Primality Testing in Polynomial Time

From Randomized Algorithms to "PRIMES Is in P"

Martin Dietzfelbinger (Autor)

Buch | Softcover

X, 150 Seiten

2004 | 2004
Springer Berlin (Verlag)
978-3-540-40344-9 (ISBN)

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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.

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.

1. Introduction: Efficient Primality Testing.- 2. Algorithms for Numbers and Their Complexity.- 3. Fundamentals from Number Theory.- 4. Basics from Algebra: Groups, Rings, and Fields.- 5. The Miller-Rabin Test.- 6. The Solovay-Strassen Test.- 7. More Algebra: Polynomials and Fields.- 8. Deterministic Primality Testing in Polynomial Time.- A. Appendix.

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)

Erscheint lt. Verlag	29.6.2004
Reihe/Serie	Lecture Notes in Computer Science
Zusatzinfo	X, 150 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 233 mm
Gewicht	250 g
Themenwelt	Mathematik / Informatik ► Informatik
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	algorithm • Algorithm analysis and problem complexity • Algorithmen • Algorithmics • algorithms • Computational Number Thoery • Computer • Computer Science • Deterministic Primality Testing • Efficient Algorithms • Efficient Primality Testing • Factorization • Number Theoretical Algorithms • Number Theory • Polynomia Time Algorithms • Primality Testing • Prime • Primzahl • randomized algorithms
ISBN-10	3-540-40344-2 / 3540403442
ISBN-13	978-3-540-40344-9 / 9783540403449
Zustand	Neuware