How is RSA used?
We now know how to encode and decode a message x. But how is this used in practice? An example:

Payments over the internet
Suppose you want to order an item from a mail order company and pay for it via the internet. Now you don't want to send your credit card number over the net to this company just like that. The chances that this number falls into the wrong hands are too big. To warrant safe communication with this company, the communication will be encoded. The company chooses two primes p and q and appropriate encoding number v, decoding number w and modulus m = pq. The numbers v and m are made public by the company. If you now place your order and want to send your credit card number, you first encode your message (including the credit card number) with the help of the encoding number v and the modulus m. Then you send the encoded message to the company. The mail order company knows the decoding number w and the modulus m and can now decode your message. However, anyone who does not know w, cannot decode your message.

How secure is RSA?
The security of RSA depends of course on the difficulty of computing the decoding number w. To find out this number it is necessary to know the two primes p and q. Once you know these primes it is a piece of cake to find w. But, as we have already seen in Chapter 1, factoring the modulus m = pq into p and q is an extremely difficult and time-consuming task (provided m is chosen sufficiently large): if one chooses two very big primes p and q, then it is almost impossible to find the factorization of the modulus m = pq. The RSA cryptosystem therefore provides excellent security.