Author: John

Angles in the spiral of Theodorus

The previous post looked at how to plot the spiral of Theodorus shown below. We stopped the construction where we did because the next triangle to be added would overlap the first triangle, which would clutter the image. But we could certainly have kept going. If we do keep going, then the set of hypotenuse […]

How to plot the spiral of Theodorus

You may have seen the spiral of Theodorus. It sticks a sequence of right triangles together to make a sort of spiral. Each triangle has a short side of length 1, and the hypotenuse of each triangle becomes the long leg of the next triangle as shown below. How would you plot this spiral? At […]

Encryption as secure as factoring

RSA encryption is based on the assumption that factoring large integers is hard. However, it’s possible that breaking RSA is easier than factoring. That is, the ability to factor large integers is sufficient for breaking RSA, but it might not be necessary. Two years after the publication of RSA, Michael Rabin created an alternative that […]

Accelerating convergence with Aitken’s method

The previous post looked at Euler’s method for accelerating the convergence of a slowly converging alternating series. Both hypotheses are necessary. The signs must alternate between terms, and applying the method to a series that is already converging quickly can slow down convergence. This post looks at Aitken’s method for speeding up the convergence of […]

Accelerating an alternating series

The most direct way of computing the sum of an alternating series, simply computing the partial sums in the terms get small enough, may not be the most efficient. Euler figured this out in the 18th century. For our demo we’ll evaluate the Struve function defined by the series Note that the the terms in […]

Data breach trends

Are data breaches becoming more or less common? This post gives a crude, back-of-the-envelope calculation to address the question. We won’t look at number of breaches per se but number of records breached. There’s a terrific visualization of data breach statistics at Information is Beautiful, and they share their data here. Note that the data […]

Beating the odds on the Diffie-Hellman decision problem

There are a couple variations on the Diffie-Hellman problem in cryptography: the computation problem (CDH) and the decision problem (DDH). This post will explain both and give an example of where the former is hard and the latter easy. The Diffie-Hellman problems The Diffie-Hellman problems are formulated for an Abelian group. The main group we […]

Magic square links and errata

Someone pointed out that what I called a knight’s tour magic square is technically a semi-magic square: the rows and columns add up to the same constant, but the diagonals do not. It turns out there are no strict magic squares formed by knight’s tours. This was proved in 2003. See a news article here. […]

Quaternion reference in the Vulgate

To contemporary ears “quaternion” refers to a number system discovered in the 19h century, but there were a couple precedents. Both refer to something related to a group of four things, but there is no relation to mathematical quaternions other than that they have four dimensions. I’ve written before about Milton’s use of the word […]

Fame, difficulty, and usefulness

Pierre Fermat is best known for two theorems, dubbed his “last” theorem and his “little” theorem. His last theorem is famous, difficult to prove, and useless. His little theorem is relatively arcane, easy to prove, and extremely useful. There’s little relation between technical difficulty and usefulness. Fermat’s last theorem Fermat’s last theorem says there are […]

Twisted elliptic curves

This morning I was sitting at a little bakery thinking about what to do before I check out of my hotel. I saw that the name of the bakery was Twist Bakery & Cafe, and that made me think of writing about twisted elliptic curves when I got back to my room. Twist of an […]

Hashing names does not protect privacy

Secure hash functions are practically impossible to reverse, but only if the input is unrestricted. If you generate 256 random bits and apply a secure 256-bit hash algorithm, an attacker wanting to recover your input can’t do much better than brute force, hashing 256-bit strings hoping to find one that matches your hash value. Even […]

May the best technology win

I’ve become skeptical of arguments of the form “X is a better technology, but people won’t quit using Y.” Comparisons of technologies are multi-faceted. When someone says “X is better than Y” I want to ask “By all criteria? There’s nothing better about Y?” When people say X is better but Y won, it’s often […]

Integral approximation trick

Here’s a simple integration approximation that works remarkably well in some contexts. Suppose you have an integrand that looks roughly like a normal density, something with a single peak that drops off fairly quickly on either side of the peak. The majority of integrals that arise in basic applications of probability and statistics fit this […]

Number of feet in a mile

Here are a couple amusing things I’ve run across recently regarding the number of feet in a mile. Both are frivolous, but also have a more serious side. Mnemonic First, you can use “five tomatoes” as a mnemonic for remembering that there are 5280 feet in a mile. “Five tomatoes” is a mnemonic for the […]

Discriminant of a cubic

The discriminant of a quadratic equation ax² + bx + c = 0 is Δ = b² – 4ac. If the discriminant Δ is zero, the equation has a double root, i.e. there is a unique x that makes the equation zero, and it counts twice as a root. If the discriminant is not zero, […]

Distribution of quadratic residues

Let p be an odd prime number. If the equation x² = n mod p has a solution then n is a square mod p, or in classical terminology, n is a quadratic residue mod p. Half of the numbers between 0 and p are quadratic residues and half are not. The residues are distributed […]

What does CCPA say about de-identified data?

The California Consumer Privacy Act, or CCPA, takes effect January 1, 2020, less than six months from now. What does the act say about using deidentified data? First of all, I am not a lawyer; I work for lawyers, advising them on matters where law touches statistics. This post is not legal advice, but my […]

Serious applications of a party trick

In a group of 30 people, it’s likely that two people have the same birthday. For a group of 23 the probability is about 1/2, and it goes up as the group gets larger. In a group of a few dozen people, it’s unlikely that anyone will have a particular birthday, but it’s likely that […]

Channel quantity and quality

Years ago, when there were a couple dozen television stations, someone [1] speculated that when we got more channels we’d also get better content. The argument was that people are more similar in the base interests than in their more refined interests. Therefore if there are only a few channels, they will all appeal to […]