# Monte Carlo methods and Markov chains – an introduction

In our short series on machine learning, we have already applied sampling methods several times. We have used and implemented Gibbs sampling, and so far we have simply accepted that the approach works. Time to look at this in a bit more detail in order to understand why it works and what the limitations of … Continue reading Monte Carlo methods and Markov chains – an introduction

# Turn on the heating – from Hopfield networks to Boltzmann machines

In my recent post on Hopfield networks, we have seen that these networks suffer from the problem of spurious minima and that the deterministic nature of the dynamics of the network makes it difficult to escape from a local minimum. A possible approach to avoid this issue is to randomize the update rule. Intuitively, we want to … Continue reading Turn on the heating – from Hopfield networks to Boltzmann machines

# 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 … Continue reading Boltzmann machines, spin, Markov chains and all that

# 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 … Continue reading Keys in the bitcoin network: the public key

# 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 … Continue reading Keys in the bitcoin network: the private key

# 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 … Continue reading A primer on elliptic curve cryptography: practice