# Primality Testing in Polynomial Time

## From Randomized Algorithms to "PRIMES Is in P"

• Martin Dietzfelbinger
Textbook

Part of the Lecture Notes in Computer Science book series (LNCS, volume 3000)

1. Front Matter
2. Martin Dietzfelbinger
Pages 1-12
3. Martin Dietzfelbinger
Pages 13-21
4. Martin Dietzfelbinger
Pages 23-53
5. Martin Dietzfelbinger
Pages 55-71
6. Martin Dietzfelbinger
Pages 73-84
7. Martin Dietzfelbinger
Pages 85-94
8. Martin Dietzfelbinger
Pages 95-114
9. Martin Dietzfelbinger
Pages 115-131
10. Martin Dietzfelbinger
Pages 133-142
11. Back Matter

### Introduction

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.

### Keywords

Number theory Prime algorithm algorithmics algorithms computer computer science

#### Authors and affiliations

• Martin Dietzfelbinger
• 1
1. 1.Technische Universität IlmenauIlmenauGermany

### Bibliographic information

• DOI https://doi.org/10.1007/b12334
• Copyright Information Springer-Verlag Berlin Heidelberg 2004
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-40344-9
• Online ISBN 978-3-540-25933-6
• Series Print ISSN 0302-9743
• Series Online ISSN 1611-3349
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