Monte Carlo integration is not as simple in practice as it is often introduced. A homework problem might as you to integrate a function of two variables by selecting random points from a cube and counting how many of the points fall below the graph of the function. This would …

About three years ago I ran across a strange coordinate system in which familiar functions lead to interesting plots. The system is called “circular coordinates” but it is not polar coordinates. This morning I was playing around with this again. Here’s a plot of f(x) = x. And here’s a …

Today’s date, US style, is 9/26/2023, and there is only one group, up to isomorphism, of size 9262023. You could verify this in Mathematica with the command FiniteGroupCount[9262023] which returns 1. For a given n, when is there only one group of size n? There are two requirements. First, n …

Simple groups are the building blocks of groups similar to the way prime numbers are the building blocks of integers. This post will unpack this analogy in two ways: How do simple groups compare to prime numbers? How does the composition of simple groups compare to the composition of prime …

The word “normal” in mathematical nomenclature does not always means “usual” or “customary” as it does in colloquial English. Instead, it might that something has a convenient property. That is the case for normal subgroups. We can do things with normal subgroups that we cannot do with other subgroups, such …

In the previous post I mentioned that a particular Mersenne prime would be unsuitable for cryptography. In fact, all Mersenne primes are unsuitable for cryptography. A prime number p is called “safe” if p = 2q + 1 where q is also a prime. Safe primes are called safe because …

Electronic computers were invented before public key cryptography. Would public key cryptography have been possible before computers? The security of RSA encryption depends on the ratio of the difficulty of factoring relative to the difficulty of multiplication. This ratio was high, maybe higher, before modern computers. Suppose the idea of …

Here’s a quote from Don Zagier that I found in Larry Rolen’s lecture notes on modular forms. There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite …

A continued fraction of the form with n terms can be written as the composition where As discussed in the previous post, a Möbius transformation can be associated with a matrix. And the composition of Möbius transformations is associated with the product of corresponding matrices. So the continued fraction at …

A function of the form where ad – bc ≠ 0 is sometimes called a fractional linear transformation or a bilinear transformation. I usually use the name Möbius transformation. In what sense are Möbius transformations linear transformations? They’re nonlinear functions unless b = c = 0. And yet they’re analogous …

Warm up The geometric mean of two numbers is the square root of their product. For example, the geometric mean of 9 and 25 is 15. More generally, the geometric mean of a set of n numbers is the nth root of their product. Alternatively, the geometric mean of a …

The Gauss map [1] is the function where ⌊y⌋ is the floor of y, the greatest integer no larger than y. I’ve written about this map a couple times before. First, I wrote about how this map is measure-preserving. Second, I wrote about the image at the top of the …

The goal of this post is to unpack a remark in [1]: … we can say this in fancier terms. Fix a field k …. We say that an elliptic curve E defined over k is that functor which … Well that is fancy. But what does it mean? Looking …

Mathematicians often speak informally about the relative simplicity of rational numbers. For example, musical intervals that correspond to simple fractions have less tension than intervals that correspond to more complicated fractions. Such informal statements can be made more precise using height functions. There are a variety of height functions designed …

If you ask someone a question and they say “yes” immediately, that gives you different information than if they pause and slowly say “yes.” The information you receive is not just the response but also the time it took to generate the response. Encryption can be analogous. The time it …

I concluded the previous post by saying elliptic curve Diffie-Hellman key exchange (ECDHE) requires smaller keys than finite field Diffie-Hellman (FFDHE) to obtain the same level of security. How much smaller are we talking about? According to NIST recommendations, a 256-bit elliptic curve curve provides about the same security as …

Diffie-Hellman key exchange is conceptually simple. Alice and Bob want to generate a shared cryptographic key. They want to use asymmetric (public) cryptography to share a symmetric (private) key. The starting point is a large prime p and a generator 1 < g < p. Alice generates a large random …

Suppose m is a large integer that you are able to factor. To keep things simple, suppose m = pq where p and q are distinct primes; everything in this post generalizes easily to the case of m having more than two factors. You can carry out calculations mod m …