Course in algebra and algorithmics. Applications to RSA and to discrete logarithm cryptology (Q2871455)

The declared main goal of Meunier's book is the preparation of candidates for the TIPE (Travail d'Initiative Personnelle Encadré) examination which in France was established as a condition to access the scientific Grandes Écoles, but this examination has been extended to qualify the attendants for admission at graduate level studies and specialization programs to teach mathematics. It is also addressed to candidates to get the CAPES (Certificat d'aptitude au professorat de l'enseignement du second degré) which enables teachers to lecture at the university level. Hence the book is of a great pedagogical nature. However it goes far beyond its declared purposes and is a complete and sound introduction to the mathematical methods in current state of the art in cryptology. Students, practitioners and researchers may profit of the book as a consulting source in the mathematics for cryptography.NEWLINENEWLINENEWLINEInitially, algebraic structures are introduced: groups, rings and fields, as well as modular arithmetic, including quadratic residues and the splitting of cyclotomic polynomials. Complexity notions are reviewed, specially the probabilistic, the most common algorithms for primality testing (Miller-Rabin, Solovay-Strassen and Pollard's algorithm), as well as some deterministic algorithms such as AKS and index calculus and sieve based procedures. When dealing with public key cryptosystems, several attacks against RSA are reviewed and the ElGamal cryptosystem is illustrated within cyclic groups arising as subgroups of vector spaces over finite fields with convolution as operation. For sure this is not a common approach but it is a fortunate illustration of a sound cryptographic platform. Finally, elliptic curves are discussed, their group structures, the properties of Frobenius operator, Weil pairing and Schoof's algorithm for trace computations. This is a complete and self contained exposition of elliptic curves, quite motivating. In the corresponding chapter, Lenstra's factorization method based on elliptic curves is shown, and a short introduction to hyperelliptic curves is provided as well.NEWLINENEWLINEThe text may serve as reference material for most practitioners of cryptography. It provides clear and concise explanations of the mathematical background of cryptographic methods, it is opening indeed the most black boxes which are assumed as such by programmers and users of Cryptography.NEWLINENEWLINEBeing a textbook, it lacks of exercises, and although the exposition is always quite fluent and of a pleasant reading, it would be helpful to state formally some exercises and problems to the reader. The author frequently is hinting at and leaving many details to be completed by the reader. Also, the book lacks of a bibliography. Along the text, several original or wider electronic sources are mentioned, and the reader is suggested to look for these references using Google. Most probably it is in accordance with the pedagogical style of the author, but it is not a canonical procedure in textbooks.NEWLINENEWLINEIn summary, the textbook is a beautiful and rather complete exposition of the mathematical methods in Cryptography and may be of benefit for students, teachers and practitioners.