Elliptic curves: point doubling

Written by Dominik Joe Pantůček on 22 března, 2018.

After the introduction of the first two simple point operations on elliptic curves in simple Weierstrass form, we can now look at some more interesting operations available to us. Last of the three „primitive“ operations specified for points of the elliptic curve is the point doubling operation. It should be the same as if we wanted to sum not two distinct but rather two equal points. As usually, I would like to encourage anyone to give me a feedback, especially about the visualizations used.

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Elliptic curves: point addition

Written by Dominik Joe Pantůček on 15 března, 2018.

Another week, another point operation on elliptic curves. This time we are about to explore a simple point addition of two different points on elliptic curve in simple Weierstrass form. Although there are still some possibilities of enhancing the visualizations, our focus is now on finishing this series to a point where we can show at least Diffie-Hellman key exchange[1].

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Elliptic curves: point negation

Written by Dominik Joe Pantůček on 8 března, 2018.

Venturing closer to our goal of using elliptic curves for digital signatures and encryption, we describe the simplest point operation on elliptic curve in Weierstrass form. This operation is called negation and it somehow resembles the negation of numbers you learned about in elementary school. We have (sort of) finalized the software stack for visualizing elliptic curves both in 2D or 3D so I would like to hear some feedback if the pictures – and especially the video – are clear and easy to understand. And now for something completely negating…

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Elliptic curves over finite fields

Written by Dominik Joe Pantůček on 1 března, 2018.

After a quick introduction to simple elliptic curves used in cryptography and finally for securing our online communication, we cannot move forward without solving one major problem. How to represent these curves, and all of their points thereof, in computer memory. Why this is a major problem? There is an infinite number of real numbers, but computers can store only a finitely-sized things – including numbers. I’d like to explain how we can overcome this inherent trouble and show you that anyone can understand these modern cryptographic fundamentals.

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Introduction to elliptic curves

Written by Dominik Joe Pantůček on 22 února, 2018.

We are about to start a journey to the realm of elliptic curve cryptography. It may seem strange at this point as why we should bother with that, but rest assured that we will eventually find out how to use this knowledge to secure our email communication. In this introduction, you can expect to see what an elliptic curve looks like, how it is defined and how it can be simplified if we want to make some practical use of it.

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