2026-08-11 22:09:46
I was reading a blog post on boat names because it was on Hacker News this morning. The post contained a link to a data set on dog names in NYC and I poked around the data a little. The top names were not at all what I expected, but then again this is limited to NYC; it’s not a sample across the US. These were the top 10 names:
I wondered if the name frequencies might fit a power-law distribution. They do not, but they follow a log-normal distribution remarkably well.
2026-08-11 20:04:24
Suppose you were able to go back in time, to an era before computers, and give someone contemporary cryptography. Encryption methods that are essentially unbreakable now would certainly be unbreakable then. But there’s a catch: not only do attackers not have computers, neither do users.
If you told someone about RSA encryption, for example, you’d lose them right after you said “First you find a couple 1000-digit primes.” But there’s no need for using 1000-digit primes if no attacker has a computer. You could use 100 digit primes. Could you use 10 digit primes? If you chose primes just big enough to make the method unbreakable by hand, could someone implement it by hand?
Kirchoff’s principle says the strength of an encryption method should depend only on keeping the key secret, not the method. If you could keep the method secret, RSA would be unbreakable because nobody thought of anything like it before computers. But to make our thought experiment more interesting, let’s suppose that an enemy has also traveled back in time. If you tell your side about RSA, he can tell his side about it as well. So we’re back to Kirchoff’s principle.
An encryption method combinining substitution and permutation would have been practical to carry our manually. The ADFGVX cipher from 1918 was a start in this direction. That idea could been extended further, with a larger substitution set and longer permutations, and with more than one round of substitution and permutation, approaching what would be come the approach used in modern symmetric encryption. Such a method might have been manually implementable without being manually breakable.
World War II was a time of transition from manual cryptography to computerized cryptography. Encryption machines were attacked by cryptanalysis machines, though these machines were general-purpose computers. If you could implement a symmetric encryption method like AES in a mechanical device, no mechanical device could break it.
You could use something like DES, simpler than AES but still unbreakable at the time. DES is considered broken because now you could throw enough compute power at it to break it by brute force, but that would not be possible with only mechanical devices.
My hunch is that the best approach would be stream ciphers. Maybe it would be practical to implement one of these by hand or with the aid of simple calculating machines. Something like PCG, which is not cryptographically secure today [1], would have been then, though I don’t know how practical it would have been to carry out PCG, say, in the 1940s.
[1] In 2020, Charles Bouillaguet, Florette Martinez, and Julia Sauvage were able to break PCG using 20,000 CPU-hours. See their paper Practical seed-recovery for the PCG Pseudo-Random Number Generator. IACR Transactions on Symmetric Cryptology. ISSN 2519-173X, Vol. 2020, No. 3, pp. 175–196.
The post Manually unbreakable cryptography first appeared on John D. Cook.2026-08-10 22:34:21
The following extraordinary paragraph comes from Knuth’s TAOCP Volume 4A, right before the last set of exercises.
The post Learning from historical mistakes first appeared on John D. Cook.Many of the exercises below ask a modern reader to find and/or to correct errors in the literature of bygone days. The point is not to gloat over how smart we are in the 21st century; the point is rather to understand that even the pioneers of a subject can stumble. One good way to learn that a set of ideas is not really as simple as it might seem to today’s computer scientists and mathematicians is to observe that some of the world’s leading thinkers had to struggle with the concepts when they were new.
2026-08-10 22:32:42
In science and engineering classes, you might describe a system using Newton’s laws and end up with a differential equation. You then solve the differential equation, analytically or numerically, to see how the solutions behave.
You might also do the opposite, especially in a mathematics class: look at what differential equation a set of functions satisfy in order to understand those functions.
Bessel functions came out of solving differential equations from astronomy. But then they turned out to be useful, not just in other areas of science, but in pure mathematics as well. So there are Bessel function users who came to the functions first and haven’t seen the differential equation they came from.
You can learn a lot about Bessel functions, and other special functions, by looking at their defining differential equation even if you’re not directly interested in the differential equation or the physical problem that motivated it.
Bessel functions satisfy
x² y″ + x y′ + (x² − n²) y = 0.
You can tell a lot about Bessel functions just by inspecting this equation without solving it. If we divide by x² and write the equation in the form
y″ + (p(x)/x) y′ + (q(x)/x²) y = 0
then p(x) = 1 and q(x) = x² − n². The indicial equation
r(r − 1) + p(0) r + q(0) = 0
reduces to
r² = n²
and so r = ±n. That alone tells us there are two solutions, one analytic at zero and one singular at zero. These are Jn and Yn respectively. It also tells us the behavior of these functions as x goes to zero and as x goes to infinity. To find out more, look up “method of Frobinius.”
The post Inverse differential equations first appeared on John D. Cook.2026-08-10 01:17:42
I was reading a book on the history of the discovery of the structure of DNA [1] and was surprised by a few passing references to Bessel functions.
According to Claude,
When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal “layer lines.” Cochran, Crick, and Vand showed mathematically that the diffracted amplitude on the n-th layer line is proportional to a Bessel function of the first kind, order n:
Jn(2π r R)
where r is the radius of the helix and R is the distance out from the center (the meridian) in the diffraction pattern.
The citation for this paragraph is a paper from 1952 [2] that amazingly is behind a paywall.
Update: Evan pointed out something fishy in the expression above: “Isn’t there a dimensional problem with J(2π r R)? Maybe 2πr/R instead?”
This was Claude’s response:
So Evan and Claude were both right. There is an implicit division happening, but crystallographers do write r R.
[1] Watson and Crick didn’t “discover DNA” as is commonly said. DNA was discovered in 1878. Watson and Crick discovered the structure of DNA in 1953.
[2] Cochran, W., Crick, F. H. C., & Vand, V. (1952). “The Structure of Synthetic Polypeptides. I. The Transform of Atoms on a Helix.” Acta Crystallographica, 5(5), 581–586.
The post DNA and Bessel functions first appeared on John D. Cook.2026-08-10 00:46:59
At the end of my post on how not to calculate cosine I said that the first step in calculating cosine, particularly cosine of a large number, would be to do range reduction. This post will present a simple range reduction method by Cody and Waite that is adequate for moderately large arguments.
If you want to compute the sine or cosine of an angle x you could start by reducing x mod 2π since that would not change the result. However, accurately reducing a number mod 2π is not trivial; that’s why range reduction is an area of algorithm development.
Even better would be to reduce x mod π/2. Reducing to a smaller range means that power series method, and other methods such as rational approximation, will be more efficient.
So suppose you can find an integer k such that
x − k π/2 = y
where 0 ≤ y ≤ π/2. Then sin(x) is ±sin(y) or ±cos(y), depending on k mod 4 equals 0, 1, 2, or 3.
from math import *
def reduced_sin(x, k):
match k % 4:
case 0: return sin(x)
case 1: return cos(x)
case 2: return -sin(x)
case 3: return -cos(x)
Now let’s set x = 500. Then k = 318 because that’s the multiple of π/2 we need to subtract to bring x into range, and the sine of x should be the negative of the sine of the reduced value y because 318 = 2 mod 4.
The following code computes sin(x) with naive range reduction
def naive_sin(x):
k = floor(x / (pi/2))
y = x % (pi/2)
return reduced_sin(y, k)
and when x = 500 the error is on the order of 1.7 × 10−14.
The value of k above is fine, but we’d like to calculate y more accurately. The following code is much better.
def Cody_Waite_sin(x):
C1 = 1686629713 / 2**30
C2 = 4701928774853425 / 2**86
k = floor(x / (pi/2))
y = (x - k*C1) - k*C2
return reduced_sin(y, k)
This will compute sin(500) to full machine precision. What kind of magic is this?
The trick is that the exact value of C1 + C2 equals π/2 to more precision than is possible in a single float [1]. You can confirm, with bc or some other extended precision software, that the difference between C1 + C2 and π/2 is roughly 2−88, while the limit of float precision is 2−52.
If we compute
y = x - k*(C1 + C2)
then we’re doing the same calculation as naive_sin and will get the same error. But if we compute
y = (x - k*C1) - k*C2
we will get a more accurate result, provided x isn’t too large.
You can use the following code to play around and see how large x can be before errors start to creep in. For small enough x, like 500, the Cody and Waite sine returns full precision. For larger x it’s better than naive sine but does not return full precision. And for large enough x it completely breaks down.
def compare(x):
y0 = naive_sin(x)
y1 = Cody_Waite_sin(x)
y2 = sin(x)
print("Naive error: ", y2 - y0)
print("Cody Waite error:", y2 - y1)
Now this may seem circular since we’re using math.sin as our gold standard. However, this function is calling the sine function on your CPU, which is using sophisticated range reduction to compute its result accurately down to the last bit, assuming you run the code on a computer that’s less than 40 years old.
The Cody and Waite algorithm is inadequate for large x, but it’s a good place to begin studying range reduction. It shows there are clever ways of squeezing out more precision than seems possible.
[1] The numerator n1 of C1 is ⌊230 π/2⌋. The numerator n2 of C2 is the solution to
286−30n1 + n2 = ⌊286 π/2⌋.
The post A simple range reduction method first appeared on John D. Cook.
