Math ∩ Programming

Table of Contents This is part of a tutorial series on the CKKS homomorphic encryption scheme.

Table of Contents This is part of a tutorial series on the CKKS homomorphic encryption scheme.

A quick update on our homomorphic encryption compiler HEIR: our partners at Belfort have a hotword detector in FHE that runs in 350ms.

Table of Contents This is part of a tutorial series on the CKKS homomorphic encryption scheme.

On 2026-08-14 I published an article on the Google Security blog with an update on HEIR, our homomorphic encryption (HE) compiler.

Here are some more whimsical OEIS sequences I came across.

Table of Contents In this tutorial series, I will introduce the CKKS homomorphic encryption scheme from the ground up, in rather intricate detail.

I went hunting for references to the OEIS in open source code, and found some weird ones.

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).

Problem: Determine if a 32-bit number is prime (deterministically) Solution: (in C++) // Bases to test.

It’s the 5th annual April Cools!

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.

Polyhedral optimization is a tool used in compilers for optimizing loop nests.

On Monday, July 14th 2025, I hosted a mini-workshop on homomorphic encryption at Google’s Portland, Oregon office.

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).

I was inspired to browse some of Edsger Dijkstra’s essays today, and came across his speech, “Under the spell of Leibniz’s Dream”.

Editor’s note: This essay was originally published on Medium on 2016-03-05.

Last month I gave a talk on the HEIR compiler project at the FHE.org conference in Sofia, Bulgaria.

It’s April Cools!

My four-year-old son has declared 36 to be the best number.

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.

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.

I’ll be at the Joint Mathematics Meeting in Seattle (starting tomorrow).

The Hyperfixed Podcast had a lovely episode recently about tape measures.

In this living document, I will document reactions to uses of homomorphic encryption by members of the public.

In my little corner of the FHE world, things have been steadily heating up.

Editor’s note: This essay was originally published in 2019.

Welcome to the 233rd Carnival of Mathematics!

This article will explain how the blog is organized at a technical level, and show how I implemented various IndieWeb features.

Kristin Lauter and her colleagues at Facebook research recently announced a project to benchmark attacks against LWE.

This is a story about a failure to apply dynamic programming to a woodworking project.

In my recent overview of homomorphic encryption, I underemphasized the importance of data layout when working with arithmetic (SIMD-style) homomorphic encryption schemes.

In my recent overview of homomorphic encryption, I underemphasized the importance of data layout when working with arithmetic (SIMD-style) homomorphic encryption schemes.

This blog now accepts webmentions.

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.

I’ve been upstreaming a bit of my compiler work to the MLIR project.

In this living document, I will list all production systems I’m aware of that use fully homomorphic encryption (FHE).

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.”

A quick note: you can use C++11 templates to detect struct fields by name and type, and statically branch on them.

In my studies of the Remez algorithm, I learned about the barycentric Lagrange interpolation formula.

I added Twitter syndication, and because I have nothing to test it with I’ll share some random life updates.

Steven Clontz informed me of an effort he’s involved in called code4math.

POSSE stands for Publish (on your) Own Site, Syndicate Elsewhere.

I’ve been learning recently about how to approximate functions by low-degree polynomials.

About two years ago, I switched teams at Google to focus on fully homomorphic encryption (abbreviated FHE, or sometimes HE).

It’s April Cools!

There’s a family of tabletop games that are based directly on a nontrivial mathematics problem.

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.

Table of Contents In the last article we lowered our custom poly dialect to standard MLIR dialects.

Table of Contents In previous articles we defined a dialect, and wrote various passes to optimize and canonicalize a program using that dialect.