## One Clock Cycle Polynomial Math

Error correction codes and cryptographic computations are most easily performed working with GF(2^n)

## Elliptic Curve Cryptography

Secure online communications require encryption. One standard is AES (Advanced Encryption Standard) from NIST. But for this to work, both sides need the same key for encryption and decryption. This is called Private Key encryption.

## Ten Little Algorithms, Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method

Today we will be drifting back into the topic of numerical methods, and look at an algorithm that takes in a series of discretely-sampled data points, and estimates the maximum value of the waveform they were sampled from.

## Polynomial Math

Elliptic Curve Cryptography is used as a public key infrastructure to secure credit cards, phones and communications links. All these devices use either FPGA's or embedded microprocessors to compute the algorithms that make the mathematics work. While the math is not hard, it can be confusing the first time you see it. This blog is an introduction to the operations of squaring and computing an inverse over a finite field which are used in computing Elliptic Curve arithmetic. ...

## Number Theory for Codes

Everything in the digital world is encoded. ASCII and Unicode are combinations of bits which have specific meanings to us. If we try to interpret a compiled program as Unicode, the result is a lot of garbage (and beeps!) To reduce errors in transmissions over radio links we use Error Correction Codes so that even when bits are lost we can recover the ASCII or Unicode original. To prevent anyone from understanding a transmission we can encrypt the raw data...

## Practical CRCs for Embedded Systems

CRCs are a very practical tool for embedded systems: you're likely to need to use one as part of a communications protocol or to verify the integrity of a program image before writing it to flash. But CRCs can be difficult to understand and tricky to implement. The first time I attempted to write CRC code from scratch I failed once. Then twice. Then three times. Eventually I gave up and used an existing library. I consider myself intelligent: I got A's...

## Ten Little Algorithms, Part 4: Topological Sort

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 2: The Single-Pole Low-Pass Filter
- Part 3: Welford's Method (And Friends)
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
- Part 6: Green’s Theorem and Swept-Area Detection

Today we’re going to take a break from my usual focus on signal processing or numerical algorithms, and focus on...

## Ten Little Algorithms, Part 3: Welford's Method (and Friends)

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 2: The Single-Pole Low-Pass Filter
- Part 4: Topological Sort
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
- Part 6: Green’s Theorem and Swept-Area Detection

Last time we talked about a low-pass filter, and we saw that a one-line...

## Ten Little Algorithms, Part 2: The Single-Pole Low-Pass Filter

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 3: Welford's Method (And Friends)
- Part 4: Topological Sort
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
- Part 6: Green’s Theorem and Swept-Area Detection

I’m writing this article in a room with a bunch of other people talking, and while sometimes I wish they would just SHUT UP, it would be...

## Ten Little Algorithms, Part 1: Russian Peasant Multiplication

This blog needs some short posts to balance out the long ones, so I thought I’d cover some of the algorithms I’ve used over the years. Like the Euclidean algorithm and Extended Euclidean algorithm and Newton’s method — except those you should know already, and if not, you should be locked in a room until you do. Someday one of them may save your life. Well, you never know.

Other articles in this series:

- Part 1:

## Linear Feedback Shift Registers for the Uninitiated, Part VII: LFSR Implementations, Idiomatic C, and Compiler Explorer

The last four articles were on algorithms used to compute with finite fields and shift registers:

- multiplicative inverse
- discrete logarithm
- determining characteristic polynomial from the LFSR output

Today we’re going to come back down to earth and show how to implement LFSR updates on a microcontroller. We’ll also talk a little bit about something called “idiomatic C” and a neat online tool for experimenting with the C compiler.

## Linear Feedback Shift Registers for the Uninitiated, Part VIII: Matrix Methods and State Recovery

Last time we looked at a dsPIC implementation of LFSR updates. Now we’re going to go back to basics and look at some matrix methods, which is the third approach to represent LFSRs that I mentioned in Part I. And we’re going to explore the problem of converting from LFSR output to LFSR state.

Matrices: Beloved Historical DregsElwyn Berlekamp’s 1966 paper Non-Binary BCH Encoding covers some work on

## Elliptic Curve Cryptography - Extension Fields

An introduction to the pairing of points on elliptic curves. Point pairing normally requires curves over an extension field because the structure of an elliptic curve has two independent sets of points if it is large enough. The rules of pairings are described in a general way to show they can be useful for verification purposes.

## Linear Regression with Evenly-Spaced Abscissae

What a boring title. I wish I could come up with something snazzier. One word I learned today is studentization, which is just the normalization of errors in a curve-fitting exercise by the sample standard deviation (e.g. point \( x_i \) is \( 0.3\hat{\sigma} \) from the best-fit linear curve, so \( \frac{x_i - \hat{x}_i}{\hat{\sigma}} = 0.3 \)) — Studentize me! would have been nice, but I couldn’t work it into the topic for today. Oh well.

I needed a little break from...

## Linear Feedback Shift Registers for the Uninitiated, Part VI: Sing Along with the Berlekamp-Massey Algorithm

The last two articles were on discrete logarithms in finite fields — in practical terms, how to take the state \( S \) of an LFSR and its characteristic polynomial \( p(x) \) and figure out how many shift steps are required to go from the state 000...001 to \( S \). If we consider \( S \) as a polynomial bit vector such that \( S = x^k \bmod p(x) \), then this is equivalent to the task of figuring out \( k \) from \( S \) and \( p(x) \).

This time we’re tackling something...

## Elliptic Curve Cryptography - Multiple Signatures

The use of point pairing becomes very useful when many people are required to sign one document. This is typical in a contract situation when several people are agreeing to a set of requirements. If we used the method described in the blog on signatures, each person would sign the document, and then the verification process would require checking every single signature. By using pairings, only one check needs to be performed. The only requirement is the ability to verify the...

## Flood Fill, or: The Joy of Resource Constraints

When transferred from the PC world to a microcontroller, a famous, tried-and-true graphics algorithm is no longer viable. The challenge of creating an alternative under severe resource constraints is an intriguing puzzle, the kind that keeps embedded development fun and interesting.

## Unraveling the Enigma: Object Detection in the World of Pixels

Exploring the realm of embedded systems co-design for object recognition, this blog navigates the convergence of hardware and software in revolutionizing industries. Delving into real-time image analysis and environmental sensing, the discussion highlights advanced object detection and image segmentation techniques. With insights into Convolutional Neural Networks (CNNs) decoding pixel data and autonomously extracting features, the blog emphasizes their pivotal role in modern computer vision. Practical examples, including digit classification using TensorFlow and Keras on the MNIST dataset, underscore the power of CNNs. Through industry insights and visualization aids, the blog unveils a tapestry of innovation, charting a course towards seamless interaction between intelligent embedded systems and the world.

## Linear Feedback Shift Registers for the Uninitiated, Part XI: Pseudorandom Number Generation

Last time we looked at the use of LFSRs in counters and position encoders.

This time we’re going to look at pseudorandom number generation, and why you may — or may not — want to use LFSRs for this purpose.

But first — an aside:

Science Fair 1983When I was in fourth grade, my father bought a Timex/Sinclair 1000. This was one of several personal computers introduced in 1982, along with the Commodore 64. The...

## Linear Feedback Shift Registers for the Uninitiated, Part IX: Decimation, Trace Parity, and Cyclotomic Cosets

Last time we looked at matrix methods and how they can be used to analyze two important aspects of LFSRs:

- time shifts
- state recovery from LFSR output

In both cases we were able to use a finite field or bitwise approach to arrive at the same result as a matrix-based approach. The matrix approach is more expensive in terms of execution time and memory storage, but in some cases is conceptually simpler.

This article will be covering some concepts that are useful for studying the...

## Elliptic Curve Cryptography - Key Exchange and Signatures

Elliptic curve mathematics over finite fields helps solve the problem of exchanging secret keys for encrypted messages as well as proving a specific person signed a particular document. This article goes over simple algorithms for key exchange and digital signature using elliptic curve mathematics. These methods are the essence of elliptic curve cryptography (ECC) used in applications such as SSH, TLS and HTTPS.

## Finite State Machines (FSM) in Embedded Systems (Part 4) - Let 'em talk

No state machine is an island. State machines do not exist in a vacuum, they need to "talk" to their environment and each other to share information and provide synchronization to perform the system functions. In this conclusive article, you will find what kind of problems and which critical areas you need to pay attention to when designing a concurrent system. Although the focus is on state machines, the consideration applies to every system that involves more than one execution thread.

## Ten Little Algorithms, Part 7: Continued Fraction Approximation

In this article we explore the use of continued fractions to approximate any particular real number, with practical applications.

## Elliptic Curve Cryptography - Security Considerations

The security of elliptic curve cryptography is determined by the elliptic curve discrete log problem. This article explains what that means. A comparison with real number logarithm and modular arithmetic gives context for why it is called a log problem.

## Finite State Machines (FSM) in Embedded Systems (Part 3) - Unuglify C++ FSM with DSL

Domain Specific Languages (DSL) are an effective way to avoid boilerplate or repetitive code. Using DSLs lets the programmer focus on the problem domain, rather than the mechanisms used to solve it. Here I show how to design and implement a DSL using the C++ preprocessor, using the FSM library, and the examples I presented in my previous articles.

## Flood Fill, or: The Joy of Resource Constraints

When transferred from the PC world to a microcontroller, a famous, tried-and-true graphics algorithm is no longer viable. The challenge of creating an alternative under severe resource constraints is an intriguing puzzle, the kind that keeps embedded development fun and interesting.

## Unraveling the Enigma: Object Detection in the World of Pixels

Exploring the realm of embedded systems co-design for object recognition, this blog navigates the convergence of hardware and software in revolutionizing industries. Delving into real-time image analysis and environmental sensing, the discussion highlights advanced object detection and image segmentation techniques. With insights into Convolutional Neural Networks (CNNs) decoding pixel data and autonomously extracting features, the blog emphasizes their pivotal role in modern computer vision. Practical examples, including digit classification using TensorFlow and Keras on the MNIST dataset, underscore the power of CNNs. Through industry insights and visualization aids, the blog unveils a tapestry of innovation, charting a course towards seamless interaction between intelligent embedded systems and the world.

## Elliptic Curve Cryptography - Multiple Signatures

The use of point pairing becomes very useful when many people are required to sign one document. This is typical in a contract situation when several people are agreeing to a set of requirements. If we used the method described in the blog on signatures, each person would sign the document, and then the verification process would require checking every single signature. By using pairings, only one check needs to be performed. The only requirement is the ability to verify the...

## Elliptic Curve Cryptography - Extension Fields

An introduction to the pairing of points on elliptic curves. Point pairing normally requires curves over an extension field because the structure of an elliptic curve has two independent sets of points if it is large enough. The rules of pairings are described in a general way to show they can be useful for verification purposes.

## Linear Feedback Shift Registers for the Uninitiated

In 2017 and 2018 I wrote an eighteen-part series of articles about linear feedback shift registers, or LFSRs:

div.jms-article-content ol > li { list-style-type: upper-roman } Ex-Pralite Monks and Finite Fields, in which we describe what an LFSR is as a digital circuit; its cyclic behavior over time; the definition of groups, rings, and fields; the isomorphism between N-bit LFSRs and the field \( GF(2^N) \); and the reason why I wrote this series