Mathematics of public key cryptography. (Q2880040)

Public key cryptography uses a great variety of mathematical tools. Conversely, the needs of cryptography have stimulated the research in different fields of mathematics. In the last years, several books have tried to present this interrelation between public key cryptography and some branches of mathematics, for example, the book [\textit{O. N. Vasilenko}, Number-theoretic algorithms in cryptography. Providence, RI: American Mathematical Society (2007; Zbl 1114.11001)], regarding number theory, or the book [\textit{H. Cohen} et al., Handbook of elliptic and hyperelliptic curve cryptography. Boca Raton, FL: Chapman \& Hall/CRC (2006; Zbl 1082.94001)], for elliptic and hyperelliptic curves.NEWLINENEWLINEIn a similar way the book under review tries to collect most of the mathematical instruments and techniques (of algebra, number theory and geometry) used in public key cryptography as well as the cryptosystems and cryptographic protocols based on them. However, the text does not seek to be exhaustive and the author indicates some missing topics such as the cryptography based on lattices, in error-correcting codes, in multivariate polynomials or in non-commutative groups, as well as quantum cryptography.NEWLINENEWLINEAfter an introductory chapter the book is structured in seven parts. Part I gathers the basics of computational arithmetic, hash functions and message authentication codes. Part II is devoted to algebraic varieties and curves, in particular elliptic and hyperelliptic curves. Part III reviews primality tests, factorization methods and the discrete logarithm problem, and Part IV gives the main problems and algorithms related to lattices.NEWLINENEWLINEPart V deals with public key cryptography based on the discrete logarithm problem, while Part VI does the same with the cryptography based on integer factorization. Finally, Part VII studies some topics (isogenies and pairings) in elliptic curves with growing applications in contemporary cryptography.NEWLINENEWLINEMany of the mathematical results are not proven or the proofs are only sketched, but the book gives numerous and pertinent references to the existing literature. It also contains many examples and exercises inserted throughout the text.NEWLINENEWLINESumming up, the book gathers the main mathematical topics related to public key cryptography and provides an excellent source of information for both students and researchers interested in the field.