https://i.kym-cdn.com/photos/images/newsfeed/000/572/078/d6d...
- Is "number theory" an appropriate title? Should it be called "linear one-way functions explained from first principles"? Or "abstract algebra"? Or "math behind modern cryptography"? Maybe. But I think I made it quite clear in the introduction what the article is about. If you were looking for something else, feel free to leave again.
- Should I have introduced modular arithmetic and the groups of integers modulo some number before finite groups in the abstract? Maybe, but I also explain my reasoning behind this order in the introduction. As mentioned there, you get to these examples by scrolling just a bit.
I believe the confusion from the OP was that if you don't already have significant mathematical maturity (in the sense of having seen bits and pieces of number theory, linear algebra, and group theory before) you'll quickly be frustrated trying to follow along as you won't be able to reason at the necessary level of abstraction.
>Is "number theory" an appropriate title
I think "number theory from first principles" is a misleading title, and I almost skipped over it because I thought it was yet another introduction to modular arithmetic, congruences, and such. Instead, the bulk of the article lays summarizes the main results in number theory/group theory that can be put into practice to construct cryptographic primitives so I think titling it "Foundations of Cryptography" might be better.
It reminds me of Kun’s primer on probability. The primer jumps to the definition of probability space and measure in the first few paragraphs: https://jeremykun.com/2013/01/04/probability-theory-a-primer...
A good primer, nonetheless. Just not necessarily accessible to majority of programmers.
But in a very clear and straightforward way. I think you are laughting at it a bit because you know what (and how big) a measure theory is so you are automatically 'importing' all that knowledge and you think the author did too. But all is required from the reader is literally what a finite set and a fuction are. Students are very often scared of 'big words' because it seem like it should be hard even if a simple definition of that word has just been given, which is a bit weird and I'm not sure where it's comming from
I like this approach a lot, it makes it easier learning harder and harder concepts later when you start this way because you are introduced to correct terminology early instead of analogies and hand-waving. Later material is harder to learn so analogies no longer work and it often feels like 'ok, this is too hard now' or like you are starting over
Any advanced thing no problem. Just do not start from the “first principle”. Please not. Please.
Your Contributions are highly admirable and easy to summarize as you have done, Focus, Notation, Intuition, and Completeness are major milestones on their own and I don't even experience the Interactivity on my browser.
I also agree with some of the other commenters who say the reader should already have more than just a bit of exposure to number theory beforehand, developing understanding of more introductory applications would be good and could probably be chosen to provide a particularly firm foundation for the march toward cryptography which gets quite advanced quite quickly.
Next, you might want to draft what could be considered a prerequisite text of your own for this masterpiece so the less initiated can get up to speed and get the most out of it. Also, I suppose your next two articles will be further advanced and could be easier to grasp for those who first have better understanding of what you have now.
What is linear? What is one-way? What is a function? What is a finite group? What is a property?
The inline links are bandages.