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12
Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
Computing discrete logarithms in real quadratic congruence function fields of large genus
 Math. Comp
, 1999
"... Abstract. The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the dif ..."
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Cited by 42 (8 self)
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Abstract. The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the difficulty of a discrete logarithm problem in these fields. In this paper, we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus. This algorithm is a generalization of similar algorithms for real quadratic number fields. 1.
Keyexchange in real quadratic congruence function fields
 Designs, Codes and Cryptography 7
, 1996
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LiDIA  A Library for Computational Number Theory  Reference Manual
, 2001
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Equivalences Between Elliptic Curves and Real Quadratic Congruence Function Fields
 In preparation
"... In 1994, the wellknown DiffieHellman key exchange protocol was for the first time implemented in a nongroup based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a s ..."
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Cited by 14 (4 self)
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In 1994, the wellknown DiffieHellman key exchange protocol was for the first time implemented in a nongroup based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a socalled infrastructure. More recently, the scheme was extended to real quadratic congruence function fields, whose set of reduced principal ideals has a similar infrastructure. As always, the security of the protocol depends on a certain discrete logarithm problem (DLP). In this paper, we show that for real quadratic congruence function fields of genus one, i.e. elliptic congruence function fields, this DLP is equivalent to the DLP for elliptic curves over finite fields. We present the explicit corresponce between the two DLPs and prove some properties which have no analogues for real quadratic number fields. Furthermore, we show that for elliptic congruence function fields, the set of redu...
Old and New Deterministic Factoring Algorithms
 In Cohen [1
, 1996
"... this paper, two more O(n ..."
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Baby Step Giant Step in Real Quadratic Function Fields
, 1995
"... The principal topic of this article is to extend Shanks' infrastructure ideas in real quadratic number fields to the case of real quadratic congruence function fields. We apply these techniques to the problem of computing the regulator R of a real quadratic congruence function field. By making ..."
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Cited by 3 (1 self)
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The principal topic of this article is to extend Shanks' infrastructure ideas in real quadratic number fields to the case of real quadratic congruence function fields. We apply these techniques to the problem of computing the regulator R of a real quadratic congruence function field. By making use of symmetries and optimized formulas we obtain a considerable improvement in the complexity of calculating R. We also apply the ideas of Lenstra and Schoof to produce an algorithm for determining the regulator unconditionally in O(q 1 5 deg(D)+" ) operations. 1 Introduction Let k = F q be a finite field of odd characteristic with q elements and let K = k(x)( p D), where D is a monic, squarefree polynomial of even degree. Such a field is known as a real quadratic congruence function field (of odd characteristic). Note that K is a Galois extension of the rational function field k(x) with Galois group f1; oeg, where oe is the Kautomorphism which takes p D to \Gamma p D. The conjugate o...
Turning Euler's factoring method into a factoring algorithm
 Bulletin of the London Mathematical Society
, 1996
"... An algorithm is presented which, given a positive integer n, will either factor n or prove it to be prime. The algorithm takes O( « 1/3+e) steps. 1. ..."
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Cited by 3 (2 self)
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An algorithm is presented which, given a positive integer n, will either factor n or prove it to be prime. The algorithm takes O( « 1/3+e) steps. 1.
THE INFRASTRUCTURE OF A GLOBAL FIELD OF ARBITRARY UNIT RANK
, 809
"... Abstract. In the past, the infrastructure of a number or (global) function field has been used for computation of units. In the case of a onedimensional infrastructure, i.e. in the case of unit rank one, one has a binary operation which is similar to multiplication, called a giant step, which was i ..."
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Cited by 2 (1 self)
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Abstract. In the past, the infrastructure of a number or (global) function field has been used for computation of units. In the case of a onedimensional infrastructure, i.e. in the case of unit rank one, one has a binary operation which is similar to multiplication, called a giant step, which was introduced by D. Shanks. In this paper, we show a general way to interpret infrastructure in the case of arbitrary unit rank, which gives a giant step. Moreover, we relate the infrastructure and the giant step to the arithmetic in the divisor class group. Finally, we give explicit algorithms in the function field case for computing, and show how the baby stepgiant step method for unit computation generalizes to the case of arbitrary unit rank. 1.
Integer Factoring
, 2000
"... Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization. ..."
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Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.