Joys of Geometry

Creating figures with math software does not feel like fabricating illustrations for science posts. It is more of a meditation on geometry. I want to literally draw every line. I am not using grid lines or rendered surfaces. I craft a parametric curve for every line. A curve is set of equations. Yet, playing with … Continue reading Joys of Geometry

Spins, Rotations, and the Beauty of Complex Numbers

This is a simple quantum state ... |➚> = α|↑> + β|↓> ... built from an up |↑> state and a down state |↓>. α and β are complex numbers. The result |➚> is in the middle, oblique. The oblique state is a superposition or the up and down base states. Making a measurement, you … Continue reading Spins, Rotations, and the Beauty of Complex Numbers

Galaxies of Diffraction

These - the arrangement of points in the image below - are covectors, sort of. I wrote about them, some time ago. They are entities dual to vectors. Eating vectors, spitting out numbers. Vectors are again 'co' to vectors; they will eat covectors. If vectors live in a space with axes all perpendicular to each … Continue reading Galaxies of Diffraction

Circles to Circles

Using stereographic projection, you create a distorted image of the surface of a sphere, stretched out to cover an infinite plane. Each point on the sphere is mapped to a point in the equatorial plane by a projection ray starting at a pole of the sphere. Draw a circle on the sphere, e.g. by intersecting … Continue reading Circles to Circles

Lines and Circles

I poked at complex function 1/z, and its real and imaginary parts look like magical towers. When you look at these towers from above or below, you see sections of perfect circles. This is hinting at some underlying simplicity. Using the map 1/z, another complex number - w=1/z - is mapped to z. Four dimensions … Continue reading Lines and Circles

Looking Back: Hacking and Defending Windows Public Key Infrastructure (ADCS)

I live at the fringes of the cybersecurity community. I have never attended infosec conferences. There will be a talk on PKI hacking at Blackhat 2021 soon: Top AD offensive security gurus are presenting comprehensive research on abusing ADCS (Active Directory Certificate Services). I only know about that, because I noticed backlinks from their article … Continue reading Looking Back: Hacking and Defending Windows Public Key Infrastructure (ADCS)

Vintage Covectors

Covectors in the Dual Space. This sounds like an alien tribe living in a parallel universe hitherto unknown to humans. In this lectures on General Relativity, Prof. Frederic Schuller says: Now comes a much-feared topic: Dual vector space. And it's totally unclear why this is such a feared topic! A vector feels familiar: three numbers … Continue reading Vintage Covectors

Injecting an EFS Recovery Agent – and Let the Virus Scanner Help You!

How can you read files encrypted with Windows's Encrypting File System if you neither have access to the owner's encryption certificate and key and nor that of a legit data recovery agent (DRA) ... but if you are a local administrator? This work is still inspired by the hackthebox machine Helpline. You were able to … Continue reading Injecting an EFS Recovery Agent – and Let the Virus Scanner Help You!