This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures on, is Cryptology.

I had a fairly standard undergraduate course on Number Theory where we learned basic cyphers and some things about encryption. However, I am hoping to talk about the relationship of elliptic curves to encryption. Is there an appropriate level book that covers this relationship? Many of the students are strong, but lack some background. Many will have some experience with number theory, but may lack Abstract Algebra and Advanced Calculus.

In the absence of a nice book talking about elliptic curves relation to cryptology, I will probably talk about the excellent book by Ash and Gross.

I was just hoping to add this topic into my mix of lectures so I thought the MO community could offer some suggestions.

EDIT: I wanted to add that Diffe-Hulman will definitely be covered as one of the main research projects will focus on it. The elliptic curves comes by request of the students, who have heard cool things about them :D

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This guy would know: mathoverflow.net/users/2784/victor-miller – Steve Huntsman May 25 '10 at 21:10
One or two lectures may be too few to explain elliptic curves. Better describe Diffie-Hellman for the multiplicative group and perhaps mention that elliptic curves is a different way of constructing groups. – Felipe Voloch May 25 '10 at 22:11
I agree with what Felipe says. However, the whole reason for using elliptic curves (as opposed to multiplicative groups) has to do with two things: 1) There are many elliptic curves over a finite field GF(q) whose order can be every number in an interval (there are a few exceptions in the case of non-prime fields). Thus giving lots of chances to avoid the Pohlig-Hellman attack. 2) The direct analogy of a factor base attack by lifting to an elliptic curve over Q (or a number field of reasonable degree) has virtually no chance of working because of height reasons. – Victor Miller May 26 '10 at 0:00

An Introduction to Mathematical Cryptography by Hoffstein, Pipher, and Silverman is an excellent book. It discusses elliptic curves.

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Nigel Smart made available a draft of the third edition of Cryptography: An Introduction for free on his page. The book contains a very gentle introduction into elliptic curves (the author doesn't assume any prior knowledge of number theory). Smart also provides very interesting historical background (e.g. how the Enigma machine was developed and used by Germans and eventually broken by the Turing algorithm.)

Another great introductory text is Elliptic curves: number theory and cryptography by Washington. It puts a stronger emphasis on number-theoretic applications but the exposition is clear and mostly elementary.

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If it's not too late, I'll mention A Course in Number Theory and Cryptography, by Neil Koblitz. It has a good, undergraduate-level treatment of elliptic curves.

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First, a terminological comment: you probably mean cryptography. Cryptology encompasses cryptography and cryptanalysis and thus has connotations of code-breaking rather than just encryption techniques.

Is the main focus of your REU is elliptic curves and you want a "practical" application? I am a bit skeptical that talking about elliptic curves cryptography in a couple of lectures is a viable plan: motivating, say, public key cryptosystems takes quite a bit of work. Combine that with elliptic curves number theory and you'll get more than reasonable to cover in so little time. (Added: I see that Felipe Voloch has made a similar comment). Having said that, here is another excellent recent book that covers everything you can possibly want (including quantum cryptography!):

Elementary Number Theory, Cryptography, and Codes by Baldoni, Ciliberto, Cattaneo.

It may also be worthwhile checking out teaching materials developed at Cornell Math Department, Numb3rs Math Activities and especially Math Explorers club.

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You may want to consider taking a look at Simon Singh's book on the history of the subject "The Code Book" link text

He also has a downloadable application which has some very nice features, for example a working model of an enigma machine, etc.

Also there's a really basic and short book by Annette Werner called "Elliptische Kurven in der Kryptographie" which may want to look at.

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