Name: A Computational Introduction to Number Theory and Algebra
ISBN: 9780511113635

A Computational Introduction to Number Theory and Algebra

Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.

What Hacker News says

cryptbeApr 2020

This is hands down the best book on applied crypto, especially for people who want to self-learn crypto. I started reading it since version 0.1, but every time I pick it up I learn something new.The fun application section in each chapter is pure gold. The exercises are challenging but super rewarding. I can't do many of them, but reading the problem statements is usually enough to make me realize I don't understand something.

When I started learning crypto, I spent a lot of time reading A Computational Introduction to Number Theory and Algebra [1] another free, high-quality book by Shoup. I did a lot of exercises, even corrected some results in the book, and at one point thought that I wanted to publish a solution manual. I emailed professor Shoup, but he said it may not be a good idea because people would copy my solutions for their homework. It turns out he's right. The blog posts that I published still get hit from time to time, and the search keyword is always solutions for NTB [2] =).

Later on, I took CS255 [3] from Boneh. I had to pay for it myself. IIRC it was $4,000 or something. It was a huge amount of money where I'm from. I took an overdraft from my bank to pay for it, the best money I ever spent! I still remember in one of the classes when Boneh was talking about the Diffie-Hellman protocol he paused to ask us, "Do you know where this was invented?" Nobody knows, then he pointed to his left and said, "In a room a few yards from here". I felt like I was part of this amazing history. There's a stream of knowledge flowing through me and maybe one day I'd discover something cool.

Boneh is simply the best teacher. He knows everything. I like it when he poses an open problem, and ends with "if you can solve this, I'll be a friend forever". He doesn't just love teaching, but also knows how to inspire and apparently can't stop explaining things.

[1] https://www.shoup.net/ntb/

[2] https://www.google.com/search?rlz=1C5CHFA_enUS851US851&ei=z0...

[3] https://crypto.stanford.edu/~dabo/cs255/

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ginnungagapFeb 2019

Having just finished a geometric group theory course I was hoping to see something about the word problem for automatic groups, or at least an algorithm to compute Dehn presentations of f.g. hyperbolic groups, but it looks like a cool resource with plenty of interesting topics.

Another book the HN crowd might like is Shoup's "a computational introduction to number theory and algebra"[0], it is much more broad and with litte focus on implementations, but a very interesting book nonetheless

[0]https://shoup.net/ntb/

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garfieldnateOct 2018

Perhaps "A Computational Introduction to Number Theory and Algebra" by Victor Shoup? This was recommended by Dan Boneh in his crypto course.

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mafribeJun 2015

I also don't think studying algebra helps you directly with programming language theory. However algebra is of indirect help if you want to learn category theory, because the latter is a direct generalisation of the former. It is rather difficult for a non-mathematician to pick up category theory from scratch without having first seen the the algebra that category theory abstracts from. Universal properties, category theory's most important concept, appears bizarre and disconnected from reality, unless you have seen it working in the much simpler settings of groups, rings, modules etc.

To that caveat, I'd like to add that neither Rotman nor Birkhoff/MacLane are ideal places to start learning algebra. There are plenty of gentler, more modern books. In terms of tomes that are available online, I can recommend Shoup's "A Computational Introduction to Number Theory and Algebra" [1].

As an aside, note that category theory is used only in the part of programming language research that is about (pure) functional programming. In other sub-fields (e.g. OO, logic programming and concurrency), category theory has not so far proven terribly useful.

[1] http://shoup.net/ntb/

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