Riassunto:
Eliminating the need for heavy number-crunching, sophisticated mathematical software packages open the door to areas like cryptography, coding theory, and combinatorics that are dependent on abstract algebra. Applications of Abstract Algebra with Maple and MATLAB®, Second Edition explores these topics and shows how to apply the software programs to abstract algebra and its related fields.
Carefully integrating Maple™ and MATLAB®, this book provides an in-depth introduction to real-world abstract algebraic problems. The first chapter offers a concise and comprehensive review of prerequisite advanced mathematics. The next several chapters examine block designs, coding theory, and cryptography while the final chapters cover counting techniques, including Pólya's and Burnside's theorems. Other topics discussed include the Rivest, Shamir, and Adleman (RSA) cryptosystem, digital signatures, primes for security, and elliptic curve cryptosystems.
New to the Second Edition
Three new chapters on Vigenère ciphers, the Advanced Encryption Standard (AES), and graph theory as well as new MATLAB and Maple sections
Expanded exercises and additional research exercises
Maple and MATLAB files and functions available for download online and from a CD-ROM
With the incorporation of MATLAB, this second edition further illuminates the topics discussed by eliminating extensive computations of abstract algebraic techniques. The clear organization of the book as well as the inclusion of two of the most respected mathematical software packages available make the book a useful tool for students, mathematicians, and computer scientists.
Contenuti:
PRELIMINARY MATHEMATICS
Permutation Groups
Cosets and Quotient Groups
Rings and Euclidean Domains
Finite Fields
Finite Fields with Maple
Finite Fields with MATLAB
The Euclidean Algorithm
BLOCK DESIGNS
General Properties
Hadamard Matrices
Hadamard Matrices with Maple
Hadamard Matrices with MATLAB
Difference Sets
Difference Sets with Maple
Difference Sets with MATLAB
ERROR CORRECTING CODES
General Properties
Hadamard Codes
Reed-Muller Codes
Reed-Muller Codes with Maple
Reed-Muller Codes with MATLAB
Linear Codes
Hamming Codes with Maple
Hamming Codes with MATLAB
BCH CODES
Construction
Error Correction
BCH Codes with Maple
BCH Codes with MATLAB
REED-SOLOMON CODES
Construction
Error Correction
Error Correction Method Proof
Reed-Solomon Codes with Maple
Reed-Solomon Codes with MATLAB
Reed-Solomon Codes in Voyager 2
ALGEBRAIC CRYPTOGRAPHY
Two Elementary Cryptosystems
Shift and Affine Ciphers with Maple
Shift and Affine Ciphers with MATLAB
Hill Ciphers
Hill Ciphers with Maple
Hill Ciphers with MATLAB
VIGENÈRE CIPHERS
Encryption and Decryption
Cryptanalysis
Vigenère Ciphers with Maple
Vigenère Ciphers with MATLAB
THE RSA CRYPTOSYSTEM
Preliminary Mathematics
Encryption and Decryption
The RSA Cryptosystem with Maple
The RSA Cryptosystem with MATLAB
A Note on Modular Exponentiation
A Note on Primality Testing
A Note on Integer Factorization
A Note on Digital Signatures
The Diffie-Hellman Key Exchange
Discrete Logarithms with Maple
Discrete Logarithms with MATLAB
ELLIPTIC CURVE CRYPTOGRAPHY
The ElGamal Cryptosystem
The ElGamal Cryptosystem with Maple
The ElGamal Cryptosystem with MATLAB
Elliptic Curves
Elliptic Curves with Maple
Elliptic Curves with MATLAB
Elliptic Curve Cryptography
Elliptic Curve Cryptography with Maple
Elliptic Curve Cryptography with MATLAB
THE ADVANCED ENCRYPTION STANDARD
Alphabet Assignment and Text Setup
The S-Box
Key Generation
Encryption
The AES Layers
Decryption
A Note on Security
AES with Maple
AES with MATLAB
PÓLYA THEORY
Group Actions
Burnside's Theorem
The Cycle Index
The Pattern Inventory
The Pattern Inventory with Maple
The Pattern Inventory with MATLAB
Switching Functions
GRAPH THEORY
The Cycle Index of Sn
The Cycle Index of Sn with Maple
The Cycle Index of Sn with MATLAB
Counting Undirected Graphs
Counting Undirected Graphs with Maple
Counting Undirected Graphs with MATLAB
Each chapter contains Computer and Research Exercises.
APPENDIX A: USER-WRITTEN MAPLE FUNCTIONS
APPENDIX B: USER-WRITTEN MATLAB FUNCTIONS
BIBLIOGRAPHY
HINTS OR ANSWERS FOR SELECTED EXERCISES
INDEX
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