Category: Math

Strong primes

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 […]

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 […]

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 […]

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 […]

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 […]

Easter and exponential sums

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 […]

Groups in categories

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 […]

What is an isogeny?

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 […]

Isogeny-based encryption

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 […]

Random projection

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 […]

Squircle perimeter and the isoparametric problem

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 […]

Taking the derivative of a muscle car

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 […]