Results 1  10
of
105
Implementing 2Descent for Jacobians of Hyperelliptic Curves
 Acta Arith
, 1999
"... . This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Qrational Weierstra point. 1. Introduction Given some curve C over Q , one w ..."
Abstract

Cited by 62 (19 self)
 Add to MetaCart
. This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Qrational Weierstra point. 1. Introduction Given some curve C over Q , one would like to determine as much as possible of its arithmetical properties. One of the more important invariants is the MordellWeil rank of its Jacobian J , i.e., the free abelian rank of J(Q ) (finite by the MordellWeil Theorem). There is no algorithm so far that provably determines this rank, but it is possible (at least in theory) to bound it from above by computing the size of a suitable Selmer group. It is also fairly easy to find lower bounds by looking for independent rational points on the Jacobian. (It can be difficult, however, to find the right number of independent points, when some of the generators are large.) With some luck, both bounds coincide, and the rank is determined. In...
Applications of Arithmetical Geometry to Cryptographic Constructions
 Proceedings of the Fifth International Conference on Finite Fields and Applications
"... Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1). To construct DLsystems we use me ..."
Abstract

Cited by 49 (1 self)
 Add to MetaCart
(Show Context)
Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1). To construct DLsystems we use methods from algebraic and arithmetic geometry and especially the theory of abelian varieties over finite fields. It is explained why Jacobian varieties of hyperelliptic curves of genus 4 are candidates for cryptographically "good" abelian varieties (Section 2). In the third section we describe the (constructive and destructive) role played by Galois theory: Local and global Galois representation theory is used to count points on abelian varieties over finite fields and we give some applications of Weil descent and Tate duality.
Arithmetic On Superelliptic Curves
 Math. Comp
, 2000
"... This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form y n = c(x) which has only one point at infinity. Divisors are represented as ideals and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique repre ..."
Abstract

Cited by 43 (4 self)
 Add to MetaCart
(Show Context)
This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form y n = c(x) which has only one point at infinity. Divisors are represented as ideals and an ideal reduction algorithm based on lattice reduction is given. We obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time. An algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field.
A Separation Bound for Real Algebraic Expressions
 In Lecture Notes in Computer Science
, 2001
"... Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, kth root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the si ..."
Abstract

Cited by 38 (3 self)
 Add to MetaCart
Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, kth root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda real. 1
A Bost–Connes–Marcolli system for Shimura varieties, in preparation
"... We construct a Quantum Statistical Mechanical system (A, σt) analogous to the BostConnesMarcolli system of [CM04] in the case of Shimura varieties. Along the way, we define a new BostConnes system for number fields which has the “correct” symmetries and the “correct ” partition function. We give ..."
Abstract

Cited by 28 (0 self)
 Add to MetaCart
We construct a Quantum Statistical Mechanical system (A, σt) analogous to the BostConnesMarcolli system of [CM04] in the case of Shimura varieties. Along the way, we define a new BostConnes system for number fields which has the “correct” symmetries and the “correct ” partition function. We give a formalism that applies to general Shimura data (G, X). The object of this series of papers is to show that these systems have phase transitions and spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature.
Index calculus in class groups of nonhyperelliptic curves of genus three, in "Journal of Cryptology", The original publication is available at www.springerlink.com
, 2007
"... We study an index calculus algorithm to solve the discrete logarithm problem (DLP) in degree 0 class groups of nonhyperelliptic curves of genus 3 over finite fields. We present a heuristic analysis of the algorithm which indicates that the DLP in degree 0 class groups of nonhyperelliptic curves of ..."
Abstract

Cited by 26 (4 self)
 Add to MetaCart
We study an index calculus algorithm to solve the discrete logarithm problem (DLP) in degree 0 class groups of nonhyperelliptic curves of genus 3 over finite fields. We present a heuristic analysis of the algorithm which indicates that the DLP in degree 0 class groups of nonhyperelliptic curves of genus 3 can be solved in an expected time of Õ(q). This heuristic result relies on one heuristic assumption which is studied experimentally. We also present experimental data which show that a variant of the algorithm is faster than the Rho method even for small group sizes, and we address practical limitations of the algorithm.
On the Equivariant Tamagawa Number Conjecture for Abelian Extensions of a Quadratic Imaginary Field
 DOCUMENTA MATH.
, 2006
"... Let k be a quadratic imaginary field, p a prime which splits in k/Q and does not divide the class number hk of k. Let L denote a finite abelian extension of k and let K be a subextension of L/k. In this article we prove the ppart of the Equivariant Tamagawa Number Conjecture for the pair (h 0 (Spe ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
Let k be a quadratic imaginary field, p a prime which splits in k/Q and does not divide the class number hk of k. Let L denote a finite abelian extension of k and let K be a subextension of L/k. In this article we prove the ppart of the Equivariant Tamagawa Number Conjecture for the pair (h 0 (Spec(L)), Z[Gal(L/K)]).
Ray Class Fields of Global Function Fields with Many Rational Places
, 1998
"... A general type of ray class fields of global function fields is investigated. Their systematic computation leads to new examples of curves over finite fields with many rational points compared to their genera. ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
A general type of ray class fields of global function fields is investigated. Their systematic computation leads to new examples of curves over finite fields with many rational points compared to their genera.
The Regular C*algebra of an Integral Domain
, 807
"... To each integral domain R with finite quotients we associate a purely infinite simple C*algebra in a very natural way. Its stabilization can be identified with the crossed product of the algebra of continuous functions on the “finite adele space ” corresponding to R by the action of the ax+bgroup ..."
Abstract

Cited by 16 (11 self)
 Add to MetaCart
To each integral domain R with finite quotients we associate a purely infinite simple C*algebra in a very natural way. Its stabilization can be identified with the crossed product of the algebra of continuous functions on the “finite adele space ” corresponding to R by the action of the ax+bgroup over the quotient field Q(R). We study the relationship to generalized BostConnes systems and deduce for them a description as