April 1, 2008 (Unique Factorization)
2. April 3, 2008 (The Euclidean Algorithm)
2. Week 2 (Chapters 1 and 2)
1. April 6, 2008 (Analysis of the Euclidean Algorithm)
2. April 8, 2008 (The Sieve of Eratosthenes and the Prime Number Theorem)
3. Week 3 (Chapter 2)
1. April 10, 2008 (Finding Large Primes - Mersenne Primes)
2. April 17, 2008 (Fast Integer Multiplication)
4. Week 4 (Chapters 3-4 and additional readings)
1. April 22, 2008 (Modular Arithmetic and Fast Powering)
2. April 24, 2008 (Modular Arithmetic and Fast Powering)
5. Week 5 (Midterm Exam)
1. April 29, 2008 (Review)
2. May 1, 2008 (Exam 1)
6. Week 6 (RSA)
1. May 6, 2008 (RSA Public Key Encryption)
2. May 8, 2008 (RSA Public Key Encryption)
7. Week 7 (Chapter 6, 7, 9)
1. May 6, 2008 (Strong Pseudoprimes and a Probabalistic Primality Test)
2. May 8, 2008 (Quadratic Reciprocity and the Solovay-Strassen Primality Test)
8. Week 8 (Chapter 9 and Handouts)
1. May 15, 2008 (Primitive Element Theorem)
2. May 17, 2008 (Deterministic Primality Testing)
9. Week 9 (Chapters 5 and 8)
1. May 20, 2008 (Introduction to Integer Factorization Algorithms)
2. May 22, 2008 (The Quadratic Sieve)
10. Week 10 (TBD)
1. May 29, 2008 (TBD)
2. June 8, 2006 (TBD)

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Programs and Worksheets

• To be added.

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Assignments and Exams

• Assignment 1 - assign1.mw (15%) - Due Sunday April 20 at midnight.
  
• Assignment 2 - assign2.mw (15 %) - Due Tue. May 20 at midnight.
  
• Assignment 3 - Assignment 3 (20 %) - TBD.
  
• Exam 1 - Studyguide 1 (20%) - in class on Thur. May 1
• Exam 2 - Midterm 2 (20%) - Take home exam to be done with Maple, due Tue. June 3 at midnight.
• In class lab (in lieu of assignment 3) - final_lab.mw, Thur. June 5.

Solutions

• None yet.