Algebra and informatics. Applications to linear error-correcting codes (Q2871454)

This is the third book in a series of very interesting textbooks of prof. \textit{P. Meunier} [Arithmétique modulaire et cryptologie. Toulouse: Cépaduès-Éditions (2010; Zbl 1228.11002)] and [Cours d'algèbre et d'algorithmique. Applications à la cryptologie du RSA et du logarithme discret. Toulouse: Cépaduès-Éditions (2012; Zbl 1308.94002)]. The announced goal for this series is to train the readers to enter into habilitation programs to teach at college studies specialised in Mathematics and Physics in France. This series of three books has an appreciable profile of pedagogical nature. It has a great didactical value and most probably it can be supported by supplementary textbooks such as [\textit{P. Colmez}, Éléments d'analyse et d'algèbre (et de théorie des nombres). Palaiseau: Les Éditions de l'École Polytechnique (2009; Zbl 1188.00001)], [\textit{D. J. Mercier}, ``Cours de géométrie: préparation au Capes et à l'agrégation'', EPU, Éditions Publibook Université : sciences, mathématiques. Publibook (2005), 4ème éd, revue, corrigée et augmentée, parue le 30 mai 2014], [\textit{D. P. Meunier}, ``Exercices et problèmes de mathématiques pour le CAPES et l'agrégation interne: Millésime 2013'', EPU, Éditions Publibook Université. Publibook (2013)].NEWLINENEWLINENEWLINEThe current book is properly a self contained textbook in Coding Theory. Its title is misleading unless one agrees to identify ``Informatics'' with ``Coding Theory''. Initially, basic results in Group Theory, Rings and Fields and Linear Algebra are succinctly quoted, including the splitting of cyclotomic polynomials into irreducible factors and the quadratic residues. Factorization methods (Berlekamp-Massey and Cantor-Zassenhaus) in polynomial fields are explained. Error-detection and error-correction linear codes are introduced, as well as the notion of generator and parity check matrices, and they are illustrated by Reed-Solomon and Hamming codes. Then, Cyclic Linear Codes are discussed including BCH codes, Golay codes and Bose codes, their parameter calculations and the coding procedures, as well as their correctness proofs, are fully explained. Then the author introduces the Goppa codes following first an algebraic approach, in characteristic 2 fields and in higher prime characteristic, and later a geometric approach in elliptic curves. It is explained the convenience to use these codes when higher values of both length and dimension of the codes are required. The decoding procedures for the introduced codes form a chapter in the book. It is rather common to introduce a code together with its own coding and decoding procedures. However, the chosen approach provides a unifying perspective of the decoding procedures. At this moment a collection of exercises is proposed to the reader, certainly with a challenging aspect in order to reinforce the whole lecture. The closing chapter consists of applications of Coding Theory to Cryptography: the McEliece cryptoscheme is introduced (generator matrices are used as ciphering tools) with Goppa codes as platforms, and the Niederreiter cryptoscheme is also introduced in full detail although it can be introduced just by ciphering through parity-check matrices. NEWLINENEWLINENEWLINEThe textbook is a valuable tool to enter into Coding Theory, motivating for the students, helpful for teachers and a great refresher for practitioners in security and communications.