J. A. Buchmann and H. W. Lenstra, Jr., Approximating rings of integers in number fields, Journ'ee de Th'eorie des Nombres de Bordeaux 6 (1994), 221--260.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....we will prove that this problem is polynomial time equivalent to that of computing the largest squarefree divisor of an integer, for which there is no efficient algorithm known at the present time. For some practical advice on how to compute such a divisor, the reader is referred to Section 7 of [4]. the model of computation that we will be using is that of a Turing machine. We say that a language L is polynomial time reducible to a language L 0 , and denote it by L L 0 , if there is a polynomial time reduction from L to L 0 in the usual sense. 10.1. Known Results. In this ....

....of integers of a number field; DISC) Computing the discriminant of a number field; OPF) Computing the product of primes dividing an integer to an odd power. The equivalence of ROI and OPF is due to Chistov [7] The reduction used in this equivalence has been improved by Buchmann and Lenstra [4]. Considering quadratic fields we see that OPF DISC. Furthermore, the definition of the discriminant shows that given the ring of integers, we can easily compute the discriminant in polynomial time which shows that DISC ROI. Another result we will need later for our investigations is the ....

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J. Buchmann and H. W. Lenstra. Approximating rings of integers in number fields. J. de Th'eorie des Nombres de Bordeaux, 6:221--260, 1994.

....both display significantly better performances than former methods. 4.1 Computing in the number field The ring of integers. During the whole algorithm, we need to work with ideals and algebraic integers. We first have to compute an integral basis of O. In general, this is a hopeless task (see [13, 2] for a survey) but for the number fields NFS encounters (small degree and large discriminant) this can be done by the so called round algorithms [16, 4] Given an order R and several primes p i , any round algorithm will enlarge this order for all these primes so that the new order b R is p i ....

Buchmann, J. A., and Lenstra, Jr., H. W. Approximating rings of integers in number fields. J. Th'eor. Nombres Bordeaux 6, 2 (1994), 221--260.

....to several other problems in computational number theory, like that of computing the largest squarefree divisor of an integer, for which there is no efficient algorithm known at the present time. For some practical advice on how to compute such a divisor, the reader is referred to Section 7 of [4]. The model of computation that we will be using is that of a Turing machine. We say that a language L is polynomial time reducible to a language L 0 , and denote it by L L 0 , if there is a polynomial time reduction from L to L 0 in the usual sense. For an integer m we 8 denote by ....

....of an integer; LSFD) Computing the largest suqare free divisor of an integer; EOF) Computing the equal order factorization of an integer. Proof: Sketch) The equivalence of ROI and OPF is due to Chistov [6] The reduction used in this equivalence has been improved by Buchmann and Lenstra [4]. That of OPF and DISC is well known. It is easy to show that LSQF, LSFD, and EOF are equivalent (see the appendix) So, it remains to show that LSQF and COND are equivalent. To see that LSQF COND, note that given m, we can compute the conductor f(K) of K = Q( p m) which, in this case, equals ....

J. Buchmann and H. W. Lenstra. Approximating rings of integers in number fields. J. de Th'eorie des Nombres de Bordeaux, 6:221--260, 1994.

....that vector space with Z n . An alternative formulation of this problem appears in Cohen s book [4, Algorithm 2.7. 2] The algorithm presented there is based on Pohlst s modification of LLL basis reduction, so we refer to it as the MLLL algorithm [15] Similar algorithms have appeared elsewhere [3]. The MLLL algorithm has some advantages, but experiments show that it is very ill suited to the application given below. The rest of this paper gives a practical application for this problem and explains how to find the desired basis. Section 2 explains the application, and Section 3 gives the ....

J. A. Buchmann and H. W. Lenstra, Jr., Approximating rings of integers in number fields, Journ'ee de Th'eorie des Nombres de Bordeaux 6 (1994), 221--260.

....we will prove that this problem is polynomial time equivalent to that of computing the largest squarefree divisor of an integer, for which there is no efficient algorithm known at the present time. For some practical advice on how to compute such a divisor, the reader is referred to Section 7 of [4]. the model of computation that we will be using is that of a Turing machine. We say that a language L is polynomial time reducible to a language L 0 , and denote it by L L 0 , if there is a polynomial time reduction from L to L 0 in the usual sense. 10.1 Known Results In this subsection ....

....of integers of a number field; DISC) Computing the discriminant of a number field; OPF) Computing the product of primes dividing an integer to an odd power. The equivalence of ROI and OPF is due to Chistov [7] The reduction used in this equivalence has been improved by Buchmann and Lenstra [4]. Considering quadratic fields we see that OPF DISC. Furthermore, the definition of the discriminant shows that given the ring of integers, we can easily compute the discriminant in polynomial time which shows that DISC ROI. Another result we will need later for our investigations is the ....

[Article contains additional citation context not shown here]

J. Buchmann and H. W. Lenstra. Approximating rings of integers in number fields. J. de Th'eorie des Nombres de Bordeaux, 6:221--260, 1994.

....n Thetan is a matrix in Hermite normal form (cf. 27, pp. 45 51] such that the elements ff j = P n i=1 h i;j i ; 1 j n, form a ZZ basis of d(A)A. We note that there are polynomial time algorithms that given O, ff 2 O and fractional ideals A; B of O determine AB, A Gamma1 and ffO (cf. [8]) We call N(A) O : d(A)A] d(A) n the norm of the ideal A. Clearly, all entries of HNF(A) are bounded by N(A)d(A) n . We know Lemma 3. For all ideals A of O and all ff 2 A, ff 6= 0, we have jN(ff)j N(A) For all fi 2 O, fi 6= 0, we have j N(fi) j = N(fiO) The set IF of (fractional) ....

J. Buchmann, H. W. Lenstra, Jr., Approximating rings of integers in number fields, in preparation.

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J. A. Buchmann and H. W. Lenstra, Jr., Approximating rings of integers in number fields, Journ'ee de Th'eorie des Nombres de Bordeaux 6 (1994), 221--260.

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J. Buchmann, H. W. Lenstra, Jr., Approximating rings of integers in number fields, in preparation.

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J. Buchmann and H.W. Lenstra, Jr., Approximating rings of integers in number fields, J. Th'eor. Nombres Bordeaux 6 (1994), no. 2, 221--260.

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Comput. Buchmann, J., Lenstra, H. (1994). Approximating rings of integers in number fields. J. Theor. Nombres Bordx., 2:221--260.

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