PLEASE BE ADVISED THAT THIS COURSE HAS BEEN CANCELLED!

Instructor: Professor Patrick M. Ingram, York University

Course Date: January 5th - April 6th, 2026
Mid-Semester Break: February 16th - 20th, 2026
Lecture Time: Mondays & Wednesdays | 2:30 PM - 4:00 PM (ET)
Office Hours: TBA

Registration Fee:

• Students from our Principal Sponsoring & Affiliate Universities: Free
• Other Students: CAD500ドル

Capacity Limit: 15

Format: Hybrid

Course Description

Public-key cryptography addresses the problem of establishing secure communication over an insecure channel, which went from a niche concern to an everyday consideration with the advent of the internet. Since the foundational contributions of Diffie and Hellman, the framework for public-key cryptography has largely come from abstract algebra, and this course surveys those foundations.

The main topics in the course are:

• Introduction/review of finite fields, their construction and uniqueness, and implementation of efficient calculations and linear algebra.
• The classical discrete logarithm problem (DLP) as a trapdoor function.
• DLP key exchange, cryptography, and signature schemes.
• Pohlig-Hellman and Index Calculus attacks on DLP cryptosystems.
• Elliptic curves, basic definitions and group law.
• The elliptic curve discrete log problem (ECDLP) as a trapdoor function.
• ECDLP versions of DLP schemes.
• Cryptanalysis of and using elliptic curves, pairing-based attacks on ECDLP.
• Schor's algorithm and quantum attacks on DLP- and ECDLP-based cryptosystems.
• Post-quantum cryptography.
• NTRU and lattice-based cryptography.
• McEliece and code-based cryptography.

Course Learning Outcomes:

Upon successful completion of the course, students should be able to:

• Explain the basic framework of cryptography, and the distinctions between private-key and public-key cryptography.
• Explain and implement computations in finite fields, and linear algebra over finite fields.
• Explain the concept of a trapdoor function, and why DLP is a viable trapdoor.
• Describe and implement Diffie-Hellman key exchange, ElGamal PKC and signatures.
• Implement Pohlig-Hellman and Index Calculus approaches to solving DLP, and explain the significance for DLP-based cryptosystems.
• Explain and implement operations on elliptic curves over finite fields, including efficient multiplication.
• Explain why the ECDLP and generalizations are viable candidates for trapdoor functions.
• Discuss generalizations of DLP to other algebraic groups (e.g., hyperelliptic curve cryptography).
• Describe and implement elliptic curve key exchange, PKC, and signatures.
• Implement pairing-based approaches to ECDLP, and explain the significance for elliptic curve cryptography.
• Explain the theoretical problems posed by quantum computing for secure DLP- and ECDLP-based cryptography.
• Describe the fundamental underlying mathematical problems in lattice-based cryptography, and implement variants of NTRU.
• Explain the mathematical basis for code-based cryptography, and implement the McEliece cryptosystem.

Course expectations: Students are expected to attend regularly, and participate in discussion, and auditing is permitted.

Prerequisites: Students should be familiar with modular arithmetic, and some basic abstract algebra (groups, rings, etc.).

Evaluation method: Homework assignments, attendance, and a project.