Damocles's sword is hanging over our widespread classical public key cryptosystems because their security is based on the belief that computing certain mathematical functions are hard tasks. However, due to the advent of quantum computing the long desired wish of code-breakers has been fulfilled.

Section 9.1 summarizes the basic terminology of cryptology. Next we introduce two fundamental cryptosystems based on symmetric key and asymmetric key architectures in Section 9.2 and Section 9.3, respectively. Both sections contain a practical example, the RSA algorithm for the former and the ElGamal algorithm for the latter. Well-tried security techniques such as public key cryptography seem to become obsolete at the dawn of the third millennium while new quantum-based solutions are emerging. We explain in Section 9.4 how to exploit quantum algorithms introduced in previous chapters of this book to break public key cryptosystems.

Fortunately defense against such quantum-assisted attacks is available by also using quantum principles. We explain them in Chapter 10.

Keeping information secret played/plays/will always play an important role the history. Battles, wars or the destiny of whole nations depended often on broken codes e.g. the fact that the Allies were in possession of the German ciphering system called ENIGMA proved to be decisive in World War II.

The science dealing with secret information is called *cryptology*.^{1} It ...

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