John D. Cook

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 […]

I mentioned on Twitter yesterday that one way to relieve the boredom of grading math papers is to explore mistakes. If a statement is wrong, what would it take to make it right? Is it approximately correct? Is there some different context where it is correct? Several people said they’d like to see examples, so […]

Karen Uhlenbeck has just received the Abel Prize. Many say that the Fields Medal is the analog of the Nobel Prize for mathematics, but others say that the Abel Prize is a better analog. The Abel prize is a recognition of achievement over a career whereas the Fields Medal is only awarded for work done […]

I was looking up something on the NIST (National Institute of Standards and Technology) web site the other day and ran across thermocouple polynomials. I wondered what that could be, assuming “thermocouple” was a metaphor for some algebraic property. No, it refers to physical thermocouples. The polynomials are functions for computing voltage as a function […]

“Unbalanced oil and vinegar” is a colorful name for a cryptographic signature method. This post will give a high-level description of the method and explain where the name comes from. The RSA encryption algorithm depends on the fact that computers can easily multiply enormous numbers, but they cannot efficiently factor the product of two enormous […]

You can construct a finite field of order pn for any prime p and positive integer n. The elements are polynomials modulo an irreducible polynomial of degree n, with coefficients in the integers mod p. The choice of irreducible polynomial matters, though the fields you get from any two choices will be isomorphic. For example, […]

The paper Issues Encountered Deploying Differential Privacy describes some of the difficulties the US Census Bureau has run into while deploying differential privacy for the 2020 census. It’s not surprising that they would have difficulties. It’s surprising that they would even consider applying differential privacy on such an enormous scale. If your data project is […]

What is the closest planet to Earth? The planet whose orbit is closest to the orbit of Earth is clearly Venus. But what planet is closest? That changes over time. If Venus is between the Earth and the sun, Venus is the closest planet to Earth. But if Mercury is between the Earth and the […]

Introductions to elliptic curves often start by saying that elliptic curves have the form y² = x³ + ax + b. where 4a³ + 27b² ≠ 0. Then later they say “except over fields of characteristic 2 or 3.” What does characteristic 2 or 3 mean? The order of a finite field is the number of […]

The US Census Bureau is convinced that traditional methods of statistical disclosure limitation have not done enough to protect privacy. These methods may have been adequate in the past, but it no longer makes sense to implicitly assume that those who would like to violate privacy have limited resources or limited motivation. The Bureau has […]

Daniel Bernstein’s Curve25519 is the elliptic curve y² = x³ + 486662x² + x over the prime field with order p = 2255 – 19. The curve is a popular choice in elliptic curve cryptography because its design choices are transparently justified [1] and because cryptography over the curve can be implemented very efficiently. This […]

Here’s something I find puzzling: why isn’t CPU time more valuable? I first thought about this when I was working for MD Anderson Cancer Center, maybe around 2002. Our research in adaptive clinical trial methods required bursts of CPU time. We might need hundreds of hours of CPU time for a simulation, then nothing while […]

If you take these chaotic-looking values for your x-coordinates and these chaotic-looking values for your y coordinates you get this image that looks more ordered. The image above is today’s exponential sum.

As I noted in this post, RSA encryption is often carried out reusing exponents. Sometimes the exponent is exponent 3, which is subject to an attack we’ll describe below [1]. (The most common exponent is 65537.) Suppose the same message m is sent to three recipients and all three use exponent e = 3. Each […]

The RSA encryption algorithm depends indirectly on the assumption that factoring the product of large primes is hard. The algorithm presented here, invented by Shafi Goldwasser and Silvio Micali, depends on the same assumption but in a different way. The Goldwasser-Micali algorithm is more direct than RSA, thought it is also less efficient. One thing […]

Gil Kalai wrote a blog post yesterday entitled “Test Your Intuition (or knowledge, or programming skills) 36.” The challenge is to evaluate the infinite product I imagine there’s an elegant analytical solution, but since the title suggested that programming might suffice, I decided to try a little Python. I used primerange from SymPy to generate […]

I wrote a while back about Base32 and Base64 encoding, and yesterday I wrote about Bitcoin’s Base58 encoding. For completeness I wanted to mention Base85 encoding, also known as Ascii85. Adobe uses it in PostScript and PDF files, and git uses it for encoding patches. Like Base64, the goal of Base85 encoding is to encode […]

A few weeks ago I wrote about base32 and base64 encoding. I’ll review these quickly then discuss base58 and its use in Bitcoin. Base32 and Base64 All three methods have the goal of compactly representing large numbers while maintaining readability. Douglas Crockford’s base32 encoding is the most conservative: it’s case-insensitive and it does not use […]

My previous post talked about the ChaCha random number generator and how Google is using it in a stream cipher for encryption on low-end devices. This post talks about how to implement ChaCha in pure Python. First of all, the only reason to implement ChaCha in pure Python is to play with it. It would […]

The ChaCha cryptographic random number generator is in the news thanks to Google’s Adiantum project. I’ll discuss what’s going on, but first a little background. The name of the project comes from a genus of fern. More on that below as well. One-time pads The one-time pad is a provably unbreakable way to encrypt things. […]

Dialog from a congressional hearing February 26, 2019. Representative Katie Porter: My question for you is whether you would be willing to share today your social security, your birth date, and your address at this public hearing. Equifax CEO Mark Begor: I would be a bit uncomfortable doing that, Congresswoman. If you’d so oblige me, […]

This post will present a couple ways to share secrets using polynomials. We have a group of n people who want to share a secret between them so that k of them will have to cooperate in order to unlock the secret. For example, maybe a committee of n = 5 wants to require the cooperation of […]

Image editor Image editing software is complicated, and I don’t use it often enough to remember how to do much. I like Paint.NET on Windows because it is in a sort of sweet spot for me, more powerful than Paint and much less complicated than Photoshop. I found out there’s a program Pinta for Linux […]

This morning I read an article by Dennis Felsing about his impressive/intimidating Linux desktop setup. He uses a lot of tools that are not the easiest way to get things done immediately but are long-term productivity investments. Remembrance of syntax past Felsing apparently is able to remember the syntax of scores of tools and programming […]

The most common way to test whether a large number is prime is the Miller-Rabin test. If the test says a number is composite, it’s definitely composite. Otherwise the number is very likely, but not certain, to be prime. A pseudoprime is a composite number that slips past the Miller-Rabin test. (Actually, a strong pseudoprime. […]

As I explained in an earlier post, a first pass at the definition of an elliptic curve is the set of points satisfying y² = x³ + ax + b. There are a few things missing from this definition, as indicated before, one being the mysterious “point at infinity.” I gave a hand-waving explanation that […]

If you want your music to have more bass, more mid-range, and more treble, then you just want the music louder. You can increase all three components in absolute terms, but not in relative terms. You can’t increase the proportions of everything. Would you like more students to major in STEM subjects? OK, what subjects […]

Linear regression Suppose you have a linear regression with a couple predictors and no intercept term: β1x1 + β2x2 = y + ε where the x‘s are inputs, the β are fixed but unknown, y is the output, and ε is random error. Given n observations (x1, x2, y + ε), linear regression estimates the parameters β1 […]

Elliptic curves are pure and applied, concrete and abstract, simple and complex. Elliptic curves have been studied for many years by pure mathematicians with no intention to apply the results to anything outside math itself. And yet elliptic curves have become a critical part of applied cryptography. Elliptic curves are very concrete. There are some […]

According to this article, Microsoft is patching Windows 7 and Windows Server 2008 to look for SHA-2 hash functions of updates. These older versions of Windows have been using SHA-1, while newer version are already using SHA-2. This is a good move, but unnecessary. Here’s what I mean by that. The update was likely unnecessary […]

Here’s an oversimplified survey of cryptographic hash functions: Everyone used to use MD5, now they use some variation on SHA. There’s some truth to that. MD5 was very popular, and remains popular years after it was proven insecure. And now variations on SHA like SHA1 and SHA256 are commonly used. But there are a lot […]

I’ve written about elliptic curve and alluded to the fact that there’s a special kind of addition for points on the curve. But I haven’t gone into details because it’s more complicated than I wanted to get into. However, there’s a special case where the details are not complicated, the so called Edwards curves. I’ll look […]

I’ve long been impressed by shell one-liners. They seem like magical incantations. Pipe a few terse commands together, et voilà! Out pops the solution to a problem that would seem to require pages of code. Are these one-liners real or mythology? To some extent, they’re both. Below I’ll give a famous real example. Then I’ll argue […]

There are an infinite number of elliptic curves, but a small number that are used in elliptic curve cryptography (ECC), and these special curves have names. Apparently there are no hard and fast rules for how the names are chosen, but there are patterns. The named elliptic curves are over a prime field, i.e. a […]

Yesterday I mentioned μRNG, a true random number generator (TRNG) that takes physical sources of randomness as input. These sources are independent but non-uniform. This post will present the entropy extractor μRNG uses to take non-uniform bits as input and produce uniform bits as output. We will present Python code for playing with the entropy extractor. (μRNG […]

If a coin comes up heads with probability p and tails with probability 1-p, the entropy in the coin flip is S = –p log2 p – (1-p) log2 (1-p). It’s common to start with p and compute entropy, but recently I had to go the other way around: given entropy, solve for p. It’s easy to come up […]

A recurring theme in math is that you may not need to do what it looks like you need to do. There may be a shortcut to where you want to go. A special case of this is that you may not need all the information that you think you need. For example, if you […]

A week ago I wrote about using some Python code to play with the sum-product theorem of Erdős and Szemerédi and its conjectured refinement. This morning I learned that the Erdős-Szemerédi theorem has been extended to finite fields. David Johnston left a comment saying that he and his colleagues used this extension to finite fields as […]

In a earlier post I introduce the Legendre symbol where a is a positive integer and p is prime. It is defined to be 0 if a is a multiple of p, 1 if a has a square root mod p, and -1 otherwise. The Jacobi symbol is a generalization of the Legendre symbol and uses the same notation. It […]

I’ve started a new Twitter account for data privacy and related topics. Twitter gave me the handle @data_tip even though that’s not what I typed in, and what I typed in is not being used. Apparently they don’t let you pick your handle any more. Data Privacy (@data_tip)

You’ll often hear Phase I dose-finding trials referred to as dose escalation studies. This is because simple dose-finding methods can only explore in one direction: they can only escalate. Three-plus-three rule The most common dose finding method is the 3+3 rule. There are countless variations on this theme, but the basic idea is that you give […]

Implementation flaws in RSA encryption make it less secure in practice than in theory. RSA encryption depends on 5 numbers: Large primes p and q The modulus n = pq Encryption key e Decryption key d The numbers p, q, and d are kept secret, and the numbers e and n are made public. The encryption method relies on the assumption that in practice one cannot […]

Supercookies, also known as evercookies or zombie cookies, are like browser cookies in that they can be used to track you, but are much harder to remove. What is a supercookie? The way I first heard supercookies describe was as a cookie that you can appear to delete, but as soon as you do, software […]

Quanta Magazine posted an article yesterday about the sum-product problem of Paul Erdős and Endre Szemerédi. This problem starts with a finite set of real numbers A then considers the size of the sets A+A and A*A. That is, if we add every element of A to every other element of A, how many distinct sums are there? If we […]

I heard the phrase “normal approximation to the Laplace distribution” recently and did a double take. The normal distribution does not approximate the Laplace! Normal and Laplace distributions A normal distribution has the familiar bell curve shape. A Laplace distribution, also known as a double exponential distribution, it pointed in the middle, like a pole […]

The CCPA, the California Privacy Protection Act, was passed last year and goes into effect at the beginning of next year. And just as the GDPR impacts businesses outside Europe, the CCPA will impact businesses outside California. The law specifically mentions probabilistic identifiers. “Probabilistic identifier” means the identification of a consumer or a device to a […]