Arkadii Slinko - Algebra for Applications: Cryptography, Secret Sharing, Error-Correcting, Fingerprinting, Compression (Springer Undergraduate Mathematics Series)

• The chapter on secret sharing (Chap. ). By using secret sharing we show how a cryptosystem like RSA can be used by an organisation to share the decryption key between members of that organisation.

• The chapter on polynomials (Chap..

• The chapter on compression of information (Chap..

The aim of a Lecturer should be, not to gratify his vanity by a shew of originality; but to explain, to arrange, and to digest with clearness, what is already known in the science

George Pryme (17811868)

This book originated from my lecture notes for the one-semester course which I have given many times in The University of Auckland since 1998. The goal of this book is to show the incredible power of algebra and number theory in the real world. It does not advance far in theoretical algebra, theoretical number theory or combinatorics. Instead, we concentrate on concrete objects like groups of points on elliptic curves, polynomial rings and finite fields, study their elementary properties and show their exceptional applicability to various problems in information handling. Among the applications are cryptography, secret sharing, error-correcting, fingerprinting and compression of information.

Some chapters of this bookand especially number-theoretic and cryptographic onesuse GAP for illustrations of the main ideas. GAP is a system for computational discrete algebra, which provides a programming language, a library of thousands of functions implementing algebraic algorithms, written in the GAP language, as well as large data libraries of algebraic objects.

If you are using this book for self-study, then, studying a certain topic, familiarise yourself with the corresponding section of Appendix A, where you will find detailed instructions how to use GAP for this particular topic. As GAP will be useful for most topics, it is not a good idea to skip it completely.

I owe a lot to Robin Christian who in 2006 helped me to introduce GAP to my course and proofread the lecture notes. The introduction of GAP has been the biggest single improvement to this course. The initial version of the GAP notes, which have now been developed into Appendix A, was written by Robin. Stefan Kohl, with the assistance of Eamonn OBrien, kindly provided us with two programs for GAP that allowed us to calculate in groups of points on elliptic curves. I am grateful to Paul Hafner, Primo Potonic, Jamie Sneddon and especially to Steven Galbraith who in various years were members of the teaching team for this course and suggested valuable improvements or contributed exercises.