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 […]

# Author: John

## Unifiers and Diversifiers

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 […]

## Internet privacy as seen from 1975

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 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 predict […]

## Impossible to misunderstand

“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 […]

## Comparing Truncation to Differential Privacy

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 […]

## Golden ratio primes

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. When φ is a large power of 2, these prime numbers are useful in cryptography […]

## Goldilocks and the three multiplications

Mike Hamburg designed an elliptic curve 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 prime in this […]

## Tricks for arithmetic modulo NIST primes

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 […]

## Elliptic curve P-384

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 […]

## Bessel function crossings

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 […]

## Orthogonal graphs

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 […]

## Fascination burnout

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 […]

## Area and volume of Menger sponge

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 […]

## Regular expression for ICD-10 diagnosis codes

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. […]

## A misunderstanding of complexity

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 […]

## Improving on the sieve of Eratosthenes

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 […]

## How category theory is applied

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 […]

## Rare and strange ICD-10 codes

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 […]

## State privacy laws to watch

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 […]

## Quantum leaps

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 […]