  Nowadays, predominant asymmetric cryptographic schemes are considered to be
secure because discrete logarithms are believed to be hard to be computed. The
algorithm of Shor can effectively compute discrete logarithms, i.e. it can
brake such asymmetric schemes. But the algorithm of Shor is a quantum algorithm
and at the time this algorithm has been invented, quantum computers that may
successfully execute this algorithm seemed to be far out in the future. The
latter has changed: quantum computers that are powerful enough are likely to be
available in a couple of years. In this article, we first describe the relation
between discrete logarithms and two well-known asymmetric security schemes, RSA
and Elliptic Curve Cryptography. Next, we present the foundations of
lattice-based cryptography which is the bases of schemes that are considered to
be safe against attacks by quantum algorithms (as well as by classical
algorithms). Then we describe two such quantum-safe algorithms (Kyber and
Dilithium) in more detail. Finally, we give a very brief and selective overview
of a few actions currently taken by governments and industry as well as
standardization in this area. The article especially strives towards being
self-contained: the required mathematical foundations to understand
post-quantum cryptography are provided and examples are given.
