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Proving primality in essentially quartic random time
 Math. Comp
, 2003
"... Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1. ..."
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Cited by 18 (0 self)
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Abstract. This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n) 4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail. 1.
Factoring into Coprimes in Essentially Linear Time
"... . Let S be a nite set of positive integers. A \coprime base for S" means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P . There is a natural coprime base for S. This paper introduc ..."
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Cited by 16 (2 self)
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. Let S be a nite set of positive integers. A \coprime base for S" means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P . There is a natural coprime base for S. This paper introduces an algorithm that computes the natural coprime base for S in essentially linear time. The best previous result was a quadratictime algorithm of Bach, Driscoll, and Shallit. This paper also shows how to factor S into elements of P in essentially linear time. The algorithms apply to any free commutative monoid with fast algorithms for multiplication, division, and greatest common divisors; e.g., monic polynomials over a eld. They can be used as a substitute for prime factorization in many applications. 1.
Detecting perfect powers by factoring into coprimes
 MATHEMATICS OF COMPUTATION
, 2006
"... This paper presents an algorithm that, given an integer n> 1, finds the largest integer k such that n is a kth power. A previous algorithm by the first author took time b 1+o(1) where b = lg n; more precisely, time b exp(O ( √ lg b lg lg b)); conjecturally, time b(lg b) O(1). The new algorithm ta ..."
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Cited by 11 (3 self)
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This paper presents an algorithm that, given an integer n> 1, finds the largest integer k such that n is a kth power. A previous algorithm by the first author took time b 1+o(1) where b = lg n; more precisely, time b exp(O ( √ lg b lg lg b)); conjecturally, time b(lg b) O(1). The new algorithm takes time b(lg b) O(1). It relies on relatively complicated subroutines—specifically, on the first author’s fast algorithm to factor integers into coprimes—but it allows a proof of the b(lg b) O(1) bound without much background; the previous proof of b 1+o(1) relied on transcendental number theory.
Integer Factorization
, 2006
"... Factorization problems are the “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the sc ..."
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Cited by 10 (1 self)
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Factorization problems are the “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.” But what exactly is the problem? It turns out that there are many different factorization problems, as we will discuss in this paper.