## Linear Feedback Shift Registers for the Uninitiated, Part V: Difficult Discrete Logarithms and Pollard's Kangaroo Method

October 1, 2017

Last time we talked about discrete logarithms which are easy when the group in question has an order which is a smooth number, namely the product of small prime factors. Just as a reminder, the goal here is to find $k$ if you are given some finite multiplicative group (or a finite field, since it has a multiplicative group) with elements $y$ and $g$, and you know you can express $y = g^k$ for some unknown integer $k$. The value $k$ is the discrete logarithm of $y$...

## Linear Feedback Shift Registers for the Uninitiated, Part IV: Easy Discrete Logarithms and the Silver-Pohlig-Hellman Algorithm

Last time we talked about the multiplicative inverse in finite fields, which is rather boring and mundane, and has an easy solution with Blankinship’s algorithm.

Discrete logarithms, on the other hand, are much more interesting, and this article covers only the tip of the iceberg.

What is a Discrete Logarithm, Anyway?

Regular logarithms are something that you’re probably familiar with: let’s say you have some number $y = b^x$ and you know $y$ and $b$ but...

## Linear Feedback Shift Registers for the Uninitiated, Part III: Multiplicative Inverse, and Blankinship's Algorithm

September 9, 2017

Last time we talked about basic arithmetic operations in the finite field $GF(2)[x]/p(x)$ — addition, multiplication, raising to a power, shift-left and shift-right — as well as how to determine whether a polynomial $p(x)$ is primitive. If a polynomial $p(x)$ is primitive, it can be used to define an LFSR with coefficients that correspond to the 1 terms in $p(x)$, that has maximal length of $2^N-1$, covering all bit patterns except the all-zero...

## Linear Feedback Shift Registers for the Uninitiated, Part II: libgf2 and Primitive Polynomials

July 17, 2017

Last time, we looked at the basics of LFSRs and finite fields formed by the quotient ring $GF(2)[x]/p(x)$.

LFSRs can be described by a list of binary coefficients, sometimes referred as the polynomial, since they correspond directly to the characteristic polynomial of the quotient ring.

Today we’re going to look at how to perform certain practical calculations in these finite fields. I maintain a Python library on bitbucket called...

## Linear Feedback Shift Registers for the Uninitiated, Part I: Ex-Pralite Monks and Finite Fields

Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.

— Évariste Galois, May 29, 1832

I was going to call this short series of articles “LFSRs for Dummies”, but thought better of it. What is a linear feedback shift register? If you want the short answer, the Wikipedia article is a decent introduction. But these articles are aimed at those of you who want a little bit deeper mathematical understanding,...

## Ten Little Algorithms, Part 6: Green’s Theorem and Swept-Area Detection

Other articles in this series:

## Donald Knuth Is the Root of All Premature Optimization

TL;DR

The idea, basically, is that even though optimization of computer software to execute faster is a noble goal, with tangible benefits, this costs time and effort up front, and therefore the decision to do so should not be made on whims and intuition, but instead should be made after some kind of analysis to show that it has net...

## Zebras Hate You For No Reason: Why Amdahl's Law is Misleading in a World of Cats (And Maybe in Ours Too)

February 27, 20171 comment

I’ve been wasting far too much of my free time lately on this stupid addicting game called the Kittens Game. It starts so innocently. You are a kitten in a catnip forest. Gather catnip.

And you click on Gather catnip and off you go. Soon you’re hunting unicorns and building Huts and studying Mathematics and Theology and so on. AND IT’S JUST A TEXT GAME! HTML and Javascript, that’s it, no pictures. It’s an example of an

## The Other Kind of Bypass Capacitor

There’s a type of bypass capacitor I’d like to talk about today.

It’s not the usual power supply bypass capacitor, aka decoupling capacitor, which is used to provide local charge storage to an integrated circuit, so that the high-frequency supply currents to the IC can bypass (hence the name) all the series resistance and inductance from the power supply. This reduces the noise on a DC voltage supply. I’ve...

## How to Succeed in Motor Control: Olaus Magnus, Donald Rumsfeld, and YouTube

December 11, 2016

Almost four years ago, I had this insight — we were doing it wrong! Most of the application notes on motor control were about the core algorithms: various six-step or field-oriented control methods, with Park and Clarke transforms, sensorless estimators, and whatnot. It was kind of like a driving school would be, if they taught you how the accelerator and brake pedal worked, and how the four-stroke Otto cycle works in internal combustion engines, and handed you a written...

## Lost Secrets of the H-Bridge, Part I: Ripple Current in Inductive Loads

July 8, 2013

So you think you know about H-bridges? They're something I mentioned in my last post about signal processing with Python.

Here we have a typical H-bridge with an inductive load. (Mmmmm ahhh! It's good to draw by hand every once in a while!) There are four power switches: QAH and QAL connecting node A to the DC link, and QBH and QBL connecting node B to the DC link. The load is connected between nodes A and B, and here is represented by an inductive load in series with something else. We...

## The Least Interesting Circuit in the World

It does nothing, most of the time.

It cannot compute pi. It won’t oscillate. It doesn’t light up.

Often it makes other circuits stop working.

It is… the least interesting circuit in the world.

What is it?

About 25 years ago, I took a digital computer architecture course, and we were each given use of an ugly briefcase containing a bunch of solderless breadboards and a power supply and switches and LEDs — and a bunch of

## Important Programming Concepts (Even on Embedded Systems) Part II: Immutability

September 14, 2014

Other articles in this series:

This article will discuss immutability, and some of its variations in the topic of functional programming.

There are a whole series of benefits to using program variables that… well, that aren’t actually variable, but instead are immutable. The impact of...

## Linear Feedback Shift Registers for the Uninitiated, Part I: Ex-Pralite Monks and Finite Fields

Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.

— Évariste Galois, May 29, 1832

I was going to call this short series of articles “LFSRs for Dummies”, but thought better of it. What is a linear feedback shift register? If you want the short answer, the Wikipedia article is a decent introduction. But these articles are aimed at those of you who want a little bit deeper mathematical understanding,...

## Another 10 Circuit Components You Should Know

October 30, 20131 comment

It's that time again to review all the oddball goodies available in electronic components. These are things you should have in your bag of tricks when you need to design a circuit board. If you read my previous posts and were looking forward to more, this article's for you!

1. Bus switches

I can't believe I haven't mentioned bus switches before. What is a bus switch?

There are lots of different options for switches:

• mechanical switch / relay: All purpose, two...

## Slew Rate Limiters: Nonlinear and Proud of It!

October 6, 2014

I first learned about slew rate limits when I was in college. Usually the subject comes up when talking about the nonideal behavior of op-amps. In order for the op-amp output to swing up and down quickly, it has to charge up an internal capacitor with a transistor circuit that’s limited in its current capability. So the slew rate limit $\frac{dV}{dt} = \frac{I_{\rm max}}{C}$. And as long as the amplitude and frequency aren’t too high, you won’t notice it. But try to...

## Oscilloscope Dreams

My coworkers and I recently needed a new oscilloscope. I thought I would share some of the features I look for when purchasing one.

When I was in college in the early 1990's, our oscilloscopes looked like this:

Now the cathode ray tubes have almost all been replaced by digital storage scopes with color LCD screens, and they look like these:

Oscilloscopes are basically just fancy expensive boxes for graphing voltage vs. time. They span a wide range of features and prices:...

## Lessons Learned from Embedded Code Reviews (Including Some Surprises)

My software team recently finished a round of code reviews for some of our motor controller code. I learned a lot from the experience, most notably why you would want to have code reviews in the first place.

My background is originally from the medical device industry. In the United States, software in medical devices gets a lot of scrutiny from the Food and Drug Administration, and for good reason; it’s a place for complexity to hide latent bugs. (Can you say “

## The CRC Wild Goose Chase: PPP Does What?!?!?!

I got a bad feeling yesterday when I had to include reference information about a 16-bit CRC in a serial protocol document I was writing. And I knew it wasn’t going to end well.

The last time I looked into CRC algorithms was about five years ago. And the time before that… sometime back in 2004 or 2005? It seems like it comes up periodically, like the seventeen-year locust or sunspots or El Niño,...

## First-Order Systems: The Happy Family

May 3, 20141 comment
Все счастли́вые се́мьи похо́жи друг на дру́га, ка́ждая несчастли́вая семья́ несчастли́ва по-сво́ему.

— Лев Николаевич Толстой, Анна Каренина

Happy families are all alike; every unhappy family is unhappy in its own way.

— Lev Nicholaevich Tolstoy, Anna Karenina

I was going to write an article about second-order systems, but then realized that it would be...