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
- 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
- To understand the number theoretic foundations of modern
cryptography and the principles behind their security.
- To implement and analyze cryptographic and number theoretic
- To be able to use Maple to explore mathematical concepts and
- 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.
- Undergraduate data structures course (CS 260)
- Courses in linear algebra (MATH 201 or equivalent), discrete
mathematics (MATH 221 or equivalent).
- 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.
- Joseph Urbano
Office Hours: T 10-12 (CLC)
- 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.
- 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.
- Maple Computer Algebra System
- Integer and polynomial arithmetic
- Euclidean algorithm and continued fractions
- Modular Arithmetic, Fermat's theorem, Chinese Remainder Theorem, Discrete Log
- Public-key cryptosystems (RSA, El Gamel)
- Side chanel attacks
- Primality testing
- Integer factorization
- Elliptic Curve Cryptography
- Lattice Cryptography
- Cryptographic protocols
- Digital cash
- Homomorphic encryption
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.
- In Class Labs and Quizzes (40%)
- Homework Assignments (60%)
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.
- 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 Application Center
- SymbolicNet --
Symbolic Mathematical Computation Information Center
- The Prime Pages
- GIMPS: The Great Internet
Mersenne Prime Search
See Piazza Discussion Forum.
Look Here for Important
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
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 Solovay-Strassen Primality Test)
- Mar. 2, 2016
(Blum Coin Flipping Protocol and Goldwasser-Micali Probabalistic Encryption)
- Week 6 (Integer Factorization - Sections 3.5-3.7 )
(Introduction to Integer Factorization Algorithms)
- Week 7 (Attacks on RSA - Chapter 3 )
(The Quadratic Sieve)
- (Attacking RSA)
- Week 8 (Diffie-Hellman Key Exchange - Chapter 2)
(El Gamal Public Key Encryption and Diffie-Hellman 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)
- 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
- Lab 7 - lab7.mw - Explores the quadratic sieve integer
- 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, Goldwasser-Micali 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
Created: 9/26/05 (revised 9/24/17) by email@example.com