**Cryptography
based on elliptic curves**

*Csirmaz
Laszlo <laci@degas.ceu.hu>*

**CEU**

There is a
hard-to-solve mathematical problem behind all public key cryptography
system. For example, RSA uses the fact that while it is relatively
simple to decide whether a number of several hundred digits is
composite or not, it is beyond hope to find its prime factors. The
so-called *discrete logarithm* problem is behind the
Diffie--Hellman method. This problem can be phrased as follows: given
the base and the result of an exponentiation, find the exponent. Here
exponentiation is, of course, repeated multiplication, but for the
multiplication we always take the remainder when dividing by a given
large prime number. The problem can be worded in a more general
settings. Given some kind of operation, called multiplication, among
several (say 10^{100} or so) objects. Choose a base *g*,
an exponent *n*, and compute *y*=*g ^{n}*. The
task is to determine the exponent