Abstract Algebra vs Number Theory?

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Discussion Overview

The discussion revolves around the relevance of abstract algebra and number theory in the context of learning cryptography. Participants explore which subject might be more beneficial for someone interested in pursuing cryptographic studies, considering both theoretical and practical aspects.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants suggest that number theory is essential for cryptography, while others argue that abstract algebra is more important.
  • One participant proposes that both subjects are valuable and recommends studying both if possible.
  • Another participant emphasizes that understanding abstract algebra can simplify many concepts in number theory, particularly regarding group theory.
  • There is a mention that a graduate-level introduction to cryptography may only require basic knowledge of number theory, such as Legendre symbols and modular arithmetic.
  • Some participants recommend self-study in number theory alongside formal education in abstract algebra, citing specific resources like the Jones and Jones book.
  • Questions are raised about the content of abstract algebra courses and their applications in cryptography, with a brief overview provided regarding topics typically covered in such courses.

Areas of Agreement / Disagreement

Participants express differing opinions on the importance of abstract algebra versus number theory for cryptography, indicating that there is no consensus on which subject is definitively more important.

Contextual Notes

Some claims about the ease of learning number theory depend on prior knowledge of proof-writing and abstract algebra concepts, which may not be universally applicable. The discussion also reflects varying levels of familiarity with cryptographic requirements.

Who May Find This Useful

Individuals interested in cryptography, abstract algebra, and number theory, particularly students considering coursework or self-study in these areas.

MathWarrior
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I was wondering if one wanted to pursue learning more about cryptography which of these classes would be the most important? Number theory of abstract algebra?
 
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Number theory of abstract algebra <-- should read number theory or abstract algebra.
 
Can't you do both? Both are quite important for crypto I think. Anyway, if you end up picking one, then I'd really suggest studying the other one too in your free time...
 
Just take a course in abstract algebra, most number theory concepts come much easier if you can operate within a group. Many ugly proofs in number theory become quite nice if you know group theory.

That said, it's not like you're in any real rush. An graduate-level intro course in cryptography won't require more than the absolute basics. Probably just legendre symbols and basic modular arithmetic.
 
I'd have to agree, I'd take the abstract algebra course. Abstract really helps open the doors to much the higher mathematics as well.
 
I'd also say that abstract algebra is more important in general. Perhaps you should consider picking up a Number Theory book and doing some self-study. I don't think Number Theory is terribly difficult to learn if you know how to write proofs and knowledge of it might help you in algebra.

If you do want to learn Number Theory, I'd recommend the Jones and Jones book. It's well-written and affordable. Plus, it gives solutions to all the problems, so you can work through the points at which you get stuck more easily. I've had a good time with it.
 
What do you learn in abstract algebra? Is it like normal algebra? How is it used in cryptography?
 
MathWarrior said:
What do you learn in abstract algebra? Is it like normal algebra? How is it used in cryptography?

Generally abstract algebra courses cover group theory, homomorphisms and isomorphisms, ring theory, field theory, and Galois theory. Basically, it introduces algebraic structures and explores solutions to polynomial equations. Of course, a first course generally won't cover anything beyond some ring theory.
 

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