J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In S'eminaire de Th'eorie des Nombres, pages 27-- 41, Paris, 1988-89.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....principal ideal a = aK[u] b v)K[u] nd its distance (a; O) That is, nd deg( where a = Notice that there exists a unique (up to constants) generator for a such that 0 deg( R. The algorithm basically follows Strategy 1 from above, and is based on similar ideas of Buchmann [11] and Abel [1] in real quadratic number elds. First, let S = fp 1 ; p 2 ; png be the factor base consisting of all split and rami ed prime ideals P i with deg P i t for some bound t: The rst stage of the algorithm consists of nding m n t smooth principal ideals, each of which yields ....

J. Buchmann, \A subexponential algorithm for the determination of class groups and regulators of algebraic number elds", Seminaire de Theorie des Nombres, Paris

....[158] 83] 159] 132] 160] 173] and [174] for an index calculus application of the number eld sieve; and [53] 58] 131] 115] and [7] for a function eld analogue. 4 DANIEL J. BERNSTEIN The same ideas are also used to compute class groups and regulators of number elds. See [87] [38], 39] 90] and [40] Acknowledgments. Thanks to Carl Pomerance for drawing my attention to the unsieveable integers in [54] Thanks to Christine Swart for her comments. 2. Multiplication and division The algorithms here use a multiplication box that computes xz given nonnegative integers x ....

Johannes Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number elds, in [80] (1990), 27-41. MR 92g:11125.

....computation time spent in using (2.9) to approximate C( is in computing the class number and regulator of the quadratic order O : We used the method described in [12] Algorithm 4. 3) The underlying strategy of this algorithm is the same as that of Hafner and McCurley [9] and its variants [7] [4], 1] 6] Suppose we have computed a factor base FB = fp 1 ; p k g consisting 8 MICHAEL J. JACOBSON, JR. AND HUGH C. WILLIAMS of invertible prime ideals such that the equivalence classes of some subset of FB generates the class group Cl of O : For v 2 Z k we de ne FB v = k ....

....Smith normal form of A; are precisely the elementary divisors of Cl : Thus, in addition to h ; we get the structure of Cl as a direct product of cyclic subgroups with very little extra e ort. This strategy can easily be extended to compute class groups and regulators of real quadratic orders [4, 1]. In this case, we compute relations of the form ( v; logj j) where FB v = i.e. generates the principal ideal FB v : We produce a generating system L 0 = f( v 1 ; logj 1 j) v 2 ; logj 2 j) v l ; logj l j)g of the extended relation lattice 0 = f( v; log j j) 2 ....

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J. Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number elds, Seminaire de Theorie des Nombres (Paris), 1988-89, pp. 27-41.

....Class Group. The first problem which must be solved in order to implement this method is to compute the structure of Cl Delta : We used the method described in [19] Algorithm 4. 3) The underlying strategy of this algorithm is the same as that of Hafner and McCurley [16] and its variants [14] [3], 1] 11] Suppose we have computed a factor base FB = fp 1 ; p k g consisting of invertible prime ideals such that the equivalence classes of some subset of FB generates Cl Delta : For v 2 Z k we define FB v = k Y i=1 p v i i where p i 2 FB: We call v a relation if FB ....

.... formed by taking the relations v i as columns, has rank k: The diagonal elements which are greater than 1 in S; the Smith normal form of A; are precisely the elementary divisors of Cl Delta : This strategy can easily be extended to compute class groups and regulators of real quadratic orders [3, 1]. In this case, we compute relations of the form ( v; logjflj) where FB v = fl) i.e. fl generates the principal ideal FB v : We produce a generating system L 0 = f( v 1 ; logjfl 1 j) v 2 ; logjfl 2 j) v l ; logjfl l j)g of the extended relation lattice 0 = f( v; log ....

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J. Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, S'eminaire de Th'eorie des Nombres (Paris), 1988-89, pp. 27--41.

....in class groups of quadratic number elds. In the case of imaginary quadratic elds, the algorithm is based on methods applied by Hafner and McCurley [HM89] to determine the structure of the class group of imaginary quadratic elds. In the case of real quadratic elds, the algorithm of Buchmann [Buc89] for computation of class group and regulator forms the basis. We employ the rigorous elliptic curve factorization algorithm of Pomerance [Pom87] and an algorithm for solving systems of linear Diophantine equations proposed and analysed by Mulders and Storjohann [MS99] Under the assumption ....

....the class group rst. On the contrary, the methods applied can be extended to compute the class group. Since Gauss, there has been continuous interest in computing class groups. Comparatively recently, the interest has also turned to the question of how to compute them eOEciently. With [HM89] and [Buc89] we would like to mention two papers which made a breakthrough by proving that this calculation can be done in time subexponential in the size of the discriminant of the eld: HM89] did this for imaginary quadratic elds, Buc89] for general number elds. Here, we will employ the methods of these ....

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J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number elds. In S#minaire de Th#orie des Nombres, pages 2741, Paris, 1988-89.

....time spent in using (2.9) to approximate C ( Delta) is in computing the class number and regulator of the quadratic order O Delta : We used the method described in [12] Algorithm 4. 3) The underlying strategy of this algorithm is the same as that of Hafner and McCurley [9] and its variants [7] [4], 1] 6] Suppose we have computed a factor base FB = fp 1 ; p k g consisting of invertible prime ideals such that the equivalence classes of some subset of FB generates the class group Cl Delta of O Delta : For v 2 Z k we define FB v = k Y i=1 p v i i where p i 2 FB: We ....

....of A; are precisely the elementary divisors of Cl Delta : Thus, in addition to h Delta ; we get the structure of Cl Delta as a direct product of cyclic subgroups with very little extra effort. This strategy can easily be extended to compute class groups and regulators of real quadratic orders [4, 1]. In this case, we compute relations of the form ( v; logjflj) where FB v = fl) i.e. fl generates the principal ideal FB v : We produce a generating system L 0 = f( v 1 ; logjfl 1 j) v 2 ; logjfl 2 j) v l ; logjfl l j)g of the extended relation lattice 0 = f( v; log ....

[Article contains additional citation context not shown here]

J. Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, S'eminaire de Th'eorie des Nombres (Paris), 1988-89, pp. 27--41.

....to extend their method to handle two large primes. Our algorithm should also be very effective in computing class groups of real quadratic orders. For each relation v one would also have to compute a minimum QUADRATIC CLASS GROUPS 9 ff such that FB v (ff) Then, the methods described in [5], 9] and [1] can be applied directly. Further experiments are currently underway in these directions. ....

J. Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, S'eminaire de Th'eorie des Nombres (Paris), 1988-89, pp. 27--41.

....is useful in many applications, including Diophantine analysis (see Newman 1972) and determining the canonical structure of Abelian groups. Recently, algorithms for the Smith normal form have been used to compute the structure of the class group of a number field (see Hafner McCurley 1989, Buchmann 1988). In a typical situation we have a finite Abelian group G in which each element a 2 G can be written as g e 1 1 g e 2 2 Delta Delta Delta g em m with respect to a set of generators g 1 ; g m 2 G. We represent a by (e 1 ; e m ) 2 Z n Theta1 , yielding a homomorphism ae : ....

J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In S'eminaire de th'eorie des nombres, Paris, 1988.

.... 1972, Chou Collins 1982) combinatorics (see Wallis et al. 1972) and determining the canonical structure of Abelian groups (see Newman 1972) Recently, algorithms for the Smith normal form have been used to compute the structure of the class group of a number field (see Hafner McCurley 1989, Buchmann 1988). It is often the case that A is sparse (lots of zero entries) and it is desirable to take advantage of this sparsity when computing the Smith form. Existing algorithms do not do this, and suffer from fill in (much like Gaussian elimination) as well as coefficient growth (see Kannan Bachem ....

J. Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In S'eminaire de th'eorie des nombres, Paris, 1988.

.... is useful in many applications, including Diophantine analysis (see Newman [12, 1972] computing the structure of finitely generated abelian groups (see Haves, Holt Rees [7, 1993] and computing the structure of the class group of a number field (see Hafner McCurley [5, 1989] and Buchmann [1, 1988]) In Section 3 we present our main result an asymptotically fast algorithm for computing Smith normal forms over ZZ d . Let A be an n Theta m matrix over ZZ d . We assume without loss of generality that n m the Smith normal form of the transpose of A will have the same invariant factors ....

Buchmann, J. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In S'eminaire de th'eorie des nombres (Paris, 1988).

....is useful in many applications, including Diophantine analysis (see Newman 1972) and determining the canonical structure of Abelian groups. Recently, algorithms for the Smith normal form have been used to compute the structure of the class group of a number field (see Hafner McCurley 1989, Buchmann 1988). In this paper we present two new probabilistic algorithms to compute the Smith normal form of an integer matrix. Their costs are substantially smaller than the previously best known (deterministic) algorithm of Hafner McCurley (1989,1991; see also Kannan Bachem 1979, Chou Collins 1982, ....

J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In S'eminaire de th'eorie des nombres, Paris, 1988.

....the constant ae. 5.1 Finding a relation The crucial part of Algorithm 3 is the creation of relations in Steps 2) and 4) We argued in Section 4. 2 that the probability of finding a relation is heuristically NB h , a claim we make precise in this section, using techniques inspired by those in [4] and [22] In a first step we determine how many exponent vectors e yield a fixed relation c: Lemma 9 Let c 2 Gamma. Then the number of vectors e 2 f0; E Gamma 1g n which yield the relation c equals the number of B smooth reduced degree zero divisors P r i=1 r i P i such that r Gamma ....

Johannes Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In [11], pages 27--41, 1990.

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J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In S'eminaire de Th'eorie des Nombres, pages 27-- 41, Paris, 1988-89.

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J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. S'eminaire de th'eorie des nombres, Paris 1988-1989, 27-41. Birkhauser, Boston Basel Berlin, 1990.

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Buchmann, J. A subexponential algorithm for the determination of class groups and regulators of algebraic number elds. In Seminaire de Theorie des Nombres (1988-89), pp. 27-41.

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J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In S'eminaire de Th'eorie des Nombres, pages 27--41, Paris, 1988-89.

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J. Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, S'eminaire de th'eorie des nombres, Paris

No context found.

J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In Seminaire de Theorie des Nombres, pages 27--41, Paris, 1988-89.

No context found.

Johannes Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In Catherine Goldstein, editor, Seminaire de Theorie des Nombres, Paris 1988--1989, volume 91 of Progress in Mathematics, pages 27--41. Birkhauser, 1990.

....Hermite normal form (see Section 2) The computation of the integer kernel of an integer matrix is necessary for the solution of important problems in computational number theory. It is, for example, a key step in the determination of a system of fundamental units of an algebraic number eld (see [1]) There are also applications to group theory, since abelian groups are Z modules (see for example [9] The problem of computing the image of (that is, the HNF basis of the image of ) has been studied extensively, for example in [4] 10] 14] 15] 11] 7] 6] 12] and [8] The ....

Buchmann, J. A subexponential algorithm for the determination of class groups and regulators of algebraic number elds. In Seminaire de Theorie des Nombres (1988-89), pp. 27-41.

.... of the product of the regulator and the class number of a number field by means of the analytic class number formula (see for example [Sha72] Len82] BW89] Coh95] JLW95] and the determination of a system of fundamental units and the regulator from a generating set of units (see for example [Buc89], PZ89] Coh95] Those computations are carried out with rational approximations of a certain precision. Therefore, roundoff errors occur and those errors have to be taken into account. Unfortunately, most algorithms which use approximations to real numbers and their implementations ignore ....

J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In S'eminaire de Th'eorie des Nombres, pages 27-- 41, Paris, 1988-89.

....Informatik Technische Universit at Darmstadt 1 Introduction Computing the class group and regulator of an algebraic number field K are two major tasks of algorithmic algebraic number theory. The asymptotically fastest method known has conjectured sub exponential running time and was proposed in [Buc89]. In this paper we show how sieving methods developed for factoring algorithms can be used to speed up this algorithm in practice. We present numerical experiments which demonstrate the efficiency of our new strategy. For example, we are able to compute the class group of an imaginary quadratic ....

....field C of complex numbers. We denote the maximal order of K by OK . The norm of an algebraic number ff will be denoted by N (ff) and the norm of an ideal a will be denoted by N (a) The class number of K will be denoted by h and the regulator by R. We briefly review the algorithm presented in [Buc89]. Let FB be a set of prime ideals over K and k = jF Bj. For e = e 1 ; e k ) 2 ZZ k ; we write FB e = k Y i=1 p e i i : By AFB we denote all algebraic numbers ff in K which, considered as principal ideals, can be represented as a power product of the ideals of the factor base ....

J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In S'eminaire de Th'eorie des Nombres, pages 27--41, Paris, 1988-89.

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J. Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, S'eminaire de Th'eorie des Nombres (Paris), 1988-89, pp. 27--41.

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J. Buchmann, A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, S'eminaire de th'eorie des nombres, Paris (1988-1989) 28-41.

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J. Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number elds. In S#minaire de Th#orie des Nombres, pages 2741, Paris, 1988-89.

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