Lenstra, A. K.; Lenstra, H. W. jun.; Lovász, László Factoring polynomials with rational coefficients. (English) Zbl 0488.12001 Math. Ann. 261, 515-534 (1982). This paper describes a polynomial-time algorithm for the factorization of primitive polynomials \(f\in \mathbb Z[X]\) into irreducible factors. The number of bit operations used by the algorithm is \(O(n^{12} + n^9(\log \vert f\vert)^3)\), where \(n\) is the degree of \(f\) and \(\vert \sum_i a_iX^i \vert = (\sum_i a_i^2)^{1/2})\). The result can be generalized to algebraic number fields and to polynomials in several variables. One of the main ingredients of the algorithm is a new basis reduction algorithm for lattices in \(n\)-dimensional space. This basis reduction algorithm can be used to find short vectors in an \(n\)-dimensional lattice. The paper briefly mentions two applications of this algorithm in diophantine approximation. It is also of importance for problems from operations research and cryptography. Reviewer: H. W. Lenstra jun. Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 71 ReviewsCited in 738 Documents MSC: 11Y16 Number-theoretic algorithms; complexity 11C08 Polynomials in number theory 11R09 Polynomials (irreducibility, etc.) 68W30 Symbolic computation and algebraic computation Keywords:polynomial-time algorithm; factorization of primitive polynomials; lattice basis reduction algorithm; diophantine approximation; operations research; cryptography PDF BibTeX XML Cite \textit{A. K. Lenstra} et al., Math. Ann. 261, 515--534 (1982; Zbl 0488.12001) Full Text: DOI EuDML References: [1] Adleman, L.M., Odlyzko, A.M.: Irreducibility testing and factorization of polynomials, to appear. Extended abstract: Proc. 22nd Annual IEEE Symp. Found. Comp. Sci., pp. 409-418 (1981) [2] Brentjes, A.J.: Multi-dimensional continued fraction algorithms. Mathematical Centre Tracts 145. Amsterdam: Mathematisch Centrum 1981 · Zbl 0471.10024 [3] Cantor, D.G.: Irreducible polynomials with integral coefficients have succinct certificates. J. Algorithms2, 385-392 (1981) · Zbl 0489.68035 [4] Cassels, J.W.S.: An introduction to the geometry of numbers. Berlin, Heidelberg, New York: Springer 1971 · Zbl 0209.34401 [5] Ferguson, H.R.P., Forcade, R.W.: Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two. Bull. Am. Math. Soc.1, 912-914 (1979) · Zbl 0424.10021 [6] Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford: Oxford University Press 1979 · Zbl 0423.10001 [7] Knuth, D.E.: The art of computer programming, Vol. 2, Seminumerical algorithms. Reading: Addison-Wesley 1981 · Zbl 0477.65002 [8] Lenstra, A.K.: Lattices and factorization of polynomials, Report IW 190/81. Amsterdam: Mathematisch Centrum 1981 · Zbl 0477.12002 [9] Lenstra, H.W., Jr.: Integer programming with a fixed number of variables. Math. Oper. Res. (to appear) [10] Mignotte, M.: An inequality about factors of polynomials. Math. Comp.28, 1153-1157 (1974) · Zbl 0299.12101 [11] Pritchard, P.: A sublinear additive sieve for finding prime numbers. Comm. ACM24, 18-23 (1981) · Zbl 0454.68084 [12] Barkley Rosser, J., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math.6, 64-94 (1962) · Zbl 0122.05001 [13] Yun, D.Y.Y.: The Hensel lemma in algebraic manipulation. Cambridge: MIT 1974; reprint: New York: Garland 1980 [14] Zassenhaus, H.: On Hensel factorization. I. J. Number. Theory1, 291-311 (1969) · Zbl 0188.33703 [15] Zassenhaus, H.: A remark on the Hensel factorization method. Math. Comp.32, 287-292 (1978) · Zbl 0383.12003 [16] Zassenhaus, H.: A new polynomial factorization algorithm (unpublished manuscript, 1981) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.