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