John D. Cook

I ran across a video this afternoon that explains that the sum of volumes of all even-dimensional unit spheres equals eπ. Why is that? Define vol(n) to be the volume of the unit sphere in dimension n. Then and so the sum of the volumes of all even dimensional spheres is But what if you […]

The AES (Advanced Encryption Standard) algorithm takes in blocks of 128 or more bits [1] and applies a sequence of substitutions and permutations. The substitutions employ an “S-box”, named the Rijndael S-box after its designers [2], an invertible nonlinear transformation that works on 8 bits at a time. There are 256 = 16 × 16 […]

I recently learned about the Linux command line utility shuf from browsing The Art of Command Line. This could be useful for random sampling. Given just a file name, shuf randomly permutes the lines of the file. With the option -n you can specify how many lines to return. So it’s doing sampling without replacement. […]

The National Security Agency has stated clearly that they believe this is the time to start moving to quantum-resistant encryption. Even the most optimistic enthusiasts for quantum computing believe that practical quantum computers are years away, but so is the standardization of post-quantum encryption methods. The NSA has also made some suggestions for what to […]

A cosmic ray striking computer memory at just the right time can flip a bit, turning a 0 into a 1 or vice versa. While I knew that cosmic ray bit flips were a theoretical possibility, I didn’t know until recently that there had been documented instances on the ground [1]. Radiolab did an episode […]

There are a couple different definitions of a strong prime. In number theory, a strong prime is one that is closer to the next prime than to the previous prime. For example, 11 is a strong prime because it is closer to 13 than to 7. In cryptography, a strong primes are roughly speaking primes […]

I saw a couple tweets this morning quoting Freeman Dyson’s book Infinite in All Directions. Unifiers are people whose driving passion is to find general principles which will explain everything. They are happy if they can leave the universe looking a little simpler than they found it. Diversifiers are people whose passion is to explore […]

Science fiction authors set stories in the future, but they don’t necessarily try to predict the future, and so it’s a little odd to talk about what they “got right.” Getting something right implies they were making a prediction rather than imagining a setting of a story. However, sometimes SF authors do indeed try to […]

“The goal is not to be possible to understand, but impossible to misunderstand.” I saw this quote at the beginning of a math book when I was a student and it stuck with me. I would think of it when grading exams. Students often assume it is enough to be possible to understand, possible for […]

Traditional methods of data de-identification obscure data values. For example, you might truncate a date to just the year. Differential privacy obscures query values by injecting enough noise to keep from revealing information on an individual. Let’s compare two approaches for de-identifying a person’s age: truncation and differential privacy. Truncation First consider truncating birth date […]

The golden ratio is the larger root of the equation φ² – φ – 1 = 0. By analogy, golden ratio primes are prime numbers of the form p = φ² – φ – 1 where φ is an integer. To put it another way, instead of solving the equation φ² – φ – 1 […]

Mike Hamburg designed an elliptic curve for use in cryptography he calls Ed448-Goldilocks. The prefix Ed refers to the fact that it’s an Edwards curve. The number 448 refers to the fact that the curve is over a prime field where the prime p has size 448 bits. But why Goldilocks? Golden primes and Goldilocks […]

The US National Institute of Standards and Technology (NIST) originally recommended 15 elliptic curves for use in elliptic curve cryptography [1]. Ten of these are over a field of size 2n. The other five are over prime fields. The sizes of these fields are known as the NIST primes. The NIST curves over prime fields […]

The various elliptic curves used in ellitpic curve cryptography (ECC) have different properties, and we’ve looked at several of them before. For example, Curve25519 is implemented very efficiently, and the parameters were transparently chosen. Curve1174 is interesting because it’s an Edwards curve and has a special addition formula. This post looks at curve P-384. What’s […]

The previous looked at the angles that graphs make when they cross. For example, sin(x) and cos(x) always cross with the same angle. The same holds for sin(kx) and cos(kx) since the k simply rescales the x-axis. The post ended with wondering about functions analogous to sine and cosine, such as Bessel functions. This post […]

Colin Wright posted a tweet yesterday that said that the plots of cosine and tangent are orthogonal. Here’s a plot so you can see for yourself. Jim Simons replied with a proof so short it fits in a tweet: The product of the derivatives is -sin(x)sec²(x) = -tan(x)/cos(x), which is -1 if cos(x)=tan(x). This made […]

Here a little dialog from Anathem by Neal Stephenson that I can relate to: “… I don’t care …” Asribalt was horrified. “But how can you not be fascinated by—” “I am fascinated,” I insisted. “That’s the problem. I’m suffering from fascination burnout. Of all the things that are fascinating, I have to choose just […]

The Menger sponge is the fractal you get by starting with a cube, dividing each face into a 3 by 3 grid (like a Rubik’s cube) and removing the middle square of each face and everything behind it. That’s M1, the Menger sponge at the 1st stage of its construction. The next stage repeats this […]

Suppose you’re searching for medical diagnosis codes in the middle of free text. One way to go about this would be to search for each of the roughly 14,000 ICD-9 codes and each of the roughly 70,000 ICD-10 codes. A simpler approach would be to use regular expressions, though that may not be as precise. […]

Iterating simple rules can lead to complex behavior. Many examples of this are photogenic, and so they’re good for popular articles. It’s fun to look at fractals and such. I’ve written several articles like that here, such as the post that included the image below. But there’s something in popular articles on complexity that bothers […]

Ancient algorithm Eratosthenes had a good idea for finding all primes less than an upper bound N over 22 centuries ago. Make a list of the numbers 2 to N. Circle 2, then scratch out all the larger multiples of 2 up to N. Then move on to 3. Circle it, and scratch out all […]

Instead of asking whether an area of mathematics can be applied, it’s more useful to as how it can be applied. Differential equations are directly and commonly applied. Ask yourself what laws govern the motion of some system, write down these laws as differential equations, then solve them. Statistical models are similarly direct: propose a […]

ICD-10 is a set of around 70,000 diagnosis codes. ICD stands for International Statistical Classification of Diseases and Related Health Problems. The verbosity of the name is foreshadowing. Some of the ICD-10 codes are awfully specific, and bizarre. For example, V95.4: Unspecified spacecraft accident injuring occupant V97.33XA: Sucked into jet engine, initial encounter V97.33XD: Sucked […]

A Massachusetts court ruled this week that obtaining real-time cell phone location data requires a warrant. Utah has passed a law that goes into effect next month that goes further. Police in Utah will need a warrant to obtain location data or to search someone’s electronic files. (Surely electronic files are the contemporary equivalent of […]

A literal quantum leap is a discrete change, typically extremely small [1]. A metaphorical quantum leap is a sudden, large change. I can’t think of a good metaphor for a small but discrete change. I was reaching for such a metaphor recently and my first thought was “quantum leap,” though that would imply something much […]

I opened a blog posts a while back by saying One of the differences between amateur and professional software development is whether you’re writing software for yourself or for someone else. It’s like the difference between keeping a journal and being a journalist. This morning I saw where someone pulled that quote and I thought […]

For the last couple years, the exponential sum of the day for Easter Sunday has been a cross. This was not planned, since the image each day is determined by the numbers that make up the date, as explained here. This was the exponential sum for last Easter last year, April 1, 2018: and this […]

The first time I saw a reference to a “group in a category” I misread it as something in the category of groups. But that’s not what it means. Due to an unfortunately choice of terminology, “in” is more subtle than just membership in a class. This is related to another potentially misleading term, algebraic […]

The previous post said that isogenies between elliptic curves are the basis for a quantum-resistant encryption method, but we didn’t say what an isogeny is. It’s difficult to look up what an isogeny is. You’ll find several definitions, and they seem incomplete or incompatible. If you go to Wikipedia, you’ll read “an isogeny is a […]

If and when large quantum computers become practical, all currently widely deployed method for public key cryptography will break. Even the most optimistic proponents of quantum computing believe such computers are years away, maybe decades. But it also takes years, maybe decades, to develop, test, and deploy new encryption methods, and so researchers are working […]

The Mathematica function ExternalEvalute lets you call Python from Mathematica. However, there are a few wrinkles. I first pasted in an example from the Mathematica documentation and it failed. ExternalEvaluate[ "Python", {"def f(x): return x**2", "f(3)"} ] It turns out you (may) have to tell Mathematica where to find Python. I ran the following, tried […]

Last night after dinner, the conversation turned to high-dimensional geometry. (I realize how odd that sentence sounds; I was with some unusual company.) Someone brought up the fact that two randomly chosen vectors in a high-dimensional space are very likely to be nearly orthogonal. This is a surprising but well known fact. Next the conversation […]

I needed a bad random number generator for an illustration, and chose RANDU, possibly the worst random number generator that was ever widely deployed. Donald Knuth comments on RANDU in the second volume of his magnum opus. When this chapter was first written in the late 1960’s, a truly horrible random number generator called RANDU […]

Programmers have an easier time scaling up than scaling down. You could call this foresight or over-engineering, depending on how things work out. Scaling is a matter of placing bets. Experienced programmers are rightfully suspicious of claims that something only needs to be done once, or that quick-and-dirty will be OK [*]. They’ve been burned […]

If you have a fixed length of rope and you want to enclose the most area inside the rope, make it into a circle. This is the solution to the so-called isoparametric problem. Dido’s problem is similar. If one side of your bounded area is given by a straight line, make your rope into a […]

I’ve been getting a lot of spam lately saying my web site does not rank well on “certain keywords.” This is of course true: no web site ranks well for every keyword. I was joking about this on Twitter, saying that my site does not rank well for women’s shoes, muscle cars, or snails because […]

Data privacy is subtle and difficult to regulate. The lawmakers who wrote the HIPAA privacy regulations took a stab at what would protect privacy when they crafted the “Safe Harbor” list. The list is neither necessary or sufficient, depending on context, but it’s a start. Extreme values of any measurement are more likely to lead […]

My newest Twitter account is Data Privacy (@data_tip). There I post tweets about ways to protect your privacy, statistical disclosure limitation, etc. I had a clever idea for the icon, or so I thought. I started with the default Twitter icon, a sort of stylized anonymous person, and colored it with the same blue and […]

As dimension increases, the ratio of volume between a unit ball and a unit cube goes to zero. Said another way, if you have a high-dimensional ball inside a high-dimensional box, nearly all the volume is in the corners. This is a surprising result when you first see it, but it’s well known among people […]

The previous post looked at what exponent makes the area of a squircle midway between the area of a square and circle of the same radius. We could ask the analogous question in three dimensions, or in any dimension. (What do you call a shape between a cube and a sphere? A cuere? A sphube?) […]

Architect Peter Panholzer coined the term “squircle” in the summer of 1966 while working for Gerald Robinson. Robinson had seen a Scientific American article on the superellipse shape popularized by Piet Hein and suggested Panholzer use the shape in a project. Piet Hein used the term superellipse for a compromise between an ellipse and a […]

The US HIPAA law only protects the privacy of health data held by “covered entities,” which essentially means health care providers and insurance companies. If you give your heart monitoring data or DNA to your doctor, it comes under HIPAA. If you give it to Fitbit or 23andMe, it does not. Government entities are not […]

Fitness monitors reveal more information than most people realize. For example, it may be possible to infer someone’s religious beliefs from their heart rate data. If you have location data, it’s trivial to tell whether someone is attending religious services. But you could make a reasonable guess from cardio monitoring data alone. Muslim prayers occur […]

I got a review copy of The Mathematics of Data recently. Five of the six chapters are relatively conventional, a mixture of topics in numerical linear algebra, optimization, and probability. The final chapter, written by Robert Ghrist, is entitled Homological Algebra and Data. Those who grew up with Sesame Street may recall the song “Which […]

I just had a client ship me a laptop. We never discussed what OS the computer would run. I haven’t opened the box yet, but I imagine it’s running Windows 10. I’ve had clients assume I run Windows, but also others who assume I run Linux or Mac. I don’t recall anyone asking me whether […]

Differential equations rarely have closed-form solutions. Some do, and these are emphasized in textbooks. For this post we want to look specifically at homogeneous second order linear equations: y ” + a(x) y‘ + b(x) y = 0. If the coefficient functions a and b are constant, then the solution can be written down in terms […]

It occurred to me recently that I rarely hear about finite rings. I did a Google Ngram search to make sure this isn’t just my experience. Source Why are finite groups and finite fields common while finite rings are not? Finite groups have relatively weak algebraic structure, and demonstrate a lot of variety. Finite fields […]

Paolo Perrone gives a nice, succinct motivation for monads in the introduction to his article on probability and monads. … a monad is like a consistent way of extending spaces to include generalized elements of a specific kind. He develops this idea briefly, and links to his dissertation where he gives a longer exposition (pages […]

Secret codes and error-correcting codes have nothing to do with each other. Except when they do! Error-correcting codes Error correcting code make digital communication possible. Without some way to detect and correct errors, the corruption of a single bit could wreak havoc. A simple example of an error-detection code is check sums. A more sophisticated […]

Many times on this blog I’ve argued that the difference between pure and applied math is motivation. As my graduate advisor used to say, “Applied mathematics is not a subject classification. It’s an attitude.” Traditionally there was general agreement regarding what is pure math and what is applied. Number theory and topology, for example, are […]