Primality Testing in Polynomial Time

1. Front Matter

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2. 1. Introduction: Efficient Primality Testing

   • Martin Dietzfelbinger

   Pages 1-12

3. 2. Algorithms for Numbers and Their Complexity

   • Martin Dietzfelbinger

   Pages 13-21

4. 3. Fundamentals from Number Theory

   • Martin Dietzfelbinger

   Pages 23-53

5. 4. Basics from Algebra: Groups, Rings, and Fields

   • Martin Dietzfelbinger

   Pages 55-71

6. 5. The Miller-Rabin Test

   • Martin Dietzfelbinger

   Pages 73-84

7. 6. The Solovay-Strassen Test

   • Martin Dietzfelbinger

   Pages 85-94

8. 7. More Algebra: Polynomials and Fields

   • Martin Dietzfelbinger

   Pages 95-114

9. 8. Deterministic Primality Testing in Polynomial Time

   • Martin Dietzfelbinger

   Pages 115-131

10. A. Appendix

    • Martin Dietzfelbinger

    Pages 133-142

11. Back Matter

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On August 6, 2002,a paper with the title “PRIMES is in P”, by M. Agrawal, N. Kayal, and N. Saxena, appeared on the website of the Indian Institute of Technology at Kanpur, India. In this paper it was shown that the “primality problem”hasa“deterministic algorithm” that runs in “polynomial time”. Finding out whether a given number n is a prime or not is a problem that was formulated in ancient times, and has caught the interest of mathema- ciansagainandagainfor centuries. Onlyinthe 20thcentury,with theadvent of cryptographic systems that actually used large prime numbers, did it turn out to be of practical importance to be able to distinguish prime numbers and composite numbers of signi?cant size. Readily, algorithms were provided that solved the problem very e?ciently and satisfactorily for all practical purposes, and provably enjoyed a time bound polynomial in the number of digits needed to write down the input number n. The only drawback of these algorithms is that they use “randomization” — that means the computer that carries out the algorithm performs random experiments, and there is a slight chance that the outcome might be wrong, or that the running time might not be polynomial. To ?nd an algorithmthat gets by without rand- ness, solves the problem error-free, and has polynomial running time had been an eminent open problem in complexity theory for decades when the paper by Agrawal, Kayal, and Saxena hit the web.

• Number theory
• Prime
• algorithm
• algorithmics
• algorithms
• computer
• computer science
• algorithm analysis and problem complexity

From the reviews:

"This book gives an account of the recent proof by M. Agrawal, N. Kayal and N. Saxena … that one can decide in polynomial time whether a given natural number is prime or composite. … It presents the background needed from number theory and algebra to make the proof accessible to undergraduates. … This concise book is written for students of computer science and of mathematics." (Samuel S. Wagstaff, Mathematical Reviews, Issue 2005 m)

"The book can logically be separated into two parts: the first covering introductory material and the second covering the AKS result itself. … Chapters … are a joy to read, and I found the proofs and explanations clear and concise. Amazingly, the material is presented in full, with complete proofs given for all results necessary for proving the main results of the book. … I would enthusiastically and wholeheartedly recommend this book … ." (Jonathan Katz, SIGACT News, Vol. 37 (1), 2006)

• Technische Universität Ilmenau, Ilmenau, Germany

  Martin Dietzfelbinger

Univ.-Prof. Dr.(USA) Martin Dietzfelbinger (b. 1956) studied Mathematics in Munich and earned his Ph.D. from the University of Illinois at Chicago. In 1992, he obtained his Habilitation at the Universität Paderborn with a thesis on randomized algorithms; in the same year he became a professor of computer science at the Universität Dortmund. Since 1998, he holds the chair for Complexity Theory and Efficient Algorithms at the Faculty of Computer Science and Automation of the Technische Universität Ilmenau, Germany. His main research interests are in complexity theory and data structures.

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