Algebra for Applications

This book examines the relationship between mathematics and data in the modern world. Indeed, modern societies are awash with data which must be manipulated in many different ways: encrypted, compressed, shared between users in a prescribed manner, protected from an unauthorised access and transmitted over unreliable channels. All of these operations can be understood only by a person with knowledge of basics in algebra and number theory.

This book provides the necessary background in arithmetic, polynomials, groups, fields and elliptic curves that is sufficient to understand such real-life applications as cryptography, secret sharing, error-correcting, fingerprinting and compression of information. It is the first to cover many recent developments in these topics. Based on a lecture course given to third-year undergraduates, it is self-contained with numerous worked examples and exercises provided to test understanding. It can additionally be used for self-study.

• BCH Codes
• Cryptography
• Cryptology
• Elgamal Cryptosystem
• Elliptic Curves
• Error-Correcting Codes
• Euler Totient Function
• Factorisation
• Fingerprinting Codes
• Finite Field
• Fitingof’s Compression Code
• Integers
• Lagrange Interpolation
• Linear Codes
• Linear Secret Sharing Scheme
• Miller-Rabin Pseudoprimality Test
• Permutation Groups
• Polynomial
• Prefix Codes
• Primality Testing

“Brief, but thoughtful, exercise sets are presented throughout the text with worked answers to all exercises provided in an appendix. Overall, this book, part of the ‘Springer Undergraduate Mathematics’ series, is an excellent introduction to modern applications of algebra and number theory for advanced students. Summing Up: Highly recommended. Upper-division undergraduates and graduate students.” (D. S. Larson, Choice, Vol. 53 (10), June, 2016)

“The different topics are well motivated and many examples are interweaves through the text as well as small programs in the computer packages GAP. Appendix A describes this package and gives instructions on how to use it. Also a list of exercises are proposed at the end of each subsection and solutions are provided at the end of the book. Biographical and bibliographical references are given as footnotes and at the end of some chapters.” (Juan Tena Ayuso, zbMATH 1345.94002, 2016)

• Department of Mathematics, The University of Auckland, Auckland, New Zealand

  Arkadii Slinko

ARKADII SLINKO is currently Professor of Mathematics at the University of Auckland. He has published extensively in a wide range of journals. His current research is in applied discrete mathematics with particular focus on the mathematics of social choice, mathematical politics, decision theory, game theory and secret sharing.

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