Re: Recommendation for learning the beauty in elliptic curves?

Paul Crowley <ciphergoth@xxxxxxxxx> wrote:
Is there a book or other text that describes the mathematics of
elliptic curves in a way that brings out the elegance? In everything I
can find about it, all of the constructions and definitions appear to
be inelegant, arbitrary, and unmotivated, and the corresponding proofs
a dull slog with no "aha" moments. I am sure however that this would
not be such an active and fruitful field of study if there were not
some deep underlying elegance to the whole thing.

I would love to read something that presented the whole matter in a
way that made that elegance clear, even if it takes longer to get to
results that are useful in cryptography. Can anyone recommend such a
text, either as a book or online?

Joseph Silverman has at least two books on elliptic curves that I've
found useful for learning about elliptic curves.

Rational Points on Elliptic Curves, with John Tate, Springer-Verlag,
UTM, 1992.

The Arithmetic of Elliptic Curves, Springer-Verlag, GTM 106, 1986.

These are geared towards number theory, of course, and there is no
cryptography in there. Beauty, elegance and depth is there, however.
The first is rather elementary, the second uses more machinery from
algebraic geometry.

I just noticed that Silverman has another book out, An Introduction to
Mathematical Cryptography,

I'm going to buy it. It isn't perfect, I think Remark 2.11 in the sample
sections is incorrect. But I expect it to be very good.

Or is there simply a hurdle of inelegance to overcome when one starts
learning about these things, and once past that hurdle the nice
results start coming?

Not if you are willing to do some fancy mathematics.

Kristian Gjøsteen