Journal of Algorithms, to appear. Final revisions are in progress. FACTORING INTO COPRIMES IN ESSENTIALLY LINEAR TIME
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Abstract:
Abstract. 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 quadratic-time 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.
Citations
| 30 | Detecting perfect powers in essentially linear time – Bernstein - 1998 |
| 4 | Jerey Shallit, Sums of divisors, perfect numbers, and factoring – Bach, Miller - 1986 |
| 1 | symposium on symbolic and algebraic computation '90, Association for Computing Machinery – International - 1990 |
| 1 | Jerey Shallit, Factor re – Bach, Driscoll - 1993 |
| 1 | Fast ideal arithmetic via lazy localization – Bernstein |
| 1 | Fast solution of Topelitz systems of equations and computation of Pade approximants – Brent, Gustavson, et al. - 1980 |
