On 2026-08-14 I published an article on the Google Security blog with an update on HEIR, our homomorphic encryption (HE) compiler. This is a companion article, in which I have no limits on word count or jargon, and I can feel free to be honest. So strap in. Assuming you won’t read the linked corporate blog post, HEIR is a compiler that converts an input program to a program that operates directly on encrypted data.
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 missed a huge opportunity to commit to his bit and actually try to submit some of his sequences before publishing the comic.
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 some of the mathematical background necessary in the formulation of the CKKS encryption scheme, specifically the polynomial ring used in the most basic version of CKKS, and the canonical embedding…
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 called mercury, which is advertised as having “an extensive library of algorithms to generate or transform numbersequences that can modulate parameters.
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, 8, 4, 63, 1, 316, …
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 exponent, int64_t modulus) { int64_t res = 1; int64_t b = base % modulus; int64_t e = exponent; while (e > 0) { if (e & 1) { // Doesn't overflow because we assume 32-bit integer inputs res = (res *…
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 convince you of their thesis: a “big idea” book.
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 is in a file called bicyclic.py. The previous article linked above covers the general concepts behind “FHE packing,” which I will assume as background knowledge for this article:
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 the representation of sets and relations and basic manipulations on them.
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 various subsets of the community, I figured it was about time to get us all in a room together.
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 possible to operate on encrypted data without seeing 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 side-effect of the hard times was the creation of a spiritual climate in which the distinction between pure and applied science had vanished: of all the things you could do, you just did the most urgent one, and the development of some urgently…
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 request the Provost for the CSU system put a blanket ban on double majors.
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 things I did. The first was how many signs were posted on the hallways describing the maternity team’s KRs and displaying charts showing that they are meeting them.
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 calls them “step squad” numbers thanks to Numberblocks.
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 such it can be rather brittle and difficult. Numerical errors and accuracy limits in these subsolvers contribute to weird edge cases that make the algorithm fail to converge.
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, routing, and packing. That’s basically how all supply chain optimization works.
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 to get into programming, mainly as field research for a future book on programming for mathematicians, the converse of A Programmer’s Introduction to Mathematics
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 this… I realized there’s no such thing as a measurement.
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