CRYPTOGRAPHY IS AN ANCIENT ART, well over two thousand years old. The need to keep certain information secret has always existed, and attempts to preserve secrets have therefore existed as well. But it is only in the last thirty years that cryptography has developed into a science that has offered us needed security in our daily lives. Whether we are talking about automated teller machines, cellular telephones, Internet commerce, or computerized ignition locks on automobiles, there is cryptography hidden within. And what is more, none of these applications would work without cryptography! The historyofcryptographyoverthepastthirtyyearsisauniquesuccessstory. The most important event was surely the discovery of public key cryptography in the mid 1970s. It was truly a revolution: We know today that things are possible that previously we hadn't even dared to think about. Dif?e and Hellman were the ?rst to formulate publicly the vision that secure communication must be able to take place spontaneously. Earlier, it was the case that sender and receiver had ?rst to engage in secret communication to establish a common key. Dif?e and Hellman asked, with the naivety of youth, whether one could communicate secretly without sharing a common secret. Their idea was that one could encrypt information without a secret key, that is, one that no one else could know. This idea signaled the birth of public key cryptography. That this vision was more than just wild surmise was shown a few years later with the advent of the RSA algorithm.