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A Subexponential Algorithm for the Determination of Class Groups and Regulators of Algebraic Number Fields
, 1990
"... A new probabilistic algorithm for the determination of class groups and regulators of an algebraic number field F is presented. Heuristic evidence is given which shows that the expected running time of the algorithm is exp( p log D log log D) c+o(1) where D is the absolute discriminant of F , wh ..."
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Cited by 51 (5 self)
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A new probabilistic algorithm for the determination of class groups and regulators of an algebraic number field F is presented. Heuristic evidence is given which shows that the expected running time of the algorithm is exp( p log D log log D) c+o(1) where D is the absolute discriminant of F , where c 2 R?0 is an absolute constant, and where the o(1)function depends on the degree of F . 1 Introduction Computing the class group and the regulator of an algebraic number field F are two major tasks of algorithmic algebraic number theory. In the last decade, several regulator and class group algorithms have been suggested (e.g. [16],[17],[18],[3]). In [2] the problem of the computational complexity of those algorithms was adressed for the first time. This question was then studied in [2] in great detail. The theoretical results and the computational experience show that computing class groups and regulators is a very difficult problem. More precisely, it turns out that even under the a...
Voronoi's Algorithm in Purely Cubic Congruence Function Fields of Unit Rank 1
 Math. Comp
"... Abstract. The first part of this paper classifies all purely cubic function fields over a finite field of characteristic not equal to 3. In the remainder, we describe a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1 and characterist ..."
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Cited by 14 (9 self)
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Abstract. The first part of this paper classifies all purely cubic function fields over a finite field of characteristic not equal to 3. In the remainder, we describe a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1 and characteristic at least 5. The technique is based on Voronoi’s algorithm for generating a chain of successive minima in a multiplicative cubic lattice, which is used for calculating the fundamental unit and regulator of a purely cubic number field. 1.
An Analysis of the Reduction Algorithms for Binary Quadratic Forms
 Voronoi's Impact on Modern Science
, 1997
"... We prove in this paper that the classical reduction algorithms for integral binary quadratic forms have quadratic running time. 1 Introduction The reduction algorithm for binary quadratic forms, invented by Lagrange, Legendre, and Gauß is one of the most fundamental and important algorithms in the ..."
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Cited by 13 (3 self)
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We prove in this paper that the classical reduction algorithms for integral binary quadratic forms have quadratic running time. 1 Introduction The reduction algorithm for binary quadratic forms, invented by Lagrange, Legendre, and Gauß is one of the most fundamental and important algorithms in the algorithmic theory of binary quadratic forms and quadratic fields. It permits to find the successive minima of positive definite binary quadratic forms, to decide the equivalence of integral binary quadratic forms, to compute the fundamental unit of real quadratic orders, to compute in the class group of quadratic fields etc. The efficiency of the methods for solving the above problems depends heavily on the efficiency of reduction. In [Lag80] Lagarias slightly modified the classical reduction algorithm and proved that his modification reduces a form f in time O(nM(n)) where n is the length of the bit string necessary to represent f (see below for a more precise definition) and M(n) is the ...
Constructing nonresidues in finite fields and the extended Riemann hypothesis
 Math. Comp
, 1991
"... Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in pol ..."
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Cited by 8 (0 self)
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Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomialtime bound holds even if k is exponentially large. More generally, assuming the ERH, in time (n log p) O(n) we can construct a set of elements
Computing Arakelov class groups
, 2008
"... Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of ..."
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Cited by 7 (0 self)
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Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann’s algorithm in this context.
Unit Computation in Purely Cubic Function Fields of Unit Rank 1, to appear
 in Proceedings of the Third Algorithmic Number Theory Symposium ANTSIII, Lecture Notes in Computer Science 1423
, 1998
"... Abstract. This paper describes a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1. The technique is based on Voronoi’s algorithm for generating a chain of successive minima in a multiplicative cubic lattice which is used for calculati ..."
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Cited by 6 (4 self)
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Abstract. This paper describes a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1. The technique is based on Voronoi’s algorithm for generating a chain of successive minima in a multiplicative cubic lattice which is used for calculating the fundamental unit and regulator of a purely cubic number field. 1
Cryptography Based on Number Fields with Large Regulator
, 2000
"... We explain a variant of the FiatShamir identification and signature protocol that is based on the intractability of computing generators of principal ideals in algebraic number fields. We also ..."
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Cited by 4 (0 self)
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We explain a variant of the FiatShamir identification and signature protocol that is based on the intractability of computing generators of principal ideals in algebraic number fields. We also
Under the assumption of the Generalized Riemann Hypothesis verifying the class number belongs to NP ∩ co–NP
 IN ANTS1: ALGORITHMIC NUMBER THEORY, EDS L.M. ADLEMAN AND MD. HUANG, SPRINGERVERLAG (BERLIN), LECTURE NOTES IN COMPUTER SCIENCE NO. 877
, 1994
"... We show that under the assumption of a certain Generalized Riemann Hypothesis the problem of verifying the value of the class number of an arbitrary algebraic number field F of arbitrary degree belongs to the complexity class NP ∩ coNP. In order to prove this result we introduce a compact represen ..."
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Cited by 2 (0 self)
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We show that under the assumption of a certain Generalized Riemann Hypothesis the problem of verifying the value of the class number of an arbitrary algebraic number field F of arbitrary degree belongs to the complexity class NP ∩ coNP. In order to prove this result we introduce a compact representation of algebraic integers which allows us to represent a system of fundamental units by (2 + log 2 (\Delta)) O(1)
A Practical Version of the Generalized Lagrange Algorithm
, 1994
"... this paper we describe an implementation of the generalized Lagrange algorithm (GLA) for computing the unit group O ..."
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Cited by 1 (0 self)
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this paper we describe an implementation of the generalized Lagrange algorithm (GLA) for computing the unit group O
CONTENTS
"... ABSTRACT. Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class grou ..."
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ABSTRACT. Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann’s algorithm in this context.