Guide to Elliptic Curve Cryptography

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1. throwaway81523Jan 2026
   It's very hard to get stuff right with the secp curves. That's one of the reasons for the move to curve25519 and similar. The book "Guide to Elliptic Curve Cryptography" by Hankerson, Menezes, and Vanstone is mostly very careful step by step instruction of how to do secp* arithmetic properly. It would still be useful to have some formal verification to help the assurance of of any particular implementation.
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2. throwaway81523Dec 2021
   Thanks, that is a good article. There are a lot of tricks to implementing ECC efficiently (besides correctly), enough that writing a competitive DIY implementation is a lot of work even if you know what you're doing. That's in contrast with RSA, where the work is all in the modular exponentiation operation, which is already carefully optimized in libraries like GMP (so just call GMP and you are done).

   Hankerson, Menezes, and Vanstone's "Guide to Elliptic Curve Cryptography" is very helpful if you are trying to implement ECC for some reason. Most of us are probably better off using an existing implementation.

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3. throwaway81523Nov 2021
   It is very easy to make mistakes in ECC implementation. If you really want to code it yourself, there's a book by Menezes, Vanstone et al ("Guide to Elliptic Curve Cryptography, I think) that gives very explicit instructions about how to do everything. I used to hack on a P256 implementation. But, it is a bit out of date by now, and IIRC it doesn't say anything about Curve25519.
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4. ReidZBMay 2014
   Mmm, certainly.

   Hoffstein et al.'s An Introduction to Mathematical Cryptography is a wonderful text, highly thought of, that really is a better choice than the above link for this sort of thing. Another shorter text is Koblitz's A Course in Number Theory and Cryptography, but I think all of the material in that is also in Hoffstein et al.'s Introduction.

   For a practical implementation-oriented treatment, Cryptography Engineering by Ferguson, Schneier, and Kohno is a good text. It's especially great as a launching point for someone interested in cryptography, especially from a systems-building perspective.

   For a mostly-rigorous treatment of the basics and theory of modern cryptography, I am a huge fan of Katz and Lindell's Introduction to Modern Cryptography; I've espoused its virtues several times here on HN. If you want the theoretical background with some application still thrown in, this is a good choice.

   For a deep theoretical treatment, requiring an already-built background in theoretical computer science and probability theory, see Goldreich's Foundations of Cryptography volumes. They are very dense, very dry works, but packed with information.

   For a specific focus on elliptic curve cryptography, Menezes et al. have published Guide to Elliptic Curve Cryptography, though elliptic curves are also treated in the above two references (Hoffstein et al. and Koblitz). Still, I appreciate the focus on ECC, which is large enough to merit entire books.

   For a reference text, Menezes et al.'s Handbook of Applied Cryptography contains a lot of information, although it's starting to show its age. Still, if you need to find a definition or something, it's a great place to look.

   For an understanding of the (roughly) state-of-the-art in block ciphers, see Rijmen and Daemen's The Design of Rijndael. (Rijndael is perhaps better known as "AES" today, modulo some minor differences; Rijmen and Daemen are the creators of Rijndael).

   For a treatment of the history of cryptology, the seminal work is Kahn's The Codebreakers. It is a massive text, but covers cryptologic history relatively well - up until recent times, anyway. For a shorter but still good text, see Singh's The Code Book.

   Then, there are areas of mathematics and theoretical computer science that are used heavily in cryptography. The above already include some references to that (quite a few of the above text include an intro to number theory section, for example), but of course, there are always the textbooks designed specifically for that field. For instance, Dummit and Foote's Abstract Algebra, Baker's Comprehensive Course in Number Theory, Papadimitriou's Computational Complexity, and so on. Those are pretty easy to find with Google, though, if you search for influential textbooks in those fields.

   This list is hopelessly incomplete, but I think it is quite a good broad survey of some different textbooks. If you studied all of the above in their entirety, you would be quite knowledgeable, at least.

   Edit: Well, uh, here at 5am I realized you were probably talking about ones that were available online or what have you. Oops. Of the above, I think only the Handbook of Applied Cryptography is (legally) free. Still, I'll leave this here for future reference.

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