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Elliptic Curves: The Great Mystery

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Re: Elliptic Curves: The Great Mystery

#11
post #4
post #3

Earlier quoted context omitted.

Even as someone with mathematical training, my eyes started skipping lines a few paragraphs in. This content is good, for sure, but you have to be in the right headspace to follow along. Quanta's material is serving a different need. Quanta tells you about it. This article tells you it. Both are valuable!

This draws you into the math instead of running away from it. I feel like a reasonably smart (and motivated) person with no knowledge of group theory or number theory could follow along here and understand the material. I feel like the people who write Quanta articles want me to remain ignorant.

There are literally hundreds of introductory level analysis textbooks. Some are very terse with challenging exercises presenting things in a definition-theorem-lemma-proof example style (baby Rudin, Lang, Kolmogorov & Fomin, Zorich). Others are pretty chatty, and give lots and lots of examples and motivating discussion (Carothers, Abbott, Pugh). Still others focus more on building (correct) intuition, have few detailed proofs, but give lots of high level vistas of the landscape. These explain how analysts think, and give more historical context (Bressoud, both of Bryant's books, WW Sawyer's introduction to numerical functional analysis). There are even some that take a somewhat more Socratic approach and relegate most of the material to the exercises (Moore & Cloud).

I claim that this diversity of math writing on a single topic is fantastic. I love Baby Rudin, and found Abbott and even Carothers to be so chatty as to be confusing, where others find both of those texts to be a breath of fresh air. Later I came to appreciate Carothers more, and by extension the higher level texts by Bryant and Sawyer.

If you don't like a particular piece of math writing, just consider that there might be others out there who benefit from it. For instance, my partner who has intense math anxiety and barely passed high school algebra can often follow them, and I've found them nice for getting a quick description of a field in math I know nothing about and the problems and methods of that field.

Anyway to be clear, this isn't to say you shouldn't be critical, just that we need more math writing, not less, and if something isn't landing for you, maybe it's landing for others.

Re: Elliptic Curves: The Great Mystery

#12
post #4
post #3

Earlier quoted context omitted.

Even as someone with mathematical training, my eyes started skipping lines a few paragraphs in. This content is good, for sure, but you have to be in the right headspace to follow along. Quanta's material is serving a different need. Quanta tells you about it. This article tells you it. Both are valuable!

This draws you into the math instead of running away from it. I feel like a reasonably smart (and motivated) person with no knowledge of group theory or number theory could follow along here and understand the material. I feel like the people who write Quanta articles want me to remain ignorant.

[deleted]

Re: Elliptic Curves: The Great Mystery

#13

This book is a beautiful step by step exposition of the conjecture accessible to anyone with a high school/secondary school level education in math/maths: https://press.princeton.edu/books/hardcover/9780691151199/el...

Subsequently, if you've taken an undergraduate level algebra class (or just have mathematical maturity and can work through some algebraic definitions) I suggest Tate and Silverman's Rational Points on Elliptic Curves. It's not super rigorous but it's good at introducing elliptic curves.

https://link.springer.com/book/10.1007/978-3-319-18588-0

Re: Elliptic Curves: The Great Mystery

#14

This book is a beautiful step by step exposition of the conjecture accessible to anyone with a high school/secondary school level education in math/maths: https://press.princeton.edu/books/hardcover/9780691151199/el...

Subsequently, if you've taken an undergraduate level algebra class (or just have mathematical maturity and can work through some algebraic definitions) I suggest Tate and Silverman's Rational Points on Elliptic Curves. It's not super rigorous but it's good at introducing elliptic curves. https://link.springer.com/book/10.1007/978-3-319-18588-0

I also really like Cassels' _Lectures on Elliptic Curves_, which is written very lucidly and actually describes in detail how to do a lot of the calculations that tend to be glossed over a bit by more theory-inclined authors.

Re: Elliptic Curves: The Great Mystery

#20

[flagged]

Do you mean you use known compromised/unreliable algorithms with a significant probability of allowing someone to steal someone's money?

The point of the OP was that Dalek library provides API to Ristretto (https://ristretto.group) which is a proper mathematical prime order group, unlike stock Ed25519 API that exposes cofactors and caused famous issues in Monero.
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