In this post, I will show you how a bitcoin transaction presented in the raw format is to be interpreted and how conversely a bitcoin transaction stored in a C++ (and later Python) object can be converted into a hexadecimal representation (a process called serialization). Ultimately, the goal of this and subsequent posts will be…More

## The Ising model and Gibbs sampling

In the last post in the series on AI and machine learning, I have described the Boltzmann distribution which is a statistical distribution for the states of a system at constant temperature. We will now look at one of the most important applications of this distribution to an actual model, the Ising model. This model was proposed…More

## Transactions in the bitcoin network

In my previous posts on the bitcoin protocol, I have described those objects that constitute participants – private and public keys and bitcoin addresses. Now we will look at those objects that represent actual transfers of bitcoins between these participants, namely at transactions. Essentially, a bitcoin transaction consists of two parts. First, a transaction contains…More

## The Boltzmann distribution

Boltzmann machines essentially learn statistical distributions. During the training phase, we present them a data set called the sample data that follows some statistical distribution. As the weights of the model are adjusted as part of the learning algorithm, the statistical model represented by the Boltzmann machine changes, and the learning phase is successful if…More

## Boltzmann machines, spin, Markov chains and all that

The image above displays a set of handwritten digits on the left. They look a bit like being sketched on paper by someone in a hurry and then scanned and digitalized, not very accurate but still mostly readable – but they are artificial, produced by a neuronal network, more precisely a so called restricted Boltzmann…More

## Keys in the bitcoin network: the public key

In my last post, we have looked in some detail at the private key – how it is generated and how it can be decoded and stored. Let us now do the same with the public key. Recall that a public key is simply a point on the elliptic curve SECP256K1 that is used by…More

## Keys in the bitcoin network: the private key

In my last post, I have shown you how arithmetic on elliptic curves can be used to create and verify digital signatures. We have seen that every party that creates a signature is represented by a private key – kept securely – and a public key, which is made available to everyone who wants to verify…More

## A primer on elliptic curve cryptography: practice

In the last post, we have looked a bit at the theory behind elliptic curves. In this post, we will now see how all this works down to earth and use Python to actually run some calculations. The first thing that we need is an explicit formula for the addition of two points on an…More

## A primer on elliptic curve cryptography: theory

Strong cryptography is at the heart of the blockchain and many other modern technologies, so it does not hurt to get familiar with the basics. In this post I will explain the foundations of one very commonly used algorithm called elliptic curve digital signature. This post will be a bit lengthy and theoretical but do…More

## What this blog is about

Over the last couple of years, I have spent a considerable part of my spare time digging deeper into some topics around computer science and mathematics, mostly driven by the desire to understand how all that really works. Many years ago, I wrote a multi-threaded Unix kernel and made it boot on my PC to…More