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
publickey 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) 8952893
email: jjohnson AT cs DOT drexel DOT edu
office hours: M 24 and W 24 (UC 139). Additional hours by appointment.
 TA
 Ken Fox
Office Hours: F 122 (CS Student Resource Center)
email: kwf26@drexel.edu
 Meeting Time
 MW 9:0010:20 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
 Wade Trappe and Lawrence Washington,
Introduction to Cryptography with Coding Theory,
2nd Edition, Prentice Hall, 2006.
 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.
 Students will be required to read several additional
papers.
Topics
 Maple Computer Algebra System
 Integer and polynomial arithmetic
 Euclidean algorithm and continued fractions
 Modular Arithmetic, Fermat's theorem, Chinese Remainder Theorem
 Symmetric key cryptosystems (DES, AES)
 Publickey cryptosystems (RSA, El Gamel)
 Side chanel attacks
 Primality testing
 Integer factorization
 Cryptographic protocols
 Digital cash
 Homomorphic encryption
Grading
 Class Participation (25%)
 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
 Maple Getting Started Guide.
 Maple Users Manual.
 Maple Introductory Programming Guide.
 Maple Advanced Programming Guide.
 Web Pages
 Waterloo Maple
 Maple Student Center
 Maple
Essentials
 Maple
Programming
 Maple Application Center
 SymbolicNet 
Symbolic Mathematical Computation Information Center
 The Prime Pages
 GIMPS: The Great Internet
Mersenne Prime Search
Look Here for Important
Announcements
See Piazza Discussion Forum.
Lectures
This list is subject to change.
 Week 1 (Introduction  Chapter 2)
 Jan. 4, 2016
(Introduction to Cryptography and Cryptanalysis)
 Jan. 6, 2016
(Polyalphabetic Substitution Cyphers)
 Week 2 (Introduction to Modern Cryptography and Basic Number Theory  Chapter 3 and 6 )
 Jan. 11, 2016
(Introduction to Modern Cryptography)
 Jan. 13, 2016 (Modular Arithmetic and Fast Powering)
 Week 3 (Modular Arithmetic, Linear Algebra and Matrices  Chapter 2 and 3)
 (Univ. Holiday) Jan. 18, 2016
 Jan. 20, 2016
(Linear Algebra and Hill Ciphers)
 Week 4 (RSA  Chapter 6 )
 Jan. 25, 2016 (RSA Public Key Encryption)
 Jan. 27, 2016 (Complexity of gcd and Powering)
 Week 5 (Attacks on RSA  Chapter 6)
 Feb. 1, 2016 (Attacking RSA)
 Feb. 3, 2016 (Timing Attacks on RSA)
 Week 6 (Symmetric Key Encryption  Chapters 4 and 5)
 (Maple Lab) Feb. 8, 2016
 Feb. 10, 2016
(Data Encryption Standard)
 Week 7 (DiffieHellman Key Exchange  Chapters 7 and 9)
 Feb. 15, 2016
(El Gamal Public Key Encryption and DiffieHellman Key Exchange)
 Feb. 17, 2016
(Collision Algorithms and the Discrete Log Problem)
 Week 8 (Primality Testing  Chapter 6)
 Feb. 22, 2016
(Unique Factorization, the Sieve of Eratosthenes and the Prime Number Theorem)
 Feb. 24, 2016
(Strong Pseudoprimes and a Probabalistic Primality Test)
 Week 9 (Probabalistic Encryption  Chapter 3 and 8)
 Feb. 29, 2016
(Quadratic Reciprocity and the SolovayStrassen Primality Test)
 Mar. 2, 2016
(Blum Coin Flipping Protocol and GoldwasserMicali Probabalistic Encryption)
 Week 10 (Integer Factorization  Chapter 6)
 Mar. 7, 2016
(Introduction to Integer Factorization Algorithms)
 Mar. 9, 2016
(Dixon's Algorithm)
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.
Created: 9/26/05 (revised 1/11/16) by jjohnson@cs.drexel.edu