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 book 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 book. 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 book include hardware implementation, side-channel attacks, lattice-based cryptography, cryptosystems based on coding theory, multivariate cryptosystems and cryptography in non-Abelian groups.