Did I lend "Theory of Numbers" to anyone?

[ciphergoth]

I can't find my copy of An Introduction to the Theory of Numbers anywhere, and I haven't been able to for some time. Did I lend it to you, and if so can I have it back please? Ta!

Update: also, opinions on books on elliptic curves solicited, for the four or five of you who might have some! See my comment below.

Comments

[phantas.livejournal.com]

Anything in particular on your mind?

[ciphergoth.livejournal.com]

Actually, there totally is and it is very likely something you can help with, I just wasn't sure you'd want me to bug you about it! As I think you know, I know just short of bugger all about elliptic curves and ECC, and I've decided it's about time I did something about it. I got a bonus recently at work and bought myself a couple of presents; after reading http://www.isg.rhul.ac.uk/~sdg/ecc.html I bought "Guide to Elliptic Curve Cryptography" by Darrel Hankerson, Alfred Menezes, and Scott Vanstone, and "Elliptic Curves: Number Theory and Cryptography" by Lawrence C. Washington.

I'm trying to read the second of these, but I'm finding it pretty hard going. Maybe this is just beyond my math barrier, but I think it might be something else.

One thing that makes it hard is that the way it is presented makes it seem as if the study of elliptic curves lacks the elegance of other fields of mathematics. Definitions such as the j-invariant, the elliptic curve group law, or the generalised Weierstrass equation itself itself are set out in a form that makes them seem arbitrary and unmotivated. And for example the proof of associativity for the group law is 13 pages of hard slog with few results along the way that are interesting in themselves and no real "aha" moments.

Is there elegance and symmetry to the definitions and mathematics behind elliptic curves? There must be - 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. But how do I find out about it? Where can I find a neat, pleasing definition of the group law, or of associativity, and so forth?

Thanks for asking, I think you were expecting a more specific question and I hope you don't mind this rather woolier one!

[wildbadger.livejournal.com]

I'd also be interested in the answer!

[phantas.livejournal.com]

I'll try to put a bit of summary afterwards and try to make it as clear as possible. :)

The short answer is: there is but taking the stuff over Finite Fields ( that is, working with a finite number of elements and, hence, fixed number of bits) makes everything various degrees more complicated. On the other end, we tend to visualize geometrical things over the Real numbers, how we see stuff. FF gets rid of all that, pretty much.

[wildbadger.livejournal.com]

Thanks! I did a maths degree years ago but never quite got my head round algebra beyond the basics of vector spaces. Analysis and topology always seemed so much more intuitive.

[phantas.livejournal.com]

Not at all! It is a good question!

If your online status at LJ jabber account is real, we can chat about that via IM as soon as I get my Lasagna (on Sundays you're allowed to have lunch at 4pm!)

[zotz]

Steve Galbraith would be a good person to ask, but I've no idea whether you know him.

[zotz]

Actually, I've just noticed that link above . . . unless there are two suspiciously similar-looking Kiwi mathematicians in Britain specialising in cryptography, that's Siouxsie Wiles' other half. Last time I saw him was at last year's Neubauten gig. He was looking well. Siouxsie was at home looking after the sprog that evening, but apparently she was well too.

[ciphergoth.livejournal.com]

I already emailed him, but apparently he's on holiday...

[phantas.livejournal.com]

you do know he's one of my supervisors, don't you?
he's going back to New Zealand (I think the position is in New Zealand) in the near future, by the way.

[phantas.livejournal.com]

Steven has the rare ability of explaining the hardest things in a way that makes them seem, well, simple. Like "Duh! why didn't I understand it on my own before?!"

... then you go back to your desk and realise that, well, the things *are* actually hard! :)

[wotw]

Silverman and Tate's book "Rational Points on Elliptic Curves" is one
of the greatest textbooks ever written.