# Basic Understanding of Elliptic curve

I want to know the basic understanding about Elliptic curve. Why it is need and when it is useful. I have searched much on internet but I am not a science student hence I am not able to understand the things. I want to implement Elliptic curve cryptography algorithm in my programming but I dont have basic understanding of Elliptic curve.

• Please suggest me some document tutorials for that – Bhavik Ambani Jan 29 '12 at 3:30
• Wouldn't trying to find a library which implements this be an immensely better way to do this? Implementing cryptographic algorithms correctly, efficiently and safely is a highly non-trivial thing, best done by people with an in-depth understanding of all the issues involved (going from the math to coding to whatnot!) – Mariano Suárez-Álvarez Jan 29 '12 at 3:34
• I guess I answered your question in the Cryptography Stack Exchange as well (crypto.stackexchange.com/questions/1768/ecc-cryptography) – Jalaj Jan 29 '12 at 4:30
• Why would you want to implement something that you don't yet understand? Also surely you answered your own question about uses if you want to implement one... – fretty May 27 '12 at 7:52
• I have heared that it is the best and faster security implementation algorithm, hence I wanted to understand that. – Bhavik Ambani May 27 '12 at 7:53

There are lot many articles to name. The theory of elliptic curves, have originated a few decades earlier, but has influenced major areas, and became a central part where one can formulate a beautiful conjectures.

1. The Arithmetic of Elliptic curves by J.Silverman.
2. Rational Points on Elliptic Curves by J.Silverman and J.Tate.
3. If you scroll down to the bottom of this page, you exactly find a set of books, that will be very useful for a complete reference.
4. Another fantastic book which gives a very gentle introduction into the subject is Lawrence C. Washington Elliptic Curves Number Theory and Cryptography, Second Edition Discrete Mathematics and Its Applications 2008 which is available online.

And if you search in google, you can in fact find lot many paper presentations given by Prof.Silverman. Do search it.

Thank you.

• I don't know anything about the references at the linked site, but I wanted to point out that the two Silverman books are not at a beginner's level. They are distinctly graduate level textbooks aimed at people with a firm grasp of algebraic number theory and algebraic geometry. – davidlowryduda May 28 '12 at 12:01
• @mixedmath: Rational points on elliptic curves by Silverman and Tate is readable at undergrad level. It has infact no algebraic geometry from whatever i have seen. – user9413 Jun 4 '12 at 8:55
• @Chandrasekhar : Yes, you are right. The arithmetic of elliptic curves is a bit complicated one for beginners, where as the rational points on the elliptic curves is really a good book to start. I do have sent the OP another book called " Elliptic Curves and Cryptography" by Lawrence Washington, which gives a very gentle introduction to the subject. – IDOK Jun 4 '12 at 9:22

At least in the field of Cryptography lately in the recent years a lot of progress has been contributed due to progress in Elliptic Curves. The study of Elliptic Curves have been done past few decades as Iyengar pointed out, but the applications to that have been only realized lately.

Read the book http://tinyurl.com/7kjywzh (at least preview the first few pages on line to get an idea). It all depends what you want to achieve. If you just want to read and understand because you have a passion, then keep digging on-line. If not, you have to choose where you are inclined to apply that.

If you are interested in the cryptographical applications of elliptic curves, Washington does a great job of introducing elliptic curves and applying them to cryptography.

Vanstone et al has a book that doesn't spend much time talking about elliptic curves but rather talks about algorithms related to elliptic curve cryptography (ECC) and their implementation.

Finally David Jao has a expository paper on ECC and Menezes an expository paper on pairing-based cryptography that is often implemented in ECC for more effective security.

For a very beginner introduction when I read this: http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/ I found a simple and practical explanation of the subject expecially from a developer point of view. Moreover there are basic python and javascript implementations. Let's say that if you intend to implement ecdsa in your software there are libraries providing secure reference implementations.