On the forefront of cryptographic research lies Elliptic Curve Theory. Traditionally, a public key cryptosytem derives its strength from the difficulty of factoring a large number into its constituent primes. However, Elliptic Curve Cryptography (ECC) derives its strength from another concept: the difficult-to-work discrete logorithms. That new problem is much more difficult than traditional ones to solve, and thus provides a more secure system of cryptography.

The most apparent applications for elliptic curve cryptography is the "smart card," as it is dubbed. These cards are designed to be carried around by people for use in financial transactions. They can be used to authenticate people with digital signatures, keep data secure from alteration during an electronic transaction, and, by the same token, keep it from prying eyes. The elliptical curve mechanism applies to smart cards for several reasons.

First, the size of the key needed in ECC is a lot smaller than needed in other systems. This is because of the difficulty in breaking the encryption; a larger key is not required because the system is so secure. In fact, in a 78-bit key, the difference is thirty-five times! That makes ECC much more practical than other systems for places where storage space is at a premium.

Another advantage of ECC is that it requires less time for encryption than do other means. In fact, previous small-scale public key uses required a special crypto-processor to do the math. With ECC that is not necessary, reducing costs and saving time.

Finally, the time it takes to generate a pair of keys for ECC is tremendously shorter than with PGP or another public key system. While it takes almost 5 seconds to produce one with RSA, the same feat can be done in .004 seconds with ECC. Again, this reduces costs.

An elliptic curve is similar to the equation for an ellipse everyone learns in Algebra 2:

(x-a)^2 + (y-b)^2 = r^2

However, the equation for an elliptic curve is somewhat more complicated:

Y^2 [ + x*y ] = x ^ 3 + a*x^2 + b

In these, x and y are variables. However, a and b are constants.

Here is a quick picture of an elliptic curve: