SYLLABUS
A rough outline of the course is available here. Lecture notes from the 2019 edition of the course are available on OCW. The 2021 edition will be similar but may include some new material on isogeny graphs and quaternion algebras relevant to post-quantum elliptic curve cryptography (post-quantum refers to protocols that are not known to be vulnerable to a quantum-polynomial-time attack, the algorithms are intended to run a classical computer).
For spring 2021 all classes and office hours will be held online. I may pre-record or post-record some lecture content for asynchronous consumption in order to allow more time for interaction during the lectures.
Text Book
There is no required text; lecture notes will be provided. We will make reference to material in the following books, all of which can be accessed electronically from MIT (see the MIT Libraries web page for offsite access).
Elliptic Curves: Number Theory and Cryptography, Second Edition, Lawrence C. Washington. (errata)
Elliptic Curves, J.S. Milne. (errata)
The Arithmetic of Elliptic Curves, Joseph H. Silverman. (errata)
Advanced Topics in the Arithmetic of Elliptic Curves, Joseph H. Silverman. (errata)
Primes of the form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication, David A. Cox. (errata)
The following two books give quite accessible introductions to elliptic curves from different perspectives. You may find them useful as supplemental reading, but we will not use of them in the course.
Elliptic Curves in Cryptography, Blake, Seroussi, and Smart.
Rational Points on Elliptic Curves, Joseph H. Silverman and John Tate.
The following references provide introductions to algebraic number theory and complex analysis; neither of these topics is an official prerequisites for this course, but we will occasionally need to make use of their results.
Algebraic number theory, J.S. Milne.
Complex analysis, Serge Lang
Software
Some of the theorems and algorithms presented in lecture will be demonstrated using Sage, a python-based computer algebra system, hosted on CoCalc (all 18.783 students will be provided free access). Most of the problem sets will contain at least one computationally-focused problem, which you will likely want to use Sage to solve, but you are free to use other packages, or roll your own code from scratch. You will be graded on your results and your mathematical explanation and analysis of your algorithm, not your code.
Problem Sets
There will be weekly problem sets, each of which typically contain three to five multi-part problems. You are not expected to solve all of the problems, you be given the option to choose a subset to turn in. Some problems are purely theoretical in nature, while others are more computationally focused; those who prefer proofs to programming (or vice versa) can choose problems that appeal to their interests.
Problem sets are to be prepared in typeset form (typically via latex) and submitted electronically as pdf-file to Gradescope. Collaboration is permitted/encouraged, but you must write up your own solutions and explicitly identify any collaborators, or simply give the name of your pset group on pset partners, as well as any resources you used that are not listed above. There will be computational problems for which the correct answer will be different for every student, based on a unique identifier derived from your student ID.
Late Policy
Late problem sets will not be accepted; your lowest problem set score is dropped when computing your grade, so you can skip one with no penalty. Please do not ask for permission to turn in a problem set late; if you find yourself short on time it is better to skip a problem set and get an early start on the next one than to put additional stress on yourself by trying to finish two problem sets in the same week.
Grading
Your grade will be primarily determined by your average problem set score, after dropping your lowest score, plus bonus points you can earn by participating in Zoom polls held in class -- you will get one point of extra credit for each question you answer (correctly or incorrectly). That might not sound like much, but over the course of the term it could be enough to make up for an entire problem set you missed.
Update: As announced in class on 3/1/2021, Problem Set 1 will be excluded when computing your grade unless your second worst score on the remaining problem sets is worse than your score on Problem Set 1, in which case your worst two problem set scores will be dropped and Problem Set 1 will be included. If you did not turn in Problem Set 1, there is no penalty and your lowest score on the remaining problem sets will still be dropped.
There is no curve, your grade will be based on a standard scale (anything over 97.5 is an A+, 92.5 to 97.5 is an A, 90 to 92.5 is an A-, etc...), so you will always know where you stand -- note this applies to graduate students as well as undergraduates, there is not a separate scale for graduate students. The problem sets generally get harder as the course progresses, so it is better to bank points early than to rely on making up lost ground later.
Disability Accommodations
Please contact Disability and Access Services as early in the term as possible, if you have not already done so.
If you already have an accommodation letter, please be sure to submit a copy to Mathematics Academic Services.
Even if you do not plan to use any accommodations, if there is anything I can do to facilitate your learning, please let me know.
