Elliptic Curve Cryptography

Question

I’m studying “Elliptic Curve Cryptography”. It seems like that; it is very hard to understand the concept of “Identity Element”.

Actually my question is why we need “Identity Element”? As far as I understood, we need “Identity Element” in order to define inverse –P of any group element P. Am I correct?

Moreover can somebody show me some introductory material on elliptic curve cryptography?

Answer 1

A lot of cryptographic proofs rely on very general mathematical concepts about "sets of objects". Some of these concepts are "groups" (Abelian Groups), "modules", "fields" and "rings". For these structured sets of objects a lot of lemmas and theorems have been derived and proofed in a very general way, once you accept the fundamental axioms as true which were used to construct them.

These structures can be constructed. You need "Elements", "Identity Elements", "Inverse Elements" and "Operations" and some "Axioms" that are assumed as always true. (Like "Use operation XY, apply it to ELEMENT and INVERSE_ELEMENT and the result will always be IDENTITY_ELEMENT.") So if you can verify about any set of objects that it fulfills the minimal pre-conditions for one of the abovely mentioned structures, then it will also fulfill all generally proven high level theorems.

For Elliptic Curves you just proof all the basic ingredients (i.e. axiomatically defined properties) are there to make them an Abelian Group, and BANG!, you've prooven that all other theorems related to Abelian Groups are also true. One of the axiomatic pre-conditions for Abelian Groups is the "identity element".

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I found this publication to be a very good introduction into Elliptic Curve cryptography, for people with some mathematical background. It comes with quite a few Java applets to play with online. Unfortunately it's German only, but maybe that helps you anyway:

http://www.warendorf-freckenhorst.de/elliptische-kurven/frame.html

Another piece of software that lets you play with all sorts of cryptographic algorithms (including Elliptic Curves) is the now open sourced "CrypTool", available in English, German and Spanish. It is suitable for anybody with interest in technical or IT things:

http://www.cryptool.de/index.php/en.html
http://www.cryptool.de/index.php/en/download-topmenu-63.html (Download)

Here is a short introduction to CrypTool in the form of a presentation:

http://www.cryptool.de/download/CrypToolPresentation-en.pdf

Edit: Here is an English-language introduction into Elliptic Curve mathematics: http://www.certicom.com/index.php/ecc-tutorial