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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
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)
Implementing Cryptographic Protocols Based on Algebraic Number Fields
"... . We show how to implement cryptographic protocols based on class groups of algebraic number fields of degree ? 2. We describe how the involved objects can be represented and how the arithmetic in class groups can be realized efficiently. Furthermore we show how to generate cryptographically sui ..."
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Cited by 1 (0 self)
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. We show how to implement cryptographic protocols based on class groups of algebraic number fields of degree ? 2. We describe how the involved objects can be represented and how the arithmetic in class groups can be realized efficiently. Furthermore we show how to generate cryptographically suitable algebraic number fields. In the final version we will give a numerical example and first timings. Right now, timings have not yet been computed (in fact, we just managed to finish the implementation) and the notation of the example (computed and written down by one of the other authors) is incomprehensible even for me. Unfortunately, I am the only one of the four authors available this week. 1 Introduction Many protocols of public key cryptography can be implemented in a finite abelian group such as the multiplicative group of a finite field [Odl85], the group of points over an elliptic curve over a finite field [Kob87] or the class group of algebraic number fields [BW88, BW89,...
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
INFRASTRUCTURE, ARITHMETIC, AND CLASS NUMBER COMPUTATIONS IN PURELY CUBIC FUNCTION FIELDS OF CHARACTERISTIC AT LEAST 5
, 2009
"... One of the more difficult and central problems in computational algebraic number theory is the computation of certain invariants of a field and its maximal order. In this thesis, we consider this problem where the field in question is a purely cubic function field, K/Fq(x), with char(K) ≥ 5. In add ..."
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One of the more difficult and central problems in computational algebraic number theory is the computation of certain invariants of a field and its maximal order. In this thesis, we consider this problem where the field in question is a purely cubic function field, K/Fq(x), with char(K) ≥ 5. In addition, we will give a divisortheoretic treatment of the infrastructures of K, including a description of its arithmetic, and develop arithmetic on the ideals of the maximal order, O, of K. Historically, the infrastructure, RC, of an ideal class, C ∈ Cl(O) has been defined as a set of reduced ideals in C. However, we extend work of Paulus and Rück [PR99] and Jacobson, Scheidler, and Stein [JSS07b] to define RC as a certain subset of the divisor class group, JK, of a cubic function field, K, specifically, the subset of distinguished divisors whose classes map to C via JK → Cl(O). Our definition of distinguished generalizes the same notion by Bauer for purely cubic function fields of unit rank 0 [Bau04] to those of unit rank 1 and 2 as well. Further, we prove a bijection between RC, as a set of distinguished divisors, and the infrastructure of C defined by “reduced” ideals, as in [Sch00, SS00, Sch01, LSY03, Sch04]. We describe the arithmetic on RC, providing new results on the baby step and giant step operations and generalizing notions of the inverse of a divisor in R [O] from quadratic infrastructures in [JSS07b] to cubic infrastructures. We also give algorithms to
Computing Arakelov class groups
, 801
"... 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. 1. Introduction. In his 1972 Boulder paper [26], Daniel Shanks observed that the quadratic forms in the principal cycle of reduced binary quadratic forms of positive discriminant exhibit a grouplike behavior. This was a surprising phenomenon, because the principal cycle itself constitutes the trivial class of the class group. Shanks called this grouplike structure ‘inside’