Last time we covered an operation in the LWE encryption scheme called modulus switching, which allows one to switch from one modulus to another, at the cost of introducing a small amount of extra noise, roughly $\sqrt{n}$, where $n$ is the dimension of the LWE ciphertext.
The Learning With Errors problem is the basis of a few cryptosystems, and a foundation for many fully homomorphic encryption (FHE) schemes. In this article I’ll describe a technique used in some of these schemes called modulus switching. In brief, an LWE sample is a vector of values in $\mathbb{Z}/q\mathbb{Z}$ for some $q$, and in LWE cryptosystems an LWE sample can be modified so that it hides a secret message $m$.
This is a draft of a chapter from my in-progress book, Practical Math for Programmers: A Tour of Mathematics in Production Software. Tip: Determine an aggregate statistic about a sensitive question, when survey respondents do not trust that their responses will be kept secret.
It’s April Cools! We’re taking back April Fools. When I was younger I had a strange relationship with alcohol, not because of any sort of trauma, but because I found it decidedly boring and disgusting to the taste. I didn’t drink in high school, didn’t enjoy parties in college, and didn’t care for tailgating or other sports-based events where drinking was common.
Previous posts in this series: Silent Duels and an Old Paper of Restrepo Silent Duels—Parsing the Construction Silent Duels—Constructing the Solution part 1 Since it’s been three years since the last post in this series, and the reason for the delay is that I got totally stuck on the implementation. I’m publishing this draft article as partial progress until I can find time to work on it again. If you haven’t read the last post, please do.
tl;dr: I’m writing a new book, sign up for the announcements mailing list. I’ve written exactly zero new technical blog posts this year because I’ve been spending all my writing efforts on my next book, Practical Math for Programmers (PMFP, subtitle: A Tour of Mathematics in Production Software). I’ve written a little bit about it in my newsletter, Halfspace. There I rant, critique, brainstorm, and wax poetic about math and software.
Lately I’ve been studying Fully Homomorphic Encryption, which is the miraculous ability to perform arbitrary computations on encrypted data without learning any information about the underlying message.
I learned of a neat result due to Kevin Ventullo that uses group actions to study the structure of hash functions for unordered sets and multisets. This piqued my interest because a while back a colleague asked me if I could think of any applications of “pure” group theory to practical computer programming that were not cryptographic in nature.
Welcome to the 197th Carnival of Mathematics! 197 is an unseemly number, as you can tell by the Wikipedia page which currently says that it has “indiscriminate, excessive, or irrelevant examples.” How deviant. It’s also a Repfigit, which means if you start a fibonacci-type sequence with the digits 1, 9, 7, and then continue with $ a_n = a_{i-3} + a_{i-2} + a_{i-1}$, then 197 shows up in the sequence.
We’re ironically searching for counterexamples to the Riemann Hypothesis. Setting up Pytest Adding a Database Search Strategies Unbounded integers Deploying with Docker Performance Profiling Scaling up Productionizing In the last article we added a menagerie of “production readiness” features like continuous integration tooling (automating test running and static analysis), alerting, and a simple deployment automation.
Recently I’ve been helping out with a linear algebra course organized by Tai-Danae Bradley and Jack Hidary, and one of the questions that came up a few times was, “why should programmers care about the concept of a linear combination?” For those who don’t know, given vectors $ v_1, \dots, v_n$, a linear combination of the vectors is a choice of some coefficients $ a_i$ with which to weight the vectors in a sum $ v = \sum_{i=1}^n a_i v_i$.