What Are Elliptic Curves and Why Are They Useful?Date: 03/04/2006 at 12:53:14 From: arun Subject: elliptic curves and related problems What are elliptic curves? Why are they important? Date: 03/04/2006 at 13:43:47 From: Doctor Vogler Subject: Re: elliptic curves and related problems Hi Arun, Thanks for writing to Dr. Math. There are many ways I could answer this question. But let's try this one: In the simplest sense, curves are solution sets of polynomial equations in two variables (say x and y). For example, the circle x^2 + y^2 = 1 is a curve, as are lines, hyperbolas, and many other shapes, although not, for example, sine waves, since these are not given by polynomial equations. Elliptic curves, in the simplest sense, are curves of the form y^2 + dxy + ey = x^3 + ax^2 + bx + c. The reason they are important is that the solutions form an Abelian group. This means that you can "add" two points on the curve and get another point on the curve. This addition is associative, commutative, and has an identity and inverses. This group law is described in Cubic Diophantine Equation in Three Variables http://mathforum.org/library/drmath/view/66650.html So sometimes elliptic curves arise in Diophantine equations, which means that you have an equation, and you want to find all integer, or all rational, solutions. There are methods for solving this kind of problem on elliptic curves, whereas not all curves (or equations) can be so solved. Other times, they arise because they are useful groups. For example, there is a method for factoring numbers that uses elliptic curves and modular arithmetic, where the elliptic curve is useful because you can create a group from the unknown factors of your number, whose size can vary somewhat. See also Factoring Algorithms http://mathforum.org/library/drmath/view/65455.html There are related algorithms for testing and verifying that a large number is prime using elliptic curves. Furthermore, there are cryptographic schemes (codes) that work on elliptic curves that are believed to be more secure than similar codes that only use regular modular arithmetic. In each case, the important fact is that the elliptic curve makes a group, and so you can use elliptic curves to come up with groups that have desirable properties which help you solve certain kinds of problems. If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ Date: 03/05/2006 at 09:50:53 From: arun Subject: Thank you (elliptic curves and related problems) Thanks, Doctor Vogler, for providing me such a useful piece of information. |
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