V8a: Mastering Elliptic Curves in Cryptography 🔐

Welcome to "V8a: Elliptic Curves," the first lecture in the five-part series on elliptic curve cryptography (ECC), part of Alfred Menezes's "Crypto 101: Building Blocks" course. This video offers a comprehensive introduction to elliptic curves, a critical component of modern cryptography, essential for securing communications in software engineering, cybersecurity, and beyond. Designed for computer science students, mathematics majors, and cybersecurity professionals, this lecture dives into foundational concepts like the "elliptic curve defining equation," "number of points," and Hasse’s Theorem, laying the groundwork for understanding ECC.

You’ll also learn about key operations such as "point addition" and "point doubling" and discover how elliptic curves form an abelian group, making them powerful for cryptographic applications. Get insights into how elliptic curves are applied in protocols like ECDH (Elliptic Curve Diffie-Hellman) and ECDSA (Elliptic Curve Digital Signature Algorithm), which are widely used in secure communication.

This video is ideal for anyone seeking to understand how elliptic curves form the basis of advanced cryptographic schemes and is especially valuable for those in applied cryptography and cybersecurity. Dive into elliptic curve cryptography with this state-of-the-art free online course on one of today’s most vital security topics!

Lecture playlist:
https://www.youtube.com/playlist?list=PLA1qgQLL41SRsHMtsb4up5fTytDeHV_yJ

Course web page: https://cryptography101.ca/crypto101-building-blocks/
The slides are available on the course web page.

"Elliptic curve cryptography" chapter from "Textbook of Applied Cryptography": https://drive.google.com/file/d/1y30ccFhMI2_btwzCADyDmwSs8758tqlU/view

Other cryptography courses: https://cryptography101.ca

Slides
00:00 Introduction
02:18 Slide 295: Elliptic curves over R
04:16 Slide 296: Elliptic curves over R (2)
04:40 Slide 297: Elliptic curves over Z89
06:24 Slide 298: Example 1
08:44 Slide 299: Elliptic curves
09:45 Slide 300: Number of points
11:10 Slide 301: Point addition
12:27 Slide 302: Point addition (2)
14:32 Slide 303: Addition rule
18:02 Slide 304: Derivation of A3 (1)
12447 Slide 305: Derivation of A3 (2)
21:10 Slide 306: Example 1 (cont'd)
23:02 Slide 307: Abelian group
25:14 Concluding remarks

Corrections
16:50 Slide 303: I should have said "or y1 = -y2" instead of "or y1 = -2"
21:06 Slide 305: There is a missing parenthesis at the end of the formula for P+Q