Unit computation in purely cubic function fields of unit rank 1

Scheidler, Renate; Stein, Andreas

doi:10.1007/bfb0054895

Cited by 5 publications

(5 citation statements)

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“…In certain cases, namely real quadratic (i.e. real hyperelliptic) function fields [33,32,11] and for certain cubic function fields of unit rank one [25,22], we can efficiently compute baby steps, inverse baby steps and giant steps (i.e. given x, y ∈ X, we can compute bs(x), bs −1 (x) and gs(x, y)), and we can efficiently compute relative distances One further fundamental property of these infrastructures is that computation of d is hard, i.e.…”

Section: One Can Interpret Finite Cyclic Groups As Discrete Cyclic In...mentioning

confidence: 99%

“…This structure behaves similar to finite cyclic groups, with the main difference that the operation corresponding to multiplication is not associative. This structure was generalized from real quadratic number fields to arbitrary number fields of unit rank one [6], and also to real quadratic function fields [33,30,32] and more general function fields [25,22]. Moreover, the key exchange protocol for infrastructures was refined [13,12] and extended to real quadratic function fields [26,10].…”

Section: Introductionmentioning

confidence: 99%

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Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures

Fontein

2008

Advances in Mathematics of Communications

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In discrete logarithm based cryptography, a method by Pohlig and Hellman allows solving the discrete logarithm problem efficiently if the group order is known and has no large prime factors. The consequence is that such groups are avoided. In the past, there have been proposals for cryptography based on cyclic infrastructures. We will show that the Pohlig-Hellman method can be adapted to certain cyclic infrastructures, which similarly implies that certain infrastructures should not be used for cryptography. This generalizes a result by Müller, Vanstone and Zuccherato for infrastructures obtained from hyperelliptic function fields.We recall the Pohlig-Hellman method, define the concept of a cyclic infrastructure and briefly describe how to obtain such infrastructures from certain function fields of unit rank one. Then, we describe how to obtain cyclic groups from discrete cyclic infrastructures and how to apply the Pohlig-Hellman method to compute absolute distances, which is in general a computationally hard problem for cyclic infrastructures. Moreover, we give an algorithm which allows to test whether an infrastructure satisfies certain requirements needed for applying the Pohlig-Hellman method, and discuss whether the Pohlig-Hellman method is applicable in infrastructures obtained from number fields. Finally, we discuss how this influences cryptography based on cyclic infrastructures. Definition 2.1. Let R ∈ R >0 be a positive real number. A cyclic infrastructure (X, d) of circumference R is a non-empty finite set X with an injective map d : X → R/RZ, called the distance function.Definition 2.2. We say that a cyclic infrastructure (X, d) of circumference R is discrete if R ∈ Z and d(X) ⊆ Z/RZ.One can interpret finite cyclic groups as discrete cyclic infrastructures as follows: Let G = g be a finite cyclic group of order m and d : G → Z/mZ be the discrete logarithm map 1 (to the base g), i.e. we have g d(h) = h for every h ∈ g . By 1 The discrete logarithm of an element h ∈ g is sometimes, in particular in Elementary Number Theory, also called the index of h with respect to g.

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Section: One Can Interpret Finite Cyclic Groups As Discrete Cyclic In...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures

Fontein

2008

Advances in Mathematics of Communications

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“…Shanks' method was first extended to function fields in works of A. Stein and H. G. Zimmer [Ste92,SZ91], Stein and Williams [SW98,SW99] and Scheidler and Stein [SS98,Sch01]. The relationship between the infrastructure in real elliptic and hyperelliptic function fields and the divisor class group in their imaginary counterparts was investigated by Stein in [Ste97], and by S. Paulus and H.-G. Rück in [PR99].…”

Section: Introductionmentioning

confidence: 99%

The infrastructure of a global field of arbitrary unit rank

Fontein

2011

Math. Comp.

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Abstract. In this paper, we show a general way to interpret the infrastructure of a global field of arbitrary unit rank. This interpretation generalizes the prior concepts of the giant-step operation and f -representations, and makes it possible to relate the infrastructure to the (Arakelov) divisor class group of the global field. In the case of global function fields, we present results that establish that effective implementation of the presented methods is indeed possible, and we show how Shanks' baby-step giant-step method can be generalized to this situation.

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“…273-304] computes the fundamental unit of a complex cubic number field by generating the "Voronoi continued fraction expansion" of the unit. An explicit implementation in purely cubic fields was given by Williams et al [9], and Williams' version was adapted to purely cubic congruence function fields of characteristic at least 5 and unit rank 1 in [4,5]. In short, the method produces a chain (θ n ) n∈N of successive minima in the maximal order of the field by starting with θ 1 = 1 and computing adjacent minima θ n of increasing absolute value such that θ n+1 = µ n θ n and µ n is the minimum adjacent to 1 in the reduced fractional principal ideal a n = (θ −1 n ) (n ∈ N).…”

Section: Introductionmentioning

confidence: 99%

“…A general introduction to function fields can be found in [6]; the purely cubic case is discussed in considerable detail in [2] and [4,5]. Let k = F q be a finite field of order q whose characteristic is not 3.…”

Section: Introductionmentioning

confidence: 99%

Purely cubic function fields with short periods

Scheidler¹

1999

Publ. Math. Debrecen

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A "function field version" of Voronoi's algorithm can be used to compute the fundamental unit of a purely cubic complex congruence function field of characteristic at least 5. This is accomplished by generating a sequence of minima in the maximal order of the field. The number of mimima computed is the period of the field. Generally, the period is very large -it is proportional to the regulator and exponential in the genus of the field -but there are classes of fields with very short periods. For several infinite families of such fields, we explicitly compute the Voronoi continued fraction expansions and the fundamental units. We also investigate the case of period length 1 where the minima in the maximal order are exactly the units of the field. Finally, we explore the connection between regulator and period and other cases of small periods and regulators.

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Unit computation in purely cubic function fields of unit rank 1

Cited by 5 publications

References 10 publications

Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures

Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures

The infrastructure of a global field of arbitrary unit rank

Purely cubic function fields with short periods

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