Algebra for Applications

1. Front Matter

   Pages i-xiv

   PDF

2. Integers

   • Arkadii Slinko

   Pages 1-39

3. Cryptology

   • Arkadii Slinko

   Pages 41-77

4. Groups

   • Arkadii Slinko

   Pages 79-127

5. Fields

   • Arkadii Slinko

   Pages 129-146

6. Polynomials

   • Arkadii Slinko

   Pages 147-170

7. Secret Sharing

   • Arkadii Slinko

   Pages 171-190

8. Error-Correcting Codes

   • Arkadii Slinko

   Pages 191-234

9. Compression

   • Arkadii Slinko

   Pages 235-255

10. Appendix A: GAP

    • Arkadii Slinko

    Pages 257-274

11. Appendix B: Miscellania

    • Arkadii Slinko

    Pages 275-279

12. Solutions to Exercises

    • Arkadii Slinko

    Pages 281-361

13. Back Matter

    Pages 363-368

    PDF

Modern societies are awash with data that needs to be manipulated in many different ways: encrypted, compressed, shared between users in a prescribed manner, protected from unauthorised access, and transmitted over unreliable channels. All of these operations are based on algebra and number theory and can only be properly understood with a good knowledge of these fields. This textbook provides the mathematical tools and applies them to study key aspects of data transmission such as encryption and compression.

Designed for an undergraduate lecture course, this textbook provides all of the background in arithmetic, polynomials, groups, fields, and elliptic curves that is required to understand real-life applications such as cryptography, secret sharing, error-correcting, fingerprinting, and compression of information. It explains in detail how these applications really work. The book uses the free GAP computational package, allowing the reader to develop intuition about computationallyhard problems and giving insights into how computational complexity can be used to protect the integrity of data.

The first undergraduate textbook to cover such a wide range of applications, including some recent developments, this second edition has been thoroughly revised with the addition of new topics and exercises. Based on a one semester lecture course given to third year undergraduates, it is primarily intended for use as a textbook, while numerous worked examples and solved exercises also make it suitable for self-study.

• public key cryptography
• Secret key cryptography
• RSA Cryptosystem
• BCH Code
• Elgamal Cryptosystem
• Shamir’s Secret Sharing Scheme
• digital signature
• Diffie-Hellman
• ideal secret sharing scheme
• Huffman compression code
• Fitingof Compression Code
• Error-Correcting Codes
• Euler Totient Function
• Fingerprinting Codes
• Lagrange Interpolation
• Linear Secret Sharing Scheme
• Miller-Rabin Pseudoprimality Test
• Prefix Codes
• Primality Testing
• Reed-Solomon Codes

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

  Arkadii Slinko

Arkadii M. Slinko is currently Professor of Mathematics at the University of Auckland, New Zealand. Before taking this position in 1993 he was a Senior Research Fellow of the interdisciplinary Institute of Systems Analysis of Russian Academy of Sciences in Moscow. He has published extensively in a wide range of journals in mathematics, computer science, economics, and politics. His current research focuses on the mathematics of social choice, game theory, and secret sharing.

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