Recent content on Math ∩ Programming
Here are some more whimsical OEIS sequences I came across. XKCD 2016 joked that “OEIS keeps rejecting my submissions,” including one that gives “Integers in increasing order of width when printed in Helvetica.” Well, two days after that comic was published (2018-07-09), Hugo Pfoertner published A316600, with a very precise definition. Then he did Arial. Randall Munroe…
Table of Contents In this tutorial series, I will introduce the CKKS homomorphic encryption scheme from the ground up, in rather intricate detail. Each article in this series corresponds to a pull request on a GitHub repository. The code for this article is in this pull request. Follow along by cloning the repository and checking out the code at the relevant commit. This first article will cover…
I went hunting for references to the OEIS in open source code, and found some weird ones. There are not one, but two live-coding music frameworks that use OEIS sequences as a source for “anything that can be sequenced” in music. I’m guessing that’s used for choosing pseudorandom melodies, interesting rhythyms, or how to overlap tracks in different ways. The first project is…
A051070 is a sequence about OEIS sequences. a(n) is the n-th term in sequence A_n (or -1 if A_n doesn’t have enough terms). So the first term in A051070 is 1 because A000001 is the number of groups of order n, and that sequence has 1 as its entry in index 1. A000002 is the Kolakoski sequence (what? For another time) and has value 2 in entry 2. The sequence continues: 1, 2, 1, 0, 2, 3, 0, 7,…
Problem: Determine if a 32-bit number is prime (deterministically) Solution: (in C++) // Bases to test. Using the first 4 prime bases makes the test deterministic // for all 32-bit integers. See https://oeis.org/A014233. int64_t bases[] = {2, 3, 5, 7}; inline int countTrailingZeros(uint64_t n) { if (n == 0) return 64; return __builtin_ctzll(n); } int64_t modularExponentiation(int64_t base, int64_t…
It’s the 5th annual April Cools! Here are my previous April Cools articles This year it’s a book review of Ben Recht’s book, The Irrational Decision: How We Gave Computers the Power to Choose For us, released Mar 10, 2026. The publishing industry has a stupid name for the subcategory of non-fiction book where a domain expert weaves factual evidence together into a story to try to…
In an earlier article, I covered the basic technique for performing matrix-vector multiplication in fully homomorphic encryption (FHE), known as the Halevi-Shoup diagonal method. This article covers a more recent method for matrix-matrix multiplication known as the bicyclic method. The code implementing this method is in the same GitHub repository as the previous article, and the bicyclic method…
Polyhedral optimization is a tool used in compilers for optimizing loop nests. While the major compilers that use this implement polyhedral optimizations from scratch,1 there is a generally-applicable open source C library called the Integer Set Library (ISL) that implements the core algorithms used in polyhedral optimization. This article gives an overview of a subset of ISL, mainly focusing on…
On Monday, July 14th 2025, I hosted a mini-workshop on homomorphic encryption at Google’s Portland, Oregon office. Though Portland is a small city, it’s becoming a hub for homomorphic encryption. Intel and Google both have a presence here, as well as the hardware startup Niobium, and a few individuals from other companies who happen to be based here. Since I had been having lunch with…
I work on homomorphic encryption (HE or FHE for “fully” homomorphic encryption) and I have written a lot about it on this blog (see the relevant tag). This article is a collection of short answers to questions I see on various threads and news aggregators discussing FHE. Facts If a service uses FHE and can respond to encrypted queries, can’t the service see your query? How is it…
I was inspired to browse some of Edsger Dijkstra’s essays today, and came across his speech, “Under the spell of Leibniz’s Dream”. It’s the sort of personal history I love to read, which gives one person’s sense of the world over a period of change. One quote that immediately struck me was his view of his country’s spirit after WWII: An important…
Editor’s note: This essay was originally published on Medium on 2016-03-05. I have made minor edits in this republishing and added a few small retrospective notes. 2010–2011 (Year 0) I had just switched my major at Cal Poly State University from computer science to math. I wanted to double major but California was in a budget crisis and a few weeks before I tried submitting my double-major…
Last month I gave a talk on the HEIR compiler project at the FHE.org conference in Sofia, Bulgaria. The video is on YouTube now, and the slides are public. I plan to write more about HEIR in the coming months, because it’s been an exciting and fulfilling ride!
It’s April Cools! Last year I wrote about parenting, in 2023 about friendship bracelets. and in 2022 about cocktails. This year it’s a bit of a meandering stroll through some ideas around mutual aid and self-reliance. Maternity wards If you walk around the maternity ward at Kaiser Permanente’s Sunnyside medical center outside of Portland, Oregon, you might notice the same two…
My four-year-old son has declared 36 to be the best number. His reason: 36 is the only number (he knows of) that is both a square and a staircase number AND an up-and-down-staircase number. “Staircase numbers” are what he calls triangular numbers (numbers that are the sum of the first $n$ integers). This name comes from the blocks he has that can be arranged into a staircase. He also…
A colleague of mine recently lent a hand implementing a polynomial approximation routine I could port to our compiler, though it wasn’t the method I was expecting. As I had written about previously, I was studying the Remez algorithm and implementing a prototype in Python. Remez approximation involves an iterated loop that alternates between root-finding and linear-system solving, and as…
Back in 2020, when I worked in the supply chain side of Google, I had a fun and impactful side project related to human-level explanations of linear programs. A linear program is a mathematical model that defines some number of variables, linear constraints, and a linear objective function. When some variables are forced to be integer (ILPs), you can solve a lot of useful problems like scheduling,…
I’ll be at the Joint Mathematics Meeting in Seattle (starting tomorrow). If you see me there, say hi! I will have a very light schedule, plenty of time for coffee chats. I’ll be attending many of the crypto sessions for the homomorphic encryption talks. And on Thursday at 3PM, I’ll be at the Code4Math booth in the exhibition hall. I’ll be chatting with math people who want…
The Hyperfixed Podcast had a lovely episode recently about tape measures. It started from “why does my tape measure seem to always be off a little bit” and went all the way to the inherent limitations of physical measurement at small scales. In there is an awesome quote by Adam Savage, “I had always had faith in the sanctity and solidity of numbers… and when I got into…
In this living document, I will document reactions to uses of homomorphic encryption by members of the public. By “member of the public,” I mean people who may be technical, but are not directly involved in the development or deployment of homomorphic encryption systems. This includes journalists, bloggers, aggregator comment threads, and social media posts. My main goal is to…
In my little corner of the FHE world, things have been steadily heating up. For those who don’t know, my main work project right now is HEIR (Homomorphic Encryption Intermediate Representation), a compiler toolchain for fully homomorphic encryption (FHE). For an extended introduction see this talk from October 2023. The primary focus of HEIR is to compile to FHE hardware accelerators. And…
Editor’s note: This essay was originally published in 2019. I have made minor edits in this republishing. There was a MathOverflow thread about mathematically interesting games for 5–6 year olds. A lot of the discussion revolved around how young age 5 really is, and how we should temper expectations because we don’t really remember what it’s like to be 5. In response to an…
Welcome to the 233rd Carnival of Mathematics! Who can forget 233, the 6th Fibonacci prime? Hey, not all numbers are interesting. Don’t ask me about the smallest positive uninteresting number. You can’t make it interesting with your feeble mind tricks! Anyway, on to the fun. Provers and Shakers The big discovery this month was a new largest known prime number, $2^{136279841} - 1$, as…
This article will explain how the blog is organized at a technical level, and show how I implemented various IndieWeb features. Table of Contents: Motivation Structure and Deployment Static search index Running scripts via GitHub Actions Social media syndication and the “shortform” section Links to syndicated versions at the end of each post Warning for a too-long first paragraph…
Kristin Lauter and her colleagues at Facebook research recently announced a project to benchmark attacks against LWE. The announcement was on the post-quanum crypto mailing list. They state: “Our approach is motivated by the need to study more carefully the effect on security of using small secrets and small error in standardized LWE settings like Kyber and Homomorphic Encryption. In…
This is a story about a failure to apply dynamic programming to a woodworking project. I’ve been building a shed in my backyard, and for one section I decided to build the floor by laying 2x4 planks side by side. I didn’t feel the need to join them with tongue-and-groove, but I did notice that using 2x4s alone wouldn’t fit the width they were supposed to fill. I also had some 2x6…
In my recent overview of homomorphic encryption, I underemphasized the importance of data layout when working with arithmetic (SIMD-style) homomorphic encryption schemes. In the FHE world, the name given to data layout strategies is called “packing,” because it revolves around putting multiple plaintext data into RLWE ciphertexts in carefully-chosen ways that mesh well with the…
In my recent overview of homomorphic encryption, I underemphasized the importance of data layout when working with arithmetic (SIMD-style) homomorphic encryption schemes. In the FHE world, the name given to data layout strategies is called “packing,” because it revolves around putting multiple plaintext data into RLWE ciphertexts in carefully-chosen ways that mesh well with the…
This blog now accepts webmentions. I used webmention.io and webmention.js for live rendering. You can see an example at the end of my old Bezier Curves post. After my initial experiments with POSSE, I’ve made a few improvements to the system. Now shortform posts are syndicated to Mastodon, Bluesky, and Twitter, and the links to the syndicated posts are automatically added to the end of each…
Table of Contents In this article I’ll show how to use PDLL, a tool for defining MLIR patterns, which itself is built with MLIR. PDLL is intended to be a replacement for defining patterns in tablegen, though there are few public examples of its use. In fact, the main impetus for PDLL is that tablegen makes it difficult to express things like: Operations that return multiple results…
I’ve been upstreaming a bit of my compiler work to the MLIR project. Yesterday, I merged in a tutorial on mlir-opt, the main debugging tool for running passes on MLIR code. This is roughly the upstreamable parts of my first MLIR tutorial entry, MLIR — Running and Testing a Lowering. Mehdi Amini also provided a lot of useful information during review that taught me some stuff I didn’t…
In this living document, I will list all production systems I’m aware of that use fully homomorphic encryption (FHE). For background on FHE, see my overview of the field. If you have any information about production FHE systems not in this list, or corrections to information in this list, please send me an email with sufficient detail allow the claim to be publicly verified. For all…
Ben Recht, a computer science professor at UC Berkeley, recently wrapped up a 3-month series of blog posts on Paul Meehl’s “Philosophical Psychology.” Recht has a table of contents for his blog series. It loosely tracks a set of lectures that Meehl gave in 1989 at the University of Minnesota. In it, he surveys of the philosophy of science, lays out a framework for scientific…
A quick note: you can use C++11 templates to detect struct fields by name and type, and statically branch on them. I first heard of this solution from breeze1990. Say I want to detect if a struct has a field size of type int. Create two template instantiations of the same name, here HasStaticSize that defaults to false. #include <type_traits> template <typename T, typename = void> struct…
In my studies of the Remez algorithm, I learned about the barycentric Lagrange interpolation formula. The context is finding a polynomial of degree at most $n$ that passes through $n+1$ points $(x_0, y_0), \dots, (x_n, y_n)$. The classical Lagrange interpolation formula is what you’d write down if you “just did it.” $$f(x) = \sum_{i=0}^n y_i \cdot \prod_{j \neq i}\frac{x -…
I added Twitter syndication, and because I have nothing to test it with I’ll share some random life updates. My daughter was born recently, which means I’m on paternity leave for a few months. Hopefully in the liminal hours of sleep training, I’ll have some time to work on my book. Or at least catch up on reading. I finally published my grandmother’s autobiography. I doubt…
Steven Clontz informed me of an effort he’s involved in called code4math. It’s described as a professional organization for the advancement of mathematical research through building non-research software infrastructure. By that he means, for example, writing software packages like Macaulay2 or databases of mathematical objects that other researchers can use to do their research. Clontz…
POSSE stands for Publish (on your) Own Site, Syndicate Elsewhere. I first heard about it from Cory Doctorow. I’m experimenting with automation to convert posts tagged shortform into Mastodon threads (I’m mathstodon.xyz/@j2kun). I’m using Hugo as a static site generator, with the source a (private) GitHub repository, and Netlify for deployments. After a deployment, Netlify calls a…
I’ve been learning recently about how to approximate functions by low-degree polynomials. This is useful in fully homomorphic encryption (FHE) in the context of “arithmetic FHE” (see my FHE overview article), where the computational model makes low-degree polynomials cheap to evaluate and non-polynomial functions expensive or impossible. In browsing the state of the art I came…
About two years ago, I switched teams at Google to focus on fully homomorphic encryption (abbreviated FHE, or sometimes HE). Since then I’ve got to work on a lot of interesting projects, learning along the way about post-quantum cryptography, compiler design, and the ins and outs of fully homomorphic encryption. If you’ve heard about FHE and you’re a software person, you’ve…
It’s April Cools! Last year I wrote about friendship bracelets and the year before about cocktails. This year it’s parenting. Parenting articles are a dime a dozen and always bury the lede behind a long story. I’ll skip that. How to think about your child and your role as a parent These are framing devices. Concrete things to do to work toward these are in the next section. I will…
There’s a family of tabletop games that are based directly on a nontrivial mathematics problem. As a casual and fun way to inaugurate my new blog (migrated from Wordpress to Hugo, after my work on getting better LaTeX mathmode support in Hugo), I thought I’d write a short listicle about them, so that I have a place to add more as I find them, as well as give the shortest canonical…
Table of Contents In this article we’ll implement a global optimization pass, and show how to use the dataflow analysis framework to verify the results of our optimization. The code for this article is in this pull request, and as usual the commits are organized to be read in order. The noisy arithmetic problem This demonstration is based on a simplified model of computation relevant to the HEIR…
Table of Contents In the last article we lowered our custom poly dialect to standard MLIR dialects. In this article we’ll continue lowering it to LLVM IR, exporting it out of MLIR to LLVM, and then compiling to x86 machine code. The code for this article is in this pull request, and as usual the commits are organized to be read in order. Defining a Pipeline The first step in lowering to machine…
Table of Contents In previous articles we defined a dialect, and wrote various passes to optimize and canonicalize a program using that dialect. However, one of the main tenets of MLIR is “incremental lowering,” the idea that there are lots of levels of IR granularity, and you incrementally lower different parts of the IR, only discarding information when it’s no longer useful for optimizations.…
Can you find a set of cards among these six, such that the socks on the chosen cards can be grouped into matching pairs? (Duplicate pairs of the same sock are OK) Spoilers: If the cards are indexed as 1 2 3 4 5 6 Then the following three subsets work: $\{ 1, 2, 4, 5, 6 \}$, $\{ 2, 3, 6 \}$, and $\{ 1, 3, 4, 5 \}$.
Table of Contents In a previous article we defined folding functions, and used them to enable some canonicalization and the sccp constant propagation pass for the poly dialect. This time we’ll see how to add more general canonicalization patterns. The code for this article is in this pull request, and as usual the commits are organized to be read in order. Why is Canonicalization Needed? MLIR…
In cryptography, we need a distinction between a cleartext and a plaintext. A cleartext is a message in its natural form. A plaintext is a cleartext that is represented in a specific way to prepare it for encryption in a specific scheme. The process of taking a cleartext and turning it into a plaintext is called encoding, and the reverse is called decoding. In homomorphic encryption, the…
Table of Contents Last time we defined folders and used them to enable some canonicalization and the sccp constant propagation pass for the poly dialect. This time we’ll add some additional safety checks to the dialect in the form of verifiers. The code for this article is in this pull request, and as usual the commits are organized to be read in order. Purpose of a verifier Verifiers ensure the…
Table of Contents Last time we saw how to use pre-defined MLIR traits to enable upstream MLIR passes like loop-invariant-code-motion to apply to poly programs. We left out -sccp (sparse conditional constant propagation), and so this time we’ll add what is needed to make that pass work. It requires the concept of folding. The code for this article is in this pull request, and as usual the commits…