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
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: MWF 3-4 W (UC 139). Additional hours by appointment.
TA
Joseph Urbano
Office Hours: T 10-12 (CLC)
e-mail: jau35@drexel.edu
Meeting Time
MWF 9:00-9:50 in Science Center (SC) 3401 Market 326
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. Jeffrey Hoffstein, Jill Pipher, and Joseph Silverman An Introduction to Mathematical Cryptography, 2nd Edition, Springer, 2014.
  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, Discrete Log
  5. Public-key cryptosystems (RSA, El Gamel)
  6. Side chanel attacks
  7. Primality testing
  8. Integer factorization
  9. Elliptic Curve Cryptography
  10. Lattice Cryptography
  11. Cryptographic protocols
  12. Digital cash
  13. Homomorphic encryption


Grading

  1. In Class Labs and Quizzes (40%)
  2. Homework Assignments (60%)
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. Students will receive the higher grade with or without a curve.

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


Look Here for Important Announcements

See Piazza Discussion Forum.


Lectures

This list is subject to change.
  1. Week 1 (Introduction - Substitution Cyphers and Classical Cryptography)
    1. Introduction to Cryptography and Cryptanalysis
    2. Polyalphabetic Substitution Cyphers
  2. Week 2 Introduction to Maple and Modular Arithmetic
    1. Introduction to Maple and Maple worksheets
    2. Modular Arithmetic
  3. Week 3 (Number Theoretic Algorithms and Modern Cryptography - Chapter 1 and 3)
    1. Euclidean Algorithm and Fast Powering
    2. Introduction to Modern Cryptography and RSA Public Key Encryption
  4. Week 4 (Primality Testing - Sections 3.4 and 3.9)
    1. (Unique Factorization, the Sieve of Eratosthenes and the Prime Number Theorem)
    2. (Strong Pseudoprimes and a Probabalistic Primality Test)
  5. Week 5 (Probabalistic Encryption - Section 3.10)
    1. (Quadratic Reciprocity and the Solovay-Strassen Primality Test)
    2. Mar. 2, 2016 (Blum Coin Flipping Protocol and Goldwasser-Micali Probabalistic Encryption)
  6. Week 6 (Integer Factorization - Sections 3.5-3.7 )
    1. (Introduction to Integer Factorization Algorithms)
    2. (Dixon's Algorithm)
  7. Week 7 (Attacks on RSA - Chapter 3 )
    1. (The Quadratic Sieve)
    2. (Attacking RSA)
  8. Week 8 (Diffie-Hellman Key Exchange - Chapter 2)
    1. (El Gamal Public Key Encryption and Diffie-Hellman Key Exchange)
    2. (Collision Algorithms and the Discrete Log Problem)
  9. Week 9 (Thanksgiving)
    1. Catch Up lecture
  10. Week 10 (Elliptic Curve Cryptography - Chapter 5)
    1. (Introduction to Elliptic Curves and Elliptic Curve Cryptography)
  11. Week 11 (Lattice Based Cryptography - Chapter 6)
    1. (Merkle-Hellman Cryptosystem)
    2. Introduction to Lattices


Programs and Worksheets


Assignments

Solutions

 
Created: 9/26/05 (revised 9/24/17) by jjohnson@cs.drexel.edu