Sage/Cocalc local server: mathcrypto.byu.edu (if broken, you can use cocalc.com)

Substitution/affine cipher solver

Continued fraction low exponent attack

**Instructor**: Paul Jenkins

**Office**: 279 TMCB, 801-422-5868

**Email**: jenkins@math.byu.edu (for most questions)

**Lecture**: 2:00-3:15 PM TTh, 3108 JKB

**Office hours**: 2:00-2:50 PM MWF or by appointment

**TA**: Darren Lund, darrenlund@mathematics.byu.edu

**TA office hours**: Th 11-11:50 in 149 TMCB computer lab, 12-1 MWF online at https://byu.zoom.us/j/95003671989?pwd=ZEN6TmIrTmdGbjluR0wvY0dPcTVCdz09

**Textbook**: Introduction to Cryptography with Coding Theory, Second Edition, Wade Trappe and Lawrence C. Washington, Pearson Prentice Hall, ISBN 0-13-186239-1. Please note the list of known errors in the textbook online at http://www.math.umd.edu/~lcw/cryptoerrata2.pdf and mark your book accordingly if you wish.

**Grading**: Homework and projects 30%, reading assignments 10%, two midterms 15% each, final exam 30%. Grades will be available on BYU Learning Suite.

**Exams**: Two midterm exams in the testing center on October 5-7 and November 19-23. A study guide for the first exam is available here. A study guide for the second exam is available here. The final exam will be a take home exam, due at the scheduled final exam time (3:00 PM, Wednesday December 16). The final exam will cover all material studied this semester.

**Homework**: Homework will be due on Tuesdays and Fridays at 5:00 PM, submitted on Learning Suite. Homework assignments will be posted on the course website. Your homework should be neat and should include enough detail that another student from the class could follow your arguments. Homework that is excessively sloppy may receive less than full credit. Late homework will not be accepted. Working in groups on homework is encouraged, but each student should write up each problem, without looking at other students’ written solutions. The three homework assignments with the lowest score will be dropped. There will also be larger projects assigned from time to time; details will be announced in class and will appear on the course website. Project scores may not be dropped.

**Prerequisites**: Math 213 (linear algebra) or equivalent, or instructor's consent. Math 371 is recommended. Some prior experience with programming and with a computer algebra system such as Sage, Maple, or Mathematica is desirable, but not essential.

**Course Description**: This is a course in the mathematics and algorithms of modern cryptography. It complements, rather than being equivalent to, the CS course on Computer Security (CS 465) and the IT course on Cybersecurity and Penetration Testing (IT 567).

**Minimal Learning Outcomes**: The student should gain a understanding of the following topics. In particular this includes knowing the definitions, being familiar with standard examples, and being able to solve mathematical and algorithmic problems by directly using the material taught in the course. This includes appropriate use of Sage, Maple, Mathematica, or other software.

- Classical systems, including
- Substitution theory
- Block ciphers
- Enigma

- Elementary number theory as follows.
- Euclid's algorithm
- Modular arithmetic and the algorithm for modular exponentiation
- Chinese Remainder Theorem
- Fermat and Euler Theorems
- Primitive roots
- Legendre and Jacobi symbols
- Elementary continued fractions
- Simple discussion of finite fields

- The DES and AES encryption standards
- RSA and its strengths and weaknesses; attacks on RSA
- Wiener's continued fraction attack on low decryption exponent

- Primality testing algorithms
- Factorization techniques
- The quadratic sieve

- Discrete logarithms, Diffie-Hellman key exchange, ElGamal

Additional topics may include, but are not limited to: Elliptic curve cryptography, birthday attacks and probability, quantum cryptography (key distribution, Shor's algorithm), hash functions, digital signatures, secret sharing, and lattices and lattice algorithms (the LLL algorithm, NTRU system, lattice attacks on RSA).

This is a 3 credit class. The BYU Catalog states that “the expectation for undergraduate courses is three hours of work per week per credit hour for the average student who is appropriately prepared; much more time may be required to achieve excellence.” Thus, an average student should expect to spend at least 6 hours per week outside of lecture on working problems, reading the textbook, reviewing concepts, and completing assignments.

**Electronic devices**: Computers will be necessary to complete some homework problems. On exams, only testing center calculators may be used. Do not use mobile phones or permit them to ring during class. Office hours will be held by the TA in the computer lab in 149 TMCB on Thursdays from 11:00 to 11:50 AM to answer computer questions.

**COVID-19:** While COVID-19 conditions persist and until further notice, students and faculty are required to wear masks at all times during class; faculty are not at liberty to waive this expectation. Students who feel sick, including exhibiting symptoms commonly associated with COVID-19 (fever; cough; shortness of breath/difficulty breathing; chills; muscle pain; sore throat; new loss of taste or smell; etc.) should not attend class and should work with their instructor to develop a study plan for the duration of the illness.

**Preventing & Responding to Sexual Misconduct**: In accordance with Title IX of the Education Amendments of 1972, Brigham Young University prohibits unlawful sex discrimination against any participant in its education programs or activities. The university also prohibits sexual harassment-including sexual violence-committed by or against students, university employees, and visitors to campus. As outlined in university policy, sexual harassment, dating violence, domestic violence, sexual assault, and stalking are considered forms of "Sexual Misconduct" prohibited by the university.

University policy requires all university employees in a teaching, managerial, or supervisory role to report all incidents of Sexual Misconduct that come to their attention in any way, including but not limited to face-to-face conversations, a written class assignment or paper, class discussion, email, text, or social media post. Incidents of Sexual Misconduct should be reported to the Title IX Coordinator at t9coordinator@byu.edu or (801) 422-8692. Reports may also be submitted through EthicsPoint at https://titleix.byu.edu/report or 1-888-238-1062 (24-hours a day).

BYU offers confidential resources for those affected by Sexual Misconduct, including the university's Victim Advocate, as well as a number of non-confidential resources and services that may be helpful. Additional information about Title IX, the university's Sexual Misconduct Policy, reporting requirements, and resources can be found at http://titleix.byu.edu or by contacting the university's Title IX Coordinator.

**Student Disability**: Brigham Young University is committed to providing a working and learning atmosphere that reasonably accommodates qualified persons with disabilities. A disability is a physical or mental impairment that substantially limits one or more major life activities. Whether an impairment is substantially limiting depends on its nature and severity, its duration or expected duration, and its permanent or expected permanent or long-term impact. Examples include vision or hearing impairments, physical disabilities, chronic illnesses, emotional disorders (e.g., depression, anxiety), learning disorders, and attention disorders (e.g., ADHD). If you have a disability which impairs your ability to complete this course successfully, please contact the University Accessibility Center (UAC), 2170 WSC or 801-422-2767 to request a reasonable accommodation. The UAC can also assess students for learning, attention, and emotional concerns. If you feel you have been unlawfully discriminated against on the basis of disability, please contact the Equal Employment Office at 801-422-5895, D-285 ASB for help.

**Honor Code**: In keeping with the principles of the BYU Honor Code, students are expected to be honest in all of their academic work. Academic honesty means, most fundamentally, that any work you present as your own must in fact be your own work and not that of another. Violations of this principle may result in a failing grade in the course and additional disciplinary action by the university. Students are also expected to adhere to the Dress and Grooming Standards. Adherence demonstrates respect for yourself and others and ensures an effective learning and working environment. It is the university's expectation, and every instructor's expectation in class, that each student will abide by all Honor Code standards. Please call the Honor Code Office at 422-2847 if you have questions about those standards.