Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I hear a lot about Elliptic Curve Cryptography these days, but I'm still not quite sure what they are or how they relate to crypto...

share|improve this question
add comment

3 Answers

up vote 18 down vote accepted

Here is a super nice powerpoint on the subject!

http://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf

share|improve this answer
12  
Looks like a pdf to me. –  Robin Chapman Sep 5 '10 at 10:38
1  
@Robin A PDF of slides from a presentation actually –  yydl Jun 22 '11 at 2:06
    
But obviously not a pdf created with Powerpoint. PP does not use Computer Modern. –  Henning Makholm Sep 9 '11 at 16:28
add comment

You can see this article as well: www.cs.umd.edu/Honors/reports/ECCpaper.pdf

share|improve this answer
add comment

The technical definition is a nonsingular projective curve of genus 1, which is an abelian variety under the group law: basially, this means that you draw the line through two points on the curve -- which can be embedded in the projective plane -- and find where that line intersects the curve again (and call that the negative of the sum).

We can always put elliptic curves in the (projectivization of the) form $y^2 = x^3 - Ax + B$.

So, the meaning of "abelian variety" is that you can add points on the elliptic curve, which is really useful; there isn't a way to do this for most objects in algebraic geometry. Then one can study things like the torsion points on an elliptic curve, with respect to this abelian group structure: it's a theorem that there are $m^2$ torsion points of order $m$ for instance, if you 're working in an algebraically closed field.

In fact, one way to think of this is that an elliptic curve is really--algebraically and topologically--a torus if you are working over the complex numbers, and the torsion points in the torus are easily determined. (Namely, a torus is algebraically the product of two copies of the unit circle.)

This also yields the theorem about torsion points for algebraically closed fields of characteristic zero via the "Lefschetz principle." (For characteristic p, you need a different argument.)

Other things one can consider include the group of points with coordinates in, say, the rational numbers (assuming the curve is defined by rational coefficients). One of the central theorems is that this group is finitely generated. The point is that the geometry of the elliptic curve leads to a rich algebraic structure.

That's a bit about elliptic curves; I know nothing about cryptography and can't comment on that.

share|improve this answer
    
This is a nice answer and I feel sheepish that I took the lazy route. Kudos. –  BBischof Jul 21 '10 at 4:39
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.