I just spent a week working intensively with 5 people from very different backgrounds, whom I had mostly never met, on a problem that I’d never seen before Monday. And it was great, though I did need to sleep for 10 hours Saturday night to even partially recover. What strange thing just happened to me? I went to the Workshop on Algebraic Geometry for Coding Theory and Cryptography at IPAM.

IPAM, the Institute for Pure and Applied Mathematics, is a math institute funded primarily by the National Science Foundation, located on the campus of UCLA. IPAM does semester-length thematic programs as well as one-week workshops. Everett Howe, Kristin Lauter, and Judy Walker organized this project-based research workshop to bring together pure and applied mathematicians and encourage collaboration in the overlap of these fields. The model was adopted from the Women in Numbers workshops (see my last blog). Participants were organized into 6 groups of 5-7 people each. A leaders or pair of co-leaders brought a research question for each group, hopefully one that could yield at least a little progress in a week of hard work. Participants did not generally have any special knowledge in their assigned problem, except for background readings sent by the leaders before the workshop. Each group was a mix of well-established researchers, early-career people like me, and often a few graduate students. The plan—work hard, learn the problem, prove preliminary results, start a collaboration that will lead to a publishable paper in the next 3-6 months.

In the teaching world, I think a lot about active learning and how can I get my students to really engage with material though problem solving. I love this workshop model because it gives professional mathematicians the chance to learn new areas in one of same ways we think our students learn best.

My group studied variants of the McEliece cryptosystem. Devised in 1978 by Robert McEliece, this public key cryptosystem relies on the difficulty of decoding a random linear error-correcting code. In its original form it has never been broken. RSA, the most widely used public key cryptosystem in the world, was also devised (publicly) in 1978, making these two the longest-standing unbroken public key cryptosystems. However, currently nobody uses McEliece because the key sizes necessary for good security were seen as impractically large. Why use McEliece when RSA works so well?

However, McEliece is increasingly relevant as technological advances make storing large keys less problematic and quantum computers loom on the horizon. Thanks to Shor’s algorithm, a large-scale working quantum computer would make RSA and other currently used cryptosystems totally insecure. Government agencies are paying attention–NIST is planning to call for proposals for Post-Quantum cryptographic standards. McEliece’s system (as well as lattice-based systems and those based on multivariate polynomial problems) has not been found vulnerable to known quantum algorithms, so is a good post-quantum candidate. Many variants of McElice’s system involve using algebraic-geometry codes or other highly structured codes. Of course, order is bad in cryptography, so these variants are often vulnerable to attacks that uncover or exploit their structure. Our group was exploring what variants of McEliece’s original system could stand up to recent attacks.

Jumping into a problem like this with a bunch of people you’ve never met, especially as an early-career person, could be horrible. If people don’t try to connect with the whole group, are afraid or unwilling to speak up, or are not really invested, the whole thing goes nowhere. My group was great, though. I credit this to our general willingness to ask questions and openly admit and discuss the things we didn’t understand. Everybody’s background knowledge was relevant to the topic but nobody felt that they should already know every piece. It was the things that we didn’t know that gave us room to work together. I think that the presence of less experienced people encouraged more experienced people to ask extra questions, if only for the sake of others. Also, we developed our own private gameshow based on making up a disguising matrix and then guessing the dimension of the Schur product of the disguised code with itself. Endless hours of fun! Just wait for the app!

It wasn’t all gameshows and group work. During the early part of the week, each group leader gave a 45-minute overview of the topic for all the workshop participants. At the end of most days, someone from each group gave a 10-minute progress report to the full workshop. The most junior person in each group gave a 20-minute summary talk on the last day. Our topic was exciting to me but I was very intrigued by these other ideas as well. We were reminded not to poach people’s problems, which I agree is an important point to make. However, we also had a lot of opportunities to talk to pretty much everyone at the workshop and engage with them a bit about the other projects. For example, I learned about potential applications of my some of my thesis work that I never would have heard of otherwise, since the papers are slightly out of my usual field. I would not have met the person doing this work at a number theory conference. I think all these talks were a good idea not only because they gave the whole group repeated exposure to each problem, but also because the talks demanded that everyone buy into their group projects and stay engaged through the whole week. Less experienced people couldn’t give up and advanced researchers couldn’t check out or leave graduate students in the dust, because the graduate students were going to speak for the whole group at the end.

With all the positive things I have to say about this week, it would be deceptive not to mention (again) that this workshop was totally exhausting. I really couldn’t keep up with emails from work or other responsibilities. I was super tired and way behind in my teaching and other research work by the end of the week. Still, it was totally worthwhile. I came in knowing a fair amount about algebraic curves over finite fields, and with some cryptography and basic coding theory background. I left with much more concrete knowledge in our problem area than I’d ever have brought home from a normal, all-talks research conference. Even more exciting, I got a much better sense of several really interesting problems involving codes, algebraic geometry and cryptography, and saw the kinds of tools that people are using for these problems. Best of all, I spent a week with 36 really cool people. Now if I can only dig out from under my grading.