Primality testing in polynomial time [electronic resource] : from randomized algorithms to "PRIMES is in P" /
Dietzfelbinger, Martin.
Berlin : Springer,
2004.
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” has a “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 mathematicians again and again for centuries. Only in the 20th century, with the advent 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 signicant size. Readily, algorithms were provided that solved the problem very efficiently 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 find an algorithm that gets by without randomness, 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.
Includes bibliographical references and index.
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 -- References -- Index.
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” has a “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 mathematicians again and again for centuries. Only in the 20th century, with the advent 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 signicant size. Readily, algorithms were provided that solved the problem very efficiently 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 find an algorithm that gets by without randomness, 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.
Algebra.
Algorithms.
Computers.
Data encryption (Computer science).
Mathematical statistics.
Numbers, Prime.
Polynomials.