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: MWF 34 W (UC 139). Additional hours by appointment.
 TA
 Joseph Urbano
Office Hours: T 1012 (CLC)
email: jau35@drexel.edu
 Meeting Time
 MWF 9:009:50 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
 Jeffrey Hoffstein, Jill Pipher, and Joseph Silverman
An Introduction to Mathematical Cryptography,
2nd Edition, Springer, 2014.
 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
 Maple Computer Algebra System
 Integer and polynomial arithmetic
 Euclidean algorithm and continued fractions
 Modular Arithmetic, Fermat's theorem, Chinese Remainder Theorem, Discrete Log
 Publickey cryptosystems (RSA, El Gamel)
 Side chanel attacks
 Primality testing
 Integer factorization
 Elliptic Curve Cryptography
 Lattice Cryptography
 Cryptographic protocols
 Digital cash
 Homomorphic encryption
Grading
 In Class Labs and Quizzes (40%)
 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
 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  Substitution Cyphers and Classical Cryptography)

Introduction to Cryptography and Cryptanalysis

Polyalphabetic Substitution Cyphers
 Week 2 Introduction to Maple and Modular Arithmetic
 Introduction to Maple and Maple worksheets
 Modular Arithmetic
 Week 3 (Number Theoretic Algorithms and Modern Cryptography  Chapter 1 and 3)
 Euclidean Algorithm and
Fast Powering

Introduction to Modern Cryptography and
RSA Public Key Encryption
 Week 4 (Primality Testing  Sections 3.4 and 3.9)

(Unique Factorization, the Sieve of Eratosthenes and the Prime Number Theorem)

(Strong Pseudoprimes and a Probabalistic Primality Test)
 Week 5 (Probabalistic Encryption  Section 3.10)

(Quadratic Reciprocity and the SolovayStrassen Primality Test)
 Mar. 2, 2016
(Blum Coin Flipping Protocol and GoldwasserMicali Probabalistic Encryption)
 Week 6 (Integer Factorization  Sections 3.53.7 )

(Introduction to Integer Factorization Algorithms)

(Dixon's Algorithm)
 Week 7 (Attacks on RSA  Chapter 3 )

(The Quadratic Sieve)
 (Attacking RSA)
 Week 8 (DiffieHellman Key Exchange  Chapter 2)

(El Gamal Public Key Encryption and DiffieHellman Key Exchange)

(Collision Algorithms and the Discrete Log Problem)
 Week 9 (Thanksgiving)
 Catch Up lecture
 Week 10 (Elliptic Curve Cryptography  Chapter 5)

(Introduction to Elliptic Curves and Elliptic Curve Cryptography)
 Week 11 (Lattice Based Cryptography  Chapter 6)

(MerkleHellman Cryptosystem)
 Introduction to Lattices
Programs and Worksheets
 Lab 1  lab1.mw  Introduces Maple programming and
data structures and reviews shift and substitution cyphers.
 Lab 2  lab2.mw  Further introduces Maple programming
and data structures, reviews modular arithmetic and introduces a generalization of the shift
cypher called an affine cypher.
 Lab 3  lab3.mw  Reviews the RSA public key cryptosystem,
provides an implementation of the RSA algorithm and asks students to use Maple's
factoring algorithm to break RSA for a modest size public key.
 Lab 4  lab4.mw  Explores various probabilistic
primality tests and reviews the theory behind them.
 Lab 5  lab5.mw  Explores quadratic reciprocity and
a coin flipping protocol.
 Lab 6  lab6.mw  Explores Dixon's integer factorization
algorithm.
 Lab 7  lab7.mw  Explores the quadratic sieve integer
factorization algorithm.
Assignments
 Assignment 1 (Vigenere Cypher) assign1.mw  Due Wed. Oct. 11 at 11:59pm
 Assignment 2 (Hill Cypher) assign2.mw  Due Wed. Oct. 25 at 11:59pm
 Assignment 3 (Square Roots, GoldwasserMicali and Zero Knowledge Proofs)) assign3.mw  Due Wed. Nov. 15 at 11:59pm
 Assignment 4 (El Gamal and Discrete Logs)) assign4.mw  Due Wed. Nov. 29 at 11:59pm
 Assignment 5 (Elliptic Curve Cryptography) assign5.mw  Due Fri. Dec. 8 at 11:59pm
 Final Assignment (Lattice Based Cryptography) assign6.mw  Due Wed. Dec. 13 at 11:59pm
Solutions
Created: 9/26/05 (revised 9/24/17) by jjohnson@cs.drexel.edu