An introduction to Cryptography

In this introduction to Cryptography, we start by giving some basic notions from elementary number theory and see how they can be applied to send ``secret'' messages. After introducing congruence theory, Fermat's little theorem and the Euler-Fermat theorem, we examine, from a historical point of view, some classic cryptographic systems. Then we give the main properties of modern cryptographic systems and, in particular, we define public key cryptography. One of the most important public-key systems (which exploits the fact that in $\Z$ primality algorithms are much ``faster'' than factorization ones) is presented: the R.S.A. cryptosystem. We also present another public key system which exploits the discrete logarithm problem and which is also consistently used as an alternative to R.S.A. (for example the U.S. government offices use it for their digital signature scheme). An analogue of the discrete logarithm problem is used in the so-called ``elliptic curves cryptosystems''. Unfortunately, the amount of mathematical theory needed to understand the elliptic curves method is too great to be explained here. So, we will say, in the last paragraph, just a few words on the differences between the discrete logarithm problem and its elliptic curve analogue. This article is geared toward the amateur interested in the subject.