A Computational Introduction To Number Theory And Algebra

الموضوع في 'قسم الرياضيات' بواسطة dedoda, بتاريخ ‏مايو 31, 2008.

  1. dedoda

    dedoda Well-Known Member

    إنضم إلينا في:
    ‏ديسمبر 6, 2007
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    PMP Trainer
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    [​IMG]Author(s) : Victor Shoup
    Publisher : Cambridge
    Year : Jun 2005
    ISBN : 0521851548
    Language : English
    Pages : 534
    File type : PDF
    Size : 4.5 MB

    Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography.

    It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.

    This is an outstanding and well-written book whose aim is to introduce the reader to a broad range of material -- ranging from basic to relatively advanced -- without requiring any prior knowledge on the part of the reader other than calculus and mathematical maturity. That the book succeeds at this goal is quite an accomplishment! ...this book is a must-read for anyone interested in computational number theory or algebra and especially applications of the latter to cryptography. I would not hesitate, though, to recommend this book even to students 'only' interested in the algebra itself (and not the computational aspects thereof); especially for computer science majors, this book is one of the best available introductions to that subject.

    Chapter 01 - Basic properties of the integers
    Chapter 02 - Congruences
    Chapter 03 - Computing with large integers
    Chapter 04 - Euclid’s algorithm
    Chapter 05 - The distribution of primes
    Chapter 06 - Finite and discrete probability distributions
    Chapter 07 - Probabilistic algorithms
    Chapter 08 - Abelian groups
    Chapter 09 - Rings
    Chapter 10 - Probabilistic primality testing
    Chapter 11 - Finding generators and discrete logarithms in Z*p
    Chapter 12 - Quadratic residues and quadratic reciprocity
    Chapter 13 - Computational problems related to quadratic residues
    Chapter 14 - Modules and vector spaces
    Chapter 15 - Matrices
    Chapter 16 - Subexponential-time discrete logarithms and factoring
    Chapter 17 - More rings
    Chapter 18 - Polynomial arithmetic and applications
    Chapter 19 - Linearly generated sequences and applications
    Chapter 20 - Finite fields
    Chapter 21 - Algorithms for finite fields
    Chapter 22 - Deterministic primality testing




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