## Applications of Abstract Algebra with MAPLEThe mathematical concepts of abstract algebra may indeed be considered abstract, but its utility is quite concrete and continues to grow in importance. Unfortunately, the practical application of abstract algebra typically involves extensive and cumbersome calculations-often frustrating even the most dedicated attempts to appreciate and employ its intricacies. Now, however, sophisticated mathematical software packages help obviate the need for heavy number-crunching and make fields dependent on the algebra more interesting-and more accessible. Applications of Abstract Algebra with Maple opens the door to cryptography, coding, Polya counting theory, and the many other areas dependent on abstract algebra. The authors have carefully integrated Maple V throughout the text, enabling readers to see realistic examples of the topics discussed without struggling with the computations. But the book stands well on its own if the reader does not have access to the software. The text includes a first-chapter review of the mathematics required-groups, rings, and finite fields-and a Maple tutorial in the appendix along with detailed treatments of coding, cryptography, and Polya theory applications. Applications of Abstract Algebra with Maple packs a double punch for those interested in beginning-or advancing-careers related to the applications of abstract algebra. It not only provides an in-depth introduction to the fascinating, real-world problems to which the algebra applies, it offers readers the opportunity to gain experience in using one of the leading and most respected mathematical software packages available. |

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### Contents

Preliminary Mathematics | 1 |

12 Cosets and Quotient Groups | 6 |

13 Rings and Euclidean Domains | 9 |

14 Finite Fields | 13 |

15 Finite Fields with Maple | 16 |

16 The Euclidean Algorithm | 18 |

Block Designs | 27 |

22 Hadamard Matrices | 31 |

73 The RSA Cryptosystem with Maple | 147 |

74 A Note on Modular Exponentiation | 150 |

75 A Note on Primality Testing | 152 |

76 A Note on Integer Factorization | 153 |

77 A Note on Digital Signatures | 154 |

78 The DiffieHellman Key Exchange | 155 |

Elliptic Curve Cryptography | 163 |

82 The ElGamal Cryptosystem with Maple | 166 |

23 Hadamard Matrices with Maple | 33 |

24 Difference Sets | 36 |

25 Difference Sets with Maple | 39 |

ErrorCorrecting Codes | 43 |

32 Hadamard Codes | 46 |

33 ReedMuller Codes | 48 |

35 Linear Codes | 53 |

36 Hamming Codes with Maple | 60 |

BCH Codes | 67 |

42 Error Correction in BCH Codes | 70 |

43 BCH Codes with Maple | 77 |

431 Construction of the Generator Polynomial | 78 |

432 Error Correction | 80 |

ReedSolomon Codes | 91 |

52 Error Correction in ReedSolomon Codes | 93 |

53 Proof of ReedSolomon Error Correction | 97 |

54 Binary ReedSolomon Codes | 101 |

55 ReedSolomon Codes with Maple | 102 |

551 Construction of the Codewords | 103 |

552 Error Correction | 105 |

56 ReedSolomon Codes in Voyager 2 | 111 |

Algebraic Cryptography | 115 |

62 The Hill Cryptosystem | 119 |

63 The Hill Cryptosystem with Maple | 124 |

64 Generalizations of the Hill Cryptosystem | 129 |

65 The TwoMessage Problem | 131 |

The RSA Cryptosystem | 139 |

71 Mathematical Prerequisites | 140 |

72 RSA Encryption and Decryption | 142 |

83 Elliptic Curves | 168 |

84 Elliptic Curves with Maple | 175 |

85 Elliptic Curve Cryptography | 178 |

86 Elliptic Curve Cryptography with Maple | 182 |

Polya Theory | 189 |

91 Group Actions | 190 |

92 Burnsides Theorem | 192 |

93 The Cycle Index | 195 |

94 The Pattern Inventory | 198 |

95 The Pattern Inventory with Maple | 203 |

96 Switching Functions | 205 |

97 Switching Functions with Maple | 208 |

Basic Maple Tutorial | 213 |

A2 Arithmetic | 214 |

A3 Defining Variables and Functions | 216 |

A4 Algebra | 217 |

A5 Case Sensitivity | 218 |

A6 Help File | 219 |

A8 Conditional Statements | 221 |

A9 Maple Procedures | 223 |

Some Maple Linear Algebra Commands | 225 |

UserWritten Maple Procedures | 231 |

C2 Chapter 7 Procedures | 234 |

C3 Chapter 8 Procedures | 235 |

C4 Chapter 9 Procedures | 236 |

Hints and Solutions to Selected Written Exercises | 243 |

248 | |

### Other editions - View all

Applications of Abstract Algebra with MAPLE Richard Klima,Neil P. Sigmon,Ernest Stitzinger No preview available - 1999 |

### Common terms and phrases

BCH code beads binary vector block design Chapter choose ciphertext colleague columns in H compute construct contains convert coset cycle index cyclic decipher the message define determine discrete logarithm distinct necklaces ElGamal cryptosystem elliptic curve encipher encryption exponent entering the following equation equivalent error correction scheme Euclidean algorithm Euclidean algorithm table Euclidean domain factors field elements finite field following command ftable group G Hadamard matrix Hamming code Hence Hill encryption method incidence matrix initial blocks insecure line intruder involutory irreducible key matrix line of communication linear code loop matrix H modulo multiplicative necklace example nonzero elements Note ordered pairs parameters parity check matrix pattern inventory permutation plaintext plaintext integers positive integer prime primitive polynomial primitive polynomial p(x received vector Reed-Muller code Reed-Solomon codes Reed-Solomon error correction rigid motions ring RSA cryptosystem secret message Section show how Maple subgroup switching functions Theorem values Written Exercise