# Chapter 20 of 24 - Public-Key Cryptography

Well that's a pretty sexy title isn't it. Shame I'm not being examined on any of it.

I got half-way through the first section on "cryptosystems based on modular arithmetic" before I decided to pick my battles wisely and return to chapter 19. As a consequence I never got to the meaty bit of the material. -namely encryption and decryption in the RSA cryptosystem.

The second section that I got nowhere near was "cryptosystems based on elliptic curves". This section looked difficult and lengthy, so I'm glad I discovered it wasn't examined before I reached it.

I'll keep this chapter in the back of my mind in case I want to revisit it at any point in the future (after my exam).

So, this brings me to the end of Book E on rings and fields. Normally at this point, I would've finished the related assignment by now, but I'm still very much stuck on the question related to splitting fields. I'll spend the next day or so trying to make as much sense as I can of it to pull out a few extra marks.

Next up is the last book, Book F, and the last four chapters on metric spaces. Home stretch...

# Chapter 19 of 24 - Fields and Geometry

Initially I had sped through this section, not paid much attention to it and moved straight onto chapter 20. See, my assignment seems to give me a choice of whether to answer a question on fields and geometry from chapter 19, or to answer a question on cryptography from chapter 20. Cryptography sounded far more interesting (and something you could easily apply in code!), so I kind of skipped chapter 19 entirely.

I got half-way thought chapter 20 and found out online that nothing from chapter 20 will actually be in the exam. The exam contributes to 80% of my final grade for this module, so picking my battles, I dropped chapter 20 and returned to chapter 19.

In an apparent change of pace, chapter 19 was laid out into just two sections. The first was revisiting field extensions, introducing some more nuanced facts about them. The second was an introduction to ruler and compass constructions, providing proofs of why you can't trisect the angle and why you can't square the circle. It was interesting, but very involved. Especially when you're left to your own devices trying to prove why other geometric constructions might not be possible.

Now for a very quick write-up of chapter 20... :/

# Chapter 18 of 24 - Fields and Polynomials

Chapter 18.

Oh, Chapter 18.

Easily the most challenging, exhausting, and the longest chapter yet.

Firstly preliminaries were covered, following by a very lengthy introduction to field extensions. This second section on field extensions was extremely frustrating as it's the only one so far that has required extra reading outside of the materials I was given. Next up was finite fields (very similar to the previous except not related to infinite fields). Then the fourth and last section was on splitting fields, which is again frustrating as the related question in the assignment seems to use language that isn't included in this section.

All in all a bit of a battle, and I still haven't been able to start the assignment question on splitting fields.

Once again, I'm running out of time, so I'll have to move onto chapter 19 and come back to this later...

# Chapter 17 of 24 - Rings and Homomorphisms

We're hopping around a bit now. Book E is back to rings and fields.

I'm falling behind rather severely with the study at the moment. Needless to say, the panic plus the speed with which I needed to work through this chapter meant that a lot of the information in this chapter didn't settle in my brain in the way I needed it to.

First section was on fields of fractions (which I'm still not terribly clear on). Second section was introducing the concept of an ideal of a ring, followed by operations on ideals. Lastly, the fourth section was on quotients and was the one that introduced homomorphisms between rings, and explained how images and kernels relate to ideals of rings.

Okay. Calm the panic.

Very very quickly jumping on to the next section...

# Chapter 16 of 24 - Open and Closed Sets

Sections in this chapter started with closed sets and open sets, which sound fairly straight-forward, but really they covered a formalisation of how sets are closed or open under a distance metric. eg, a set in two dimensions may be closed conventionally, but may not be closed under an unusual distance metric.

Next up were closures, interiors, boundaries, and the size of sets. Again, normally this would be fairly simple. -less so when looking at these in terms of the distance metric.

This chapter also included a little bonus seventh section introducing topology which looked great (and I've always wanted to learn more about) but unfortunately this section was not assessed, and I didn't have the time for extra reading. It seems this degree isn't just about learning maths, but specifically about learning maths fast.

So that brings Book D to a close!