John D. Cook

The nth prime is approximately n log n. For more precise estimates, there are numerous upper and lower bounds for the nth prime, each tighter over some intervals than others. Here I want to point out upper and lower bounds from a dissertation by Christian Axler on page viii. First, define Then for sufficiently large […]

Nines and sigmas are two ways to measure quality. You’ll hear something has four or five nines of reliability or that some failure is a five sigma event. What do these mean, and how do you convert between them? Definitions If a system has fives nines of availability, that means the probability of the system […]

If someone tells you repeatedly that something isn’t hard, maybe it’s just hard. Monads A post by Gilad Bracha got me thinking about this. He says Last time I looked, the Haskell wiki listed 29 tutorials on [monads]. … Could it just be that people just have a hard time understanding monads? If so, what […]

Regular expressions are challenging, but not for the reasons commonly given. Non-reasons Here are some reasons given for the difficulty of regular expressions that I don’t agree with. Cryptic syntax I think complaints about cryptic syntax miss the mark. Some people say that Greek is hard to learn because it uses a different alphabet. If […]

Here’s kind of an unusual question: What is the density of integers that have an even number of prime factors with an exponent greater than 1? To define the density, you take the proportion up to an integer N then take the limit as N goes to infinity. It’s not obvious that the limit should […]

Last year Adam Roberts had some fun with Finnegans Wake [1], seeing how little he could edit it and turn it into something that sounded like Return of the Jedi. I wrote a blog post where I quantified the difference between the original and the parody using Levenshtein distance, basically how many edits it takes […]

A common attempt to protect privacy is to truncate dates to just the year. For example, the Safe Harbor provision of the HIPAA Privacy Rule says to remove “all elements of dates (except year) for dates that are directly related to an individual …” This restriction exists because dates of service can be used to […]

If an inheritance is to be divided per stirpes, each descendant gets an equal share. If a descendant has died but has living descendants, his or her share is distributed by applying the rule recursively. Example For example, suppose a man had two children, Alice and Bob, and stipulates in his will that his estate […]

Steve Gibson’s Security Now is one of the podcasts I regularly listen to, and so I’ve been hearing him talk about his SQRL for a while. This week he finally released SQRL: Secure Quick Reliable Login. You can read more about SQRL in the white paper posted on the GRC web site. Here’s a tease […]

The reason businesses have employees rather than contracting out everything is to reduce transaction costs. If a company needs enough graphics work, they hire a graphic artist rather than outsourcing every little project, eliminating the need to evaluate bids, write contracts, etc. Some things are easier when no money has to change hands. But some […]

Trott’s constant is the unique number whose digits equal its continued fraction coefficients. Uniqueness assumes the number is expanded into a simple continued fraction, i.e. one with all numerators equal to 1. See OEIS sequence A039662. More continued fraction posts Best rational approximations to π Continued fraction cryptography Normal hazard continued fraction

Suppose you have two pseudorandom bit generators. They’re both fast, but not suitable for cryptographic use. How might you combine them into one generator that is suitable for cryptography? Coppersmith et al [1] had a simple but effective approach which they call the shrinking generator, also called irregular decimation. The idea is to use one […]

Aatish Bhatia posted something interesting on Twitter: if you define multiplication on Rock, Paper, Scissors to be the winner of a match, the result is commutative but not associative. Here's a neat thing about the algebra of Rock, Paper, Scissors. If you define 'multiplication' as the game's winner, then it's commutative, i.e. P x R […]

It occurred to me for the first time this morning that the words liminal and subliminal must be related, just after reading an article by Vicki Boykis that discusses liminal spaces. I hear the two words in such in different contexts—architecture versus psychology—and hadn’t thought about the connection until now. If I were playing a […]

Yesterday I was at a wedding, and a vase broke in the aisle shortly before the bridal party was to enter. Guests quickly picked up the pieces, but the vase left a pool of water on the hard floor. A security guard ran (literally) for a mop and cheerfully picked up the water. He could […]

I’ve been unable to get some R libraries to install on my Linux laptop. Two libraries in particular were tseries and tidyverse. The same libraries installed just fine on Windows. (Maybe you need to install Rtools first before installing these on Windows; I don’t remember.) I use conda all the time with Python, but I […]

This morning as a sort of experiment I decided to look back at all my blog posts written on May 30 each year. There’s nothing special about this date, so I thought it might give an eclectic cross section of things I’ve written about. *** Last year on this day I wrote about Calendars and […]

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 […]