Open Access
Four primality testing algorithms
René Schoof
- 01 Jan 2008
- pp 101-126
TL;DR: In this expository paper, the first test is very efficient, but is only capable of proving that a given number is either composite or ‘very probably’ prime, while the second and fourth primality tests are at present most widely used in practice.
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Abstract: In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or ‘very probably’ prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost, but not quite polynomial time. The fourth algorithm exploits elliptic curves. Its running time is difficult to estimate, but it behaves well in practice.
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Citations
The arithmetic of number rings
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TL;DR: In this paper, the main structural results on number rings are described, i.e., integral domains for which the eld of fractions is a number eld. Whenever possible, the algorithmically undesirable hypothesis that the number ring in question be integrally closed is avoided.
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On practical aspects of the Miller-Rabin Primality Test
TL;DR: The dependence between the length of testing numbers and the number of rounds of MRPT sufficient to give the correct answer is studied and some recommendations how to choose a suitable set of bases at which MRPT runs more efficiently are given.
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Basic algorithms in number theory
Joe Buhler,Stan Wagon +1 more
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References
•Book
A Course in Computational Algebraic Number Theory
Henri Cohen
- 01 Jan 1993
TL;DR: The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
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•Book
Introduction to Cyclotomic Fields
Lawrence C. Washington
- 01 Jan 1982
TL;DR: In this paper, Dirichlet characters were used to construct p-adic L-functions and Bernoulli numbers, which are then used to define the class number formula.
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PRIMES is in P
TL;DR: In this paper, an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite is presented. But the algorithm is not deterministic in the sense that
Factoring integers with elliptic curves
TL;DR: This paper is devoted to the description and analysis of a new algorithm to factor positive integers that depends on the use of elliptic curves and it is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2.
Probabilistic algorithm for testing primality
TL;DR: The algorithm has the feature that when it determines a number composite then the result is always true, but when it asserts that a number is prime there is a provably small probability of error.
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