Cryptography Reference
In-Depth Information
1
Introduction
Cryptography is an interdisciplinary field of great practical importance. The subfield of
public key cryptography has notable applications, such as digital signatures. The security
of a public key cryptosystem depends on the difficulty of certain computational problems in
mathematics. A deep understanding of the security and efficient implementation of public
key cryptography requires significant background in algebra, number theory and geometry.
This topic gives a rigorous presentation of most of the mathematics underlying public key
cryptography. Our main focus is mathematics. We put mathematical precision and rigour
ahead of generality, practical issues in real-world cryptography or algorithmic optimality.
It is infeasible to cover all the mathematics of public key cryptography in one topic. Hence,
we primarily discuss the mathematics most relevant to cryptosystems that are currently
in use, or that are expected to be used in the near future. More precisely, we focus on
discrete logarithms (especially on elliptic curves), factoring based cryptography (e.g., RSA
and Rabin), lattices and pairings. We cover many topics that have never had a detailed
presentation in any textbook.
Due to lack of space some topics are not covered in as much detail as others. For
example, we do not give a complete presentation of algorithms for integer factorisation,
primality testing and discrete logarithms in finite fields, as there are several good references
for these subjects. Some other topics that are not covered in the topic include hardware
implementation, side-channel attacks, lattice-based cryptography, cryptosystems based on
coding theory, multivariate cryptosystems and cryptography in non-Abelian groups. In
the future quantum cryptography or post-quantum cryptography (see the topic [
48
]by
Bernstein, Buchmann and Dahmen) may be used in practice, but these topics are also not
discussed in this topic.
The reader is assumed to have at least a standard undergraduate background in groups,
rings, fields and cryptography. Some experience with algorithms and complexity is also
assumed. For a basic introduction to public key cryptography and the relevant mathematics
the reader is recommended to consult Smart [
513
], Stinson [
532
] or Vaudenay [
553
].
An aim of the present book is to collect in one place all the necessary background
and results for a deep understanding of public key cryptography. Ultimately, the text
presents what I believe is the “core” mathematics required for current research in public
key cryptography and it is what I would want my PhD students to know.
