Building on the prior lesson of the Division Algorithm and Modular Arithmetic, we move onto Prime Factorization and Euclid’s Algorithm. These are not only extremely interesting facets of math and number theory, but one of the most utilized pieces of math in the modern digital world. Prime Factorization is fundamental to almost all encryption, and it is by the sheer scale of the math that allows it to be so — for even simple encryption breaking, a conventional computer could take over a hundred years running full bore before it could crack some types of encryption. This is also what makes things like Quantum Computers so terrifying — they have the capacity to break prime factorization, and thus, our entire modern digital security infrastructure. ...

I am currently enrolled in WGU, a fully online college where I am getting a Computer Science Bachelor’s degree. To help solidify the information I learn there, I am going to reiterate things here so that I can express things in the form of teaching - a tool others can use, if not a way to find my own shortcomings or areas I need to improve upon. Recently, I finished a section of a Discrete Mathematics II course on Algorithms. This section focused on the structure of what an algorithm is, asymptotic growth of functions, analysis of algorithms, and advanced analysis of algorithms. Generally, the topics discussed in this section are not complicated - but can seem that way prior to the reading. ...

More often than not, the first mathematical object we are exposed to is Integers. Integers are whole numbers, things like 1, 2, 3. These are simple, and relatively profound in their ability to represent quantities and otherwise unrelated topics. In fact, we have studied integers, known as Number Theory, for thousands of years with no real discernable purpose — until recently. That is to say, not all science comes with some clear endgoal, but it is the search in of itself that provides us with dividends. Number Theory is one such case — where we have built out a system that allows us to encrypt large amounts of data using nothing more than prime factorization for exceptionally large numbers — something that up until today, was almost entirely pointless. ...

Motivation To better understand the nature of simple library calls like “printf()”, I wanted to build a logger with minimal library usage. In doing so, the goal was to force me to research system call level functions that can help build the intuition behind function calls I might otherwise take for granted. This is not intended to be a reusable library, but simply as a learning scaffold. The Prototype Below is the full experiment (35 LOC). The goal was for a working model, not for performance or API design. ...