Since 1976 the raise of public key cryptosystems (such as RSA) has led
to the development of Computational Number Theory. Its aim is to
solve simple problems such as factoring numbers into primes, as quickly
as possible.
In this article, we survey the algorithmic solutions given to three
problems: how to prove a number prime; how to factor a number; how to
compute discrete logarithms in finite fields.
We also stress the relationships between the high level of mathematics
involved (elliptic curves, number fields, etc.) and the use of new
computational means (parallel machines, networks of workstations).