## Factoring polynomials with rational coefficients.(English)Zbl 0488.12001

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.

### MSC:

 11Y16 Number-theoretic algorithms; complexity 11C08 Polynomials in number theory 11R09 Polynomials (irreducibility, etc.) 68W30 Symbolic computation and algebraic computation
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### 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)
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