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Finite Fields and Applications (Student Mathematical Library)

5.0 out of 5 stars 1 customer review
ISBN-13: 978-0821844182
ISBN-10: 0821844180
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Editorial Reviews

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"This book gives a quick, clear introduction to finite fields and discusses applications in combinatorics, algebraic coding theory, and cryptography. ... The work contains more than 100 exercises, appendixes on number theoretic and algebraic background, exercise hints, and a list of 65 references." ---- CHOICE Magazine

"The book is very well written and most stimulating. In addition, it is well suited for self-study. The student having a course using this book or anyone studying it will be substantially rewarded. On the whole, the authors succeed in providing a nice introduction to an active field of research." ---- Mathematical Reviews

"Altogether, this undergraduate text is an extremely carefully written introduction to the subject for beginners, which stands out by its remarkable features: instructional mastery, lucidity, diversity, profundity, topicality, and user-friendly determination. Being a lovely invitation to this current topic of mathematical (and interdisciplinary) research, the book is perfectly suitable both for classroom use and for individual study." ---- Zentralblatt MATH
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Product Details

  • Series: Student Mathematical Library (Book 41)
  • Paperback: 175 pages
  • Publisher: American Mathematical Society (December 23, 2007)
  • Language: English
  • ISBN-10: 0821844180
  • ISBN-13: 978-0821844182
  • Product Dimensions: 0.5 x 5.5 x 8 inches
  • Shipping Weight: 8 ounces (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Best Sellers Rank: #860,201 in Books (See Top 100 in Books)

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By Man Kam Tam on May 17, 2008
Format: Paperback Verified Purchase
To transmit photos from Mars back to earth, to download files from Internet where the server resides a few thousand miles away, and to send email globally are fascinating. Thus to understand the mathematics which made "... possible for the person receiving a message to detect and correct errors that have arisen during the transmission process" is interesting. "The detection of errors is accomplished by noticing that the received sequence is not a codeword ... For some code, it is possible for the receiver to determine, with high probability, the intended message when the received sequence is not a codeword. Such codes are ... called error-correcting codes." Gary L. Mullen and Carl Mummert's "Finite Field and Applications" introduces the error-correcting codes (algebraic coding theory) and the related mathematics. The book has four chapters. They are: finite fields, combinatorics, algebraic coding theory, and cryptography.

The chapter of algebraic coding theory includes (a) basic properties of codes (linear code, parity-check digits, parity-check matrix, parity-check equation, and systematic form), (b) bounds for parameters of codes (Hamming bound, Plotkin bound, Singleton bound, and Gilbert-varshamov bound), (c) decoding methods (nearest neighbor decoding and syndrome decoding method), (d) code constructions (Hamming, cyclic, BCH, and Goppa codes), (e) codes and combinatorial designs, and (f) codes and Latin squares.

On chapter four, symmetric key cryptography (automatic teller machines over public phone lines), public key cryptography (the RSA cryptosystem, Double-round quadratic enciphering, the Diffie-Hellman system, Elliptic curves and elliptic curve cryptography), and threshold scheme (to split a piece of secret information among several individuals) are introduced.
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