Tópico(s): Analytic Number Theory Research
2012 - Polish Academy of Sciences | Acta Arithmetica
... Janicic (GCLC), Anders Jensen (Gfan), Markus Schmies (jtem), Michael Pohst (KANT/KASH), Dave Saunders (LinBox), Dan Grayson (Macaulay2), ...
Tópico(s): Formal Methods in Verification
2008 - Taylor & Francis | International Journal of Computer Mathematics
Florian Heß, Sebastian Pauli, Michael Pohst,
Tópico(s): Computability, Logic, AI Algorithms
2006 - Springer Science+Business Media | Lecture notes in computer science
Florian Heß, Sebastian Pauli, Michael Pohst,
Let $\mathbf {k}$ be a global field with maximal order $\mathfrak o_{\mathbf k}$ and let ${\mathfrak {m}}_{0}$ be an ideal of $\mathfrak o_{\mathbf k}$. We present algorithms for the computation of the multiplicative group $(\mathfrak o_{\mathbf k}/{\mathfrak {m}}_{0})^*$ of the residue class ring $\mathfrak o_{\mathbf k}/{\mathfrak {m}}_{0}$ and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order ...
Tópico(s): Finite Group Theory Research
2003 - American Mathematical Society | Mathematics of Computation
István Gaál, Páter Olajos, Michael Pohst,
Abstract We consider the existence of power integral bases in composites of polynomial orders of number fields. We prove that if the degree of the composite field equals the product of the degrees of its subfields and the minimal polynomials of the generating elements of the polynomial orders have a multiple linear factor in their factorization modulo q, then the composite order admits no power integral bases. As an application we provide several examples including a parametric family of "simplest ...
Tópico(s): Advanced Differential Equations and Dynamical Systems
2002 - Taylor & Francis | Experimental Mathematics
An efficient algorithm is given for the resolution of relative Thue equations. The essential improvement is the application of an appropriate version of Wildanger's enumeration procedure based on the ellipsoid method of Fincke and Pohst. Recently relative Thue equations have gained an important application, e.g., in computing power integral bases in algebraic number fields. The presented methods can surely be used to speed up those algorithms. The method is illustrated by numerical examples.
Tópico(s): Polynomial and algebraic computation
2001 - American Mathematical Society | Mathematics of Computation
We develop an algorithm for computing all generators of relative power integral bases in quartic extensions K of number fields M. For this purpose we use the main ideas of our previously derived algorithm for solving index form equations in quartic fields (I. Gaál, A. Pethő, and M. Pohst, 1993, J. Symbolic Comput.16, 563–584; 1996, J. Number Theory57, 90–104). In this way we reduce the problem to the resolution of a cubic and several corresponding quartic relative Thue equations over M. These equations ...
Tópico(s): Coding theory and cryptography
2000 - Elsevier BV | Journal of Number Theory
Claus Fieker, A. Jurk, Michael Pohst,
Let $\mathbb {Q}\subseteq \mathcal {E}\subseteq \mathcal {F}$ be algebraic number fields and $M\subset \mathcal {F}$ a free $o\varepsilon$-module. We prove a theorem which enables us to determine whether a given relative norm equation of the form $|N_{\mathcal {F}/\mathcal {E}}(\eta )| = |\theta |$ has any solutions $\eta \in M$ at all and, if so, to compute a complete set of nonassociate solutions. Finally we formulate an algorithm using this theorem, consider its algebraic complexity and give some examples.
Tópico(s): Polynomial and algebraic computation
1997 - American Mathematical Society | Mathematics of Computation
Mario Daberkow, Claus Fieker, Jürgen Klüners, Michael Pohst, Katherine Roegner, M. Schörnig, K. Wildanger,
The software packageKANT V4for computations in algebraic number fields is now available in version 4. In addition a new user interface has been released. We will outline the features of this new software package.
Tópico(s): History and Theory of Mathematics
1997 - Elsevier BV | Journal of Symbolic Computation
Jürgen Klüners, Michael Pohst,
The purpose of this article is to determine all subfields Q(β) of fixed degree of a given algebraic number field Q(α). It is convenient to describe each subfield by a pair (h,g) of polynomials in Q[t] resp. Z[t] such thatgis the minimal polynomial of β = h(α). The computations are done in unramifiedp-adic extensions and use information concerning subgroups of the Galois group of the normal closure of Q(α) obtained from the van der Waerden criterion.
Tópico(s): Polynomial and algebraic computation
1997 - Elsevier BV | Journal of Symbolic Computation
We consider the totally real cyclic quintic fields K n = Q ( ϑ n ) K_{n}=\mathbb {Q}(\vartheta _{n}) , generated by a root ϑ n \vartheta _{n} of the polynomial f n ( x ) = x 5 + n 2 x 4 − ( 2 n 3 + 6 n 2 + 10 n + 10 ) x 3 + ( n 4 + 5 n 3 + 11 n 2 + 15 n + 5 ) x 2 + ( n 3 + 4 n 2 + 10 n + 10 ) x + 1. \begin{multline*} f_{n}(x)=x^{5}+n^{2}x^{4}-(2n^{3}+6n^{2}+10n+10)x^{3}\ +(n^{4}+5n^{3}+11n^{2}+15n+5)x^{2}+(n^{3}+4n^{2}+10n+10)x+1. \end{multline*} Assuming that m = n 4 + 5 n 3 + 15 n 2 + 25 n + 25 m=n^{4}+5n^{3}+15n^{2}+25n+25 is square free, we compute explicitly an integral ...
Tópico(s): Coding theory and cryptography
1997 - American Mathematical Society | Mathematics of Computation
Tópico(s): Coding theory and cryptography
1996 - Springer Science+Business Media | Lecture notes in computer science
István Gaál, Attila Pethö, Michael Pohst,
LetQ1, Q2∈Z[X, Y, Z] be two ternary quadratic forms andu1, u2∈Z. In this paper we consider the problem of solving the system of equations[formula]According to Mordell [12] the coprime solutions of[formula]can be presented by finitely many expressions of the formx=fx(p, q),y=fy(p, q),z=fz(p, q), wherefx, fy, fz∈Z[P, Q] are binary quadratic forms andp, qare coprime integers. Substituting these expressions into one of the equations of (1), we obtain a quartic homogeneous equation in two variables. If ...
Tópico(s): Cryptography and Residue Arithmetic
1996 - Elsevier BV | Journal of Number Theory
We give an efficient algorithm for the resolution of index form equations, especially for determining power integral bases, in sextic fields with an imaginary quadratic subfield. The method reduces the problem to the resolution of acubic relative Thue equationover the quadratic subfield. At the end of the paper we give a table containing the generators of all power integral bases in the first 25 fields of this type with smallest discriminant (in absolute value).1991 Mathematics Subject Classification: ...
Tópico(s): Coding theory and cryptography
1996 - Elsevier BV | Journal of Symbolic Computation
István Gaál, A. Pethö, Michael Pohst,
Let m, n be distinct square-free rational integers and let K=Q(√m, √n). Combining Baker-type inequalities with a suitable version of the Baker-Davenport reduction method we give a computational algorithm for determining all elements with minimal index in such number fields.
Tópico(s): Algebraic Geometry and Number Theory
1995 - Elsevier BV | Journal of Number Theory
Albert Schwarz, Michael Pohst, Fredéric Diaz,
All algebraic number fields F of degree 5 and absolute discriminant less than $2 \times {10^7}$ (totally real fields), respectively $5 \times {10^6}$ (other signatures) are determined. We describe the methods which we applied and list significant data.
Tópico(s): Chaos-based Image/Signal Encryption
1994 - American Mathematical Society | Mathematics of Computation
Albert Schwarz, Michael Pohst, Fredéric Diaz,
All algebraic number fields F of degree 5 and absolute discriminant less than 2 × 10 7 2 \times {10^7} (totally real fields), respectively 5 × 10 6 5 \times {10^6} (other signatures) are determined. We describe the methods which we applied and list significant data.
Tópico(s): Algebraic Geometry and Number Theory
1994 - American Mathematical Society | Mathematics of Computation
Wilhelm Plesken, Michael Pohst,
Integral laminated lattices with minimum 4 which are generated by vectors of minimum length are constructed systematically together with their automorphism groups. All lattices obtained lie in the Leech lattice.
Tópico(s): Semiconductor Lasers and Optical Devices
1993 - American Mathematical Society | Mathematics of Computation
Johannes Buchmann, David Ford, Michael Pohst,
With the mixed-type case now completed, all algebraic number fields of degree 4 with absolute discriminant > 10 6 > {10^6} have been enumerated. Methods from the totally real and totally complex cases were used without major modification. Isomorphism of fields was determined by a method similar to one of Lenstra. The T 2 {T_2} criterion of Pohst was applied to reduce the number of redundant examples.
Tópico(s): Analytic Number Theory Research
1993 - American Mathematical Society | Mathematics of Computation
István Gaál, Attila Pethö, Michael Pohst,
In this paper we reduce the problem of solving index form equations in quartic number fields K to the resolution of a cubic equation F (u, v) = i and a corresponding system of quadratic equations Q1 (x, y, z) = u, Q2(x, y, z) = v, where F is a binary cubic form and Q1, Q2 are ternary quadratic forms. This enables us to develop a fast algorithm for calculating "small" solutions of index form equations in any quartic number field. If, additionally, the field K is totally complex we can combine the two forms ...
Tópico(s): Cryptography and Residue Arithmetic
1993 - Elsevier BV | Journal of Symbolic Computation
István Gaál, A. Pethö, Michael Pohst,
In this paper we develop a method for computing all small solutions (i.e. with coordinates of absolute value <107) of index form equations in totally real biquadratic number fields. If the index form equation is not solvable, this will also be recognized by our algorithm in most cases. As an application we present all such solutions in quadratic extensions K of Q(√5) of discriminant DKQ < 63000 and of Q(√2) of discriminant DKQ < 39000.
Tópico(s): Algorithms and Data Compression
1991 - Elsevier BV | Journal of Number Theory
Itamar Gal, Attila Peth�, Michael Pohst,
Tópico(s): Cryptography and Residue Arithmetic
1991 - Birkhäuser | Archiv der Mathematik
Michael Pohst, Jacques Martinet, Fredéric Diaz,
The minimum discriminant of totally real octic algebraic number fields is determined. It is 282,300,416 and belongs to the ray class field over Q(√2) of conductor (7 + 2 √2): F = Q(√α) for α = (7 + 2 √2 + (1 + √2) √7 + 2 √2)/2. There is—up to isomorphy—only one field of that discriminant. The next two smallest discriminant values are 309,593,125 and 324,000,000. For each field we present a full system of fundamental units and its class number.
Tópico(s): Coding theory and cryptography
1990 - Elsevier BV | Journal of Number Theory
Johannes Buchmann, Michael Pohst,
In this paper we describe how the LLL-algorithm can be used to compute a basis of a lattice L in R n from a system of k generating vectors and a lower bound for the lengths of the non zero vectors in L. The algorithm which we present is proved to be polynomial time in n + k and the size of the input data. The algorithm is applied to the problem of finding multiplicative relations between units of algebraic number fields. Numerical results show that our method works very efficiently.
Tópico(s): Cryptography and Data Security
1989 - Springer Science+Business Media | Lecture notes in computer science
W Brauer, P Brinch, Hansen Gries, D Luckham, C Moler, Amir Pnueli, G Seegmijller, Josef Stoer, N. Wirth, Teo Mora, Alfonso Miola, Gianna X al Organizers, Roma Cioni, Mirella Schaerf, Alain Poli, A. Poll, T. Beth, J Caimet, Josep Dı́az, A. G. Hearn, Joos Heintz, L. Huguet, Hideki Imai, Remco Loos, H Liineburg, Harold Mattson, E Abhyankar, E. F. Assmus, Giorgio Ausiello, Elwyn R. Berlekamp, Martin Bossert, Ernest F. Brickell, Peter Bundschuh, Paul Camion, Jacques Calmet, Michel Carral, Michael Clausen, Gary Cohen, B. Courteau, R Desq, Giuliana Dettori, P Duhammel, R Dvomicich, Andrea R. Ferro, Willi Geiselmann, R. H. Goodman, A Heam, I Heintz, Erich Kaltofen, H Ltineburg, J. Massey, John A. Michon, Gavin R. Norton, Andrew Odlyzko, E. Olcayto, Francesco Parisi-Presicce, P. Piret, Michael Pohst, Marco Protasi, T. Recio, Luigi Robbiano, Sei-ichiro SAKATA, Andrzej Salwicki, Ruti Segal, Mervyn Singer, Peter T. C. So, E Stricldand, Carlo Enrico Traverso, JH van Lint, H. Ward, Volker Weispfenning, Jan Wolf, J. Wolfmann, W. Wolfowicz, Bostwick F. Wyman,
Tópico(s): Polynomial and algebraic computation
1989 - Springer Science+Business Media | Lecture notes in computer science
The reduction algorithm of Lenstra et al. (1982) is modified in a way that the input vectors can be linearly dependent. The output consists of a basis of the lattice generated by the input vectors as well as non-trivial linear combinations of O by the input vectors if those are linearly dependent.
Tópico(s): Advanced Data Storage Technologies
1987 - Elsevier BV | Journal of Symbolic Computation
The standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of Z m {{\mathbf {Z}}^m} in a suitable box. However, it suffices to consider those x ∈ Z m {\mathbf {x}} \in {{\mathbf {Z}}^m} which lie in a suitable ellipsoid having a much smaller volume than the box. We show in this paper that searching through that ellipsoid is in many cases much more efficient. If combined with an appropriate reduction procedure our method allows to do ...
Tópico(s): Cryptography and Data Security
1985 - American Mathematical Society | Mathematics of Computation
Tópico(s): Advanced Mathematical Identities
1983 - Springer Science+Business Media | Lecture notes in computer science
A new method of determining algebraic number fields with discriminants of small absolute value is developed that avoids lengthy considerations of subfields. As an application all minimum discriminants of sixth degree fields are computed.
Tópico(s): Coding theory and cryptography
1982 - Elsevier BV | Journal of Number Theory
Michael Pohst, Hans Zassenhaus,
The new method for efficient computation of the fundamental units of an algebraic number field developed by the authors in an earlier paper is considerably improved with respect to (Section 1) utilization to best advantage of the element of choice inherent in the method and the mastery of the linear programming techniques involved, (Section 2) ideal factorization, and (Section 3) the determination of sharper upper bounds for the index of U ε {U_\varepsilon } in U F {U_F} .
Tópico(s): Polynomial and algebraic computation
1982 - American Mathematical Society | Mathematics of Computation