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.

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.

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.

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.

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.

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.

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.

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.