This is a guest post by my colleague Adam Lelkes. The goal of this primer is to introduce an important and beautiful tool from probability theory, a model of fair betting games called martingales. In this post I will assume that the reader is familiar with the basics of probability theory. For those that need to refresh their knowledge, Jeremy’s excellent primers (1, 2) are a good place to start.

So far on this blog we’ve given some introductory notes on a few kinds of algebraic structures in mathematics (most notably groups and rings, but also monoids). Fields are the next natural step in the progression. If the reader is comfortable with rings, then a field is extremely simple to describe: they’re just commutative rings with 0 and 1, where every nonzero element has a multiplicative inverse.

Last time we saw a geometric version of the algorithm to add points on elliptic curves. We went quite deep into the formal setting for it (projective space $ \mathbb{P}^2$), and we spent a lot of time talking about the right way to define the “zero” object in our elliptic curve so that our issues with vertical lines would disappear.

I’m pleased to announce that another paper of mine is finished. This one just got accepted to MFCS 2014, which is being held in Budapest this year (this whole research thing is exciting!). This is joint work with my advisor, Lev Reyzin. As with my first paper, I’d like to explain things here on my blog a bit more informally than a scholarly article allows.

Last time we looked at the elementary formulation of an elliptic curve as the solutions to the equation $$y^2 = x^3 + ax + b$$ where $ a,b$ are such that the discriminant is nonzero: $$-16(4a^3 + 27b^2) \neq 0$$ We have yet to explain why we want our equation in this form, and we will get to that, but first we want to take our idea of intersecting lines as far as possible.

This is a guest post by my friend and colleague Adam Lelkes. Adam’s interests are in algebra and theoretical computer science. This gem came up because Adam gave a talk on probabilistic computation in which he discussed this technique. Problem: simulate a biased coin using a fair coin.

Finding solutions to systems of polynomial equations is one of the oldest and deepest problems in all of mathematics. This is broadly the domain of algebraic geometry, and mathematicians wield some of the most sophisticated and abstract tools available to attack these problems. The elliptic curve straddles the elementary and advanced mathematical worlds in an interesting way.

This is a guest post by my friend and colleague Adam Lelkes. Adam’s interests are in algebra and theoretical computer science. This gem came up because Adam gave a talk on probabilistic computation in which he discussed this technique. Problem: Simulate a fair coin given only access to a biased coin.

With all the recent revelations of government spying and backdoors into cryptographic standards, I am starting to disagree with the argument that you should never roll your own cryptography. Of course there are massive pitfalls and very few people actually need home-brewed cryptography, but history has made it clear that blindly accepting the word of the experts is not an acceptable course of action.

A few awesome readers have posted comments in Computing Homology to the effect of, “Your code is not quite correct!” And they’re right! Despite the almost year since that post’s publication, I haven’t bothered to test it for more complicated simplicial complexes, or even the basic edge cases! When I posted it the mathematics just felt so solid to me that it had to be right (the irony is rich, I know).